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EEL 3552 Lab Manual (Valencia West) - (ECE) at UCF

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1. ae 4 56 Where f t is the minimum value of the message signal before any dc is added and A is the amount of dc added Note that m 100 for A f t There are two forms of demodulation used in this lab 1 Synchronous The modulated waveform is multiplied by another carrier of hopefully identical frequency and phase to the AM carrier High frequency terms that result from the multiplication are removed with a filter and only the message is left Note that the carrier must be regenerated for an actual receiver 2 Envelop Detection When m 10096 peak detection sample and hold can be used Sample and hold means that the amplitude of the signal at a given point in time is sampled and held until the next sample pulse arrives The FET used in the AM modulator and demodulator Figure 4 25 and Figure 4 26 is used as a variable resistor where the resistance between the drain D and the source S varies proportional to the voltage applied to the gate G Since there is a square wave varying from 0 to V volts been applied to the gate the resistance across the source and the drain can be thought as either an open R when V 2 V or a short R20 for V 20 Analysis of the circuit shows a gain of V V 1 for R differential amplifier configuration and a gain of V V 1 when R 0 inverting amplifier configuration 6 Procedure 6 1 AM Modulation 1 Build the modulator shown in Figure 4 25
2. CD RW drive ae m om 0 0 amo o mito D DOUDOU o DOUDD e Control panel A K o o CO e Ground terminal Channel inputs On Standby Probe comp terminals Switch Auxiliary input a CURSORS turn cursors on and off Push DEFAULT SETUP to return settings to factory default values Push AUTOSET to automatically set up TRIGGER gt POSITION the vertical horizontal and trigger ES controls based on selected channels EDGE SOURCE COUPLING SLOPE Select the multipurpose knobs to cHi bc Por adjust parameters selected from MultiView cH2 Ac MG the screen interface Push a fine Zoom NF RE button to toggle between normal EF E and fine adjustment with the UE mE LE corresponding multipurpose NORNI o 7 5 i c ER G2 CD VERTICALL Push MultiView Zoom to add a magnified graticule to the display Push HORIZ or VERT to assign the multipurpose knobs to the horizontal or vertical scale and position parameters Turn the channel displays on and off Vertically scale position or change the input termination b Figure 1 1 a Front Panel b Control Panel Tektronix User Manual reference 1 2 PREPATATION a Calculate the amplitude in dBV of a 1 KHz 2 volt peak to peak sine wave b Calculate the peak to peak voltage of a 10 dBV 4 KHz sine wave EXPERIMENT 1 TIME DOMAIN DISPLAY a Turn on the Oscilloscope and allo
3. lt lt Thus we obtain an undistorted version of the transmitted signal Figure 4 24 illustrates the frequency response of a VSB filter that selects the lower sideband and a vestige of the upper sideband 4 30 In practice the VSB filter is designed to have some specified phase characteristics To avoid distortion of the information signal the VSB filter should be designed to have a linear phase over its passband o o o Figure 4 23 VSB filter characteristics Figure 4 24 Frequency response of VSB filter for selecting the lower sideband of the message signals 4 31 2 11 Why modulation Modulation is the process by which a property of a parameter of a signal in proportion to a second signal The primary reason for using modulation in communication is 1 To raise up the frequency of a signal to reduce the wavelength such that a relatively small antenna can transmit or receive the signal 2 To separate base band signals in frequency or time so that more than one signal can be transmitted on the same channel 3 To have the base band signal transformed for ease of transmission The degree to which a signal is modulated is measured by a modulation index which will have a different physical significance for each type of modulation Every form of modulation has advantages and disadvantages when compared to the others FM has more strengths than weaknesses compared to the others Delta Modulation has m
4. 0 001 uF 200 ohm C2 R1 1uF 100K V 13 6 XR2206 VCO 6V 6V Figure 5 5 b Generation of FM signal d Use the V found as the amplitude of the input signal and find Af on the oscilloscope Compare Af eax calculated and measured Display only the FM signal on the oscilloscope and trigger on the rising edge of the waveform such that you see the following Figure 5 6 it will look like a ribbon fnax 1 tmin fmin 1 tmax Afpeak fmax 7 fmin 2 Figure 5 6 Frequency components of the modulated signal This ribbon displays all frequencies in the FM signal at once The minimum and maximum frequencies can be easily detected and directly measured Recall that Af a is only half of the pea peek to peak frequency swing What parameters determine the bandwidth of an FM signal e View the frequency domain waveform to obtain modulation indices of 2 4 5 52 8 65 These are zero carrier amplitude indices Include calculations to verify your results in your lab report 7 2 Demodulation a Build the FM demodulator utilizing the LM565N PLL shown in Figure 5 7 Connect the output of the FM modulator shown in Figure 5 5 b to the input of the FM demodulator b Apply a sinusoidal message signal and observe the demodulated message signal Sketch both waveforms How do they compare Hint for the Tektronix Oscilloscope An FM demodulator is typically followed by a filter to block the carrier
5. Name them say S1 C1 and X respectively Use the ft function for this You have to create your own ft function The code for this function is given below Make sure that you saved this function as a separate Matlab file with the same name ft m in the Matlab working directory function X ft x fs n2 length x n 2 max nextpow2 fs nextpow2 n2 X1 fft x n fs X X1 1 n 8 An example has been given below how you find the fourier transform using this Matlab function You can write the following line of code in your main program to find fourier transform of a signal S1 ft signal fs For good demodulation of the received signal the carrier at the receiver end needs to be of the same frequency and phase as the carrier of the transmitter end In practice this may not be the case 4 33 7 Carrier at the receiver demodulator is given by the following equation carrier2 sin 2n f delta t phi Where delta is the frequency error and phi is the phase error Initially assume that both of them are Zero 8 For demodulation multiply the received signal by the carrier and name it x1 and then do a low pass filtering For low pass filtering use the following piece of Matlab code b fir1 24 500 fs The first parameter gives you the order of the filter The second parameter is the cutoff frequency Here it is 200 Hz x_demod conv2 x1 b same Here b is the coefficie
6. Acos w t min SON sg omen 4 44 1s EROR Gg aa where f t is the normalized version of f r Note that f t has minimum value of 1 In the above equation we define the modulation index m to be equal to the ratio min f x T 4 45 Actually the aforementioned definition is applicable independently of whether f t has minimum value of unity or not 2 10 Vestigial Sideband Suppressed Carrier VSB SC The stringent frequency response requirements on the sideband remover of a SSB SC system can be relaxed by allowing a part called vestige of the unwanted sideband to appear at the 4 28 output of the modulator Thus we simplify the design of the sideband filter at the cost of a modest increase in the channel bandwidth required to transmit the signal The resulting signal is called vestigial sideband suppressed carrier VSB SC system The suppressed carrier name sterns from the fact that one more time no pure carrier is sent over the channel To generate a VSB SC signal we begin by generating a DSB SC signal and passing it through a sideband filter with frequency response H v as shown in Figure 4 21 In the time domain the VSB SC signal may be expressed as mt f t eos o h t 4 46 where A t is the impulse response of the VSB SC filter In the frequency domain the corresponding expression is M o Fle 0 F o A o 4 47 Sideband filter H f VSB SC signal Figure 4 21 Generation of a VSB
7. e 2 16 27 where T The above set of functions is a complete orthonormal set Therefore any function f t defined over an interval of length T can be expanded as a linear combination of the t s In particular f Yc4 0 Y GEHE Q 17 The coefficients C in the above expansion are chosen according to the rules specified by the Orthogonality Theorem That is C f FO ed 2 18 The above equation is denoted as the Exponential Fourier Series expansion of the signal f t Actually in most instances the exponential Fourier series expansion of a signal is provided by the following equation jus gt kg 2 19 where 1 jn ot n te at 2 20 Our original goal has now been accomplished since we can now derive the amplitude frequency plot of f t by plotting the magnitude of F with respect to frequency and we can also derive the phase frequency plot of f r by plotting the phase of F with respect to frequency Note that in general the F s are complex numbers and a complex number F can P of the exponential Fourier series expansion of f t and the coefficients of the trigonometric be written as the product of its magnitude F times e One can show that the coefficients Fourier series expansion of f t are related as follows 2 7 a jb 1 lt n lt o 2 21 The following are some comments The exponential Fourier series and the trigonometric Fourier series expans
8. 0 It is therefore apparent that the first term in equation 4 22 is removed by the low pass filter in Figure 4 6 provided that the cut off frequency of this filter is greater than but less than 2o This is satisfied by choosing o gt At the filter output we then obtain a signal given by o r AA cos d r 4 23 The demodulated signal o t is therefore proportional to f t when the phase error is a constant The amplitude of this demodulated signal is maximum when 0 and is minimum 4 10 zero when The zero demodulated signal which occurs for i represents the quadrature null effect of t he coherent detector Thus the phase error in the local oscillator causes the detector output to be attenuated by a factor equal to cos As long as the phase error is a constant the detector output provides an undistorted version of the original baseband signal f t In practice however we usually find that the phase error varies randomly with time due to random variations in the communications channel The result is that at the detector output the multiplying factor cos also varies randomly with time which is obviously undesirable Therefore provision must be made in the system to maintain the local oscillator in the receiver in perfect synchronism in both frequency and phase with the carrier wave used to generate the DSB SC modulated signal in the transmitter 2 7 Quadrature Carri
9. Setting the oscilloscope Trigger Coupling to HF REJ will help sync on the modulation frequency c Apply a square wave message signal and observe the demodulated message signal Sketch both waveforms How do they compare Why is the demodulated sinusoidal message more faithfully reproduced than the demodulated square wave 5 23 Voc No Connection Input No Connection Input No Connection VCO Output No Connection Phase Comparator Voc VCO Input VCO Control Voltage Timing Capacitor Timing Resistor Phase Lock Loop Pin Assignment 6V 6V 0 1 uF 0 1uF Output 0 001 2uF LM565 Phase Lock Loop 0 0022uF 1 1 0 1uF 1 Figure 5 7 PLL as an FM Demodulator 0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 5 2 5 4 5 6 5 8 1 00 0 99 0 96 0 91 0 85 0 77 0 67 0 57 0 46 0 34 0 22 0 11 0 00 0 10 0 19 0 26 0 32 0 36 0 39 0 40 0 40 0 38 0 34 0 30 0 24 0 18 0 11 0 04 0 03 0 09 0 10 0 20 0 29 0 37 0 44 0 50 0 54 0 57 0 58 0 58 0 56 0 52 0 47 0 41 0 34 0 26 0 18 0 10 0 01 0 07 0 14 0 20 0 26 0 30 0 33 0 34 0 35 0 33 0 31 Table 5 1 Bessel Functions of the first kind J x J 0 02 0 04 0 08 0 11 0 16 0 21 0 26 0 31 0 35 0 40 0 43 0 46 0 48 0 49 0 48 0 47 0 44 0 41 0 36 0 31 0 25 0 18 0 12 0 05 0 02 0 09 0
10. 16KO C 0 01uF Ri 10KQ Ro 12 KO Figure 3 16 4 order low pass filter R 10KO Ri 1 5 KQ 3 26 Calculating Filter Parameters The following procedures will help you understand how to measure filter parameters on the scope or AC voltmeter 7 2 Cutoff frequency a From the function generator obtain a 2 volt low frequency 10 or 100 Hz range sine wave It is important to use a sine wave because there is only one harmonic in a sinusoid and you only want to test the transfer function at one point in frequency at time Connect the function generator output to the filter input b Connect the filter output to the oscilloscope See Figure 3 18 or AC voltmeter input See Figure 3 17 c Vary the input frequency by a factor of 2 or 3 either way and make sure the output level doesn t vary significantly If it doesn t then your input frequency is within the pass band When you know you are within the pass band measure the output voltage level or dB level This will be your reference level d Vary the input frequency until the output level drops 3 dB below the reference level On the AC voltmeter this can be read directly On the oscilloscope 3 dB is equivalent to the voltage level reduced by a factor of 707 or at 3 dB point Vout 3dB 707 Vout ref Note this frequency This is your cutoff frequency 7 3 Passband gain a Using the function generator generate a 2 volt sine wave at a frequency that you know
11. Fourier series expansion and make appropriate remarks 2 What is the difference between an energy signal and a power signal Give two examples of each 5 Implementation In this lab you will be introduced to the relationship between pulse shapes and their spectra There are some important concepts to recognize when performing the experiment such as 1 When a pulse becomes narrower in the time domain its energy becomes more spread out in frequency 2 The envelope of the spectrum of a periodic signal is the Fourier transform of the pulse shape To illustrate the first concept given above consider the Fourier Transform of a rectangular pulse shown in Figure 2 11 It can be found from the frequency domain graph Figure 2 11 that 90 of the energy is concentrated within the main lobe in frequency As the pulse width is decreased in time or T decreases the width of the main lobe in the frequency domain spreads thus spreading out the energy of the signal To illustrate the second concept recall that when two signals are convolved in the time domain their transforms are multiplied in the frequency domain Any periodic signal can be represented mathematically described as the basic pulse shape convolved with a time sequence of pulses Thus the spectrum of the periodic signal can be derived as shown in Figure 2 12 6 Procedure EQUIPMENT TDS5052B Oscilloscope Function Generator You should bring a floppy disk to store your wave
12. Re eR te C RR t rer oa EM picea eos Cc ES ee E EET En OC Td i 0 1000 2000 3000 4000 5000 Frequency Hz Figure 3 10 Butterworth Filters How the frequency response changes with order 3 19 Filters are key components in communication systems An ideal filter will only allow a specified set of frequencies to pass from input to output with equal gain However no filter is ideal and as a result there are many types of filters that are used in modeling an ideal filter closely in one or more aspects Filters have many applications besides separating a signal form surrounding noise or other signals Some of these applications are listed below 1 Integration of a signal 2 Differentiation of a signal 3 Pulse shaping 4 Correcting for spectral distortion of a signal 5 Sample and hold circuits Two Butterworth low pass filters LPF s will be used in this lab a first order and a fourth order Butterworth filter The general transfer function of a Butterworth LPF is 2 A IH e x 2n 1 a Where the filter cutoff frequency and n the order of the filter The actual transfer functions of the filters that are going to be used in this lab are 1 3 20 1 RCS ve 1st order LPF IH s 4 order LPF jt TE R R RCS E AE cs ics S15 es jJ 1 1 T Where o 1 RC for both filters 3 20 Two applications for low pass filters will be illustrated
13. assumptions that 1 the DC value of f t is zero and ii the bandwidth of the information signal is much smaller than the carrier frequency Consequently A Power of f t Power of m t 4 40 DES 4 40 For the special case where f t Kcos a t we can conclude that 2 25 2 2 2 Poudroto s ke cum 4 41 2 4 2 2 4 27 It is worth noting that from the power dedicated to transmit the modulated signal A i 2 goes to the transmission of the pure carrier and the rest goes to the transmission of the useful signal If we define the efficiency of our system as the ratio of the power dedicated to the transmission of the useful signal versus the power dedicated to the transmission of the modulated signal we conclude that the efficiency of our DSB LC system for a sinusoidal information signal is equal to 2 m m 2 Efficiency 7 4 42 The best we can do is to let 77 assume its maximum value this happens when m 1 Then 1 0 33 33 4 43 So 67 of the available transmitter power goes to the carrier alarming quantity Usually we do not know K so we might operate at less than 33 efficiency The notion of the modulation index can be extended to all signals as long as they are normalized so that their maximum negative value is unity To be more specific consider the DSB LC signal corresponding to an information signal whose maximum negative value is different than unity Then m t Acos a t f t cos a t
