Dynamic Signal Analysis Basics
Contents
1. CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS where g t is the window weighting function and T is the window duration The data analyzed x t are then given by x t w t x t where x t is the original data and x t is the data used for spectral analysis A window in the time domain is represented by a multiplication and hence is a convolution in the frequency domain A convolution can be thought of as a smoothing function This smoothing can be represented by an effective filter shape of the window i e energy at a frequency in the original data will appear at other frequencies as given by the filter shape Since time domain windows can be represented as a filter in the frequency domain the time domain windowing can be accomplished directly in the frequency domain In most DSA products rectangular Hann Flattop and several other data windows are used Rectangular Window w k 1 O lt k lt N 1 Hann Window w k 0 5 1 cos 22k N 1 0 lt k lt N 1 Because creating data window attenuates a portion of the original data a certain amount of correction has to be made in order to get an un biased estimation of the spectra In linear spectral analysis an Amplitude Correction is applied in power spectral measurements an Energy Correction is applied See the sections below for details Linear Spectrum A linear spectrum is the Fourier transform of windowed time domain data The linear spectrum is useful for ana2. This correction will make the peak or RMS reading of a sine wave at specific frequency correct regardless of which data window is applied For example if a 1 0 volt amplitude 1kHz sine wave sampled at 6 4kHz is analyzed with a Linear Spectrum with Hann window you will get following the spectral shape E SIG0035_BLOCK ch1 1 500 1 t t 0 032 0 032 0 033 0 034 0 034 0 035 Time seconds T T E SIG0035_APS chI 20 000 T 0 000 20 000 40 000 60 000 80 000 dB v 0 peak 100 000 120 000 t 0 000 500 000 1000 000 1500 000 2000 000 Frequency Hz m Figure 1 Sine wave with Hanning window applied to the spectrum The top picture is the digitized time waveform The sine wave is not smooth because of the low sampling rate relative to the frequency of the signal However the well known Nyquist principle indicates that the frequency estimate from the FFT will be accurate as long as the sampling rate is SS COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 7 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS more than twice of the signal frequency The frequency spectrum of the period signal will show the accurate frequency and level Note for a more accurate sample of the time waveform a higher sampling rate is required Figure 2 illustrates a windowing function applied to a pure sine tone SI60035_APS ch1 1 200 lt
3. total harmonic distortion phase match amplitude flatness etc In recent years time domain data acquisition devices and DSA instruments have gradually converged together More and more time domain instruments such as oscilloscopes can do frequency analysis while more and more dynamic signal analyzers can do long time data recording DSA uses various different technology of digital signal processing Among them the most fundamental and popular technology is based on the so called the Fast Fourier Transform FFT The FFT transforms the time domain signals into the frequency domain To perform FFT based measurements however you need to understand the fundamental issues and computations involved This Chapter describes some of the basic signal analysis computations discusses antialiasing and acquisition front end for FFT based signal analysis explains how to use windowing functions correctly explains some spectrum computations and shows you how to use FFT based functions for some typical measurements In this Chapter we will use standard notations for different signals Each type of signal will be represented by one specific letter For example G stands for a one side power spectrum while H stands for a transfer function The following table defines the symbols used in this Chapter Cyx Coherence function between input signal x and output signal y Gxx Auto spectral function one sided of signal x Gyx Cross spectral functi
4. in parallel as shown in Figure 18 The frequency response function FRF of this mechanical system is also shown output 10 output input 19 input 102 L L L LLLIL L L L LLLILIL 10 10 102 Frequency Hz m Figure 18 SDOF system and their frequency response The differential equation of motion for this system is given by mi cx kx f t The natural frequency wn and damping ratio can be calculated from the system parameters as C we and 2w m m where m is the mass k is the spring stiffness and c is the damping coefficient The natural frequency w is in units of radians per second rad s The typical units displayed on a digital signal analyzer are in Hertz Hz The damping ratio C can also be represented as a percent of critical damping the damping level at which the system experiences no oscillation This is the more common understanding of modal damping Figure 18 illustrates the response of a SDOF system to a transient excitation showing the effect of different damping ratios COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 26 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS u t in t sec 0 00 Wn 5 0 rad s Uy 20 0 in sec 4 00 00 0 80 1 60 2 40 3 20 m Figure 19 Step response of a SDOF system with different damping ratios A SDOF system with light damping factor will have longer oscillation in a transient process This is why the exponenti
5. mW into a load of 50 for radio frequencies where 0 dB is 0 22 Vrms or 600 Q for audio frequencies where 0 dB is 0 78 Vrms The picture below shows a sine wave with 1V amplitude displayed in dB Because the reference is 1Vpk it shows the peak value of this sine wave as OdB COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 27 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS m Figure 20 Show a 1Vpk sine signal in frequency domain with dB scaling Another display format is called Log or LogMag The Log display shows the signal scaled logarithmically with the grid values and cursor readings in actual engineering value The picture below shows the same signal in LogMag m Figure 21 A 1Vpk sine signal in frequency domain with LogMag scaling When dB reference is not specified the dB reference is 1 0 engineering unit In acoustics application the dB reference for the sound pressure value is set to 20uPa The same input signal will result in different dB readings when dB reference is changed Fv vI COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 28 CRYSTAL Instruments 8 TRANSIENT CAPURE AND HAMMER TESTING TRANSIENT CAPTURE AND HAMMER TESTING Transient Capture In the previous Chapters of this manual we have discussed how the acquisition mode can be defined in the CSA Editor and selected on the CoCo device This chapter will demonstrate how to use CoCo to conduct ha
