Signal Processing and Control
Contents
1. you may be able to use various algorithms to optimize them Some of these optimization algorithms are available in LabVIEW Finding the ap propriate constants is called tuning the PID Practical PID controllers work by calculating the error e y y between the process vari able y and the setpoint integrating and differentiating the error scaling by the appropriate constants and summing to get a drive signal PIDs can be implemented in analog or digital Analog PIDs output a continuous voltage Digital PIDs operate in a loop they measure the process variable compute the drive and output it with a DAC and loop back to measuring the process variable Consequently the drive from a digital PID is a series of steps Whether implemented in analog or digital the system has a finite response time If analog the response time is set by the bandwidth of the circuitry If digital the response time is set by the speed of the ADC and DAC the computation time of the calculation and by any other tasks that might be running on the processor that distract the processor s attention The sum of all these times is called the service interval As a rule of thumb PID controllers work well if their bandwidth is a factor of ten higher than the frequency with which changes occur in the process Software implementations are easier to tune and can be easier to integrate into a bigger sys tem On dedicated hardware software PID loops can run quite fa2. 1 Besampling a rigidly periodic waveform 2 Use an ADC whose bandwidth exceeds the bandwidth of the signal that you wish to recover 3 Use an ADC that acquires samples by a First obtaining an analog sample of the instantaneous signal level b Second presenting this analog sample to the analog to digital converter Note that the actual signal may change during the conversion time but this will not affect the conversion since the signal being converted is the previ ously obtained analog sample not the current signal Acquiring and storing an analog sample may seem difficult but in fact is easy to do with a common circuit called a sample and hold On command the sample remem bers the analog value at its input typically by storing the value in a capacitor If your system does meet these requirements you can acquire an accurate sample of a wave form by following these steps 1 After receiving a trigger indicating the beginning of your waveform acquire one sample set Note that the sample rate can be well less than twice the signal fre quency 2 After receiving a second trigger wait for a delay interval less than the time between samples and then acquire a second sample set Interleave this second set with the first set offsetting each point by the initial interval 3 After receiving a third and successive triggers acquire and interleave more data sets each offset by a different amount The offsets can be evenly and increme
3. Fourier transforms find the spectral content of a signal the amplitude and phase of the signal as a function of frequency You are all familiar with Fourier series from your math and physics classes Fourier transforms are quite similar In a Fourier series you consider only the harmonics of the pe riodic wave of period T that you are analyzing T 2 A rosin 22 a A T In taking the Fourier transform you analyze all times and all frequencies F fe dt The Fourier transform of a sinusoid is a delta function centered at the frequency of the sinusoid 2 Real time signals are those which you receive as in infinite train of periodic samples and which you wish to analyze as your receive each sample Other techniques like Fourier Transforms can be better when you receive your entire sample before you begin processing 3 Tunable filtering can be achieved by multiplying an incoming signal at frequency f by a variable frequency sine wave at frequency fy The beat frequency at f f 1is then passed through a sharp fixed frequency filter The filter is tuned by changing f This common technique is called super heterodyning and is used in most radios and TVs See http en wikipedia org wiki Superhet 4 A series of amplifiers in which each amplifier is fed by the output by the previous amplifier in the chain Last Revision August 2007 Page 3 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved
4. becomes in creasingly painful Pattern Matching As you saw in Exercise 10 2 humans are remarkably good at recognizing some sorts of signals in the presence of noise Computer algorithms exist that can sometimes do the same The simplest is the well known linear least squares algorithm If you know your data lies on a line why retain the indi vidual data points By fitting the data to a line you can average most of the noise away The linear least squares algorithm is easily generalized to fit polynomial coefficients to data or even linear combinations of nonlinear functions More powerful methods are needed to fit nonlinear equations One of the most common is the Levenberg Marquardt algorithm which will fit one or more unknown coefficients in a nonlinear expression to a data set Given a reasonable guess for the unknown coef ficients it can work remarkably well The algorithm is quite difficult to program fortunately Lab VIEW comes with an implementation Improving your ADC The techniques discussed above all assume that the data from your ADC is near perfect that the sample rate 1s high and that quantization is unimportant It 1s sometimes possible to improve your sample data when these conditions are not met Increasing the Effective Sample Rate Equivalent Time Sampling A technique called equivalent time sampling ETS can sometimes be used to increase the ef fective sampling rate of your ADC For ETS to work your system must