14. by F o f e at 4 11 exists and the original signal can be obtained from its FT by following the rule f t eni F a e do 4 12 2m oe where a if f t is continuous at t j fert if f t is continuous at t OS 2 Observation 1 The Fourier transform F of a signal f t is in general a complex function Its magnitude F o and phase represent the amplitude and phase of various frequency components of f r Observation 2 If the independent variable in the FT of a signal is chosen to be the frequency f instead of the angular frequency we have the following Fourier transform pair X f e atre 7 lt 4 14 and f X f e qf 4 15 o0 2 3 Fourier Series to Fourier Transform Example Let fr r be a periodic signal depicted in Figure 4 2 i To To 2 7 2 0 7 2 To 2 To Figure 4 2 A periodic signal fro t The Fourier series expansion of this signal of period T is given by the following equation oo iis gt d 4 16 where l Ty 2 jn ot F gs lo f the dt 4 17 and insit 4 18 We can compute F s as follows EST L jn2m IT c s jn2m IT s T Eu Je dis i I Je E 2l T pens n jz sin nzt T T T Jn2z 4 19 nat T T e The line spectrum of the signal f t is shown in Figure 4 3 for c T k Note that the separation of two consecutive lines in the line spectrum of f t is equal to 2z T Hence as T gets larger and larger
15. complex exponential within the passband of the filter will actually produce an output the passband is the range of frequencies from 12710 rads to 127710 rads Hence using the already derived formula that gives the output of an LTI system due to a periodic input we can write g t 2 pirra 2 4 ieni0 r 2 2 _ 2 il6ni0 r 2 m m T 2 il6ni0 r 2 2 4 2 pillonio rs2 2 2 s lori0 r 2 3 19 m m T 3 16 2 8 Filters Filtering has a commonly accepted meaning of separation something is retained and something is rejected In electrical engineering we filter signals usually frequencies A signal may contain single or multiple frequencies We reject frequency components of a signal by designing a filter that provides attenuation over a specific band of frequencies and we retain components of a signal through the absence of attenuation or perhaps even gain over a specified band of frequencies Gain may be defined as how much the input is amplified at the output Filters are classified according to the function they have to perform Over the frequency range we define pass band and stop band Ideally pass band is defined as the range of frequencies where gain is 1 and attenuation is 0 and stop band is defined as the range of frequencies where the gain is 0 and the attenuation is infinite Filters can be mainly classified as low pass high pass band pass and band stop filters A low pass filter can be characterized by the property t
16. cursors appear in the upper waveform both outside of the zoomed frequency range Use the mouse to drag one of the cursors into the zoomed frequency range and position it at the peak of the 1 KHz signal Does the amplitude agree with the value that you calculated in the Preparation i Change the frequency of the Function Generator to 2 KHz Does this accurately represent the input signal On the Toolbar click File click Save As and save to your floppy disk Note Do not ever save to the Local Disk C j Change the frequency of the Function Generator to 4 KHz Does the frequency domain waveform confirm that the signal is 4 KHz k Decrease the amplitude of the Function Generator output to 10 dBV as measured in the frequency domain display Turn the time domain on again by pressing the CH1 button Turn the Zoom off by pressing the Zoom icon Read the peak to peak voltage measurement Does this measurement agree with the value that you calculated in the Preparation 1 4 REPORT In your report describe what you have learned in this experiment Compare your experimental measurements with the theoretical calculations Remember to insert the picture that you saved as part of your report Write all conclusions REFERENCES 1 5 EXPERIMENT 2 Periodic Signal Spectra 1 Objective To understand the relationships between time waveforms and frequency spectra 2 Theory 2 1 Motivation Consider for a moment the special class
17. for all practical purposes their duration can be assumed to be equal to zero Examples of such phenomena are a hammer blow a very narrow voltage or current pulse etc In the precise mathematical sense the impulse signal denoted by r is not a function signal it is a distribution or a generalized function A distribution is defined in terms of its effect on another function usually called test function under the integral sign The impulse distribution or signal can be defined by its effect on the test function glt which is assumed to be continuous at the origin by the following relation faos 2 ee a 0 0 otherwise a This property is called the sifting property of the impulse signal In other words the effect of the impulse signal on the test function olt under the integral sign is to extract or sift its value at the origin As it is seen r is defined in terms of its action on glt and not in terms of its value for different values of t One way to visualize the above definition of the impulse function is to think of the impulse function as a function determined via a limiting operation applied on a sequence of well known 3 9 signals The defining sequence of signals is not unique and many sequences of signals can be used such as 1 Rectangular Pulse 5 t lim ul 5 gi 2 Triangular Pulse 3 Two sided exponential 4 Gaussian Pulse 1 z t t P oC imr Se T0 The ab
18. i e as f r becomes less and less periodic looking the line spectrum of f r becomes denser and denser Also note that the envelope of line spectrum of fr r is the Fourier transform within a constant of proportionality of the aperiodic pulse of width 7 centered at 7 0 It can be shown that most of the power of f r is located within the frequency interval 2z 1 2x c This indicates to us that as we make the signal faster in the time domain by decreasing the duration 7 of the pulses the frequency content of the signal gets expanded in the frequency domain This statement is a manifestation of what is called in communication theory the time bandwidth product It can be shown that the time bandwidth product is constant that is if we expand the signal in the time domain we shrink the frequency content of the signal in the frequency domain and vice versa sine xt 2 o xt 2 0 Figure 4 3 The line spectrum of the signal f ro t 2 4 Fourier Transform Properties The Fourier transform of an aperiodic signal is as useful as the Fourier series of a periodic signal As a reminder the Fourier series of a periodic signal allows us to compute the output of a LTI system due to a periodic input without having to compute the convolution Furthermore the Fourier series of a periodic signal provides us with information pertaining to the frequency content of the signal Similarly the Fourier transform of a signal
19. is inversely proportional to the square of the harmonic s number For example if the 1st harmonic s magnitude is A then the 3rd harmonic s magnitude will be A 9 the 5th harmonic s magnitude will be A 25 etc a Setthe Function Generator Duty Cycle knob to Cal and switch the function to triangular wave Set the frequency to 1 KHz Determine the amplitude and frequency of the 1st 3rd and 5rd harmonics b Determine the location of zeros Save the waveform 7 Calculations and Questions 1 Compare the waveforms you obtained for rectangular waves on the basis of harmonic location zero location and peak spectrum amplitude i e first harmonic level Explain the results on the basis of the pulse width and signal frequency 2 Use your data to show that the spectrum of a triangular wave is equal to the spectrum of a square wave squared 3 The spectrum of the square wave should decay proportional to 1 where 2f The spectrum of the triangular wave should decay proportional to 1 oy Verify this using your measurements 2 19 Time Domain A rect t Ty T 2 0 T2 F A rect t T ATSin zfT xfT MAIN LOBE Figure 2 10 Rectangular Function and its frequency representation Fourier transform 2 20 Time Domain Frequency Domain A rect t To ATsin nfT nf T T 2 0 T2 X X T gt 0 T2 A T gt a T 0 Ta T2 E Tot Y 2 2 ft F f Figure 2 11 Mathematic representation of
20. of AM demodulator the envelop detector no carrier waveform is required and the synchronous demodulator carrier waveform is needed Modulator Channel Demoodulator LPF f t f t Figure 4 27 AM Block Diagram 6 3 Envelope detection 1 Connect the envelope detector of Figure 4 28 2 Adjust the oscillator for a DSB LC modulated waveform using sinusoidal modulation with f 100 Hz Connect V t to the oscilloscope input m 3 Set C to the minimum value of the capacitor substitution box noting the output waveform as C is increased Sketch the optimum demodulated waveform Compare V t with the modulating waveform V t Comment 4 Repeat part 3 for a triangular modulating signal 5 Now adjust for a DSB SC waveform comparing V t with the modulating waveform V t Does the envelope detector demodulate DSB SC 4 38 MODULATOR Figure 4 28 Envelope detector 6 4 Synchronous detection 1 Connect the synchronous detector of Figure 4 29 2 Adjust for a DSB LC signal Compare V t with V t Sketch the waveforms Has the message been recovered 3 Adjust for a DSB SC signal Again compare V t with V t Sketch the waveforms Is it possible to retrieve the original message Explain 4 Is it possible to demodulate a DSB LC signal A DSB SC signal C Carrier l i R 11K Local l R 10 K l Oscillator S ees ees C 0 001 uF Figure 4 29 Synchronous Demodulation 7 Questions an
21. of variables in the above equation In particular let us substitute t with x Then 1 re re ems 5 29 The above integral cannot be evaluated in closed form but it has been extensively calculated for various values of s and most n s of interest It has a special name called the nth order Bessel function of the first kind and argument This function is commonly denoted by the symbol J 8 Therefore we can write C J B 5 30 As a result puru Y Ben 5 31 n oo often called the Bessel Jacobi equation If we evaluate the real and imaginary parts of the right hand side of equation 5 31 we will be able to calculate term A and term B respectively It turns out that if we substitute these values for term A and term B in the original equation for m r see equation 5 25 we will end up with m t gt J Acosta no t 5 32 The property of the Bessel function coefficients actually Property 1 that led us to the above results is listed below Some additional properties of the Bessel function coefficients are also listed 1 J B is real valued 2 J B J_ B for n even 5 7 3 J B J_ B for n odd 4 Y2 1 n o0 The advantage of equation 5 32 compared to the original equation 5 24 that defined the FM signal m r is that now through equation 5 32 we can compute the FT of the signal m r It will consist of an infinite sequence of impulses located at posit
22. output amplitude RM Cancel Help Apply Set the simulation stop time at say 0 002 from the simulation gt parameters menu Run the simulation from the simulation gt start menu Then double click on the scopes to see the time domain signals Do the simulation with square wave input signal Is the demodulated signal the faithful reproduction of the input signal Why 5 Implementation In linear FM frequency modulation the instantaneous frequency of the output is linearly dependent on the voltage at the input Zero volts at the input will yield a sinusoid at center frequency f at the output The equation for a FM signal is m t 7 cos o t g t 5 45 Where t g t f t instantaneous phase and the instantaneous angular frequency is 5 16 o t a dst 5 46 In linear FM the instantaneous frequency can be approximated by a straight line o t v k f t 5 47 Where k isa constant and f t is the input signal 6 flo k f r at 0 t 27k f r at 5 48 m r coso t 27k f f at 5 49 Note that the amplitude of an FM signal never varies For a sinusoid input and positive k the modulated FM waveform will relate to the input as shown in figure 5 1 Depending on the VCO k can be positive or negative and f can also vary and is a function of external timing resistor and capacitor values The relationship between input voltage and output frequency at any given point
23. shifts the phase of the sinusoidal signal by 90 degrees We want to be able to characterize this block in the frequency domain To do so let us identify the Fourier transforms of an input signal to this block equal to cos a t and the output signal produced by this block equal to cos t 90 sin a t Obviously FT sin a t j25 o ja o 0 4 27 m FT cos a t 26 c z e 4 28 Based on the above equations it is straightforward to derive the result that 4 17 F T sin a t jsignum o F T cos a t 4 29 where the function signum is equal to 1 when c gt 0 and equal to 1 when c lt 0 Equation 4 29 tells us that the transfer function of the block designated by the notation 90 is equal to jsignum o Now we are ready to generalize the construction of an upper sideband signal using the block diagram of Figure 4 13 for the case where the information signal f t is of arbitrary nature i e not necessarily sinusoidal In Figure 4 14 we provide a block diagram for the construction of the upper sideband signal of a modulated wave m t when the information signal f t is arbitrary Comparing Figures 4 13 and 4 14 we see that they are identical where in both figures the block designated by the notation 90 corresponds to a system with transfer function equal to jsignum o In Figure 4 15 we show in a pictorial fashion why the block diagram of Figure 4 14 works What is worth n
24. so far sends the signal m t over the channel with Fourier transform shown in Figure 4 9 One important observation regarding the Fourier transform of m t is that is consists of two sidebands the upper sideband and the lower sideband There is no need to send both sidebands over the channel since the FT of the information signal f r can be constructed from either the upper or the lower sideband The single sideband suppressed carrier SSB SC that we intend to discuss in the following takes advantage of the above observation by sending only one of the sidebands over the channel 4 13 Multiplier Antenna f r cos a a COS wel fit cos wet f r b F fit cos wer f AS aad ine Lower Upper x T 4 TN Gs 2h ae e TK 0 We m iw Lo 3v n Figure 4 9 An amplitude modulation suppressed carrier transmission Hence the SSB SC system differs from the DSB SC system because just before the modulated wave m t is ready to head off for the channel one of the sidebands is chopped off The block diagram of an SSB SC system is shown in Figure 4 10 Furthermore the sequence of Fourier transform plots shown in Figure 4 11 upper sideband demonstrate that the system works The only problem with the block diagram of the system shown in Figure 4 10 is that it requires the utilization of a perfect filter that eliminates completely the frequencies a little bit below c and passes without distortion the frequenci
25. square wave that we will be using in our lab show how the spectrum of the modulated signal will look like 2 How can we demodulate a signal of this kind Explain and illustrate all your answers 5 Implementation By this time you already know that in amplitude modulation the amplitude of the carrier frequency varies with the amplitude of the desired message signal Typically the carrier will be a cosine or some sine wave but in this lab a square wave is used as the carrier for simplicity in implementation Amplitude Modulation is implemented by multiplying the message signal with the carrier as in the theory part which can be expressed as m t f t c t f t cos o t For a cosine carrier 4 54 The end result with sinusoidal modulation is such that the spectrum of X f remains unchanged but is removed in frequency and centered about w For any general carrier the multiplication of two signals in time yields convolution in frequency A DSB AM signal is described by the equation m t f t c t Cf t A cos o 1 4 55 Ais the average value of the signal or the dc component Typically a message signal will have no dc component When A gt 0 the m t is a DSB LC Large Carrier AM signal In this case the dc term must be added to the message signal The case when A 0 is called Suppressed Carrier DSB SC 4 35 AM The modulated index for AM is a measure of how large A is with respect to the amplitude of f r