6. second to 24 hours Assume the AverageT is 1 hour without moving linear average in a 24 hours period you can only get 24 readings This is not very useful With moving averaging you can get the readings in every 1 second for the linear averaging of the past 1 hour COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 22 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Exponential Averaging In exponential averaging records do not contribute equally to the average A new record is weighted more heavily than old ones The value at any point in the exponential average is given by yin y n 1 1 a x n a where y n is the nth average and x n is the nth new record ais the weighting coefficient Usually ais defined as 1 Number of Averaging For example in the instrument if the Number of Averaging is set to 3 and the averaging type is selected as exponential averaging then a 1 3 The advantage of this averaging method is that it can be used indefinitely That is the average will not converge to some value and stay there as is the case with linear averaging The average will dynamically respond to the influence of new records and gradually ignore the effects of old records Exponential averaging simulates the analog filter smoothing process It will not reset when a specified averaging number is reached The drawback of the exponential averaging is that a large value may embed too much memory into
7. the averaged cross spectrum between the input channel x and output channel y Gxx is the averaged auto spectrum of the input Either power spectrum power spectral density or energy spectral density can be used to compute the FRF because of the linear relationship between input and output Using the cross power spectrum method instead of simply dividing the linear spectra between input and output to calculate the FRF will reduce the effect of the noise at the output measurement end as shown below input output Lani System true Hyx P lt noise estimated lt observed x Hyx observed y m Figure 9 Frequency response function computation The frequency response function has a complex data format You can view it in real and imaginary or magnitude and phase display format The coherence function is defined as 2 c 1216 j G G where Gyxis the averaged cross spectrum between the input channel x and output channel y Gxx and Gyy are the averaged auto spectrum of the input and output Either power spectrum power spectral density or energy spectral density can be used here because of the linear relationship between input and output so that any multiplier factors will be cancelled out COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 14 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Coherence is a statistical measure of the how much of the output is caused by the input The maxi
8. will be different even though the same spectrum type is used Spectrum Types selection only applies to Power Spectrum and Linear Spectrum signals Spectrum Types do not apply to transfer functions phase functions or coherence functions Cross Spectrum Cross spectrum or cross power spectrum density is a frequency spectrum quantity computed using two signals usually the excitation and response of a dynamic system Cross spectrum is not commonly used by its own Most often it is used to compute the frequency response function FRF transmissibility or cross correlation function To compute the cross power spectral density Gyx between channel x and channel y Step 1 compute the Fourier transform of input signal x k and response signal y k N 1 Sx gt x k w k e J 2ztkn N n 0 N 1 sy gt y k w k e 2m n 0 COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 13 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Step 2 compute the instantaneous cross power spectral density Syx Sx SyT Step 2 average the M frames of Sxx to get averaged PSD Gxx Gyx Average Syx Step 3 Compute the energy correction and double the value for the single sided spectra Gyx 2 Gyx EnergyCorr Frequency Response and Coherence Function The cross power spectrum method is often used for estimating the frequency response function FRF between channel x and channel y The equation is Hy Gyx G where Gyxis
9. 00Hz Each sine cycle would have 8 integer points If 1024 data points are acquired then 128 complete cycles of the signal are captured In this case with no window applied you still can get a leakage free spectrum Figure 11 shows a sine signal at 1000 Hz with no leakage resulting in a sharp spike Figure 12 shows the spectrum of a 1010 Hz signal with significant leakage resulting in a wide peak The spectrum has significant energy outside the narrow 1010 Hz frequency Itis said that the energy leaks out into the surrounding frequencies m Figure 11 Sine spectrum with no leakage m Figure 12 Sine spectrum with significant leakage Several windowing functions have been developed to reduce the leakage effect The picture below shows a Flattop window applied to the same sine signal with frequency 1010Hz COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 16 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS m Figure 13 Sine spectrum with Flattop windowing function When Flattop window is used the leakage effect is reduced Both the sine peak and noise floor can be seen now However such data windowing operation also makes the spectrum peak fatter and less accurate In the rest of the sections we will discuss how to choose different data windows Data Window Formula In this section we will describe the math formula that we used for each data window Uniform window rectangular w k 1 0 Uniform is
10. 1 000 ooo EUpk Linear Scaling 0 600 m 9 400 x gt 0 200 0 000 0 000 1000 000 2000 000 3000 000 Frequency Hz 000 0 000 TE 5 000 display in dB EUpk 10 000 a 46 000 20 000 l 950 000 1000 000 1050 000 1100 000 Frequency Hz m Figure 2 Hanning windowing function applied to a pure sine tone The top picture is displayed in EUpk i e the peak of the spectrum is scaled to the actual 0 peak level which is 1 0 in this case The bottom picture shows the same signal with the dB scale applied Since we use 0dB as reference the 1 0 Vpk is now scaled to 0 0 dB With the dB display we can see frequency points around the peak causing by the Hanning window The linear spectrum is saved internally in the complex data format with real and imaginary parts Therefore you should be able to view the real and imaginary parts or amplitude and phase of the spectrum Power Spectrum Spectral analysis is popular in characterizing the operation of mechanical and electrical systems A type of spectral analysis the power spectrum and power spectral density PSD is especially popular because a power measurement in the frequency domain is one that engineers readily accept and apply in their solutions to problems Single channel measurements auto power spectra and two channel measurements cross power spectra both play important roles In power spectrum measurements window a
11. 3 Sine spectrum with Flattop windowing function 17 a Figure 14 Spectral shape of common windowing functions 19 a Figure 15 Window frequency response showing main lobe and side lobes 20 a Figure 16 Illustration of moving linear average 22 a Figure 17 Illustration of overlap processing 25 a Figure 18 SDOF system and their frequency response 26 a Figure 19 Step response of a SDOF system with different damping ratios 27 a Figure 20 Show a 1Vpk sine signal in frequency domain with dB scaling 28 a Figure 21 A 1Vpk sine signal in frequency domain with LogMag scaling 28 a Figure 22 Transient capture operation on CoCo 29 a Figure 23 Illustration of a typical impact test and signal processing 30 a Figure 24 Typical impact test data Top left shows excitation force impulse time signal top right shows response acceleration time signal and bottom shows FRF spectrum 31 ss xsKsF uIEsa COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE3 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS FREQUENCY ANALYSIS Basic Theory of FFT Frequency Analysis Introduction DSA often referred to Dynamic Signal Analysis or Dynamic Signal Analyzer depending on the context is an application area of digital signal processing technology Compared to general data acquisition and time domain analysis DSA instruments and math tools focus more on the dynamic aspect of the signals such as frequency response dynamic range
12. CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Dynamic Signal Analysis Basics James Zhuge Ph D President Crystal Instruments Corporation 4633 Old Ironsides Drive Suite 304 Santa Clara CA 95054 USA Wwww go ci com Part of CoCo 80 User s Manual ssaIOEcImnIi COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE1 CRYSTAL Instruments DYNAMIC SIGNAL ANALYSIS BASICS Table of Contents FREQUENGY ANALYSIS i sccsvecessvesccesgethetetsccnedesactbenebeceuadenvicebexetaatived ssaustedsdeisbadeasestoasbacukedaaevbantdbeceuadenvadbeceteatandens 4 Basic Theory of FFT Frequency Analysis U 4 Tal xele Ue 0 o tierce epecreerte rete rereectre A E rere treet T E T recente peer E crt rer 4 Fourier Transform ainiaan araa uaaa eaaa aaa aa ae ra Aaaa aaa A E ERa a e AEO ERREA 5 Data WImdOWI 0OL E rer T E T A E E O TT A eer rer er 5 Linear Spectr isinira aeaa ma uuu uQ q au ai uum u au aaraa aE 6 Power Spectr wa aoa a aan yun haat a nia nna ah AeA eae WOR a edt 8 SPSCIUM TYPOS u s sn araa anaa ana eara ade cited aa qk aqha aae a daaa 9 GrOSS SDSCUUDSQ sussaaqusspusapuyyaqakasyaqayaaykqaspantadqtusyaqauuayyielopuqtpadabuzuyadasyaykqqupuaykieclopustyadatazuyadakuaqyaqupuakhio bikan 13 Frequency Response and Coherence Function 14 Data WINdOW SelCt On ii uuu u a aus qiypauquk usya haste q as aaaea iat aa Steen ee eee eee 15 Leakage Effet rnini nu aula neg qulu Casma da ua auqa t
13. I I j I l j v 7 J U m Figure 10 Illustration of a non periodic signal resulting from sampling If there are two sinusoids with different frequencies leakage can interfere with the ability to distinguish them spectrally If their frequencies are dissimilar then the leakage interferes when one sinusoid is much smaller in amplitude than the other That is its spectral component can be hidden or masked by the leakage from the larger component But when the frequencies are near each other the leakage can be sufficient to interfere even when the sinusoids are equal strength that is they become undetectable crnvI COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 15 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS There are two possible scenarios that leakage does not occur The first is that when the whole time capture is long enough to cover the complete duration of the signals This can occur with short transient signals For example in a hammer test if the time capture is long enough it may extend to the point where the signal decays to zero In this case data window is not needed The second case is when a periodic signal is sampled at such a sampling rate that is perfectly synchronized with the signal period so that with a block of capture an integer number of cycles of the signal are always acquired For example if a sine wave has a frequency of 1000Hz and the sampling rate is set to 80
14. aging must be performed with on a triggered event so that the time signal of one average is correlated with other similar measurements Without time synchronizing mechanism averaging in the Linear Spectrum domain makes no sense Linear spectrum averaging is also called Vector averaging It averages the complex FFT spectrum The COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 23 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS real part is averaged separately from the imaginary part This can reduce the noise floor for random signals since they are not phase coherent from time record to time record Power Spectrum Averaging is also called RMS Averaging RMS averaging computes the weighted mean of the sum of the squared magnitudes FFT times its complex conjugate The weighting is either linear or exponential RMS averaging reduces fluctuations in the data but does not reduce the actual noise floor With a sufficient number of averages a very good approximation of the actual random noise floor can be displayed Since RMS averaging involves magnitudes only displaying the real or imaginary part or phase of an RMS average has no meaning and the power spectrum average has no phase information Table 1 gives a summary of the averaging methods described above m Table 1 Summary of Averaging Methods Power Spectrum Averaging Averaging No statistical spectral Statistical spectral estimate for estimate for determinist
15. al window may be chosen to reduce the leakage effect in its spectral analysis dB and Linear Magnitude Most often amplitude or power spectra are shown in the logarithmic unit decibels dB Using this unit of measure it is easy to view wide dynamic ranges that is it is easy to see small signal components in the presence of large ones The decibel is a unit of ratio and is computed as follows dB 10logio Power Pref where Power is the measured power and Pref is the reference power Use the following equation to compute the ratio in decibels from amplitude values dB 20logio Ampl Aref where Ampl is the measured amplitude and Aref is the reference amplitude When using amplitude or power as the amplitude squared of the same signal the resulting decibel level is exactly the same Multiplying the decibel ratio by two is equivalent to having a squared ratio Therefore you obtain the same decibel level and display regardless of whether you use the amplitude or power spectrum As shown in the preceding equations for power and amplitude you must supply a reference for a measure in decibels This reference then corresponds to the 0 dB level Different conventions are used for different types of signals A common convention is to use the reference 1 Vrms for amplitude or 1 Vrms squared for power yielding a unit in dBV or dBVrms In this case 1 Vrms corresponds to 0 dB Another common form of dB is dBm which corresponds to a reference of 1
16. applied More details will be discussed in the next sections for averaging operation Spectrum Types Several Spectrum Types are given for both Linear Spectrum and Power Spectrum measurements in CoCo and EDM The concept of spectrum type is explained below in detail First let s consider the signals with periodic nature These can be the signals measured from a rotating machine bearing gearing or anything that repeats In this case we would be interested in amplitude changes at fundamental frequencies harmonics or sub harmonics In this case you can choose a spectrum type of EUpk EUpkpk or EUms A second scenario might consist of a signal with a random nature that is not necessarily periodic It does not have obvious periodicity therefore the frequency analysis could not determine the amplitude at certain frequencies However it is possible to measure the r m s level or power COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 9 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS level or power density level over certain frequency bands for such random signals In this case you must select one of the spectrum types of EU ms Hz or EU ms sqrt Hz which is called power spectral density or root mean squared density A third scenario might consist of a transient signal It is neither periodic nor stably random In this case must select a spectrum type as EU S Hz which is called energy spectrum In ma