5. has an impedance of 10000hms You sample for 10mS What is the smallest signal that you can detect Background Signal Processing Overview Most real world signals are contaminated by noise A frequently used figure of merit is the signal Va to noise ratio S The S for clean signals is much larger than one as S approaches one the Noise signal fades into the noise as shown in the figure below 1 This book available in several versions for different computer languages is the standard refer ences on numerical algorithms Like Horowitz amp Hill it is informative chatty opinionated and funny Almost all physicists own a copy Chapters of the book can be downloaded free from http library lanl gov numerical index html This book is listed as a supplement read it if you want to learn more about the numerical techniques Last Revision August 2007 Page 1 of 22 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Signal to Noise 10 signal to Noise 2 6 6 4 4 ae 0 gt OF on 2 0 2 4 4 5 6 00 O02 O4 O8 1 0 00 O02 04 06 O08 1 Time Time Signal to Nose 1 Signal to Noise 0 5 6 i 4 w i T C oat c ae oo i 2 wo 4 i 6 00 02 04 06 08 1 0 0 0 02 04 06 O08 1 0 Time Time Fortunately we can often extract the signal from the noise There are three primary techniques for recovering the signa
6. Physics 111 BSC Laboratory Lab 11 Signal Processing and Control In the real world any sample has a finite length consequently the Fourier transform becomes i F a fe dt o l FO Here T is the length of the sample not the period of the wave as it was for a Fourier series The sample may include many wave periods The transform of a sinusoid with frequency is T F a elmer dt 0 1 lon The magnitude of this transform for a wave with frequency 2 and T 81s 0 8 0 6 F 0 4 0 2 0 0 Instead of the delta function at 2 that we would get from an infinitely long sample the trans form is a finite width pulse centered at 2 The width of the pulse to the first zeroes is easy to find if we assume that a Aq Then F o e 1 JAoT The first zeroes of this function occur at AoT 27 or Ao 1 fT 1 N where w 2zf and N is the number of cycles of the wave in one sample Because the pulse width is finite we cannot readily determine the precise frequency of the original signal Consider a signal that 1s actually the sum of two equal amplitude sine waves of frequency and Aq The plots below show this sig nal in green compared to a pure signal of frequency in red for various values of Ao A 0 025 Aala 0 05 A a 0 1 Aw 9 2 1 0 L 0 0 tb Oo a lt 0 10 20 30 40 O 10 20 30 40 O 10 20 30 40 O 10 20 30 40 T T r T Las
7. University of California at Berkeley Physics 111 Laboratory Basic Semiconductor Circuits BSC Lab 11 Signal Processing and Control 2007 by the Regents of the University of California All rights reserved Reading Wells amp Wells Entire Book on LabView Horowitz amp Hill Chapter 15 Press Teukolsky Vetterling and Flannery Numerical Recipes Chapters 12 13 and 15 5 LabVIEW Analysis Concepts Manual Chapters 3 and 4 Located in the BSC Share LabVIEW PID Control Toolset User Manual Chapters 1 4 Located in the BSC Share In this lab you will learn how to extract signals from noise and how to use a PID controller Several LabVIEW programs are mentioned in this lab writeup Many of these programs can be downloaded from http socrates berkeley edu phylabs bsc LV_Programs and can also be found on your lab computers in Atlas2 D 111 Lab BSC Share Two versions of the programs are typically available for download an executable version that should run without LabVIEW but re quires a large download from National Instruments which should occur automatically and only needs to be done once and should run on PC s Mac s and Linux boxes and the original LabVIEW source code which requires LabVIEW The LabVIEW Analysis Concepts Manual can be downloaded from Atlas2 D 111 Lab BSC Share LAB 11 Pre lab questions 1 You seek to detect a signal of intrinsic frequency spread Af 10 Hz with a room temperature de tector that
8. are averaged the smoothing operation average samples for 20mS Explain your reasoning to the TAs F Control The following exercises use a computer controlled PID loop to control a magnetic levitator The levi tator suspends a steel ball underneath an electromagnet The magnet field is controlled by the com puter based on feedback from a position sensor The sensor measures the position of the ball by shining an infrared light beam between the ball and the magnet The ball partially occludes the light beam from the amount of light that passes by the sensor can determine the ball s position The computer ADC cannot put out enough current to drive the electromagnet directly We provide a small circuit that amplifies the ADC output The circuit also drives the infrared source and ampli fies the signal from the light sensor The circuit needs 0 12 and 24V power from the breadboard power supplies Make sure you ground the OV Hook up the appropriately labeled leads black is ground to AI7 and AOO The switch on the magnet holder can be used to disconnect the electro magnet 11 10 Before using the Levitator calibrate the light sensor by building up a chart of ball position vs light level Write a routine Position Calibrator vi to build the table The routine Position Calibra tor Template vi has most of the code prewritten Finish the routine by following the comments in its block diagram Use the Calibrator as follows 1 Use the switc