26. the constraint that 4 lt t 1 lt 5 or 5 1 6 Then is easy to show that f t Jae t 1 t 2 1 3 10 t 2 Case D f t r is to the right of f t and f overlap f partially from the right see Figure 3 2d In order for Case D to be valid we have to satisfy the constraint that 5 1 6 or 6 lt t lt 7 Then is easy to show that de 5 2 7 1 3 11 Case E f t r is to the right of f z and f and f do not overlap see Figure 3 2e In order for Case E to be valid we have to satisfy the constraint that 6 lt t 1 or t gt 7 Then is easy to show that f t 0 3 12 Hence combining all the previous cases we get 0 fort lt 4 t 4 for4 lt t lt 5 f 11 for5 lt t lt 6 3 13 7 t for6 lt t lt 7 0 fort 7 3 4 A plot of the function f t is shown in Figure 3 2f 0 1 t 2 2 t 3 4 5 6 Figure 3 2 a Convolution under Case A Figure 3 2 b Convolution under Case B Figure 3 2 c Convolution under Case C 3 5 Case D fit 1 fe Figure 3 2 d Convolution under Case D Figure 3 2 e Convolution under Case E f l Figure 3 2f Function f r The result of convolution of f t and f t 3 6 Based on our previous computations we can state certain rules pertaining to the convolution of two rectangles These rules can be proven following the technique laid out in the previous example These rules can be used to find the convolution of two rectangle pulses wi
27. 14t e 2 42 T 3 5 7 3 Power Parseval s Relation 3 1 Definitions The power P of a real signal f t is defined as follows P lim f Pee rat 2 43 Ty oo T 2 13 It is not difficult to show that if the signal f t is periodic with period T then its power P can be computed as follows T 2 P f ra 2 44 Parseval s relation for periodic signals says that the power content of a periodic signal is the sum of the power contents of its components in the Fourier series representation of the signal In particular if the exponential Fourier series coefficients of f t is given by the following equation fi Ren 2 45 then Parseval s theorem says that the power P of f t is equal to 2 2 YF n n oo 2 46 F is the magnitude square of the complex number Fn An alternative n In the above equation expression of Parseval s theorem for periodic signals says that the power of a periodic signal f t is equal to ay DI b 2 47 n l where the a s and b s in the above equation are the trigonometric Fourier series coefficients of the periodic signal f r Besides being an alternative way of calculating the power of a periodic signal Parseval s relation allows us to calculate the amount of power of a periodic signal that is allocated to each one of its harmonic components 3 2 Example 3 a Find the power contained in the first harmonic of the p
28. 15 0 20 J 0 01 0 02 0 03 0 05 0 07 0 10 0 13 0 16 0 20 0 24 0 27 0 31 0 34 0 37 0 40 0 42 0 43 0 43 0 43 0 42 0 40 0 36 0 33 0 28 0 23 0 17 J4 0 01 0 01 0 01 0 02 0 03 0 05 0 06 0 08 0 11 0 13 0 16 0 19 0 22 0 25 0 28 0 31 0 34 0 36 0 38 0 39 0 40 0 40 0 39 0 38 J 0 01 0 01 0 02 0 02 0 03 0 04 0 06 0 07 0 09 0 11 0 13 0 16 0 18 0 21 0 23 0 26 0 29 0 31 0 33 0 35 5 25 Js 0 01 0 01 0 01 0 02 0 02 0 03 0 04 0 05 0 06 0 08 0 09 0 11 0 13 0 15 0 18 0 20 0 22 J 0 01 0 01 0 01 0 02 0 02 0 03 0 03 0 04 0 05 0 07 0 08 0 09 0 11 Ji 0 01 0 01 0 01 0 01 0 02 0 02 0 03 0 04 0 05 0 01 0 01 0 01 0 01 0 02 0 01
29. Fourier transform of m t 4 21 fi t fx t 2T m t Figure 4 16 DSB LC Signals for two information signals f t and f t F o Wc Om We Qct Om Oc Om We OctOm 0 Figure 4 17 FT of a DSB LC signal Consider now the special case where the information signal f t is of the form f t K cos o t 4 33 4 22 Based on our previous discussion the maximum amplitude of the pure carrier that we need to add to the DSB SC modulated signal to make it a legitimate DSB LC signal must satisfy the following inequality AK 4 34 Then K m t ii t cost cos o t 4 35 We usually define K 4 36 ka 4 36 the modulation index of the system Since m s maximum value is one and its minimum value is Zero it is often given in percent In Figure 4 18 we show the DSB LC signal for the case of an information signal that is sinusoidal and for various m values 1 e m 1 m lt land m 1 The case m is not allowed because then we are not going to be able to recover the information signal f t In Figure 4 19 we show a modulated wave for an arbitrary information signal f t and a modulation index m 1 or m gt 1 4 23 COS Wy A cos wel mA COS wml COS wel Figure 4 18 Effects of varying modulation indixes 4 24 f mc A 2 min f r Envelope of 1 Envelope of o f P is i l Figure 4 19 Importance of sufficient carrier i
30. LABORATORY MANUAL DEPARTMENT OF ELECTRICAL amp COMPUTER ENGINEERING UNIVERSITY OF CENTRAL FLORIDA EEL 3552 Signal Analysis amp Communications Revised UCF Valencia West January 2012 Table of Contents Safety Rules and Operating Procedures i ii Experiment 1 Spectrum Analysis 1 1 1 5 Experiment 2 Periodic Signal Spectra 2 1 2 21 Experiment 3 Low Pass Filter 3 3 30 Experiment 4 Amplitude Modulation 4 4 42 Experiment 5 Frequency Modulation 31 5 25 mO Safety Rules and Operating Procedures Note the location of the Emergency Disconnect red button near the door to shut off power in an emergency Note the location of the nearest telephone map on bulletin board Students are allowed in the laboratory only when the instructor is present Open drinks and food are not allowed near the lab benches Report any broken equipment or defective parts to the lab instructor Do not open remove the cover or attempt to repair any equipment When the lab exercise is over all instruments except computers must be turned off Return substitution boxes to the designated location Your lab grade will be affected if your laboratory station is not tidy when you leave University property must not be taken from the laboratory Do not move instruments from one lab station to another lab station Do not tamper with or remove security straps locks or other security d
31. SC signal To determine the frequency response characteristics of the filter let us consider the demodulation of the VSB SC signal m t We multiply m t by the carrier component cos a t and pass the result through an ideal lowpass filter as shown in Figure 4 22 Thus the product signal is e t m t cos a t 4 48 or equivalently M a a M o o 4 49 4 29 If we substitute for M v from equation 4 47 into equation 4 48 we obtain Mio clo 20 Florio o j 4 50 ta V F o 20 H o o The lowpass filter rejects the double frequency components and passes only the components in the frequency range le 0 Hence the signal spectrum at the output of the ideal lowpass filter is 0o Flotte o H o o 4 51 VSB SC signal o t Lowpass filter cosoxt Figure 4 22 Demodulation of the VSB SC signal m t We require that the information signal at the output of the lowpass filter be undistorted Therefore the VSB SC filter characteristic must satisfy the condition H o 0 H o to constant e Eo 4 52 The condition is satisfied by a filter that has the frequency response characteristic shown in Figure 4 23 We note that H v selects the upper sideband and a vestige of the lower sideband It has an odd symmetry about the carrier frequency o in the frequency range O O0 lt lt Where o is a conveniently selected frequency that is some small fraction of i e
32. We want to be able to derive a formula for the bandwidth of an FM signal m t for an arbitrary information signal f t Let us revisit the approximate FM signal bandwidth formula 5 33 derived for a sinusoidal information signal B X Yo 2ak a 5 38 The second equality in 5 38 is obtained by substituting the value of with its equal Now let us pay a closer look at the two terms involved in the evaluation of the approximate bandwidth B The first term ak is the maximum frequency deviation of the instantaneous frequency olt from the carrier frequency it is often denoted by Aw The second term o is the maximum frequency content of the information signal f t Keeping these two clarifications in mind we now define the approximate bandwidth of an FM signal f t to be equal to B 2 o o 5 39 5 10 where Ao is the maximum frequency deviation from the carrier frequency and o is the maximum frequency content of the information signal f r It is not difficult to show that for an arbitrary information signal f t Ao k max f t Furthermore to find we first need t to compute the FT of the information signal f r Hence based on the above equation we can claim that the approximate bandwidth of an FM signal is computable even for the case of an arbitrary info signal It is worth pointing out that the bandwidth formula given above has not been proven to be true for FM signals produced by arbitrary info
33. a periodic signal in the time and frequency domain 2 21 EXPERIMENT 3 Low Pass Filter 1 Objective To observe some applications of low pass filters and to become more familiar with working in the frequency domain 2 Theory 2 1 Systems A communication system consists of three major components the transmitter the channel and the receiver The transmitter and the receiver are comprised of a cascade of black boxes that accept input signals produce output signals and they are referred to as systems This section is devoted to understanding what a system does and clarifying the various types of systems used in the transmitter and receiver for a communication system Definition A system is a rule for producing an output signal g t due to an input signal f r If we denote the rule as T then s t TIF o 3 1 An electric circuit with some voltage source as the input and some current branch as the output is an example of a system Note that for two systems in cascade the output of the first system forms the input to the second system thus forming a new overall system If the rule of the first system is T and the rule of the second system is T then the output of the overall system due to an input f t applied to the first system is equal to g t such e r T r Fol 3 2 There are a variety of classifications of systems that owe their name to their properties In this section we will focus only in the classifi
34. ab introduction Show that this transfer function when applied to your square wave will closely predict the output amplitude you measured Note that this transfer function can be reduced to a constant multiplied by the integral of the input signal Also note that your DC level may vary from that calculated because the DC level will depend on at what point on the square wave the integration began 3 30 Laboratory 4 Amplitude Modulation 1 Objective To understand the amplitude modulation 2 Theory 2 1 From Fourier series to the Fourier Integral The Fourier series is a means for expanding a periodic signal in terms of complex exponentials This expansion considerably decreases the complexity of the description of the signal and simultaneously this expansion is particularly useful when we analyze LTI systems We can extend the idea of the Fourier series representation from periodic signals to the case of nonperiodic signals That is the expansion of a nonperiodic signal in terms of complex exponentials is still possible However the resulting spectrum is not discrete any more In other words the spectrum of nonperiodic signals covers a continuous range of frequencies To be able to demonstrate our point consider for a moment the periodic signal shown in Figure 4 1 T T2 2 0 1 2 T 2 T Figure 4 1 A periodic signal f1 t We call this periodic signal f t to remind ourselves that we are dealing with a periodic signal of pe
35. al impedance R The charging time constant r R JC must be short compared with the carrier period 27 0 that is r R lt lt 2 4 37 O so that the capacitor C charges rapidly and thereby follows the applied voltage up to the positive peak when the diode is conducting On the other hand the discharging time constant R C must be long enough to ensure that the capacitor discharges slowly through the load resistor R between positive peaks of the modulated wave but not so long that the capacitor voltage will not discharge at the maximum rate of change of the modulating wave that is 27 lt lt RC lt lt 2b 4 38 e e 4 m where o is the information signal bandwidth The result is that the capacitor voltage or detector output is nearly the same as the envelope of the AM wave as shown in Figure 4 20 One of the serious flaws of a DSB LC system is that it wastes transmitter power To illustrate that consider the DSB LC signal of equation 4 31 It is easy to show that Power of m t 2 Power of Acos a t Power of f t cos a t Power of 2AF rcos o r 4 39 The power of Acos o t is equal to A 2 while the power of f t cos t can be shown to be equal to the one half the power of f t the latter result is valid under the legitimate assumption that the bandwidth of the information signal is much smaller than the carrier frequency o Finally the power of J2Af t cos a t is equal to zero under the
36. allows to compute the output of a LTI system due to an input without having to compute the convolution Also the Fourier transform of the signal contains information about the frequency content of the signal Equations 4 11 and 4 12 or equations 4 14 and 4 15 are rarely used for the computation of the Fourier transform of signals Normally the Fourier transform of a signal is computed by utilizing Fourier transforms of well known signals see Table 4 1 and Fourier Transform Properties see Table 4 2 The Linearity Property says that the Fourier Transform FT is a linear operation and if We know the FT of two signals We can easily compute the FT of any linear combination of these two signals The Scaling Property says that by contracting expanding the signal in the time domain through multiplication with an appropriate time constant results in the expansion contraction of the FT of the signal through multiplication with the inverse of the constant this is another ramification of the time bandwidth product principle Note that contraction expansion of the signal in the time domain corresponds to making the signal faster slower The Delay Property illustrates something that you might have suspected all along Delaying a signal in the time domain does not change the frequency content of the signal The frequency content of the signal is determined by looking at the magnitude of the FT of the signal The delay property reiterates tha
37. and Figure 4 26 2 Connect the oscilloscope to the function generator output or the message oscillator output V t and to the modulator output V t using direct coupling DC throughout Set V t to a sinewave with f 100 Hz in 4 36 3 4 3 6 Adjust the dc offset and amplitude of the message of the message oscillator to yield a DSB LC modulated waveform having a modulation index m 75 Sketch V t and V t Change V t to a triangular waveform and sketch V t and V rt Adjust the dc offset of the message oscillator to zero thus producing DSB SC Again sketch V t and V t first for V t sinusoid and then V t triangular Obtain once again the DSB SC signal for V t sinusoidal and f 100 Hz Observe and sketch the spectra of V tr and V r using the spectrum analyzer To see the modulated output you may adjust the frequency display of the spectrum analyzer to the value equal to the value of the carrier frequency Repeat part 5 for DSB LC Furthermore using the spectrum analyzer determine the modulation index m Vin Vo Message impul Carrier 2 input S R 11K Figure 4 25 AM Modulation R C 1500 pF R 11K To FET gate input C Figure 4 26 Carrier generator 4 37 6 2 Amplitude Demodulation The overall general communication system block diagram is given in Figure 4 27 You have already built the modulator As a channel use a small length of wire There are two types
38. and negative values If we now look at the positive peaks of m t we see that a line through them produces the information signal f t On the other hand if we look at the positive peaks of m t a line through them produces the information signal f t only when f t is positive and it produces the negative of f t when the signal is negative The reason that the positive peaks of m t and m r are important is because we can design an inexpensive receiver that traces these positive peaks Hence when the signal f r is always positive this receiver will be able to reproduce the information signal f t from m t this is not the case though when the signal m r assumes positive and negative values Unfortunately information signals can be positive or negative To remedy the problem of a negative information signal we can add a constant to the signal that is larger than or equal to the most negative signal value Hence now m t A f t cos t Acos a t f t cos a t 4 31 The Fourier transform of m t is shown in Figure 4 17 and it is provided by the following expression M o Az o c c Srle Flo o 4 32 By looking at Figure 4 17 we can justify the name DSB LC for this communication system The Fourier transform of the modulated wave m t has both of the sidebands upper and lower Furthermore the presence of the pure carrier Acos a t is evident by the presence of the two impulses in the
39. aracteristics of Modulation MODULATION 1 AMPLITUDE AM 2 FREQUENCY FM 3 PULSE WIDTH PWM 4 PULSE AMPLITUDE PAM 5 PULSE CODED PCM 6 DELTA Am 7 DELTA SIGMA A Em CHARACTERISTICS Disregarding noise original signal can be reproduced exactly at the receiver Very susceptible to noise of all kinds Can transmit and decode wide band width signals Disregarding noise original signal can be reproduced exactly at receiver Not very susceptible to noise Limited bandwidth Similar to fm except some accuracy in reproducing the message is lost Easy to demodulate LPF Similar to AM except some accuracy is lost Can be used in multiplexing Only discrete voltage levels can be encoded Repeaters can be used to reduce noise in long distance transmission since the code is binary Inefficient use of time because it requires synchronization bits Can handle fast slew rates Does not need synchronization Simple 2 bit code Cannot accurately encode fast slew rates Same as delta except it can handle fast slew rates 4 42 Laboratory 5 Frequency Modulation 1 Objective To understand the principles of frequency modulation and demodulation 2 Theory 2 1 Introduction A sinusoidal carrier c t Acos w t 0 has three parameters that can be modified modulated according to an information signal f r 1 Its amplitude A which leads us to the class of systems designated as amplitude modulat