17. can be used to determine the modal properties of the device such as the natural frequencies and damping ratios In addition the data can be exported to third party modal analysis software to compute mode shapes An impact hammer test is the most common method of measuring FRFs The hammer imparts a transient impulsive force excitation to the device The impact is intended to excite a wide range of frequencies so that the DSA can measure the vibration of the device across this range of frequencies The bandwidth or frequency content of the excitation input depends on the size and type of impact hammer that is used The dynamic force signal is recorded by the DSA After the impact the device vibrations are measured with one or more accelerometers or other sensor and COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 29 CRYSTAL Instruments 8 TRANSIENT CAPURE AND HAMMER TESTING recorded by the DSA The DSA then computes the FRF by comparing the force excitation and the response acceleration signals Impact testing is depicted in Figure 23 Time h Ae ee Impulse a R Real Modal Para Parameters G ve oe Ft Damping a ae Mode Shape m Figure 23 Illustration of a typical impact test and signal processing The following equipment is required to perform an impact test 1 An impact hammer to excite the structure With CoCo we recommend using an impact hammer with IEPE output which all
18. ess des q Canaan pasasaq 15 Data WIrdow Formula uuu sesati iadaaa rere etree Perrier secre cyer perro rerrerrer ere rter tere oer err rrr 17 How to Choose the Right Data Window I nn 18 Guidelines of Choosing Data Windows U a 21 Averaging Techniques u u a alus a fatness gcd ade aetna ect a Ape aE aaa Alvis a Aaaa eked ENa 21 Linear AV ORANG aaesevisters ceed vats cece veke detect cde aeaieie daad dave cet vatk deere deter denarii eee eae 21 Moving n ar Average aaan aaan aaa anaa aAa a aaa a aaraa aA Ea a Aaa i ES 22 Exponential Av Grain cect secs cucss cchewct ces teateacencces iiA E EEEE aAA AEE NE AE NETEN AOS AEO eee tied 23 Peak Hold uu annua pu au qaa qd Sasak gaa E EE 23 Linear Spectrum versus Power Spectrum Averaging U n 23 Spectrum Estimation EmMOr ssia n a aa uu uuu ua au qua aa SEESE 24 Overlap Processing sugkuna va Need a Ma ah Qua eae ent pa Qua ah ea eee ek et byk 25 Single Degree of Freedom System L a 26 OB and Linear Magnitude u u uu usa akaqaayunqaqaypucyadayaaqyaqakuwykiadapuaukuauaauzyyalokaqiyaqabuauyadapuaupaaauu ayka doka auqa 27 TRANSIENT CAPTURE AND HAMMER TESTING ceccecesceseeeeeeeeeeeeeeeeesaeeaeeeceesaesaesaeeseesaesaeeaseesesaneaseees 29 Transient Oz o 10 sa umum noetan etree ree amupash petak a as errr terre er peer upaya tree pert EAEE peer unus a as Piscoya 29 impact Hammer Testi
19. form oaa COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 18 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Rectangular sti TI Magnitude dB i r i Magnitude dB g 05 Magnitude dB 8 8 Magnitude dB 8 m 0 5 0 35 42 Sample Frequency m Figure 14 Spectral shape of common windowing functions It can be seen that the spectral shape of the data window is always symmetric The spectral shape can be described as a main lobe and several side lobes csaOP m Vx COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 19 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS 6dB Peak side lobe level Main lobe width Frequency m Figure 15 Window frequency response showing main lobe and side lobes The following table lists the characteristics of several data windows Frequency Characteristics of Data Windows Window 3 dB Main Lobe 6 dB Main Maximum Width bins Lobe Width Side Lobe bins Level dB Uniform none 0 9 1 2 13 Hanning 1 4 2 0 32 Hamming 1 3 1 8 43 Blackman 1 6 2 3 58 Flattop 2 9 3 6 44 Main Lobe The center of the main lobe of a window occurs at each frequency component of the time domain signal By convention to characterize the shape of the main lobe the widths of
20. h averaging a standard capability in all modern FFT analyzers FRFs should be measured using at least 4 impacts per measurement Since one or two of the impacts during the measurement process may be bad hits too hard causing saturation too soft causing poor coherence or a double hit causing distortion in the spectrum an FFT analyzer designed for impact testing should have the ability to accept or reject the result of each impact after inspecting the impact signals An accept reject capability saves a lot of time during impact testing since you don t have to redo all measurements in the averaging process after one bad hit 4 Modal Damping Estimation The width of the resonance peak is a measure of modal damping The resonance peak width should also be the same for all FRF measurements meaning that modal damping is the same in every FRF measurement A good analyzer should provide an accurate damping factor estimate CoCo uses a curve fitting algorithm to estimate the damping factor The algorithm reduces the inaccuracy caused by the poor spectrum resolution or noise 5 Modal Frequency estimation The analyzer must provide capability of estimating the resonance frequencies CoCo uses an algorithm to identify the resonance frequencies based on the FRF References To understand the topics of this article we found the following three books to be very helpful 1 Julius S Bendat and Allan G Piersol Random Data Analysis and Measurement P
21. ic signals with random characteristics signals only Signal must have periodic Applicable to both pure random and components mixed random periodic signals Requires a synchronized Does not require a synchronized trigger in fixed relation to the trigger signals Spectrum Estimation Error You may wonder how much confidence we should have when we take the spectral measurement This is a academic topic that can go very deep First you must classify your signal types If you are measuring a deterministic signal with very few averaging the spectrum estimation can be very accurate If the signal has a random nature with partially random or significant measurement noise more averaging must be used Assume the time data is captured from a stationary random process and we calculate various spectra using window FFT and averaging techniques how much we can trust the measured spectra can be measured by a statistical quantity standard deviation Here are a few useful equations to compute the standard deviation of the spectra when linear averaging is used Functions being Standard Deviation estimated Auto spectrum Gxx 1 vn Cross spectrum Gyx 1 Cyx vn a Function 1 Cyx J2 Cyx vn COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 24 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Frequency Response be a oe Function Hyx yU Cyx Cyx V2n where nis the average nu