9. e times out of twenty the signal will be between 10 and 19mV and will be converted to 20mV Thus the average of the converted signals will be 9mV thereby recovering the original sig nal level If the sample rate is much higher than the desired signal frequency the averaging can be done by passing the converted signal through a digital low pass or smoothing filter If the signal is periodic and multiple instances of the signal are available the signal can be fur ther improved by averaging the instances The ideal noise level is about half a quantization step Below this noise level the signal will not fluctuate between steps and above this level the noise itself introduces error Control Most modern experiments are controlled by computers computers set magnetic field levels regulate temperatures move probes control timing open valves turn on sources etc Control functionality can be simple or complicated the list below describes some of the most common control methodolo gies On Off Controllers Epitomized by a light switch a typical experimental function would be controlling the state of a vacuum valve On off controls are trivial to program they are usu ally implemented by controlling the state of a digital bit which in turn controls an electronic switch On Off controls are all or nothing you cannot use them to reach a particular level or state Continuous Controllers A stovetop burner valve exemplifies of a continuo
10. e Short Term Maximum Service Interval the Long Term Maximum Service Interval and the Average Service Interval The average service interval should be slightly greater than lms If it is much greater the ball will drop Use your mouse to move the Basic Controller vi front panel window This will distract your computer and the ball will drop Open the block diagram Every iteration through the while loop adjusts the magnet current once F Add a time delay that slows the loop down thereby increasing the service time How much time can you add to the service interval before the ball drops Now add code that inserts a delay every time you push a front panel Boolean control How long a time delay can the Levitator now tol erate Why is the maximum tolerable continuous delay different from the maximum tolerable in termittent delay Calculate how long is takes the ball to freefall an appropriate distance How does this time compare to the times you observe 11 12 Look at the DAC output on the scope How large are the excursions Change the size of the proportional gain What happens to the size of the excursions How large and small can you make the gain Restore the proportional gain back to 10 Now change the derivative time Td Over what limits does the ball stay under control Restore Td back to 100u Change the integral time Ti to zero Does the ball levitate How far is it from the setpoint position 11 13 Now run the program Impulse Response
11. e strength of the drive is pro portional to the difference between the setpoint and the process variable Proportional con trol is appropriate when you wish to quickly approach a setpoint and then asymptote to it smoothly Consider a mass m hanging on a spring k of natural length y driven by a proportional con trol to a setpoint y The constant of proportionality for the control is C The equation of motion for the system is d m z mg yy C Y J 1 1 In steady state the solution of this equation is 20 E Ys Ys Aaa k 1 2 1 C As C gets large the process variable y will approach the setpoint y e Proportional Differential Controllers While Eq 1 2 correctly predicts the equilibrium state of the spring mass system the system as modeled by Eq 1 1 will never reach equilib rium Because the system possesses inertia the md y dt term in the equation of motion it will oscillate around the equilibrium To satisfactorily control the system we need to add damping 2 d d m z mg Kk y yp C y y 2CT R 1 3 Note the damping constant T has units of time The system is critically damped when k C T m C 1 4 and the system will converge to the setpoint quickly if it is near critical damping However critical damping is not always the best choice if some ringing can be tolerated the system will react more quickly if it is underdamped 1 Two definitions will c