40. ated on the wall near the door Proceed to Student Health Services if needed ii Laboratory 1 SPECTRUM ANALYSIS OBJECTIVE Analyze the spectral content of a simple signal EQUIPMENT TDS5052B Oscilloscope Function Generator You should bring a floppy disk to store your waveforms BACKGROUND A waveform representing amplitude as a function of time is called a time domain display It is also possible for a waveform to represent amplitude as a function of frequency This is called a frequency domain display A Spectrum Analyzer is an instrument which can display the frequency domain of a signal However the TDS5052B Oscilloscope has the capability of producing both time domain and frequency domain displays see Figure 1 1 A sine wave is the simplest signal for spectral analysis The amplitude of the sine wave can be determined on the vertical scale and the frequency can be determined on the horizontal scale The units of amplitude used in this experiment will be dBV which is dB relative to 1 VRMS 0dBV z1VRMS according to the formula ABV doie 5 ema Vref where Vsignal is the RMS voltage of the signal and Vref 1 volt RMS Another common unit of amplitude used for spectrum analysis is dBm which is dB relative to 1 milliwatt 0dBm 1mW according to the formula sinless ImW where Psignal is the power of the signal in milliwatts 1 1 1 Front Panel of TDS 5052B s Floppy disk drive
41. atements is to take the derivative of f t with respect to time In particular f ano cos not 2 3 From the above equation we observe that as n increases the maximum rate of variation of f t with respect t time increases We can represent each one of the f t functions in a little bit different way than the one used in Figure 2 2 That is we can represent these functions by plotting the maximum amplitude of f r i e a versus the angular frequency of the sinusoid i e n The product of these two values gives you the maximum rate of variation for the signal under consideration In Figure 2 3 we are representing the functions f t f t and T t following the aforementioned convention Obviously larger values for the location of the plotted amplitudes imply faster varying time signals also the larger the amplitudes plotted at a particular location the faster the corresponding signals vary with respect to time fit AR A PF Figure 2 2 Plots of f t and f t Representation of f t f t f t Oo 209 30 0 Figure 2 3 Reresentation of fi t f2 t f3 0 Now consider a simple example of an amplitude modulated signal 2 3 f t A 1 cose t coso t O lt lt a 2 4 In Figure 2 4 we plot the aforementioned signal f t As we can see from the figure the amplitude varies slowly between 0 and 2A Its rate of variation is given by the modulating frequency o is referred to as t
42. c with period 7 Do then the output of this system is periodic with the same period Observation 2 Only the frequency components that are present in the input of a LTI system can be present in the output of the LTI system This means that a LTI system cannot introduce 3 14 new frequency components other than those present in the input In other words all systems capable of introducing new frequency are either nonlinear systems or time varying systems or both 2 7 Frequency transfer Function Example Let f t be a signal as shown in Figure 3 6 This signal is passed through a system filter with transfer function as shown in Figure 3 7 Determine the output g t of the filter Figure 3 6 input signal f t to the filter in Frequency Transfer Function Example 3 15 127 x10 12x x10 ZH o T 2 n 2 Figure 3 7 Transfer function of the filter in Frequency Transfer Function Example We first start with the Fourier series expansion of the input This can be obtained by following the Fourier series formulas It can be shown that f t 45 CI ospr 1 10 r Z epa Z gpr n 0 2 emot 2 je 2 now 2 nono 621071 j6z10 t 10z10 t j10z10 t PP omg PEUT gl UT p e Tt T 3 18 3x 3x 5a 5a To find the output of the system we need to find the output of the system due to every complex exponential involved in the expansion of f t A closer inspection of H w though tells us that only the
43. cation of systems into the linear versus nonlinear categories and the time invariant versus time variant categories If a system is linear then the principle of superposition applies The output of a system with rule T that satisfies the principle of superposition exhibits the following property T a f t anf e l aT f e l T am e l 3 3 3 1 Where a and a are arbitrary constants A system is nonlinear if it is not linear A linear system friendly system is usually described by a linear differential equation of the following form a t e a e 0 atg C a t f 8A Where g r designates the output of the system while f t designates the input of the system In the above equation g r denotes the k th time derivative of the function g r Virtually every system that you consider in your circuit classes e g first order R C R L circuit second order R L C circuits is an example of linear systems Any circuit that has components whose v i curve is nonlinear e g a diode is likely to be a nonlinear system Another useful classification of systems is into the categories of Time Invariant versus Time Varying systems A system is time invariant if a time shift in the input results in a corresponding time shift in the output Quantitatively a system is time invariant if g t t T r 1 3 5 For any tand any pair of f t and g t where f t denotes an input to the system and g t denote
44. d 2 36 is equal to the signal f t see Figure 2 6 in the interval z 2 z 2 Also the RHS s of equation 2 27 and 2 36 are periodic with period z Hence the plot of Figure 2 8 is produced by reproducing the plot of Figure 2 7 every z units of time Figure 2 8 Plot of the RHS of equation 2 27 2 3 2 Example 2 Evaluate the trigonometric Fourier series expansion of the periodic signal f t which is depicted in Figure 2 9 f t T 17 2 5 4 7 4 1 1 2 4 t 1 Figure 2 9 plot of f t Let us concentrate on one period of the signal that is the interval 2 7 2 Knowing that T z we conclude that 2z T 2 The trigonometric Fourier series of a signal f t is therefore of the form f t a Ya cos 2nt yo sin 2nt 2 37 n l n l 2 12 where f fled r0 2 38 lt fs t cos 2ntdt 2 7 4 2 7 4 2 zj2 i zr i5 cos2nt dt i fe 2ntdt ml cos 2nt dt 22 sin nz 2 2 39 nz 2 2 rr be Iz f r sin 2ntdt 0 2 40 The last equation is a result of the fact that f r is even and sin2nt is odd Consequently f t sin2nt is odd and whenever an odd function is integrated over an interval which is symmetric around zero the result of the integration is zero Based on the above equations we can write f t der cos 2nt 2 41 nz Also if we write out a couple of terms from the above equation we get 4 1 1 1 f t A cosa cos 6t cos10t cos
45. d Calculations 1 Why must m be less than 100 for envelope detection 2 Compare Envelope Detection to Synchronous Detection and list at least one advantage and one disadvantage of each 4 39 TABLE 4 1 Some Selected Fourier Transform Pairs f t 1 e u t 2 te u t 3 z 4 et 120 5 sgn t 6 j at 7 u t 8 5 t 9 1 10 e 11 cos ot 12 sin ot 13 rect t z 14 sa we 2 2a 15 Y sa wr a 16 A t c 17 sa wi 2 P 2m 18 cos zt z rect t r 9 2W cos Wt A 1 QWt z 20 6 t F O l a jo a je 2a a o o o 2 ov27e 2 jo sen o z c jo 1 2z o 2z o Fo z e e e o jz e o 0 o tSalor 2 rect a W rect 2W t Sa r 2 P A o W 2r cos z 2 z l oc n cos zo 2W recr eo 2W Q o where 22 T 4 40 TABLE 4 2 Fourier Transforms Properties Operation Linearity Superposition Complex conjugate Scaling Delay Frequency translation Amplitude modulation Time convolution Frequency convolution Duality time frequency Symmetry even odd Time differentiation Time integration f t e f t f t cos aot A rac ff F o a F o a F lo F 0 a e a e F a F o o 1 1 5 Flo 0 gt F o ox F o F 0 sf WF udu 2af o real imaginary F v F o joF a x Feo ero where F 0 L f t at Table 4 3 Ch
46. e c t as follows m t c t f t A cos a t f t 4 20 Consequently the modulated signal m t undergoes a phase reversal whenever the formation signal f t crosses zero as indicated in Figure 4 4 The envelope of a DSB SC modulated signal is therefore different from the information signal a Baseband signal b DSB SC modulated wave Figure 4 4 The information signal f t and modulated signal m t The Fourier transform of m t can easily be determined from the Fourier transform properties of Table 4 2 That is Mo 7 A F o o F o o 4 21 For the case when the baseband signal f r is bandlimited to the interval c c as in Figure 4 5 a we thus find that the spectrum of m t is as illustrated in Figure 4 5 b Except for a change in scale factor the modulation process simply translates the spectrum of the baseband signal by Note that the transmission bandwidth required by DSB SC modulation is equal to 2 signal f t where stands for the maximum frequency content of the m F o F 0 Om 0 Om A M o V A F 0 Oc c Figure 4 5 a Spectrum of baseband signal b Spectrum of DSB SC modulated wave The baseband signal f t can be uniquely recovered from a DSB SC wave m t by first multiplying m t with a locally generated sinusoidal wave and then low pass filtering the product as in Figure 4 6 It is assumed that the local oscillator signal is exactl
47. ear term proportional to the derivative of the information signal f r So we can say that the PM case is in reality the FM case but with an information signal being the derivative of the actual information signal In other words if we have a device that produces FM signals we can make it to produce PM signals by giving it as an input the derivative of the information signal The above equation also tells us that the FM case is in reality a PM case with the modulating signal being the integral of the information signal Also if we have a device that produces PM we can make it to produce FM by providing to it as an input the integral of the information signal Hence what it boils down to is that we need to discuss either PM or FM and not both We choose to focus on FM which is used to transmit baseband analog signals such as speech or music PM is primarily used in transmitting digital signals Our primary focus in the examination of FM signals will be the analysis of its frequency characteristics Although it has been a straightforward task to find the Fourier transform of an AM signal the same is not true for FM signals Let us again consider the general form of an FM signal m t Acos a t cos k g t Asin o t sin g r 5 15 Where g t fdu If we are able to find the FT of cos k g r and sin k e r we can produce the FT of the signal m t without a lot of effort The FT of cos k f g t and sin k f g t cannot be f
48. en as well 5 13 untitled E x File Edit view Simulation Format Tools Osaa sese 2c is Dizcrete wb Dizcrete time Fhasze locked YCO loop Ready 100 lode45 E Signal Generator Block Path Signal Generator Simulink gt Source Discrete time VCO Communications Block set gt Comm Sources gt Controlled Sources Phase Lock Loop Communications Block set gt Synchronization Scope Simulink gt Sink Just drag and drop the blocks you need for your simulation in your work window Click the left mouse button hold and drag to connect the blocks by wire The parameters for the individual module have been given as an example To set the parameters of a block just double click on it 5 14 Signal Generator Wave form sine Amplitude 1 Frequency 1000 Discrete time VCO Block Parameters Discrete Time VCO 5 15 Phase Lock Loop Block Parameters Phase Locked Loop ox Phase Locked Loop mask link Implement a phase locked loop The three outputs are the lowpass filter output the phase detector output and the voltage controlled oscillator VEO output The input must be a sample based scalar signal r Parameters Lowpass filter numerator 3 0002 0 40002 Lowpass filter denominator ri 67 46 2270 9 40002 YEO input sensitivity Hz e YEO quiescent frequency Hz 1009 WCO initial phase rad o VCD
49. er Multiplexing The quadrature null effect of the coherent detector may also be put to good use in the construction of the so called quadrature carrier multiplexing or quadrature amplitude modulation QAM This scheme enables two DSB SC modulated waves resulting from the application of two physically independent information signals to occupy the same channel bandwidth and yet it allows for the separation of two information signals at the receiver output It is therefore a bandwidth conservation scheme Message signal Message Multiplexed signal fit Product m t Multiplexed p modulator signal A cos axt 90 phase shifter A Sin ct 1 f t Product signal p modulator a Product Yy Ac Ao fit modulator ___ a es L ilter Ac COS Wct 90 phase shifter A Sin ct V5 Ac A f t Product Low pass modulator J 9 filter LL b Figure 4 8 Quadrature carrier multiplexing system a Transmitter b Receiver 4 12 A block diagram of the quadrature carrier multiplexing system is shown in Figure 4 8 The transmitter part of the system shown in Figure 4 8 a involves the use of two separate product modulators that are supplied with two carrier waves of the same frequency but differing in phase by 90 degrees The transmitted signal m t consists of the sum of these two product modulator outp
50. eriodic signal f r of Example 1 2 14 From the trigonometric Fourier series expansion of f t see equation 2 27 we observe that the first harmonic of f t is equal to rus cos 2r 2 48 3x It is not difficult to show that the amount of power contained in the first harmonic is equal to 4 8s I 2 as Parseval s relation predicts b Find the power contained in the first and second harmonics of the periodic signal f t of Example 1 From the trigonometric Fourier series expansion of f t see equation 2 27 we observe that the first and second harmonic of f r is equal to 25 eos ara 5 nsn 2 49 3m 15z It is not difficult to show that the amount of power contained in the first two harmonics is equal o eF lose 2 2 as Parseval s relation predicts c Find the power contained in the DC component of the periodic signal f t of Example 1 From the trigonometric Fourier series expansion of f t see equation 2 27 we observe that the DC component of f t is equal to 2 z Hence the amount of power contained in the DC component is equal to 2 z 2 15 4 Pre lab Questions 1 Find both the trigonometric and the exponential Fourier series expansion of the signal f t in Figure 2 9 Plot the amplitude frequency for the signal Identify the first three harmonics and their amplitude Find the power content from the time domain representation of the signal and from the first three harmonics of the
51. es a little bit above c we are referring only to the positive frequencies of the upper sideband signal To relax the requirement for such a perfect filter we will consider another way of generating the upper sideband version of the signal m t f 4 14 Sideband m t Remover Channel o t Low pass gt _ gt Amplifier filter cosw t Figure 4 10 Block diagram of a SSB SC system 4 15 20c Om Om 20e 0 ew Figure 4 11 FT s at various stage of the SSB SC system of Figure 4 10 4 16 For the sake of simplicity consider first the case where the information signal is sinusoidal i e f t cos t Then the Fourier transform of m t cos c t cos o t is given in Figure 4 12 From Figure 4 12 it is easy to identify the upper sideband of m t in the frequency domain also the time domain representation of this upper sideband is readily defined as follows 1 m t zolo tO y 4 25 or m t c0s 7 1 c05 0 sin c 1 sin t 4 26 M o upper sideband lower sideband Figure 4 12 Fourier transform of m t cos t In Figure 4 13 we depict a way of constructing the upper sideband signal of equation 4 26 without having to resort to the design of perfect filters In Figure 4 13 there is a system designated by a block with the 90 notation inside it We know what this block does to an input signal that is sinusoidal it
52. es can be proven rigorously They have been verified above for an example case In the special case where the pulse rectangle f t and f t are defined to be nonzero over an interval of the same width convolution f t turns out to be a triangle Rules 2 and 5 stated above for the convolution of two rectangular pulsed extend for the case of the convolution of arbitrary shaped finite duration pulses Some useful properties of the convolution are listed below These properties can help us compute the convolution of signals that are more complicated than the rectangular pulses 1 Commutative Law f O f f 0 fO 2 Distributive Law fO 50 foO f fO fo fo 3 Associative Law 3 8 LOMO LOEO fo fo 4 Linearity Law o f O FO 9 ef 0 FO Where o in the above equation is a constant 2 5 Impulse Function The impulse function showed up in the discussion of the convolution integral and linear time invariant system We said then that the convolution integral allows us to compute the output of a linear time invariant system if we know the system s impulse response The impulse response of a system is defined to be the response of the system due to an input that is the impulse function located at t 0 The impulse or delta function is a mathematical model for representing physical phenomena that take place in a very small time duration so small that it is beyond the resolution of the measuring instrument involved and