22. igger delay are used to capture the transient signal for FRF processing It is important to capture the entire short transient signal it in the sampling window of the FFT analyzer To insure that the entire signal is captured the analyzer must be able to capture the impulse and impulse response signals prior to the occurrence of the impulse with the pre trigger Force amp Exponential Windows Two common time domain windows that are used in impact testing are the force and exponential windows These windows are applied to the signals after they are sampled but before the FFT is computed in the analyzer The force window is used to remove noise from the impulse force signal Ideally an impulse signal is non zero for a small portion of the sampling window and zero for the remainder of the window time period Any non zero data following the impulse signal in the sampling window is assumed to be measurement noise CoCo has a unique way to implement the force window This was discussed in the data windowing section in the previous chapter The exponential window is applied to the impulse response signal The exponential window is used to reduce leakage in the spectrum of the response COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 31 CRYSTAL Instruments 8 TRANSIENT CAPURE AND HAMMER TESTING 3 Accept Reject Because accurate impact testing results depend on the skill of the operator FRF measurements should be made wit
23. ime waveform f frequency variable j complex number X Fourier transform of x t Mathematically the Fourier Transform is defined for all frequencies from negative to positive infinity However the spectrum is usually symmetric and it is common to only consider the single sided spectrum which is the spectrum from zero to positive infinity For discrete sampled signals this can be expressed as N 1 X k gt x k e JF 2rkn N n 0 where x k samples of time waveform n running sample index N total number of samples or frame size k finite analysis frequency corresponding to FFT bin centers X k discrete Fourier transform of x k In most DSA products a Radix 2 DIF FFT algorithm is used which requires that the total number of samples must be a power of 2 total number of samples in FFT 2 where m is an integer Data Windowing The Fourier Transform assumes that the time signal is periodic and infinite in duration When only a portion of a record is analyzed the record must be truncated by a data window to preserve the frequency characteristics A window can be expressed in either the time domain or in the frequency domain although the former is more common To reduce the edge effects which cause leakage a window is often given a shape or weighting function For example a window can be defined as w t gt T 2 lt t lt T 2 0 elsewhere Sa COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 5
24. lyzing periodic signals You can extract the harmonic amplitude by reading the amplitude values at those harmonic frequencies An averaging technique is often used in the time domain when synchronized triggering is applied Or equivalently the averaging can be applied to the complex FFT spectra Because the averaging is taking place in the linear spectrum domain or equivalently in the time domain based on the principles of linear transform averaging make no sense unless a synchronized trigger is used Most DSA products use the following steps to compute a linear spectrum Step 1 First a window is applied x t w t x t where x t is the original data and x t is the data used for the Fourier transform Step 2 COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 6 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS The FFT is applied to x t to compute X k as described above Step 3 Averaging is applied to X k Here Averaging can be either an Exponential Average or Stable Average Result is Sx Sx Average X k Step 4 To get a single sided spectrum double the value for symmetry about DC An Amplitude Correction factor is applied to Sx so that the final result has an un biased reading at the harmonic frequencies Sx 2 e Sx AmpCorr where AmpCorr is the amplitude correction factor defined as N 1 AmpCorr gt w k k 0 where w k is the window weighting function
25. mber in linear averaging The transfer function is computed in the cross power spectrum method as presented earlier Assume a signal is random and has an expected power spectral density at 0 1 V7 Hz The goal of a measurement is to average a few power spectra and to estimate such an expected value If the average number is 1 meaning with no average the standard deviation of the error of such a measurement will be 100 When we average two frames of auto power spectra the standard deviation of the error will become 70 7 When the average number is increased to 100 the standard deviation of the error of the reading is 10 This means that the reading is likely in the neighborhood of 0 1 0 01 V Hz Now if this signal has a deterministic nature say a sine wave the spectral estimation error will only be applied to the random portion i e the noisy portion of this signal Overlap Processing To increase the speed of spectral calculation overlap processing can be used to reduce the measurement time The diagram below shows how the overlap is realized Signal Captured in the Time Domain g U Acquired Signal Data Transformed into FFT Frames No Overlap Processing Acquired Signal Post Processed with Overlap FFTs 1024 Samples 1024 Samples 1024 Samples FFTs Overlap Samples Overlap Interval Samples 1024 Samples 1024 Samples m Figure 17 Illustration of overlap processing As shown in
26. mmer testing Hammer testing refers to impact or bump testing that is conducted using an impact hammer to apply an impulsive force excitation to a test article while measuring the response excitation from an accelerometer or other sensor This type of measurement is a transient event that usually requires triggering averaging and windowing First let s briefly review the Transient Capture function on CoCo Transient Capture is one of the most common used functions for dynamic data acquisition In CoCo the Transient Capture is implemented by setting up the Acquisition Mode Acquisition Mode defines how to transform the time streams into block by block time signals It sets the trigger and the overlapping processing Before the Acquisition Mode stage the instrument acts as a data recorder while after the Acquisition Mode it is acts as a signal analyzer Data recorder Signal Analyzer m Figure 22 Transient capture operation on CoCo Besides Acquisition Mode you must first enable at least one time stream as a trigger candidate in the CSA Editor Trigger candidates are those time streams that can be selected as a trigger source The names of these trigger candidates will be passed to the CoCo During runtime one of the trigger source candidates must be selected as the trigger source Impact Hammer Testing Typically impact hammer testing is conducted with a signal analyzer to measure FRFs of the device under test The FRFs