12. erential Controllers From Eq 1 2 we see that for any finite C the equilibrium is shifted away from the setpoint If we need to hit the setpoint precisely we can add an integral term 2 d y d C P E EE OC E E 1 5 T g k y y C y y y 710 y 1 5 The integral term integrates the difference between the process variable and the set point and drives this error to zero Like T the integral constant T has units of time The proportional integral differential controller or PID controller generally works very well and is used in many systems It is not limited to systems modeled by linear second or Last Revision August 2007 Page 9 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control der differential equations like Eq 1 5 For instance the linear error e y y used in Eq 1 5 can be replaced by any odd function of e The most difficult step in deploying a PID controller is determining the constants C T and T There is no foolproof method that always gives the best values For that matter there is no single set of best values the best values depend on the desired response time and your ring and overshoot tolerances If the system is very simple you may be able to calculate the values otherwise you will have to guess them or find them by trial and error Once you get values that keep the system stable
13. estions help Did you need to go outside the course materials for assistance What additional materials could you have used What did you like and or dislike about this lab What advice would you give to a friend just starting this lab The course materials are available over the Internet Do you a have access to them and b prefer to use them this way What additional materials would you like to see on the web Last Revision August 2007 Page 17 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved
14. ghts reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control thermostat until some setpoint is reached Then the control turns off the device until the process variable falls below the setpoint at which point the device is turned back on Most feedback controls incorporate hysteresis they turn on only after the process variable has fallen slightly below the setpoint and remain on until the process variable rises slightly above the setpoint Without hysteresis the device under control may stutter on and off Stuttering can be inefficient and may damage the device The hysteretic range on a house thermostat 1s typically one or two degrees on a water heater it can be as large as 15 de grees Feedback controls are common in experiments and are used to accomplish tasks like keeping a process at a specified temperature or a fluid level at a specified height But feedback con trols are too crude for many applications e Proportional Controllers When you merge onto a highway you gun your engine in the acceleration lane until your speed approaches the traffic speed and then ease off as you match your speed to the traffic speed If you were to accelerate at low power you would reach the end of the acceleration lane before you attained sufficient speed and if you did not ease off toward the end your attempt to match the traffic speed would be very jerky This control methodology is an example of proportional control th
15. h to turn the magnet off 2 Place the 0 405in ball on the brass screw and turn the screw until the ball hits the magnet 3 Record the light signal and position by pushing the Save Measurement button The vi will ask for the position enter the number of turns by which you have lowered the ball Zero for the first measurement 4 Lower the ball by turning the screw through a fixed angle Use 45degree increments for the first 360degrees and 90 degrees thereafter You can eyeball the angles The screw pitch is 24 turns per inch subsequent programs convert the number of turns to mm 5 Go back to step 2 and repeat until light signal saturates at 10V 6 When asked save the calibration data in a file If necessary you may use an editor to cor rect any mistakes you may have made entering the number of turns As you turn the screw be aware of backlash when you reverse directions on any screw or gear posi tioner the first 10 to 20 degrees may not result in any lateral movement You can feel the backlash when you reverse directions the screw will be easy to turn at first and becomes significantly harder when the threads re engage If you watch the light signal you can will observe that the screw does not move laterally at first Because of backlash you should take your measurements while turning the screw in one direction only In fact after you touch the ball to the magnet at the beginning of the calibration and reverse Last Revision Augu
16. ift registers and the Build Array operator EE You will also need to sort the arrays which can be done with the following code sample The middle block is the Sort 1D Array operator and the remaining operators pack and unpack the arrays into arrays of clusters While demonstrating your routine to the TA s explain how this code sample works Last Revision August 2007 Page 13 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control F Dithering 11 9 25 The routine Dither vi demonstrates the virtues of dithering The routine generates a sine wave with sample rate 10kHz superimposes noise and quantizes the total signal to integer lev els It then passes the signal through a 20mS smoothing filter and finally averages multiple in stance of the signal Starting with a Signal Amplitude of 0 75 and a Noise Amplitude of 0 run the routine to see the quantized smoothed and smoothed and averaged signal Then increase the noise until you get a reasonable facsimile of the sine wave How much noise do you need Now decrease the Signal Am plitude to 0 1 and again find the minimum noise level that extracts the signal What happens when you make the noise large Does the signal degrade Now set the to Average to 1000 Approximately how small a signal can you detect How would you predict this value Hint Remember that even before the instances