53. evices Do not disable or attempt to defeat the security camera ANYONE VIOLATING ANY RULES OR REGULATIONS MAY BE DENIED ACCESS TO THESE FACILITIES I have read and understand these rules and procedures I agree to abide by these rules and procedures at all times while using these facilities I understand that failure to follow these rules and procedures will result in my immediate dismissal from the laboratory and additional disciplinary action may be taken Signature Date Lab Laboratory Safety Information Introduction The danger of injury or death from electrical shock fire or explosion is present while conducting experiments in this laboratory To work safely it is important that you understand the prudent practices necessary to minimize the risks and what to do if there is an accident Electrical Shock Avoid contact with conductors in energized electrical circuits Electrocution has been reported at dc voltages as low as 42 volts 100ma of current passing through the chest is usually fatal Muscle contractions can prevent the person from moving away while being electrocuted Do not touch someone who is being shocked while still in contact with the electrical conductor or you may also be electrocuted Instead press the Emergency Disconnect red button located near the door to the laboratory This shuts off all power except the lights Make sure your hands are dry The resistance of dry unbroken skin is relatively
54. f piecewise continuous real time signals defined over an interval 0 7 Let us also consider the set of signals 1 fio RS 2 6 Pin Fens n ot 2 7 do e sin ny n 12 2 8 where 2z T The aforementioned signals see equations 2 6 2 7 and 2 8 are orthonormal They also constitute a complete set of functions These two properties allow us to state that an arbitrary signal in S can be expressed as a linear combination of the signals in equations 2 6 2 8 such that f t Coho S Cph 6 6 2 9 n l n l and the coefficients in the linear expansion above can be computed through the following equations C MOS 2 T C 7 J f eos noqat n 12 2 10 2 Tay C yf sin nevotat n 1 2 The above expansion due to Fourier is referred to as the Trigonometric Fourier Series expansion of a signal f e A more common form for the trigonometric Fourier series expansion of a signal is given below oo oo f r a Van COSN ot gt b sinnogt 2 11 n l n l where pT a 7 f u 2 12 2 pT a 7 f f t cosna tdt n212 2 13 2 pT ne f f t sinn tdt n242 2 14 By a simple trigonometric manipulation we get that f r ag gt Na b cos nogt 0 0 aran 2 15 n l a n The above equation puts f t in the desired form In other words we have now expressed f r as a linear combination of sinusoids and as a result we can represent this signal f r by plotting the amplit
55. forms 2 16 6 1 Sine Wave In this section you will observe when the signal frequency of a sine wave is shifted the spectral line will also shift in frequency a Turn on the Oscilloscope and allow a minute for the instrument to boot and stabilize Then press Default Setup to clear settings made by other students Turn on the Function Generator and connect the output to Channel 1 input of the oscilloscope Adjust the function generator for the oscilloscope to display 10 cycles of a 1 KHz 0 5 volt peak sine wave with zero DC offset Do not change the Horizontal scale setting of the oscilloscope during the remainder of the experiment Do not change the amplitude or DC offset settings of the Function Generator during the remainder of the experiment Note Be sure the Function Generator Duty Cycle knob is set to Cal b Use the mouse to click the Math Button on the Toolbar and select Spectral Setup to open the Control Window Under the Create tab click the Magnitude 1con and click Channel 1 as source Click Apply c Click the Mag tab and select Linear for the vertical scale factor Click the Scale set the vertical scale to 100 mv rms per division and click Enter Click Apply Click OK to close the Control Window d Press the Zoom icon Set the Zoom Position to 2 5 and the Zoom Factor to 20 e Determine the amplitude and frequency displayed for the spectrum of the sine wave Save the waveform to include in your report f Change
56. going such that J 2 lt 0 01 for n gt naa This criterion is often called the 1 criterion for the evaluation of bandwidth If we find that N nax 1S the minimum index n that does not violate the 1 criterion then we can claim that the to exclude all terms of the infinite sum with index n max approximate bandwidth of our signal according to the 1 criterion is B 2n 0 5 34 5 9 It is worth mentioning that the evaluation of bandwidth based on equation 5 33 corresponds to the bandwidth of your FM signal according to a 10 criterion The above procedure followed for the evaluation of the FT of m t can be extended to the cases where the information signal f t is a sum of sinusoidal signals or a periodic signal In particular if f t a cos t a cos a 1 5 35 then the phase of our FM signal m t is provided by the following equation k k olt ot 2 sin o t 4 sin o 1 5 36 0 0 Omitting the details we arrive at a representation of the signal m t such that m r 2 AM J BDI B cos no ka t 5 37 n 0 k 0o where a k o and B a k o As we can see from the above equation we now have impulses at o n t ko as well as o t no t ko Most of the aforementioned discussion regarding the FT and the bandwidth of an FM signal m r is based on the assumption that the information signal f t is a sinusoid a sum of sinusoids or a periodic signal
57. hat the pass band extends from frequency o 0 to o to where is known to be the cutoff frequency See Figure 3 8 a A high pass filter is the complement of a low pass filter Here the previous frequency range 0to o 1s the stop band and the frequency range from o o to positive infinity and q to negative infinity is the pass band See Figure 3 8 b A band pass filter is defined as the one where frequencies from to are passed while all other frequencies are stopped See Figure 3 8 c 3 17 1 lt B gt Low pass a 0 0 O e Ho 1 High pass b 1 lt B gt Band pass c 0 0 0 0 a o0 o o Ho 1 Band stop d 1 All pass p e 0 oO Figure 3 8 The magnitude of the frequency transfer function of low pass high pass band pass band stop and all pass In all cases the responses are ideal 3 18 A band stop filter is the compliment of the band pass filter Frequencies from o to o are stopped here and all other frequencies are passed See Figure 3 8 d In the Figure 3 9 below a practical non ideal 6 order Butterworth low pass filter is shown In Figure 3 9 the various filter characteristics such as pass band cut off frequency are clearly indicated In figure 3 10 we are illustrating how the filter characteristics change with the order of the filter 5 T T Passband 6 order Low pass Butterworth Filter gain 0 oo p E poss pee 3 dB point p eem cialis i
58. have to transform this 4 6 electrical signal into an electromagnetic wave and send it to destination B The way to achieve that is by using an antenna Knowing though that the maximum frequency content of a voice signal is around 15 KHz we come to the conclusion that the antenna required must be many miles long A reasonable size antenna could adapt the signal to the channel only if the signal s lower frequency is higher than 800 KHz Hence in our case where a voice signal needs to be transmitted using a reasonable size antenna it suffices to raise the frequency content of the voice signal to the vicinity of 800 KHz To accomplish the aforementioned goal we ought to multiply the voice signal with a sinusoidal signal The result of this multiplication is to latch our f t the voice signal on the amplitude of the sinusoid or to modulate the original amplitude of the sinusoid with f t This procedure gives rise to a class of systems that are called amplitude modulation systems We distinguish four major classes of amplitude modulation systems Double Sideband Suppressed Carrier Systems DSB SC Single Sideband Suppressed Carrier Systems SSB SC Double Sideband Large Carrier Systems DSB LC and Vestigial Sideband Suppressed Carrier Systems VSB SC 2 6 Double Sideband Suppressed Carrier System DSB SC Basically double sideband suppressed carrier DSB SC modulation arises by multiplying the information signal f t and the carrier wav
59. he bandwidth of the FM signal m t is also finite To investigate the bandwidth of an FM signal thoroughly we will discriminate two cases of FM signals The case of Narrowband FM and the case of Wideband FM 2 6 Narrowband FM Consider again the FM signal m t given by the following equation m t Acos o t coslk g t A sin o r sin k g t 5 18 The terms for which FT is difficult to evaluate are cosk g t and sink f g t Each one of these terms has a Taylor series expansion involving infinitely many terms Let us see what happens if each one of these terms is approximated only by their first term in the Taylor series expansion Then cosk g t 1 5 19 sin k g t k g t Obviously if we make the above substitutions in equation 5 18 we get m t Acos t Ak g t sin o r 5 20 5 5 The advantage of the above equation is that we can evaluate its FT and consequently the FT of m r It is not difficult to see that the bandwidth of m r is approximately equal to 2 times the bandwidth of f t f t is the information signal Hence when the above approximations are accurate we are generating an FM signal whose bandwidth is approximately equal to the bandwidth of an AM signal Since in most cases an FM signal will occupy much more bandwidth than an AM signal the aforementioned type of FM signal is called narrowband FM In the sequel we are going to identify quantitatively conditions under which we ca
60. he carrier frequency The amplitude variations form the envelope of the complete signal and represent any information being transmitted Note that A 1 cos o t cos et Acos ot costo 0 t cos a a X 2 5 The representation of the aforementioned signal using the new conventions is illustrated in Figure 2 5 As we observe from Figure 2 5 larger carrier frequency corresponds to a faster varying signal also larger modulating frequency results in a faster varying signal Apa Figure 2 4 Plot of an amplitude modulated signal f t O0 Om O0 Otm Figure 2 5 Representation of the amplitude modulated signal f t The previously discussed signals vary in some sinusoidal fashion They do not carry any information but we have used them to focus on an alternative representation of the signal that conveys to us the information of how fast the signal changes with respect to time Another variation of the Fourier series representation of a signal represents the signal as a linear combination of sines and cosines of appropriate frequencies As a result the Fourier series representation of the signal which is an extension of the alternative representation of the 2 4 simple signals that we discussed above gives us an idea of how fast the signal changes with respect to time This is the second reason that the Fourier series representation of a signal is so useful 2 2 Fourier Series Let our space S of interest be the set o
61. high and thus reduces the risk of shock Skin that is broken wet or damp with sweat has a low resistance When working with an energized circuit work with only your right hand keeping your left hand away from all conductive material This reduces the likelihood of an accident that results in current passing through your heart Be cautious of rings watches and necklaces Skin beneath a ring or watch is damp lowering the skin resistance Shoes covering the feet are much safer than sandals If the victim isn t breathing find someone certified in CPR Be quick Some of the staff in the Department Office are certified in CPR If the victim is unconscious or needs an ambulance contact the Department Office for help or call 911 If able the victim should go to the Student Health Services for examination and treatment Fire Transistors and other components can become extremely hot and cause severe burns if touched If resistors or other components on your proto board catch fire turn off the power supply and notify the instructor If electronic instruments catch fire press the Emergency Disconnect red button These small electrical fires extinguish quickly after the power is shut off Avoid using fire extinguishers on electronic instruments Explosions When using electrolytic capacitors be careful to observe proper polarity and do not exceed the voltage rating Electrolytic capacitors can explode and cause injury A first aid kit is loc
62. iia LEES Meere Bees pees kasaan Dee P eee pameni k D ness iiie Dein Reef Rie dien NE D ES P PEE Deges E 2 eee eae Phi ser EEG Di ies ESR ear eae PRONTO ESR ESES Es cri E ss SE G ui l Passband i i ano OESR PRESS Meo oe NCC Ue ee ERES HERE dh RES RR ERR _ dB i j i i 25lL SEE poe EE poe PNE DD RN NE N i i i i 30 deck punoni decr panoni Calan dean os perc hace ET 1 PENER S o ON nae Las 4 Cut off Frequency 40 L l 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency Hz Figure 3 9 6 th order Low pass Butterworth Filter The following Figure 3 10 shows how the order of a filter changes the frequency response The higher the order the closer the filter behaviors as 1deal 1 2 T T T T idc kae Butterworth Filter La L How the frequency response changes with ord r 1 ee 3 nr Ager ce eras a ee Oe qucd MM KCN E CN er eng l pi ARA R 7 I Q gr Leere Ferne S Lore cede e Pee re E efe e Eee eene t Ws RNC presses TOME Gutoff frequency 3 rd EE hausse n a auensn b n Gain l Q G e aa diate iip pause 2ndorder ee AA ee deu DONEC esas be tees en 4 i penne ere QE EE NB NE NEN J i i pre Seem I 24th rder n 6thorder 0 2 d pe te ete a ie
63. imitations for most VCO s 1 The input voltage must be small usually there is an attenuating circuit at the input 2 The bandwidth is limited for a linear frequency to voltage relationship 5 3 PLL FM demodulator The phase locked loop PLL is used to demodulate FM The phases of the input and feedback signals are compared and the PLL works to make the phase difference between the two signals equal to zero Figure 5 4 shows a basic block diagram of a PLL 5 18 Phase Comparator Figure 5 4 Phase Locked Loop PLL block diagram The VCO in the feedback is an FM modulator and the center frequency can be set equal to the center frequency of m t The equation for m t is m t cos w t 27k flajda 5 54 where k f a da 0 1 and the equation of e t is e r coslo t k F a da 5 55 where k flajda 0 t Since each signal has the same center frequency w the phase comparator compares the instantaneous values of 0 t and 0 t The difference in phase is transformed into a DC voltage level proportional to the phase difference and then amplified to yield F t This voltage is the input for the VCO in the feedback As a result the difference between 6 t and 0 t will be made smaller This process occurs continually such that 6 t 6 t at all times Substituting for 0 1 0 t k F a da k f a da and r f t 5 56 d 5 19 where f t was the original message signal and F t i
64. in this lab 1 A LPF used as an integrator 2 A LPF used as a first harmonic isolator square to sine conversion 2 8 1 Integrator The Laplace representation of integration is 1 s where s j A first order low pass filter can approximate an integrator when gt gt ox then V V 1 s a For an n order low pass filter the transfer function will approximate n integrators in series at Vo s 1 Va s 7 s o y gt gt 0 Or For simplicity in this lab a first order LPF will be used to perform a single integration on a signal 2 8 2 Square to sine Converter Any periodic signal can be expressed as an infinite sum of orthogonal sinusoids For orthogonality each sinusoid or harmonic must have a different frequency and frequency must be an integer multiple of the frequency of the periodic signal For example if the frequency of a square wave is f then the n harmonic will have a frequency of n f where n is any integer from one to infinity You will see that some of these harmonics will have zero or negative amplitude The harmonics of a square wave of frequency f are shown in Figure 3 11 Not to scale the DC component of the square wave is assumed to be equal to zero Figure 3 11 The harmonics of a square wave form of frequency f 3 2 If the square wave excites the input of the low pass filter such that the first harmonic is in the pass band of the filter transfer function and the re
65. in time is duc doce 5 50 where k is positive for figure 5 1 and has units of Hz Volt Let f t Acos t then m ak m DL t 7 os 5 51 m The peak frequency deviation from is ak 2z radian sec and the total peak to peak deviation is 2 ak f 27 The modulation index f for this signal is ak 2o L 5 52 p f 5 52 Note that 7 will vary for each frequency component of the signal 5 17 5 1 Spectrum of FM The spectrum of an FM signal is described by Bessel functions As shown in section 2 7 for a single frequency constant amplitude message the spectrum of m r is m f 2 3 J B cos o t no t 5 53 ak n where 6 FM modulation index and J 5 ferao Which is the n 1 m order Bessel function evaluated at 2 Therefore for each frequency at the input there are infinite number of spectral components at the output with the amplitude or each component determined by the modulation index P The amplitude of the higher order terms will decrease on the average such that there will be a limited bandwidth where most of the energy is concentrated 52 VCO FM modulator A voltage controlled oscillator VCO converts the voltage at its input to a corresponding frequency at its output This is accomplished by a variable reactor varactor where the reactance varies with the voltage across it The varactor is part of a timing circuit which sets the VCO output frequency There are two l