27. mplitude correction is used to get un biased final spectrum amplitude reading at specific frequency In PSD or energy spectral density ESD measurements window energy correction is always used to get an un biased spectral density or energy reading To compute the spectra listed above the instrument will follow these steps COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 8 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Step 1 A window is applied x k w k x k where x k is the original data and x k is the data used for a Fourier transform Step 2 The FFT is applied to x t to compute Sx N 1 Sx x k e2nkn N Next the so called periodogram method is used to compute the spectra with area correction Using Sx Step 3 Calculate the Power Spectrum Sxx Sx Sx AmpCorr Or calculate the Power Spectral Density Sx Sx T EnergyCorr Or calculate the Energy Spectral Density Sx Sx T EnergyCorr where T is the time duration of the capture The symbol is for complex conjugation EnergyCorris a factor for energy correction which is defined as 1 N 1 EnergyCorr z2 w k k 0 Nis the total number of the samples and w k is window function For any power spectral measurement of the three types listed above the EU is automatically chosen as EU ms because only EU ms has a physical meaning related to signal power After the power spectra are calculated the averaging operation will be
28. mum coherence is 1 0 when the output is perfectly correlated with the input and zero when there is no correlation between input and output Coherence is calculated by an average of multiple frames When it is computed for only one frame then the coherence function has a meaningless result of 1 0 due to the estimation error of the coherence function The coherence function is a non dimensional real function in the frequency domain You can only view it in the real format Data Window Selection Leakage Effect Windowing of a simple signal like a sine wave may cause its Fourier transform to have non zero values commonly called leakage at frequencies other than the frequency of this sine This leakage effect tends to be worst highest near sine frequency and least at frequencies farthest from sine frequency The effect of leakage can easily be depicted in the time domain when a signal is truncated As shown in the picture after data windowing truncation distorted the time signal significantly hence causing a distortion in its frequency domain Actual Input i f L f l j i j WE I f j l E o or time l f j i l l l I 1 t f f Lf J 1 J Windowed Input IA A i Nor Periodic f f L p T P i time I l l I l J Assumed Input i n n j i a j 1 i j y l J f ili J Hf oH rr
29. ndow Averaging Techniques Averaging is widely used in spectral measurements It improves the measurement and analysis of signals that are purely random or mixed random and periodic Averaged measurements can yield either higher signal to noise ratios or improved statistical accuracy Typically three types of averaging methods are available in DSA products They are Linear Averaging Exponential Averaging and Peak Hold Linear Averaging In linear averaging each set of data a record contributes equally to the average The value at any point in the linear average in given by the equation Sum of Records A d verage N N is the total number of the records The advantage of this averaging method is that it is faster to compute and the result is un biased However this method is suitable only for analyzing short signal records or stationary signals since the average tends to stabilize The contribution of new records eventually will cease to change the value of the average Usually a target average number is defined The algorithm is made so that before the target average number reaches the process can be stopped and the averaged result can still be used asTI COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 21 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS When the specified target averaging number is reached the instrument usually will stop the acquisition and wait for the instruction for anothe
30. ng onmin da dasictateshacatudasictadevss ides aeaa Eaa EA aa Aaaa aR A nd Pana aa ER ERa ae aar 29 Impact Test Analyzer Settings res uyu saa squysadayaankqaassaskaqaypapkayasquykadaykuykqqasaaykaukypapiaqayquykadasukyaqdaskuskawka payak 31 ReferenCE Sasiia aaae aa aa aa E aA T Eaa ENE AIEEE SENES 32 EPI COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 2 CRYSTAL Instruments DYNAMIC SIGNAL ANALYSIS BASICS Table of Figures a Figure 1 Sine wave with Hanning window applied to the spectrum 7 a Figure 2 Hanning windowing function applied to a pure sine tone 8 a Figure 3 Flow chart to determine measurement technique for various signal types 10 a Figure 4 A sine wave is measured with EUpk spectrum unit The sine waveform has a 1V amplitude 11 a Figure 5 A sine wave is measured with EUrms spectrum unit The peak reading is 0 707V The sine waveform has a 1V amplitude 11 a Figure 6 A sine wave is measured with EUrms spectrum unit The peak reading is 0 5 V The sine waveform has a 1Vamplitude 12 a Figure 7 White noise with 1 volt RMS amplitude displays as 100 u Vins Hz 12 a Figure 8 Random signal with 1 volt RMS amplitude and Energy Spectrum Density format 13 K Figure 9 Frequency response function computation 14 a Figure 10 Illustration of a non periodic signal resulting from sampling 15 a Figure 11 Sine spectrum with no leakage 16 a Figure 12 Sine spectrum with significant leakage 16 a Figure 1
31. no leakage effect will occur then do not apply any window in the software select Uniform As discussed before this only occurs when the time capture is long enough to cover the whole transient range or when the signal is exactly periodic in the time frame If the goal of the analysis is to discriminate two or multiple sine waves in the frequency domain spectral resolution is very critical For such application choose a data window with very narrow main slope Hanning is a good choice If the goal of the analysis is to determine the amplitude reading of a periodic signal i e to read EU px EUpkok EUrms Or EUms the amplitude accuracy of a single frequency component is more important than the exact location of the component in a given frequency bin choose a window with a wide main lobe Flattop window is often used If you are analyzing transient signals such as impact and response signals it is better not to use the spectral windows because these windows attenuate important information at the beginning of the sample block Instead use the Force and Exponential windows A Force window is useful in analyzing shock stimuli because it removes stray signals at the end of the signal The Exponential window is useful for analyzing transient response signals because it damps the end of the signal ensuring that the signal fully decays by the end of the sample block If the nature of the data is has a random nature or unknown choose Hanning wi
32. ny applications the nature of the data cannot be easily classified Care must be taken to interpret the data when different spectrum types are used For example in the environmental vibration simulation a typical test uses multiple sine tones on top of random profile which is called Sine on Random In this type application you have to observe the random portion of the data in the spectrum with EU Hz and the sine portion of the data with EU Figure 3 shows a general flow chart to choose one of the measurement techniques and spectrum types for linear or auto spectrum Classify the nature of data Periodic narrowband Random broadband Transient broadband Power RMS Power Energy Linear Power Spectrum Spectrum Spectrum Spectrum Spectrum Dana Dena Sx Bey ensity ensity SxSx T Sqrt PSD SxSx TT Y Y Y Y Y Averaging Y Y Y Y Y Window amplitude correction Window energy correction Select one of the spectrum Y Y Y type EUpk EUpkpk EUrms 3 gt EUrms EUrms Hz EUrms sqrt Hz EUrms S Hz m Figure 3 Flow chart to determine measurement technique for various signal types The following figures illustrate the results of different measurement techniques on a 1 volt pure sine tone The figures include RMS Peak or Peak Peak value for the amplitude or power value corresponding to i