17. in the a high gain amplifier chain If for instance you have a luV signal masked by 100uV noise and you use an amplifier chain with a gain of one million the noise would be amplified to 100V saturating the final amplifier which is unlikely to be able to output much more than 10V You would not be able to detect your now 1V signal However if you use a filter early in your amplifier chain you can get rid of most of the noise and prevent it from saturating the chain In applications where analog filtering is not required digital filtering is often superior Digital fil ters are far more flexible can be made arbitrarily sharp are trivially tunable and their topology RC Butterworth Chebyshev Bessel etc is easily changeable However digital filters are compu tationally intensive and their use is limited to relatively low frequencies Moreover digital filters necessarily operate on digitized waveforms and are subject to all of the limitations Nyquist theo rem quantization of the digitization process Digital filters are difficult to design Fortunately there are many canned routines available includ ing an extensive suite in LabVIEW We need not worry too much about the differences between the types of digital filters refer to Chapter 8 of the LabVIEW Analysis Concepts manual for more infor mation In particular Figure 3 25 of this reference shows a handy flowchart for picking the correct filter type Fourier Transforms
18. k on Restore the com mon plots to plotting a simple line What is the highest frequency plotted What would you expect How does the highest frequency scale with the sampling rate Record your answers in your lab book 11 6 Now set the 60Hz Comb Amplitude to 1 Turn off autoscaling and expand the FFT frequency axis to view the 75Hz signal Change the Signal Frequency how close can you bring it to 60Hz and still differentiate your signal from the 60Hz Comb Signal How does this depend on the of Samples 11 7 Turn the 60Hz Comb off set the White Noise Amplitude to 1 and change One Shot to Cycle Repetitively How small a signal can you pick out of the noise C Averaging 11 8 25 The routine Repetitive Source vi outputs a 1000sample unit amplitude triangular wave with superimposed noise If the boolean Phase Coherent is true the waveform is truly repetitive if it is false the waveform drifts in time Program a routine that will call this vi to Average times and average the resulting instances The front panel of your routines should resemble Last Revision August 2007 Page 11 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control ie z Averaging Analyzer vi Front Panel am se apicaton Font Eo SE a Number Averaged Waveform Graph 1000 Plot 0 AF as to Average 1000 Noise amplitude J10 Phase Coherent Am
19. l 1 Bandwidth narrowing 2 Averaging 3 Pattern matching There are many ways to implement each of these techniques we will explore the most important Bandwidth Narrowing Generally signals are narrow band while noise is either wide band like Johnson noise or narrow band like 60Hz hum but of a different frequency than the signal frequency By narrowing the bandwidth of the signal we can diminish the noise thereby improving the signal to noise ratio For example it is not uncommon to detect a sine wave of frequency f and amplitude A that has been con taminated by Johnson noise The signal to noise ratio will be A 4k RTB where B is the bandwidth that we accept Making B small will increase the signal to noise ratio if we could make B arbitrarily small we could recover any size signal Unfortunately we cannot make B arbitrarily small as there are at least two limits on the size of B 1 All signals have some intrinsic frequency spread Af and the bandwidth cannot be made smaller than this spread 2 The accuracy to which a signal frequency can be determined is inversely proportional to the length of time that you measure the signal or alternately to the number of waveform cycles N that are contained in your sample For example assume you anticipate measuring a sig nal near 1kHz and you sample the signal for 100mS You would collect approximately 100 cycles in this time Thus you cannot determine the frequency of what y
