Frequency Resolution Effects on FRF Estimation: Cyclic Averaging vs. Large Block Size

Transcription

1 Frequency Resolution Effects on FRF Estimation: Cyclic Averaging vs. Large Block Size Allyn W. Phillips, PhD Andrew. Zucker Randall J. Allemang, PhD Research Assistant Professor Research Assistant Professor Structural Dynamics Research Laboratory Department of Mechanical, Industrial and Nuclear Engineering University of Cincinnati Cincinnati, Ohio U. S. A. ABSRAC It is always desirable to use acquired data in the most effective way possible. his paper explores acquiring several contiguous blocks of data and then processing the data in various ways. he first method will be the traditional approach of utilizing each contiguous block, in whole, as one RMS spectral average. Other approaches will involve dividing the contiguous block up into subsets which will then be cyclic averaged. he advantages and disadvantages of each approach are discussed in terms of total test time, processing time, data quality, and memory usage which becomes significant for moderate to large (N i xn o ) test situations. Experimental results are shown for typical structural cases. Nomenclature N = ime blocksize. N s = Frequency blocksize. N avg = Number of RMS spectral averages. N c = Number of cyclic averages. N i = Number of inputs. N o = Number of outputs. Δ f = Frequency resolution (Hz.).. Introduction he averaging of signals is normally viewed as a summation or weighted summation process where each sample function has a common abscissa. Normally, the designation of history is given to sample functions with the abscissa of absolute time and the designation of spectra is given to sample functions with the abscissa of absolute frequency. he spectra are normally generated by Fourier transforming the corresponding history. In order to generalize and consolidate the concept of signal averaging as much as possible, the case of relative time can also be considered. In this way relative history can be discussed with units of the appropriate ev ent rather than seconds and a relative spectrum will be the corresponding Fourier transform with units of cycles per event. his concept of signal averaging is used widely in structural signature analysis where the event is a revolution. his kind of approach simplifies the application of many other concepts of signal relationships such as Shannon s sampling theorem and Rayleigh s criterion of frequency resolution. 2. Background he process of signal averaging as it applies to frequency response functions is simplified greatly by the intrinsic uniqueness of the frequency response function. Since the frequency response function can be expressed in terms of system properties of mass, stiffness, and damping, it is reasonable to conclude that in most realistic structures, the frequency response functions are considered to be constants just like mass, stiffness, and damping. his concept means that when formulating the frequency response function using H, H 2,orH v algorithms, the estimate of frequency response is intrinsically unique, as long as the system is linear and the noise can be minimized or eliminated. In general, the auto- and cross-power spectrums are statistically unique only if the input is stationary and sufficient averages have been taken. Generally, this is never the case. Nevertheless, the estimate of frequency response is valid whether the input is stationary, non-stationary, or deterministic. he concept of the intrinsic uniqueness of the frequency response function also permits a greater freedom in the testing procedure. Each function can be derived as a result of a separate test or as the result of different portions of the same continuous test situation. In either case, the estimate of frequency response function will be the same as long as the time history data is acquired simultaneously for the auto- and cross-power spectrums that are utilized in any computation for frequency response or coherence function. Generally, averaging is utilized primarily as a method to reduce the error in the estimate of the frequency response function(s). his error can be broadly considered as noise on either the input and/or the output and can be considered to the sum of random and bias components. Random errors can be effectively minimized through the common approach to averaging. Bias errors, generally, cannot be effectively minimized through the common approach to av eraging. Cyclic av eraging, however, does minimize the bias error caused by the truncation of the time domain signals in conjunction with the use of a discrete Fourier transform (DF). his error is commonly known as the leakage error. he approaches to signal averaging for these different situations vary only in the relationship between each sample function used.