33. on one sided between input signal x and output signal y Hyx Transfer function between input signal x and output signal y k Index of a discrete sample Rxx Auto correlation function of signal x Ryx Cross correlation function between input signal x and output signal y Sx Linear spectral function of signal x Sxx Instantaneous auto spectral function one sided of signal x Syx Instantaneous cross spectral function one sided between input signal x and output signal y t Time variable x t Time history record X f Fourier Transform of time history record COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 4 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Fourier Transform Digital signal processing technology includes FFT based frequency analysis digital filters and many other topics This chapter introduces the FFT based frequency analysis methods that are widely used in all dynamic signal analyzers CoCo has fully utilized the FFT frequency analysis methods and various real time digital filters to analyze the measurement signals The Fourier Transform is a transform used to convert quantities from the time domain to the frequency domain and vice versa usually derived from the Fourier integral of a periodic function when the period grows without limit often expressed as a Fourier transform pair In the classical sense a Fourier transform takes the form of X f f roenan dt 00 where x t continuous t
34. ows the hammer to be connected directly to the analyzer without extra signal conditioning 2 One or multiple accelerometers that are fixed on the structure Again IEPE accelerometers can be used directly with CoCo without additional signal conditioning 3 Coco Signal Analyzer 4 The CoCo can be used to extract the resonance frequencies and damping factors of the structure In addition third party software can be used to extract modal shapes and animate the vibration modes A wide variety of structures and machines can be impact tested Of course different sized hammers are required to provide the appropriate impact force depending on the size of the structure small hammers for small structures large hammers for large structures Realistic signals from a typical impact test are shown in Figure 10 COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 30 CRYSTAL instruments Amplitude 8 TRANSIENT CAPURE AND HAMMER TESTING T T T 0 500 1 000 1 500 2 000 2 500 3 000 Frequency Magnitude dB o 500 1 000 1 500 2 000 2 500 3 000 Frequency Figure 24 Typical impact test data Top left shows excitation force impulse time signal top right shows response acceleration time signal and bottom shows FRF spectrum Impact Test Analyzer Settings The following settings are used for impact testing 1 Trigger Setup including trigger level and pre tr
35. r collection of data acquisition Moving Linear Averaging In a regular Linear Average the data rate of the output of the averaging operator is only 1 N of that of the original signal Therefore more averages takes longer to compute Thus averaging will increase the time of the measurement To reduce the time a Moving Linear Averaging can be used Moving Linear Averaging uses overlapped input data points to generate more than 1 N results within a period of time Moving linear average has the advantage that the resulted trace update time can be much shorter than the linear averaging period Moving Linear Average is computed by N 1 1 yin gt 2 x n J j 0 Where x k is the input data with sampling rate of T y n is the output data with Trace Update rate deltaT AverageT is the period of Linear Average and N is the total samples used for Linear Average N AverageT T The Moving Linear Averaging is illustrated in Figure 16 Assume the averaging period is AverageT but the progressive time for each averaging operation is delta T the output buffer will have a data rage of deltaT instead of Average T x k Saved every T N AverageT deltaT AverageT N AverageT y n saved every deltaT m Figure 16 Illustration of moving linear average The Moving Linear Average is useful in many situations For example in Sound Level Meter Leq is defined as a linear averaged value over a long period of time say 1
36. rocedures 2nd Edition Wiley Interscience New York 1986 2 Julius S Bendat and Allan G Piesol Engineering Applications of Correlation and Spectral Analysis 2nd Edition Wiley Interscience New York 1993 3 Sanjit K Mitra and James F Kaiser Ed Handbook for Digital Signal Processing Wiley Interscience New York 1993 COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 32
37. rum type is suitable for narrowband signals COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE11 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Mag 2 RMS 0 500 Asine wave with 1Vpk displayed in EUrms Hann window applied 0 400 0 300 0 200 0 100 900 000 950 000 1000 000 1050 000 1100 000 Frequency Hz m Figure 6 A sine wave is measured with EUrms spectrum unit The peak reading is 0 5V2 The sine waveform has a 1V amplitude EU Hz Power Spectrum Density The EU Hz is the spectrum unit used in power spectrum density PSD calculations The unit is in engineering units squared divided by the equivalent filter bandwidth This provides power normalized to a 1Hz bandwidth This is useful for wideband continuous signals EU Hz really should be written as EU ms Hz But probably due to the limitation of space people put it as EU Hz m Figure 7 White noise with 1 volt RMS amplitude displays as 100 u Vims Hz Figure 7 shows a white noise signal with 1V ms amplitude or 1V in power level The bandwidth of the signal is approximately 10000 Hz and the V Hz reading of the signal is around 0 0001 V Hz The 1 V RMS can be calculated as follows 1 Vims sqrt 10000Hz 0 0001 V Hz EU S Hz Energy Spectrum Density The EU S Hz displays the signal in engineering units squared divided by the equivalent filter bandwidth multiplied by the time duration of signal This spectr
38. the average result If there is a transient large value as input it may take a long time for y n to decay On the contrary the contribution of small input value of x n will have little impact to the averaged output Therefore exponential average fits a stable signal better than a signal with large fluctuations Peak Hold This method technically speaking does not involve averaging in the strict sense of the word Instead the average produced by the peak hold method produces a record that at any point represents the maximum envelope among all the component records The equation for a peak hold is yin MAX j x n j Peak hold is useful for maintaining a record of the highest value attained at each point throughout the sequence of ensembles Peak Hold is not a linear math operation therefore it should be used carefully It is acceptable to use Peak Hold in auto power spectrum measurement but you would not get meaningful results for FRF or Coherence measurement using Peak Hold Peak hold averaging will reset after a specified averaging number is reached Linear Spectrum versus Power Spectrum Averaging Averaging can be applied to either linear spectrum or power spectrum If you want to reduce the spectral estimation variance use power spectral averaging If you want to extract repetitive or periodic small signals from a noisy signal you can use triggered capture and average them in linear spectral domain Linear Spectrum aver