20. larify this discussion A process variable is some measured value that charac terizes the system the setpoint is the desired level of the process variable Last Revision August 2007 Page 8 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Critical Damping Underdamped Overdamped Position Time If the system has some initial momentum critical damping can lead to overshooting If over shooting cannot be tolerated the system must be overdamped Critical Damping Underdamped Overdamped Position The inertia in mechanical systems is explicit but even non mechanical systems exhibit iner tia like phenomena For instance thermostatically controlled heating systems have a form of inertia Typically the thermostat is not immediately adjacent to the heater When the tem perature at the thermostat exceeds the setpoint the thermostat will turn the heater off but the area near the heater will be warmer than the area near the thermostat Heat will spread away from the heater and the temperature at the thermostat will continue to rise until the temperature equalizes everywhere Thus the system has inertia just like the spring mass system In general any system with time delays will have inertia like properties Adding damping through a differential term is frequently beneficial e Proportional Integral Diff
21. ncy f 2z k T Thus discretizing the time discretizes the frequency into steps of 1 T the longer the sample the better resolved the frequency This is just a restatement of the uncertainty principle discussed above From Lab 11 we know that a signal sampled at frequency f N T cannot represent a frequency higher than the Nyquist frequency f 2 The Nyquist frequency corresponds tok N 2 No higher k has any physical meaning The Nyquist Theorem is actually derived via Discrete Fourier Trans forms Discrete Fourier Transformers would have little practical value if they could not be evaluated quickly Casually one might think that the evaluating the transform would require evaluating N 2 exponentials N 2 for each of the N samples Thus the transform of a signal represented by a 100 000 samples would appear to take 10 000 times longer to evaluate than a signal represented by 1000 samples As the accuracy of the transform increases with the sample length this would be very unfortunate Fortunately there are clever algorithms which reduce the evaluation time scaling to NIn N a 100 000 sample representation takes only 665 times longer than a 1000 sample repre sentation These algorithms are called Fast Fourier Transforms or FFTs They are generally at tributed to J W Cooley and J W Tukey who published an implementation in 1965 but many oth ers including Gauss had discovered similar algorithms The FFT algorithm works mos
22. ntally spaced or can be random Increasing the resolution Dithering ADCs add quantization noise to low level signals For example the near zero levels of a 10 bit ADC with a range of 10V are 59 39 20 0 20 and 59mV The ADC cannot represent signals in between these levels This is particularly problematic for signals that are less than 10mV the ADC will convert the signal to the constant OmV and the signal will disap Last Revision August 2007 Page 6 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control pear entirely Paradoxically adding random noise will improve the conversions If the noise is sufficiently large it will cause the signal to fluctuate between the ADC quantization levels The average of the levels reported by the ADC will be the signal level This tech nique is called dithering and is quite commonly used Noise adding circuitry may be incor porated in the ADC or you may have to add noise yourself Often the natural noise in your system is often enough to dither As and example consider a steady 9mV signal Without noise the aforementioned ADC will always convert this signal to OmV Add 10mV noise however and the signal will range be tween lmV and 19mV with occasional larger excursions Roughly eleven times out of twenty the signal will be between 1 and 10mV and will be converted to zero while nin
23. ou measure to better than about 1 N 1 of 1kHz or 10Hz nor can you make the bandwidth any smaller The derivation of this uncertainty relation is given below The two most important Bandwidth narrowing techniques are filtering and Fourier Transforming Filtering narrows the bandwidth directly attenuating the out of band noise Fourier Transforming also narrows the bandwidth but by a more subtle method While it does not attenuate the noise it allows you to concentrate on the narrow portion of the spectrum that contains your signal thereby ignoring the majority of the noise which is spread through the entire spectrum Last Revision August 2007 Page 2 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Filtering Filtering is best used for real time signals and can be implemented in analog or digital software Analog filters can be quite simple can work quite well and are often sufficient But sharp analog filters are very complicated and most analog filters cannot be tuned i e their rolloff frequencies cannot be changed without physically replacing components There are some applications that de mand analog filters for instance e Anti aliasing You must attenuate the frequencies above the Nyquist frequency be fore the signal has been digitized e Amplifier chains It is not uncommon for noise to swamp the later amplifiers