66. ing AM systems 2 Its frequency which leads us to a class of systems designated as frequency modulating FM systems 3 Its phase 0 which leads us to a class of systems designated as phase modulating PM systems We have already discussed the class of AM systems In the sequel we focus on the class of FM and PM systems Note that we can write that c t Acos o t 0 Acos 0 r 5 1 where 0 r is often called the angle of the sinusoid That s why FM and PM systems are sometimes referred to as angle modulating systems 2 2 Preliminary notions of FM and PM Systems Consider the carrier c t Acos o t t 0 We can write c t Acos 0 r 5 2 5 1 where we call O t the instantaneous phase of the carrier If we differentiate O t with respect to time t we get a time function that we designate by c t and we call it the instantaneous frequency That is ol 57 5 3 It is easy to see that the above definition of instantaneous frequency makes sense if we apply it to a pure carrier c t Acos o t 0 because then we get oXt o 5 4 4 From equation 5 3 above we see that if you have the instantaneous phase of a sinusoid you can compute its instantaneous frequency by differentiation Furthermore if you know the instantaneous frequency olt of a sinusoid you can compute its instantaneous phase by integration as follows A t olut 0 5 5 Obviously if we start with olt we get tr 7 v
67. ion were introduced for a signal f t defined over the interval 0 7 Actually the formulas provided are valid for any signal f t defined over any interval of length T the starting and the ending points of they interval are immaterial as far as the length of the interval is equal to T We expanded so far a signal f t defined over an interval of length T in terms of a linear combination of sinusoids Trigonometric Fourier Series expansion or complex exponentials Exponential Fourier Series expansion It is worth noting that the trigonometric or exponential Fourier series expansion of a signal defined over an interval T is a periodic signal of period T that is it repeats itself with period T Hence if the signal of interest f t is not periodic the trigonometric or exponential Fourier series expansion of the signal is only valid for the interval over which the signal is defined Due to the periodicity nature of the Fourier series expansion Fourier series is primarily used to represent signals of periodic nature Signals of aperiodic nature can also be represented as a sum of sinusoids or complex exponentials but this sum is in reality an integral and it is designated by the name Fourier Transform The Fourier transform of an aperiodic signal will be introduced later as an extension of the Fourier series of a periodic signal 2 5 Examples 2 3 1 Example 1 a Evaluate the trigonometric Fourier series expansion of a
68. ions c n where n is an integer In reality though no matter how big 2 is the significant J B s will be only for indices n lt ff 1 Hence the approximate bandwidth of your FM signal when the information signal is of sinusoidal nature is given by the following equation B 2 B 10 5 33 In Figure 5 2 various plots of the Bessel function coefficients J are shown As we can see these plots verify our claim above that J 8 become small for indices n gt 1 In Figures 5 3 we show the FT of signals m t for various 2 values Figure 5 2 Plot of Bessel function of the first kind J 2 5 8 Wc Q Wc p 2 ua Ao a Ao p 2 in Oc e Wc B 5 ar Ao a Ao p ll i a NM Oc o Oc B 10 jaa Ao 7 4 Ao B 10 hin LU ct ei ele D tas 63 Wc We a Om Ao gt eE Aw I Figure 5 3 Magnitude line spectra for FM waveforms with sinusoidal modulation a for constant m b for constant Ao In Table 5 1 the values of the Bessel function coefficients J are shown for various values We can use the values of the Table 5 1 to evaluate the bandwidth of the signal m t as follows We are still operating under the assumption that the information signal is of sinusoidal nature As a result expression 5 32 is a valid representation of our signal m r Let us now impose the criterion that for the evaluation of the bandwidth of the signal m t we are
69. ll be easy to prove that the transfer function of a system is the Fourier transform of the impulse response of the system We have already stated that the impulse response of a linear time invariant system describes the system completely We can make a similar statement about the transfer function of a linear time invariant system That is if we know the transfer function of a LTI system we can compute the output of this system due to an arbitrary input Since we are still operating in the context of Fourier series expansions and since Fourier series expansions are most appropriate for periodic signals consider for a moment a periodic input f t applied to the system of equation 3 14 Obviously f t has a Fourier series expansion whose form is given below f t Y perm 3 16 n oo It is not difficult to demonstrate that in this case the output g t of our system will be equal to g t YH no F e 3 17 n oo Equation 3 17 validates out claim that the frequency transfer function of a linear time invariant system is sufficient to describe the output of the system due to an arbitrary input at least for the case of an input which is periodic After the Fourier transform is introduced we will be able to extend this result to aperiodic inputs as well Looking at equation 3 17 more carefully we can make a number of observations f NE WT 27 Observation I If the input to a linear time invariant system is periodi
70. lution by f t That is RO 0 0 nra 3 7 f t 0 1 2 3 4 5 6 Figure 3 1 two rectangular pulses to be convolved We want to compute the convolution of f t and f t It seems that we have to compute the convolution integral for infinitely many instances of time f see equation 3 7 A more careful observation though allows us to distinguish the distinct range over which the convolution integral needs to be evaluated One way of finding the distinct t range is by remembering that equation 3 7 tells us that we need to slide the rectangle f with respect to the stationary rectangle f and identify the product of these two pulses over the common interval of overlap It is not difficult to show that the sliding rectangle f actually f t t is nonzero over the 7 interval t 2 7 1 We now distinguish five cases Case A f t r isto the left of f r and fand f do not overlap see Figure 3 2a In order for Case A to be valid we have to satisfy the constraint that t 1 3 or t 4 Then is easy to show that f t 0 3 8 3 3 Case B f t T is to the left of f c and f overlap f partially from the left see Figure 3 2b In order for Case B to be valid we have to satisfy the constraint that 3 1 4 or 4 lt r lt 5 Then is easy to show that f az 1 4 3 9 Case C f t r is completely overlapping with see Figure 3 2c In order for Case C to be valid we have to satisfy
71. lysis waveform in dBV with a reference voltage offset of Volt e From the Math Menu select Spectral Set Up Create the Magnitude spectra by clicking in the Magnitude and channel 1 icons respectively e Click Mag Tab and click the dB icon for vertical scale factor e At the reference menu Set the Level to OdB This will place the M1 label at the upper side of the screen Set the Level offset to 1 volt Reference voltage for dBV scale Set the scale to 10 dB 10dB per division in the vertical scale d Clear the time domain waveform turning OFF Channel 1 e Using Zoom option from the Multiview section Adjust the Multipurpose Knobs to Factor of 10 narrows the width of the zoom window to 12 5KHz and to a Position of 596 place the left side limit of the zoom window to OdB You may need to press the HORIZ bottom to re adjust your measurements You should see the first six nonzero harmonics of the square wave displayed Record the amplitude in dBV and frequency of each harmonic You can use the Cursor in cursor window type 3 28 7 6 Integrator a Generate a square wave of amplitude 5V peak and frequency 10x o ten times the cutoff frequency Apply this signal to the input of the filter from Figure 3 15 the first order filter b Sketch the output of the filter from the oscilloscope recording signal peak to peak amplitude 7 7 Square to Sine conversion First Harmonic Isolation a Obtain a 2 volt peak square wa
72. m at f 0 where you can subtract the DC back out 2 Assuming an ideal filter with a cutoff frequency of c c 2z T show graphically in the frequency domain why the first harmonic of the square wave should be the only harmonic at the output 3 Suppose you have an input signal with frequency components from 0 to 2 KHz Is it possible to design a filter which will produce an output with frequency components 4 to 6 KHz Why or why not 4 What is the difference between dB octave and dB decade 5 Component In this lab we will use OpAmp module TL084 which is shown in figure 3 14 3 24 TL084 Out 1 14 Out Amp Amp In2 13 In 4 1 t In 3 12 In Vcc4 11 VEE In5 10 In Amp Amp In6 9 In 3 2 Out 7 8 Out Figure 3 14 Pin layout of OpAmp module TL084 6 Circuit Diagrams The circuit diagram of the first order and fourth order low pass filter used in this lab are shown in Figure 3 15 and Figure 3 16 respectively 3 25 7 Procedure 7 1 Filters a Build the lowpass filter shown in Figure 3 15 b Using either the oscilloscope AC voltmeter or spectrum analyzer determine the cutoff frequency of the filter 3 dB from the passband the passband gain and the filter rate of rolloff in dB decade or dB octave c Build the lowpass filter shown in Figure 3 16 and repeat step b Keep both circuit built for the remainder of the lab Figure 3 15 First order low pass filter Vo R
73. maining harmonic are outside of the pass band then the output of the low pass filter will be a single tone signal sinusoid 3 Simulation We will use Matlab simulation to see the response of our filter circuits For simulation you will need the transfer functions of both filter circuits shown in figure 3 15 and 3 16 The transfer functions were given earlier in this manual See equation 3 20 and 3 21 In both filters R 1 6kQ and C 0 0luF For the second filter R 10kQ R 12kQ R 10KO R 1 5kQ and C 0 01uF With the above values for resistors and capacitors the transfer functions will have the following form notice that the coefficients are in descending order of s 1 1 RC s Hy 7 2 2x1 5 RC s 2 65 RC s 3 48 RC s 2 65RCs 1 A Construct a frequency array as following w 0 100 100000 Construct the coefficient vectors as follows b coefficients of numerator separated by comma a coefficients of denominator separated by comma Use the MATLAB freqs function to generate frequency response of the corresponding filter Plot the frequency response of both filters in a single graph using the following a Matlab function Note The freqs function is not available in all MatLab installations It is installed in all EE Labs at UCF 3 22 Apply the above procedure for both the filters Note that the frequency response you obtained from the MATLAB simulation that is gain versus ang
74. me of the signals that we are interested in are energy type signals that may exist for positive or negative time it is wise to choose o 0 This discussion indicates that if the input signal v t in equation 2 1 is of the form e then we can readily find the output v t of the system Furthermore if we can express the input signal v t as a linear combination of complex exponential signals of the forme we can still find without difficulty the output v t of the system the reason being that the system in 2 1 is linear and as a result if the input v r is a linear combination of complex exponentials then the output vlt can be expressed as a linear combination of the outputs of the complex exponentials involved in the input One version of the Fourier series expresses an arbitrary signal like V t as a linear combination of complex exponentials This is the first reason that the Fourier series representation of a signal is so useful A close relative of a complex exponential e are the signals cosct and sin ot remember p p g Euler s identity e cos t jsin ct Consider now the following set of familiar signals y J g g f t asinc t fa r asin 20 t fs r 2 asin3o t 2 2 Tr r asinno t In Figure 2 2 we show a plot of f t and f t More generally we can say that as n increase 2 2 the rate of variation of f t increases as well One way to quantify the above qualitative st
75. n DSB LC waveform C Ri Output AM wave s t C Figure 4 20 Envelop Detector a Circuit Diagram 4 25 05 AN a i i Amplitude IE Will EAH 1 A H b uM TUNE h XL 10 i it i E 0 8 i X i i dee ae wl nmt wal o2 1 c Figure 4 20 Envelop detector a Circuit diagram b AM wave input c Envelop detector output As we emphasized before the information signal f t can be recovered from a DSB LC signal by utilizing a peak envelope detector An envelope detector consists of a diode and a resistor capacitor RC filter see Figure 4 20 The operation of this envelope detector is as follows On a positive half cycle of the input signal the diode is forward biased and the capacitor C charges up rapidly to the peak value of the input signal When the input signal falls below this value the diode becomes reverse biased and the capacitor C discharges slowly through the load resistor R The discharging process continues until the next positive half cycle When the input signal becomes greater than the voltage across the capacitor the diode conducts again and 4 26 the process is repeated We assume that the diode is ideal presenting resistance r to current flow in the forward biased region and infinite resistance in the reverse biased region We further assume that the AM wave applied to the envelope detector is supplied by a voltage source of intern
76. n call an FM signal narrowband These conditions will be a byproduct of our discussion of wideband FM systems 2 7 Wideband FM To illustrate the ideas of wideband FM let us start with the simplest of cases where the information signal is a single sinusoid That is f t acos a t 5 21 Then the instantaneous frequency of your FM signal takes the form o t o k a cos a t 5 22 Integrating the instantaneous frequency olt we obtain the instantaneous phase e t m k olt ot sino 5 23 Consequently the FM modulated wave is k m r Acolo M 5 24 The quantity k a c is denoted by and it is referred to as the modulation index of the FM system Let us now write the above expression for the signal m r in a more expanded form m t Acos o t cos B sin o t Asin a t sin Z sin o 5 25 To simplify our discussion from now on we will be referring to the quantity cos B sin o t as term A and to the quantity sin Z sin a t as term B Terms A and B are the real and the imaginary part of the following complex exponential function PELIS 5 26 5 6 ma Ln f PRA which is a periodic function with period The above function can be expanded as an e m exponential Fourier series as follows eb sint Y C e 5 27 n oo where the coefficients C are calculated by the equation On 7 Om m oo e sin corner qy 5 28 Ag non Let us now make a substitution
77. nd click Duty Cyc Click Freq Click Close Duty cycle and frequency should now be displayed on the right edge of the screen Change the duty cycle of the 1 KHz rectangular wave to 10 Hint Pull the Duty Cycle Knob to invert the waveform for duty cycles of less than 50 Changing the duty cycle also changes the frequency so you will also need to readjust the frequency until you attain both 1 KHz and 10 duty cycle Notice that a DC component appears in the frequency domain display The DC component of the signal from the Function Generator is negative However the DC component observed in the frequency domain display is positive because the magnitude is the rms value which is always positive Determine the amplitude and frequency of the 1st 2nd 3rd 4th and 5th harmonics Determine the location of zeros Save the waveform Change the duty cycle of the 1 KHz rectangular wave to 25 Determine the amplitude and frequency of the 1st 3rd 5th 6th and 7th harmonics Determine the location of zeros Save the waveform Change the duty cycle of the 1 KHz square wave to 9096 Determine the amplitude and frequency of the 1st 2nd 3rd 4th and 5th harmonics Determine the location of zeros Save the waveform s duty cycle tw tp 100 Figure 2 10 Duty cycle of a square wave signal 2 18 6 3 Triangular Wave The triangular wave like the square wave only contains odd numbered harmonics The magnitude of the harmonics