39. the main lobe at 3 dB and 6 dB below the main lobe peak describe the width of the main lobe The unit of measure for the main lobe width is FFT bins or frequency lines The width of the main lobe of the window spectrum limits the frequency resolution of the windowed signal Therefore the ability to distinguish two closely soaced frequency components increases as the main lobe of the smoothing window narrows As the main lobe narrows and spectral resolution improves the window energy spreads into its side lobes increasing spectral leakage and decreasing amplitude accuracy A trade off occurs between amplitude accuracy and spectral resolution Side Lobes Side lobes occur on each side of the main lobe and approach zero at multiples of f N from the main lobe The side lobe characteristics of the smoothing window directly affect the extent to which adjacent frequency components leak into adjacent frequency bins The side lobe response of a strong sinusoidal signal can overpower the main lobe response of a nearby weak sinusoidal signal Maximum side lobe level and side lobe roll off rate characterize the side lobes of a smoothing window The maximum side lobe level is the largest side lobe level in decibels relative to the main lobe peak gain COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 20 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS Guidelines of Choosing Data Windows If a measurement can be made so that
40. the same as no window function Hamming window 21k N 1 w k 0 53836 0 46164 cos Hann window 21k w k 0 5 0 5 cos 1 The Hann and Hamming windows are in the family known as raised cosine windows are respectively named after Julius von Hann and Richard Hamming The term Hanning window is sometimes used to refer to the Hann window but is ambiguous as it is easily confused with Hamming window Blackman window COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 17 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS 21k 41k w k 0 84 0 5 cos 0 08 cos NLI N 1 fork 0 N 1 Flattop window l 1 1 93 ane ioo Bk ie w COS Taa COST 1 cos 1 8r 0 032 cos fork 0 N 1 Kaiser Bessel window k 1 0 1 24 E E T w 1 y cos 1 x cos 1 fork 0 N 1 6k 0 00305 cos N Exponential Window The shape of the exponential window is that of a decaying exponential The following equation defines the exponential window w _ AG mfp fork 0 N 1 where N is the length of the window w k is the window value and final is the final value of the whole sequence The initial value of the window is one and gradually decays toward zero How to Choose the Right Data Window In this section we will discuss how to choose the data window Figure 14 shows the spectral shape of four typical windows corresponding to their time wave
41. this picture when a frame of new data is acquired after passing the Acquisition Mode control only a portion of the new data will be used Overlap calculation will soeed up the calculation with the same target average number The percentage of overlap is called overlap ratio 25 overlap means 25 of the old data will be used for each spectral processing 0 overlap means that no old data will be reused Overlap processing can improve the accuracy of spectral estimation This is because when a data window is applied some useful information is attenuated by the data window on two ends of each block However it is not true that the higher the overlap ratio the higher the spectral estimation COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 25 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS accuracy For Hanning window when the overlap ratio is more than 50 the estimation accuracy of the spectra will not be improved Another advantage to apply overlap processing is that it helps to update the display more quickly Single Degree of Freedom System This section briefly discusses the single degree of freedom SDOF system as background for the frequency response function and damping estimation methods The vibration nature of a mechanical structure can be decomposed into multiple relatively independent Single Degree Of Freedom systems Each SDOF system can be modeled as a mass fixed to the ground by a spring and a damper
42. ts amplitude Notice these readings can only be applied to a periodic signal If you applied these measurement techniques to a signal with random nature the spectrum would not be a meaningful representation of the signal EUpk or EUpkpk The EU and EU pk displays the peak value or peak peak value of a periodic frequency component at a discrete frequency These two spectrum types are suitable for narrowband signals COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 10 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS 1 200 1 000 Asine wave with 1Vpk displayed in EUpk Hann window applied Mag Vv O peak a 8 0 200 950 000 1000 000 1050 000 1100 000 Frequency Hz m Figure 4 A sine wave is measured with EUpk spectrum unit The sine waveform has a 1V amplitude EU ms The EU ms displays the RMS value of a periodic frequency component at a discrete frequency This spectrum type is suitable for narrowband signals 0 800 Asine wave with 0 700 1Vpk displayed in EUrms Hann window 0 600 Mag RMS o o 6 8 o 0 200 0 100 950 000 1000 000 1050 000 1100 000 Frequency Hz m Figure 5 A sine wave is measured with EUrms spectrum unit The peak reading is 0 707V The sine waveform has a 1V amplitude EU ms Power spectrum The EUms displays the power reading of a periodic frequency component at a discrete frequency This spect
43. um type provides energy normalized to a 1Hz bandwidth or energy spectral density ESD It is useful for any signals when the purpose is to measure the total energy in the data frame Figure 8 shows a random signal with a 1 volt RMS level in the ESD format R sOEII COPYRIGHT 2009 CRYSTAL INSTRUMENTS ALL RIGHTS RESERVED PAGE 12 CRYSTAL instruments DYNAMIC SIGNAL ANALYSIS BASICS AytoPowerSpect 0 000 5000 000 10000 000 Frequency Hz m Figure 8 Random signal with 1 volt RMS amplitude and Energy Spectrum Density format The ESD is calculated as follows Values for ESD values of PSD Time Factor were the Time Factor Block size Af and Af is the sampling rate block size Notice that in EU Hz or EU S Hz EU really means the RMS unit of the EU i e EUms It should also be noted that since a window is applied in time domain which corresponds a convolution in the linear spectrum we cannot have both a valid amplitude and correct energy correction at the same time Use Figure 3 to select appropriate spectrum types In a Linear Spectrum measurement a signal is saved in its complex data format which includes both real and imaginary data Then is averaging operation applied to the linear spectrum In a Power Spectrum measurement the averaging operation is applied to the squared spectrum which has only real part Because of different averaging techniques the final results of Linear Spectrum and Power Spectrum
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