24. plitude oO c D LN l l l l l l 200 400 600 B00 1000 Time Explore the effects of varying the number of instances on recovering the signal from various amounts of noise Show that threshold for successfully recovering the signal scales appropriately Report the results in you lab notebook D Pattern Matching 11 8 25 The routine Fit Source vi produces a wave modeled by sin b x 0 5 ae b x 0 5 with superimposed noise Write a routine that uses the Curve Fitting Express vi to fit data to this equation Your routine s front panel should resemble Last Revision August 2007 Page 12 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Fit Analyzer vi Front Panel File Edit Operate Tools Browse Window Help Best fit best fit _ Signal E Fita 0 90584 2 sE 20 Fit b OISE A 19 340 0 5 Amplitude Exercise the routine with different combinations of the noise and parameters a and b How well does it work E Equivalent Time Sampling 11 9 25 Duplicate the functionality of the Equivalent Time Analyzer vi Run the program several times until you understand its behavior The routine calls a subroutine Equivalent Time Source vi which outputs a sample of sine wave offset by a known Time Delay You will need to accumulate x and y data into arrays which you can do with sh
25. st But in the Windows environment operating system distractions limit the PID loop speed to about 1kHz Hard ware implementations can be much faster In the lab A Bandwidth Narrowing Filters 11 1 Use the program Filter vi to generate a sine wave with the Signal Amplitude set to 1 the Signal Frequency set to 75 and the Modulation Frequency set to 0 1 Turn of all noise Pass the signal through a Butterworth filter of Filter Order 3 with a High Pass Cutoff of 80 anda Low Pass Cutoff of 70 Study the synchronicity of the filtered signal do the modulation minima of the filter signal occur at the same time as the minima of the unfiltered signal Decrease the band pass how does the synchronicity behave Now study the amplitude of the filtered signal How narrow can you make the bandpass before the amplitude of the signal decreases Change the Modulation Frequency to 1 Now how narrow can you make the bandpass How does the minimum acceptable bandpass scale with the modulation frequency Can you explain this scaling Record your answers in your lab book 11 2 Restore the original default parameters Turn on the 60Hz Comb Noise which adds noise at 60Hz harmonics What happens to the unfiltered signal What about the filtered signal How low can you make the signal amplitude and still recover the original signal Decrease the bandpass to 8 See the LabVIEW PID Control Toolset User Manual Last Revision August 2007 Page 10 of 17 2007 Cop
26. st 2007 Page 14 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control direction you should turn the screw until it engages before recording any measurements It may take a little practice to find the engagement point Backlash occurs because the threads and gears do not mesh perfectly This is deliberate if they did mesh perfectly friction would make them very hard to turn 11 11 Turn the magnet switch back on The levitator requires your computer s concentrated atten tion Exit out of all extraneous programs especially programs that require periodic servicing like email browsers music players etc Turn the screw several turns out of the way Run the Basic Controller vi The first time you run the vi it will ask you for the location of the your calibration data Place the ball on the brass screw and then gingerly lift the ball into place with two fingers being careful not to block the light beam with your fingers The ball should levitate While watching the ball change the Setpoint in 0 1mm steps You should be able to see the ball move up and down Find the range over which the Levita tor can hold the ball Now watch the Chart on the vi The white curve shows the position of the ball The green curve shows the drive signal and is proportional to the magnetic field The red curve plots the service interval Three indicators display th
27. t Revision August 2007 Page 4 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control The spectra are indistinguishable for Aw a 0 025 and nearly so for A a 0 05 Aside from an unimportant change in the amplitude the spectra for Aw 0 1 are distinguishable only by the details of the secondary peaks surrounding the main peak the width of the main peak is very close to that of the pure signal Only when Aw a 0 2 are the spectra easily distinguishable Yet the untransformed original signals are readily distinguishable Thus given only the spectrum we can not determine the frequency to better than about Aw q 0 1 one quarter of the width to the first zero calculated above In essence we have an uncertainty principle the longer we sample a signal the greater the time uncertainty the better we know the frequency of the signal Discrete Fourier Transforms The expression given above assumes that the signal f t is continuous Since our signals are ac quired by an ADC they are actually sets of discrete samples Thus we must use a summation rather than an integral to calculate the Fourier Transform FO gt fer where N is the number of samples taken over some time T each sample f n is taken at time nT N By analogy to the argument of the exponential in the continuous transforms each k corre sponds to a freque