78. need to find the maximum with respect to time of jo Where f t is the time derivative of f t Actually for PM systems Aw k max f r 5 43 One last comment to conclude our discussion of angle modulation systems It can be shown that the power of an FM or PM signal of the form m t Acos 6 t 5 44 is equal to A 2 3 Pre lab Questions 1 Provide a paragraph where you compare AM and FM modulation advantages disadvantages 2 Why is the use of FM more preferred than PM Explain your answer Hint Compare the frequency deviations in both cases 3 Give a short qualitative justification of the fact that FM is more noise immune than AM 4 Simulation We are going to do the simulation in Simulink of Matlab To run the Simulink enter the simulink command in the Matlab command window It should look as it is shown in the following Open a new window using the left icon 5 12 Bk Simulink WY Communications Blockset Wil Control System Toolbox Wil DSP Blockset Wil Fuzzy Logic Toolbox Wii NCD Blockset Wii Neural Network Blockset WY MPC Blocks BA Power System Blockset Wi stateflow Wii Simulink Extras Wil System ID Blocks Fhe pF CRORE This is the simulink3 library Construct the following block diagram for the FM modulation and demodulation simulation The following blocks will be necessary for your simulation The paths of the blocks have been giv
79. now displayed The orange M1 label on the left edge of the screen indicates the OdB level The orange label at the bottom left of the screen now says Math 1 20 0 dB 25 0KHz This means each vertical division is 20dBV and each 1 3 horizontal division is 25 0 KHz Thus the frequency spectrum is displayed from 0 Hz on the left to 250 KHz on the right g Press the Zoom icon that is located in the Multiview section of the Oscilloscope Control Panel Zoom allows detailed analysis of a narrower frequency range The waveform in the upper portion of the screen displays the spectrum from 0 to 250 KHz and the lower waveform is the frequency range that is being expanded A box appears in the upper waveform to identify the range that is being expanded Adjust the upper Multipurpose knob to set the Position to 1 and adjust the lower Multipurpose knob to set the Factor to 50 A Zoom Factor of 50 results in a display of 25KHz 50 500Hz per division The white label at the bottom edge of the screen confirms that the Zoom display is set to 20 dB and 500 Hz per division Setting the position to 1 sets the frequency at the center of the screen to 1 of 250 KHz 2 5 KHz Thus the Zoom is now set to display from 0 Hz to 5 KHz The 1 KHz signal should now be visible in the lower waveform h Use the mouse to click the Cursors button on the toolbar Select Cursors On Click the Cursors button again select Cursor Type and then click Waveform Two
80. nt of the filter So to find the filtered output we did the convolution of x1 and b 9 Find the Fourier transform of carrier2 and x demod Name them C2 and S2 10 Plot the following signal message signal before modulation versus time pu modulated signal versus time x demod demodulated signal versus time SV message signal in frequency domain CI carrier at transmitter end in frequency domain X modulated signal in frequency domain C2 carrier at the receiver end in frequency domain S2 demodulated signal in the frequency domain You may use following Matlab functions to plot all of them together Subplot Plot 11 Repeat the simulation using square wave carrier at both ends What happens if you use different wave shapes Explain Actually in the hardware experiments that follow you will be using a square wave carrier 12 Repeat the above simulation with a sinc function as the message signal 13 Repeat the above simulation with phi 7 4 7 2 in step 7 What happens Explain What happens if delta 200 400 Hz 4 34 4 Pre lab questions Non sinusoidal carrier Consider a case of double sideband suppressed carrier amplitude modulation DSB SC AM where instead of having a sinusoid to modulate our signal we use some other periodic signal p t with Fourier series expansion p t X P e ov 4 53 1 By providing a qualitative example i e assume a specific p t say
81. of systems that are called linear time invariant systems When a signal is applied to this system the output can be expressed in terms of linear differential equations with constant coefficients An example of such a system is shown in Figure 2 1 The differential equation that describes the input output relationship in such a system is given below d R R a OS O Tw 2 1 L v t v t Figure 2 1 A Simple linear Time invariant system For a finite input v r we expect to solve this differential equation to find the corresponding output v r The difficulty involved in this task is that the input has to be a very simple signal One way out from this complication is to express the input as a linear combination of simpler to deal with inputs The choice of simpler inputs will affect the difficulty which we may encounter 2 in solving the problem There is an intriguing possibility regarding the simpler input signals Let us for a moment assume that we choose as an input a function that repeats itself under the operation of differentiation We can show that such a function at the input will yield the same function as an output multiplied by some algebraic polynomial involving the parameters of the function and the parameters of the differential equation A function that demonstrates this behavior that is to repeat itself under the operation of differentiation is the function e where s o jo is a complex number Since so
82. ore weakness than strengths but all have their applications Some modulation techniques and their characteristics are listed in the table 4 3 2 12 Aliasing Aliasing occurs in most forms of modulation when the sampling rate or the carrier frequency is not large enough compared to the maximum message frequency Once the spectra have overlapped there is no way to separate the two signals The nyquist theorem states that for a sampled signal such as PAM the sampling frequency f must be at least twice the maximum frequency of the signal to prevent aliasing from occurring 3 Simulation In this simulation we are going to see the amplitude modulation and demodulation with different parameters Let us consider a sinusoidal message signal of 100 Hz and a sinusoidal carrier in the transmitter of 1000 Hz f 100 f 1000 Modulation index m 0 5 Sampling frequency fs 16384 4 32 MODULATION 1 Construct the time array as following t 0 1 fs 0 05 2 Construct the message signal as signal sin 2zft 3 Calculate the required dc offset from modulation index definition A abs min signal m 4 Construct the carrier at the transmitter end as carrier sin 27f t 5 Modulated output of the transmitter is given by x signal A carrier1 Notice a dot before the multiplication sign In Matlab it represents multiplication element by element 6 Find the Fourier transform of signal carrier and x
83. oting is that the output of a 90 degree shift system is called the Hilbert transform of the input Furthermore if we designate by f t the input to a 90 degree shift system the output Hilbert transform of this system is designated by f t Finally it is easy to show that 4 30 4 18 COSOmt m t 7 Cos c m t sinogt Figure 4 13 Block diagram of a system that generates the upper sideband of m t f t 2cosost c t cosoxt Y F Wc t 2 F O We f t b 2 Katos j 2 Kond Paes F A sgn o e F oto OVENS A sgn o ox F o o Figure 4 14 Block diagram of a system that generates the upper sideband of m t 4 19 Fo Om Om 6 A F W_ t F a o Qc Om We QctOm Qc Om We OctOm Q A 5 sgn aoax F 2 sgn Wc F o ox Mc Om Oc Oc Oct Om O Figure 4 15 FT at various stages of the block diagram of Figure 4 14 4 20 2 9 Double Sideband Large Carrier DSB LC System The main idea behind this system is money Its claim to fame comes from the use of a very cheap receiver called peak or envelope detector To understand this detector let us first look at the modulated signal m t for two examples of information signals f t In Figure 4 16 we show m t for an information signal f t and m t for an information signal f t The difference between the two information signals f t and f t is that f r is always positive while f r assumes positive
84. ound for any g t and in fact it has been found for few f t s To get a deeper insight consider cos k f e r and expand it in terms of its Taylor series kis kisi kal 2 4 6 cos k g t 1 5 16 5 4 If the signal f r is known its Fourier transform F v is also known and the Fourier transform of g t can be computed In particular from well known Fourier transform properties we can deduce that g t gt G o Glo g t gt Glo Glo Glo Glo 5 17 Y Based on the above two equations 5 16 and 5 17 we can state that to compute the Fourier transform of cos k amp t for an arbitrary signal g t becomes a formidable task we need to compute a lot of convolutions Furthermore it seems that the bandwidth of cos k amp t 1S infinite note that every time we multiply a signal with itself in the time domain we double its bandwidth in the frequency domain Hence if the bandwidth of g t is the bandwidth of g t is 2o the bandwidth g r is 4o the Taylor series expansion of cos k f e r contribute equally to the determination of the signal and so on In reality though not all the terms in m m cos k f g t Notice that in the Taylor series expansion of cos k f g t the coefficients multiplying the powers of g t get smaller and smaller This observation will leads us into the conclusion that the bandwidth of the terms cos k f g t and sin k f g t is indeed finite and as a result t
85. ove functions are depicted in Figure 3 4 l t l t t 2 W2 T T a b 3 10 It I t c d Figure 3 4 Function sequence definition of impulse function a rectangular pulses b triangular pulses c two sided exponentials d Gaussian pulses The impulse function that we have defined so far was positioned at time 0 In a similar fashion we can define the shifted version of the impulse function Hence the function 5 t t designates an impulse function located at position time t and it is defined as follows f oleate t a _ t a ty b otherwise The impulse function has a number of properties that are very useful in analytical manipulations involving impulse functions They are listed below Area Strength The impulse function t has unit area That is b 6 4 1 a lt t lt b Amplitude r 1 0 forallt t Graphic representation See Figure 3 5 d t Figure 3 5 Function 6 t Symmetry Time Scaling Multiplication by a time function fot 1 f otr amp Relation to the Unit Step Function The unit step function is the function defined by l E 0 t lt to ult 1 It is not difficult to see that fale 1 jar U t And that 3 12 amp r 1 4 U 4 1 Convolution Integral f o f f t d t t f t t Similar to the definition of r we can define 6 t 6 t H
86. riod T We can write T T 5 0 t3 4 1 0 otherwise where f t is equal to the periodic signal f r over the interval T 2 T 2 and zero otherwise Obviously f t has a Fourier series expansion such that Y Fen 4 2 With af Fal proge zl f re qi 4 3 n T 2 Also since f t and f t are equal over the interval T 2 7 2 we can write that where F o TF 4 5 and Q n0 4 6 Let us now define 27 AO 0 1 0 0 4 7 Then from equation 4 7 we get 2 A jot f D gt TOs 4 8 Let now T goto or equivalently Aw goes to zero It is easy to see that the equation 4 8 becomes 1 ral F oy do 4 9 f t Also it is not difficult to show that f te dt 4 10 Equations 4 9 and 4 10 are referred to as the Fourier Transform pair that allows us to transform the aperiodic time signal f t into the frequency signal F lo and vice versa Based on the aforementioned discussion we can now formally introduce the Fourier transform theorem 2 2 Fourier Transform Theorem If the signal f t satisfies certain conditions known as the Dirichlet conditions namely IE t is absolutely integrable on the real line that is irat oo 2 The number of maxima and minima of f t in any finite interval of the real line is finite 3 The number of discontinuities of f t in any finite interval of the real line is finite then the Fourier transform FT of f t defined
87. s its corresponding output A system is time varying if it is not time invariant As we emphasized before a linear system is described by a linear differential equation of the form shown in equation 3 4 If in the differential equation 3 4 the coefficients a 0 lt k lt n are constants and not functions of time then we are dealing with a linear time invariant system Otherwise we are dealing with a linear time varying system Examples of linear time varying systems from your circuit classes were circuits consisting of R L and C components and involving one or more switches that were ON or OFF at special instances 2 2 The Convolution Integral The convolution integral appears quite often in situations where we deal with a linear time invariant system LTI If this system is excited by an input f t and the impulse response of the system is A t then the output g r of the system is equal to the convolution of f r and h r as illustrated below r n free ynle ar fat 66 Note that the impulse response of a linear time invariant system is defined to be the output of the system due to an input that is equal to an impulse function located at t 0 In this manual we are going to demonstrate the convolution of two rectangular pulses 3 2 2 3 Convolution Examples Consider two rectangular pulses f and f depicted in Figure 3 1 We are going to find the convolution of these two rectangular pulses We designate the convo
88. s the demodulated output 6 Equipment TDS5052B Oscilloscope Function Generator You should bring a floppy disk to store your waveforms 7 Procedure 7 1 Modulation a Build the modulator as shown in Figure 5 5 a b Determine the constant k from the following 1 Use your triple output power supply to apply the input voltages specified in the following table and record the output frequency V Volts Fou KHz 0 0 1 0 2 0 3 0 4 0 5 0 6 0 2 Plot f vs V Draw the best straight line through these points The slope of this line out in is k gd Note that k f has units of Hz Volts What is the measured value of k f 5 20 3 According to the XR2206 data sheet the expected voltage to frequency conversion gain isk ke Hertz Volts Calculate the expected value of kr Assuming the resistors have a tolerance of 10 and the capacitors have a tolerance of 20 how do the measured and calculated values compare c Add input coupling capacitor C to the circuit as shown in Figure 5 5 b Set the modulation frequency to fm 2 KHz Fill in the following table using the following equations Af peak fm OK 5 57 Val AP uk 5 58 D modulation index Calculated Af Hz Calculated V Measured Af peax Hz 4 5 5 25 9 10 5 12 3 5 2 6V C 0 001 uF 200 ohm 13 6 XR2206 VCO Vem 6V Soy Figure 5 5 a VCO Determination of modulation constant K 6V C
89. se the latter approach because it is easier Recall that f t 24 Ya cos 2nt 2 28 m n 1 where n l n l a 2 E 1 CU 2 29 m 2n 1 2n 1 Due to Euler s identity we can write j2nt j2nt cos 2nt lt lt lt 2 30 Using Euler s identity in equation 2 30 we get E a j2nt lt a j2nt Za Dae 5 e 2 31 m n l n l 2 Notice also that the exponential Fourier series expansion of the signal f t has the following form 2 10 f Y Fel F Y Fei 53 Fie 2 32 n l n l Comparing one by one the terms of equations 2 31 and 2 32 we deduce the following identities F a 2 33 F 2 34 fls 2 35 In particular f i ru P pn iss gn iss p VAR Based on equations 2 33 through 2 35 we can plot the amplitude frequency and the phase frequency plots for the signal f t In the case where the coefficients F are real we can combine these two plots into one plot this plot is denoted as the line spectrum of f t The line spectrum of a signal f t corresponds to the plot of F s with respect to frequency The line spectrum of the signal f t in this example is depicted in Figure 2 7 2 5 2 E TU 3x s T 2 2 35 2 2 352 T Wo Wo gt 300 00 Qo 300 15x 15x Figure 2 7 Line Spectrum of f t of figure 2 6 c Plot the right hand sides RHS s of equations 2 27 and 2 36 The plot is shown in Figure 2 8 Note that the RHS s of equation 2 27 an
90. signal f t which is depicted in Figure 2 6 2 8 f t bii 1 2 n 2 t 1 Figure 2 6 plot of f t cost for 71 2 1 2 and zero elsewhere The interval of interest is 2 7 2 Hence T z and as a result o 22 T 2 The trigonometric Fourier series of a signal f t is therefore of the form f t a Ya cos 2nt 53 b sin 2nt 2 22 n l n l e z 2 2 uf n lt cos tdt 2 23 2 pal2 f t cos 2ntdt lt f cost cos2ntdt T 2 AZ z 2 e r re pus 4 er 2 24 2 erj2 2 pan b e f t sin 2ntdt lt ff gosesin 2ntdt 0 2 25 The last equation is a result of the fact that f t is even and sin2nt is odd Consequently f t sin 2nt is odd and whenever an odd function is integrated over an interval which is symmetric around zero the result of the integration is zero Based on the above equations we f t as Z E JU U Je 2nt 2 26 can write T 4 z 2n 1 2n 1 Also if we write out a couple of terms from the above equation we get 2 2 2 2 f t cos2t COS 4t cos 6t 4 2 27 m 3 15 35 b Find the exponential Fourier series of the signal f t in Figure 2 6 We can proceed via two different paths We can either find directly the exponential Fourier series expansion of f t by applying the pertinent formulas or we can use the trigonometric Fourier series expansion already derived to generate the exponential Fourier series expansion We choo