2 Since the Fourier transform is a linear function, there is no theoretical difference between the use of histories or spectra. (Practically, though, there are precision considerations.) With this in mind, the signal averaging useful to frequency response function measurements can be divided into three classifications: Asynchronous Synchronous Cyclic hese three classifications refer to the trigger and sampling relationships between sample functions. 2. Asynchronous Signal Averaging he classification of asynchronous signal averaging refers to the case where no known relationship exists between individual sample functions. he FRF is estimated solely on the basis of the intrinsic uniqueness of the frequency response function. In this case, the power spectra (least squares) approach to the estimate of frequency response must be used since no other way of preserving phase and improving the estimate is available. In this situation, the trigger to initiate digitization (sampling and quantization) takes place in a random fashion dependent only upon the equipment availability. he triggering is said to be in a free-run mode. 2.2 Synchronous Signal Averaging he synchronous classification of signal averaging adds the additional constraint that each sample function must be initiated with respect to a specific trigger condition (often the magnitude and slope of the excitation). his means that the frequency response function can be formed as a summation of ratios of X(ω ) divided by F(ω ) since phase is preserved. Even so, the power spectra (least squares) approach is still the preferred FRF estimation method due to the reduction of variance and the usefulness of the ordinary coherence function. he ability to synchronize the initiation of digitization allows for use of nonstationary or deterministic inputs with a resulting increased signal to noise ratio and reduced leakage. Both of these improvements in the frequency response function estimate are due to more of the input and output being observable in the limited time window. he synchronization takes place as a function of a trigger signal occurring in the input (internally) or in some event related to the input (externally). An example of an internal trigger would be the case where an impulsive input is used to estimate the frequency response. All sample functions would be initiated when the input reached a certain amplitude and slope. A similar example of an external trigger would be the case where the impulsive excitation to a speaker is used to trigger the estimate of frequency response between two microphones in the sound field. Again, all sample functions would be initiated when the trigger signal reached a certain amplitude and scope. 2.3 Cyclic Signal Averaging he cyclic classification of signal averaging involves the added constraint that the digitization is coherent between sample functions. his means that the exact time between each sample function is used to enhance the signal averaging process. Rather than trying to keep track of elapsed time between sample functions, the normal procedure is to allow no time to elapse between successive sample functions. his process can be described as a comb digital filter in the frequency domain with the teeth of the comb at frequency increments dependent upon the periodic nature of the sampling with respect to the event measured. he result is an attenuation of the spectrum between the teeth not possible with other forms of averaging. his form of signal averaging is very useful for filtering periodic components from a noisy signal since the teeth of the filter are positioned at harmonics of the frequency of the sampling reference signal. his is of particular importance in applications where it is desirable to extract signals connected with various rotating members. his same form of signal averaging is particularly useful for reducing leakage during frequency response measurements and also has been used for evoked response measurements in biomedical studies. A very common application of cyclic signal averaging is in the area of analysis of rotating structures. In such an application, the peaks of the comb filter are positioned to match the fundamental and harmonic frequencies of a particular rotating shaft or component. his is particularly powerful, since in one measurement it is possible to enhance all of the possible frequencies generated by the rotating member from a given data signal. With a zoom Fourier transform type of approach, one shaft frequency at a time can be examined depending upon the zoom power necessary to extract the shaft frequencies from the surrounding noise. he application of cyclic averaging to the estimation of frequency response functions can be easily observed by noting the effects of cyclic averaging on a single frequency sinusoid. Figures and 2 represent the cyclic averaging of a sinusoid that is periodic with respect to the observation time period. Figures 3 and 4 represent the cyclic averaging of a sinusoid that is aperiodic with respect to the observation time period. By comparing Figure 2 to Figure 4, the attenuation of the nonperiodic signal can be clearly observed. 3. heory of Cyclic Averaging In the application of cyclic averaging to frequency response function estimates, the corresponding fundamental and harmonic frequencies that are enhanced are the frequencies that occur at the integer multiples of Δ f. In this case, the spectra between each Δ f is reduced with an associated reduction of the bias error called leakage.

3 fundamental frequency resolution Δ f Rayleigh Criteria (Δ f = ). of the spectra via the he Fourier transform of a history is given by: Figure. Contiguous ime Records (Periodic Signal) X(ω ) = + x(t) e jω t dt () Using the time shift theorem of the Fourier transform, the Fourier transform of the same history that has been shifted in time by an amount t is: Figure 2. Av eraged ime Records (Periodic Signal) X(ω ) e j ω t = + x(t + t ) e jω t dt (2) For the case of a discrete Fourier transform, each frequency in the spectra is assumed to be an integer multiple of the fundamental frequency Δ f =. Making this substitution in Equation (2) (ω = n 2 π with n as an integer) yields: + X(ω ) e jn2 π t = x(t + t ) e jω t dt (3) Figure 3. Contiguous ime Records (Non-Periodic Signal) Note that in Equation (3), the correction for the cases where t = Nwith N is an integer will be a unit magnitude with zero phase. herefore, if each history that is cyclic averaged occurs at a time shift, with respect to the initial average, that is an integer multiple of the observation period, then the correction due to the time shift does not effect the frequency domain characteristics of the averaged result. All further discussion will assume that the time shift t will be an integer multiple of the basic observation period. he signal averaging algorithm for histories averaged with a boxcar or uniform window is: x(t) = N c N c Σ x i(t) (4) i= where: x i (t) = ime history, average i N c = Number of cyclic averages x(t) = Cyclic averaged time history Figure 4. Av eraged ime Records (Non-Periodic Signal) he first observation to be noted is the relationship between the Fourier transform of a history and the Fourier transform of a time shifted history. In the averaging case, each history will be of some finite time length which is the observation period of the data. Note that this time period of observation determines the For the case where x(t) is continuous over the time period N c, the complex Fourier coefficients of the cyclic averaged time history become: C k = x(t) e jω t dt (5)