28. t naturally on samples lengths which are powers of 2 However sam ple lengths which are factorable into the products of powers of small primes such as N 2 3 5 can also be transformed efficiently stick to sample lengths which can be so represented If necessary pad your sample with zeros to extend it to one of these lengths In practice sample lengths below one million can be converted remarkably quickly Longer sample are slower than would be predicted by N In N scaling because of cache misses Once the sample gets large enough to require virtual memory disk accesses conversion is painfully slow 5 The convention used here has no overall multiplying constant other conventions are used that multiply this expression by some factor related to pi Last Revision August 2007 Page 5 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Averaging Experiments frequently produce multiple instances of the same signal Averaging the instances can improve the signal to noise ratio if the signal itself is repetitive and the noise uncorrelated from in stance to instance Then the noise will diminish as 1 VN where Nis the number of instances Note that the signal must be truly repetitive otherwise it will average away just like the noise Also note that because of the square root dependence successive factor of two noise reductions
29. us control a control that lets us set the state of the driven object to a continuous set of levels A typical experimental function would be controlling the strength of a magnetic field Continuous con trols are commonly implemented by using the voltage from a DAC to control the target de vice a power supply for instance to control the current through a magnet A continuous control allows some systems like a magnet to reach a well defined state Most systems however are not so strictly controlled Sometimes a system will appear to be stably set to some level but a perturbation or change in system can run it out of control For in stance a stove top burner cannot be programmed to reach and hold a particular tempera ture Temporary control can be achieved by setting a pot of water on stove burner The wa ter and pot will get no hotter than 100C But once the water boils away the temperature will rise uncontrollably Having once melted through an aluminum pot which requires a temperature of 660C I am all too familiar with this scenario Feedback Controllers A feedback control adds feedback to a On Off control a thermostat is a familiar feedback control The control turns on some device a heater in the case of a 6 A smoothing filter outputs a running average of its inputs Its action is similar to that of a low pass filter Last Revision August 2007 Page 7 of 17 2007 Copyrighted by the Regents of the University of California All ri
30. vi This vi briefly changes the setpoint and plots the response Scan the derivative time Td over the range that the ball stays levitated How does the response vary When does the ball oscillate How does the stabilization time change with Td Last Revision August 2007 Page 15 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Picture of levitator with feedback control Last Revision August 2007 Page 16 of 17 2007 Copyrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control Physics 111 BSC Student Evaluation of Lab Write Up Now that you have completed this lab we would appreciate your comments Please take a few moments to answer the questions below and feel free to add any other comments Since you have just finished the lab it is your critique that will be the most helpful Your thoughts and suggestions will help to change the lab and improve the experiments Please be as specific as possible using both sides of the paper as needed and turn this in with your lab re port Thank you Lab Number Lab Title Date Which text s did you use How was the write up for this lab How could it be improved How easily did you get started with the lab What sources of information were most least helpful in getting started Did the pre lab qu
31. yrighted by the Regents of the University of California All rights reserved Physics 111 BSC Laboratory Lab 11 Signal Processing and Control the minimum values that you found in 11 1 How small a signal can you recover How well does the filter remove the 60Hz noise Record your answers in your lab book 11 3 25 Restore the original default parameters Turn off the 60Hz Comb Noise and turn on the White Noise which adds 0 045 V VHz noise For bandwidths of 10 1 and 0 1Hz how much noise should get through the filter What is the smallest signal you should be able to observe Con firm your predictions by running the program B Bandwidth Narrowing FFTs 11 4 Use the program FFT Analyzer to explore Fourier Transforms With a Signal Frequency of 75 Signal Amplitude of 1 Noise Amplitude and 60Hz Comb Amplitude of 0 Sampling Rate of 1000 and of Samples of 10000 turn off autoscaling on the FFT Frequency axis and expand the axis to view the peak at 75 Use the Common Plots option on the FFT graph to turn on the display of the individual data points How wide is the peak in Frequency How wide is the peak in points How wide would you predict it to be Change the Signal Frequency in increments of 0 01Hz How does the peak width change Now change the of Samples How does the width change Does it conform to your predictions Record your answers in you lab book 11 5 Using the same program turn the FFT frequency axis autoscaling bac
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