91. signals but it has been verified experimentally in a variety of cases Equation 5 39 is referred to as Carson s formula rule for the evaluation of the bandwidth of an FM signal and from this point on it can be applied freely independently of whether the information signal is of sinusoidal nature or not One of the ramifications of Carson s rule is that we can increase the bandwidth of an FM signal at will by increasing the modulation constant k or equivalently by increasing the peak frequency deviation Ac One of the advantages of increasing the bandwidth of the FM signal is that larger bandwidths result in FM signals that exhibit better tolerances to noise Unfortunately the peak frequency deviation of an FM signal is constrained by other considerations such as a limited overall bandwidth that needs to be shared by a multitude of FM users For example in commercial FM the peak frequency deviation is specified to be equal to 75 KHz Let us now say a word about PM From our previous discussions the form of a PM signal produced by a sinusoidal modulating signal is as follows m t Acoslo k a cos a t 5 40 and the instantaneous frequency is equal to olt o k ao sino 5 41 Hence o ak o 5 42 That is A depends on This is considered a disadvantage compared to commercial FM where Aq is fixed Carson s rule is also applicable for PM systems but to find the peak frequency deviation of a PM system you
92. t 5 6 Phase and frequency modulations are techniques that modify the instantaneous phase and frequency respectively of a sinusoid in a way dictated by an information signal f r 2 3 Phase Modulation Here the information signal f t is placed as a linear term in the instantaneous phase of the carrier That is Olt a t O k f t 5 7 where k is a constant of the modulating device Hence the PM modulated signal is equal to m r Acos a t 8 k f r 5 8 2 4 Frequency Modulation Here the information signal gets inserted as a linear term into the instantaneous frequency of the carrier That is oXt o k f t 5 9 5 2 where k is a constant due to the modulator In this case the instantaneous phase is equal to 6 o 6 k f feur 5 10 and as a result the FM modulated signal looks like m t Acos o O k f roy 5 11 A plot of FM and PM signals is shown in Figure 5 1 nm AAA PPAP PAP PAE Figure 5 1 Examples of frequency and phase modulation 5 3 Now let us discuss FM and PM simultaneously and get a better insight into their similarities and differences For simplicity assume that 0 0 Then m t Acos o t k f r 5 12 and m t Acos t k frena 5 13 Let us take m t and find its instantaneous frequency or Indeed o k ZO olt 0 k P 5 14 The above equation tells us that in the PM case the instantaneous frequency has a lin
93. t the generalized derivatives of t by the following equation jor in G t 9 a lt O lt b E otherwise We can generalize this result to ac ty b EN foot iy 64 a 0 otherwise For even values of n t is even and for odd values ofn 8 t is odd 2 6 The Frequency Transfer Function Consider a linear time invariant system which is described by the following differential equation Ya TEU d el Y 2 JU 3 14 m 0 In the above equation f t represents the input to the system and g t represents the output of a f t dt the system The expression denote 0 th derivative of f t same convention holds for g t to the above system is equal to e Then we prove t 2 H w e with Assume for a moment that the input f t CIN that the output of the system is equal to g S x mo S lt x 3 15 S V S lt 3 X 9 II MeMa 3 M o In actuality we are contending that if the input to the system is a complex exponential function e the output of the system will be the same exponential function times a constant H v which depends on the input parameters and the system parameters the a s and b s when we are referring to H v as being a constant we mean that it is independent of time The constant H v is called the transfer function of the system When the Fourier transform is introduced it wi
94. t a time shift will produce a phase shift in the FT of the signal but it will leave the magnitude of the FT unaltered The Convolution Property assures us that we do not need to perform convolution to produce the output of a LTI system due to an arbitrary input simply multiplying the FT of the input and the FT of the impulse response of the system allows us to compute the FT of the output of the system The Time Differentiation and Time Integration properties demonstrate that making the signal faster slower differentiate integrate in the time domain results in an expanded contracted frequency spectrum FT The Duality Property provides us with a new FT pair every time a FT pair is computed In particular if F is the FT of f the duality property says that F would a FT that is proportional to the mirror image of f with respect to the 0 axis Finally the Amplitude Modulation property tells us that we can shift the frequency content of a signal at an arbitrary frequency by multiplying the signal with a sinusoid AM systems are based on this property of the FT 2 5 Introduction to Communication Systems The primary function of most communication systems is to transmit information from point A to point B Consider for example the situation where we have to transmit human voice from point A to point B and the channel between points A and B is the air The human voice at point A is first converted via a microphone to an electrical signal Then we
95. the Function Generator frequency to 5 KHz Determine the amplitude and frequency displayed and save the waveform to include in your report 6 2 Square and Rectangular Waves A rectangular wave contains many sine wave harmonic frequencies When the reciprocal of the duty cycle either the positive or negative duty cycle is a whole number the harmonics corresponding to multiples of that whole number will be missing For example if the duty cycle is 50 then 1 0 5 2 Thus the 2nd 4th 6th etc harmonics will be missing i e zero A rectangular wave with a 50 duty cycle is a square wave For the square wave the magnitude of the harmonic will be inversely proportional to the harmonic s number For example if the magnitude of the Ist harmonic is A then the magnitude of the 3rd harmonic is A 3 the 5th harmonic s magnitude is A 5 etc 2 17 Change the Function Generator frequency to 1 KHz and function to square wave Determine the amplitude and frequency displayed for the 1st 3rd 5th and 7th harmonics in the spectrum of the square wave Identify the zeros of the spectrum envelope i e the frequency locations at which harmonic amplitudes are zero Save the waveform to include in your report The duty cycle of a pulse is measured as illustrated in Figure 2 10 The TDS5052B Oscilloscope can automatically measure duty cycle Click Measure on the Toolbar and click Measurement Setup Under Source click Channel 1 Click the Time tab a
96. thout actually having to compute the convolution 2 4 Golden rules for the Convolution of two Rectangle Pulses Consider a rectangular pulse f r with amplitude A over the interval xx and another rectangular pulse f t with amplitude A over the interval y y Denote their convolution by f t Then 1 2 The function f t is a trapezoid The starting point of the trapezoid is at position x y The first breakpoint of the trapezoid is a At position x y if pulse f t is of smaller width than pulse f t b At position x y if pulse f t is of smaller width than pulse f t The second breakpoint of the trapezoid is a At position x y if pulse f t is of smaller width than pulse f t b At position x y if pulse f t is of smaller width than pulse f r The end point of the trapezoid is at position x y The maximum amplitude of the trapezoid is a Equal to A A x a ifthe f t is of smaller width b Equal to A A y y if the f t is of smaller width A plot of f t Jy t and f t is shown in Figure 3 3 The technique described above to compute the convolution of two rectangular pulses can be generalized in a trivial way to compute the convolution of arbitrary shaped finite duration pulse 3 7 Xjtyi X y2 X2 Yy1 X2 y2 Figure 3 3 Convolution of two Rectangular Pulse All of the above rules pertaining to the convolution of two rectangular puls
97. to be within the filter passband Connect the function generator output to the filter input b Measure the output voltage V The filter passband gain is equal to V V V 2 See Figure 3 18 7 4 Rate of rolloff a Use the function generator to generate a sinusoid at a frequency that is higher than the cutoff frequency of the filter maybe by a factor of 2 or so Record the input amplitude V or measure in dB on the AC voltmeter you may want to use an input amplitude of 1 volt or more since the output will be attenuated 3 27 b Connect the function generator output to the filter input Measure the filter output in volts V or dB Calculate V V and convert to dB 201log V V or subtract the input dB level from the output dB level This will give you the attenuation in dB at your selected frequency c Repeat steps 1 and 2 with a frequency that is a factor of 2 higher than the first frequency used If you take the difference between the two calculated values of attenuation in dB this will be your rate of rolloff in dB octave If you want to convert N dB octave corresponds to N 20 6 dB decade 7 5 Square Wave Spectrum a Use the function generator to generate a square wave of 0 2 volts amplitude and 1KHz frequency and connect its output to Cannel 1 input of the oscilloscope b Set up the display of the Oscilloscope in order to see 10cycles of the square wave c Generate a spectrum ana
98. ude of each one of its sinusoids at the corresponding angular frequency location This will give rise to the amplitude frequency plot of f t Observe though that in equation 2 15 each sinusoid involved in the expansion of f r has a phase associated with it Hence in order to completely describe the signal f r we need to draw another plot that provides the information of the phase associated with each sinusoid This plot is denoted as the phase frequency plot and it corresponds to a plot of 0 at the location o It is worth mentioning that the aforementioned expansion of f r in terms of sinusoidal signals of frequencies that are integer multiples of the frequency is often referred to as the expansion of f t in terms of its harmonic components The frequency o is called the fundamental frequency or first harmonic while multiples of the first harmonic frequency are referred to as the second harmonic third harmonic and so on The corresponding signals in the expansion are named first harmonic component second harmonic component third harmonic component and so on The component a in the above expansion is called the DC component of the signal f t Since we shall frequently be interested in the amplitudes a b s and phases 6 s of our signal f t it would be simpler to obtain these directly from f t rather than by first finding a and b In order to achieve this let us consider the functions r
99. ular frequency in radian You can also plot gain versus frequency Hz In that case you have to use the f w 2m vector as the parameter in plot function The simulated gain versus frequency Hz plot for both filters 2 order and 6 order is shown below as Figure 3 12 Frequency response of Butterwoth Filter 3 T i 1 i 2 8 emm DEDE EE pue ieee NONO Tha A dee quo oe CRISUCNONEN ET pui areca PUT EE iai 6th orae MEME oo ee 2 4 oe ep pe T m ge ee pe ee ee pee eee a ee ce a en ue ice DENS Pate Fe re Beet ee eee fee ty il Blea cpt Mali og lll te toy la oe pd OG alle i ny Ns Rae TN UR P 1 4 JL AA ADU Ll AS lE AL nnns Gan 1 2ndorder gt gt p77 ps OS E dE E hee eee ee ee ee eee eee eee X pH o ee e eel ET E 1 1 Qa pasama ers dS ponens 17777 ee 1 L 1 1 o o 2000 4000 6000 8000 10000 12000 14000 16000 Frequency Hz Figure3 12 The simulated gain versus frequency Hz plot for both filters 2 order and 4 order Compare the response of two filters After the hardware experiment compare the experimental results with the simulated results 3 23 4 Prelab Questions 1 Calculate the Trigonometric Fourier Series of a square wave with the parameters shown assume it is periodic with periodic T Figure 3 13 a periodic square wave with period T You may find it easier to add a DC level of A volts This will only change your spectru
100. uts as shown by m t A f t cos a t A f t sin 1 4 24 where f t and f t denote the two different information signals applied to the product modulators Thus m t occupies a channel bandwidth of 2a centered at the carrier frequency where c is the maximum frequency content of the signals f t and f t The receiver part of the system is shown in Figure 4 8 b The multiplexed signal m t is applied simultaneously to two separate coherent detectors that are supplied with two local carriers of the same frequency but differing in phase by 90 degrees The output of the top E 1l detector is ZAA f t whereas the output of the bottom detector is 55 cf t For the system to operate satisfactorily it is important to maintain the correct phase and frequency relationships between the local oscillators used in the transmitter and receiver parts of the system To maintain this synchronization we may use a Costas receiver Another commonly used method is to send a pilot signal outside the passband of the modulated signal In the latter method the pilot signal typically consists of a low power sinusoidal tone whose frequency and phase are related to the carrier wave c t at the receiver the pilot signal is extracted by means of a suitably tuned circuit and then translated to the correct frequency for use in the coherent detector 2 8 The Single Sideband Suppressed Carrier System SSB SC The DSB SC system discussed
101. ve with frequency f 75 x f from the function generator b Apply the signal from a to the input of the filter in Figure 3 15 Sketch the output signal in both the time and frequency domains scaling axes c Repeat b with the filter in Figure 3 16 89 95 ja naaoao FUNCTION GENERATOR Oooo AC MILIVOLTMETER Figure 3 17 Measuring cutoff frequency using AC Voltmeter 3 29 80195 j obeoddo ooo O O O Qooo 9 om om 0 j0 o m bo 0 0 D000 CY Figure 3 18 Measuring passband using the oscilloscope 8 Calculations and Questions 1 Convert the values you measured from the spectrum analyzer in dBV to volts and compare to your Fourier Series Coefficients from 1 Recall that the measured amplitudes are RMS amplitudes so the values must be multiplied by the square root of 2 2 Compare your results from section IV parts b and c on the basis of filter gain and filter roll off rate that you measured 3 Using the waveform shown in question 1 Integrate in time to show that the integral of a square wave is a triangular wave What peak amplitude do you calculate Recall that you have to solve an indefinite integral to get an answer in functional form 4 When the first harmonic of the input signal is well beyond the filter cutoff frequency a first order low pass filter cutoff frequency a first order low pass filter approximates an integrator shown in Integrator section of l
102. w a minute for the instrument to boot and stabilize Then press Default Setup to clear settings made by other students Set the Vertical scale to 500 mv and the Horizontal scale to 1 0 ms per division b Turn on the Function Generator and connect the output to Channel 1 input of the oscilloscope c Setthe Function Generator for a sine wave output with a frequency of 1 KHz Adjust the amplitude to 2 volt peak to peak with zero DC offset The Oscilloscope should now display 10 cycles of a sine wave Do not change the Horizontal scale from 1 0 ms through the remainder of this experiment Note Displaying fewer cycles in the time domain display would result in lower frequency resolution for the frequency domain display Displaying less than 4 cycles would result in poor frequency resolution when the frequency domain of the sine wave is displayed 2 FREQUENCY DOMAIN DISPLAY a Use the mouse to click the Math Button on the Toolbar and select Spectral Setup to open the Control Window b Under the Create tab click the Magnitude icon and click Channel 1 as source Click Apply c Click the Mag tab Select dB for the vertical scale factor Click the Reference Level Offset set it to 1000 mV 1 Volt and click enter Click Apply d In this experiment a Gaussian window will be used which is the default setting e Click OK to close the Control Window f Turn the time domain display off by pressing the CH1 button The frequency domain is
103. y coherent or synchronized in both frequency and phase with the carrier wave c t used in the product modulator to generate f t This method of demodulation is known as coherent detection or synchronous demodulation Product Low pass m 0 modulator filter o e t o t A cos ct Local oscillator Figure 4 6 Coherent detection of DSB SC modulated wave It is instructive to derive coherent detection as a special case of the more general demodulation process using a local oscillator signal of the same frequency but arbitrary phase difference measured with respect to the carrier wave c t Thus denoting the local oscillator signal by Ae cos o t and using equation 4 1 for the DSB SC wave we find that the product modulator output in Figure 4 7 is e r A cos a t 9 m r A A cos o t cos o t f t 4 22 AA cos 20 t d f r A A cos df t id E 5 A A F 0 cosd V A A F 0 Figure 4 7 Illustrating the spectrum of a product modulator out put with a DSB SC modulated wave as input The first term in equation 4 22 represents the DSB SC modulated signal with a carrier frequency of 20 whereas the second term is proportional to the baseband signal f t This is further illustrated by the spectrum Fourier Transform E v shown in Figure 4 7 where it is assumed that the baseband signal f t is limited to the interval O

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