4 C k = N c N c Σ x i(t) e jω t dt (6) i= Finally: C k = N c Nc i= Σ x i(t) e jω t dt (7) Since x(t) is a continuous function, the sum of the integrals can be replaced with an integral evaluated from to N c over the original function x(t). herefore: C k = N c N c x(t) e jω t dt (8) he above equation indicates that the Fourier coefficents of the cyclic averaged history (which are spaced at Δ f = ) are the same Fourier coefficients from the original history (which are spaced at Δ f = N c ). Figures 5 through 8 show the cyclic averaging effect in the frequency domain for the cases of, 2 and 4 averages with a uniform window applied to the data. Figures 9 through 2 show the cyclic averaging effect in the frequency domain for the cases of, 2 and 4 averages with a Hanning window applied to the original contiguous data.. hese figures demonstrate the effectiveness of cyclic averaging in rejecting nonharmonic frequencies. Practically, these figures also demonstrate that, based upon effectiveness or the limitations of the dynamic range of the measured data, a maximum of 6 to 32 averages is recommended. Realistically, 4 to 8 cyclic av erages together with a Hanning window provides a dramatic improvement in the FRF estimate Figure 6. Cyclic Averaged (N c = 2) Window Uniform Window Figure 7. Cyclic Averaged (N c = 4) Window Uniform Window Figure 5. Cyclic Averaged (N c = ) Window Uniform Window

5 Figure 8. Cyclic Averaged (N c = 4) Window Uniform Window Figure. Cyclic Averaged (N c = 2) Window Hanning Window Figure 9. Cyclic Averaged (N c = ) Window Hanning Window Figure. Cyclic Averaged (N c = 4) Window Hanning Window

6 Figure 5. Contiguous ime Records with Hanning Window (Random Signal) Figure 2. Cyclic Averaged (N c = 4) Window Hanning Window he results of cyclic averaging of a general random signal with the application of a uniform window are shown in Figures 3 and 4. Likewise, the results of cyclic averaging of a general random signal with the application of a Hanning window are shown in Figures 5 and Figure 3. Contiguous ime Records (Random Signal) Figure 4. Av eraged ime Records (Random Signal) Figure 6. Av eraged ime Records with Hanning Window (Random Signal) 4. Practical Example Examples from two different lightly damped structures will be used to discuss the points of this paper. he two structures tested consist of a circular plate and a body-in-white. Both stuctures were tested with a random excitation via two shaker inputs and the responses were measured with accelerometers. Each structure was tested under two different digital signal processing techniques. he first technique used cyclic averaging with N c cyclic averages, N avg spectral averages, and N time lines. In comparison, the second method used the same N avg spectral averages, no cyclic av erages, and increased the number of time lines to N c xn so that a large blocksize resulted and the total test time for both sets of measurements were the same. As a result of the larger blocksize the frequency resolution was increased for this measurement set as compared to the first measurement set by a factor of N c. As stated the total test time for both techniques was the same since each case required the collection of NxN c xn avg time lines. he question to be answered is whether cyclic averaging provides better results than the same measurement with a finer frequency spacing and what are the advantages/disadvantages of one technique over the other. When the frequency axis of the cyclic averaged test is extracted from the finer frequency/larger blocksize run and overlaid the results are very comparable (a plot is not shown since the two different FRF s can not even be distinguished). his is a result of the comb filtering of cyclic averaging and in the abscence of other measurement noise it is expected that the two cases would exactly overlay one another. An analytical study will be conducted to verify if this is indeed the result. If all the frequency lines of the finer frequency/larger blocksize run are kept and the results of the

7 two techniques are compared, as shown in Figures 7 and 8 for the circular plate and Figures 9 and 2 for the body-in-white, the effects of cyclic averaging on leakage can be observed. A dramatic improvement in coherence as a result of cyclic averaging eliminating leakage is seen in the comparison of the circular plate data. An improvement, though minimal, is also present in the body-in-white data. Had cyclic averaging not been employed, the results would have looked like those presented in Figures 2 and 22, for the circular plate and body in white respectivily, which are the results from a pure random excitation in which the same amount of data was taken but cyclic averaging was not implemented. 8 2 Input: Output: Input: Output Figure 9. Body in White 8 Input: Output: Figure 7. Lightly Damped Circular Plate Input: Output Figure 2. Body in White 8 3 Input: Output Figure 8. Lightly Damped Circular Plate Figure 2. Lightly Damped Circular Plate

8 Input: Output: dramatically reduces both the memory requirements and the computational effort. his reduction may be trivial for small test, but substantial for moderate to large tests. It is also important to remember that if the results are going to be used for the estimation of modal parameters, it is much more important to have frequency response functions accurate at the measured frequencies than to have a large number of frequencies calculated. For large tests using current polyreference modal parameter estimation techiniques, the additional frequency lines do not contribute significantly the the results. In fact, the measurements are often decimated before inclusion. However, if there is leakage at the resonances in the large block FRFs that has been eliminated in the cyclic averaged data, the results of using the large block FRFs will bias the solution of the modal parameters. 6. References Figure 22. Body in White Besides improved data quality through the minimization of leakage, cyclic averaging offers two other noteworthy advantages. In comparison to the large blocksize case, cyclic averaged data requires less processing time. his is a direct result of the number of frequency lines being a factor of N c smaller when compared to the large blocksize case. For larger N i xn o cases this translates into N c xn i xn o lines which need not be processed when calculating FRF s. he second advantage, amount of memory required to save results, is also a direct result of this factor of N c differnece in the number of frequency lines. Since cyclic averaged results do contain a factor of N c less frequency lines when compared to the large blocksize case it requires less space to store the results. For large N i xn o tests this means a savings of N i xn o x N 2 x(n c ) complex values. Assuming standard data storage ( complex number requires 6 btyes), a moderately sized test (N i = 4, N o = 256, N s = 24, N = 248) would require approximately 35 Megabytes of storage. If the blocksize is increased by a factor of 4 (N i = 4, N o = 256, N s = 496, N = 892), the in memory required storage increases proportionately to almost 4 Megabytes. However, in contrast, the use of 4 cyclic averages (N i = 4, N o = 256, N s = 24, N = 892) requires only the additional storage for the raw time series, the spectral storage requirements remain the same (about 38 Megabytes). Note also that the computational time also increases proportionately. One advantage of the larger blocksize is that through its smaller Δ f spacing a more accurate reading of the resonance frequncies maybe obtained. Also closely spaced modes may become more visible. However, if the data is going to be utilized in an appropriate parameter estimation routine, the routine should be able to pinpoint the resonance frequencies and closely spaced modes without this increased frequency resolution. [] Bendat, J.S.; Piersol, A.G., Random Data: Analysis and Measurement Procedures, John Wiley and Sons, Inc., New York, 97, 47 pp. [2] Bendat, J. S., Piersol, A. G., Engineering Applications of Correlation and Spectral Analysis, John Wiley and Sons, Inc., New York, 98, 32 pp. [3] Otnes, R.K., Enochson, L., Digital ime Series Analysis, John Wiley and Sons, Inc., New York, 972, 467 pp. [4] Hsu, H.P., Fourier Analysis, Simon and Schuster, 97, 274 pp. [5] Allemang, R. J., "Investigation of Some Multiple Input/Output Frequency Response Function Experimental Modal Analysis echniques", Doctor of Philosophy Dissertation, University of Cincinnati, Mechanical Engineering Department, 98, 358 pp. [6] Potter, R.W., "Compilation of ime Windows and Line Shapes for Fourier Analysis", Hewlett-Packard Company, 972, 26 pp. [7] Allemang, R.J., Phillips, A.W., "Cyclic Averaging for Frequency Response Function Estimation", Proceedings, International Modal Analysis Conference, 996, 8pp. 5. Conclusions he results of this study indicate that while cyclic averaging and increased frequency produce essentially the same result for the frequencies in common, however, the use of cyclic averaging