Engineering Acoustics ME 413
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This 5 page Class Notes was uploaded by Roman Jaskolski on Friday October 23, 2015. The Class Notes belongs to ME 413 at University of Idaho taught by Michael Anderson in Fall. Since its upload, it has received 22 views. For similar materials see /class/227897/me-413-university-of-idaho in Mechanical Engineering at University of Idaho.
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Date Created: 10/23/15
Frequency Decomposition Spectral Analysis Prepared by Michael J Anderson Last Revised February 6 2008 Introduction In this section we consider spectral analysis of acoustic data that is recorded with a digital instrument Two reasons necessitate this type of an analysis The rst is that we must be careful to account for the process of representing a continuous signal with sampled points The second reason is that most sources of sound ie quotnoisequot is random in nature as opposed to single frequency harmonic In this circumstance account must be made of the frequency dependent sensitivity of the measuring instrumentation as well as the sensitivity of human perception of sound In general we have as data N evenly spaced data points xnAt n0N 1 where At is the sampling interval A shortened notation for the data points xnAt is to simply refer to them as xquot The reciprocal of the sampling interval NAP is known as the sampling frequency The units given to the sampling frequency are Hz in this case samplessecond instead of cyclessecond The time interval TNAt is known as the fundamental data period while N is usually referred to as the data record length quot of the Root Mean Square of a Signal from Sampled Data Points In acoustics as in other branches of science and engineering the root mean square value of a signal is important Recall for a signal xt the RMS value is defined as T 1 2 xm x dr T 0 where the overbar indicates a time average of the signal xt J xt dt 0 The RMS value of a sampled signal xquot n0N 1 can be estimated by approximating the above integral using a quotbar graphquot approximation to the integral as T N71 N71 1 2 1 2 1 2 ers xt x dtm n x At n x TJ NM E x N EM 0 quot0 n0 where again the overbar indicates the average of the sampled data xquot ie The last expression in the approximated integral is the standard deviation of the data points xquot To be more precise the statistical de nition of the standard deviation for a set of points xquot is but the difference between N and N 1 in the denominator makes little difference in its numerical value for large N Discrete Fourier Transform For N data points xquot the determination of the N coefficients Xk k0lN Nl Xk E xneXP393927ma k0N quot0 is known as the complex form of the Discrete Fourier Transform DFT The coefficients Xk which in general are complex valued are known as the DFT coefficients Boldface type is used to indicate that the DFT coefficients are in general complex valued They are entirely determined by the data xquot In fact the inverse DFT process is capable of recovering the data xquot from the DFT coefficients Xk The inverse DFT known as the IDFT is N71 x 2Xkexpf n 01N 1 kl Together the DFT and its inverse the IDFT comprise a transform pair which is the discrete version of the continuous Fourrier transform used on continuous signals If the data record length N is an even power of 2P were p is an integer eg 2P is 2 481632 etc a very efficient way can be used to compute the coefficients Xk This algorithm is known as the Fast Fourier Transform FFT The N coefficients Xk are in general complex For real data xquot the coefficients Xmz1XN1 are complex conjugates of the coefficients X1XN21 For this reason only the coefficients X0 through XN2 are needed for analysis of real data In addition the coefficient X0 and X N2 are entirely real so that the N data points xquot map into N independent numbers in the set of coefficients X0 XNz Reconstructing the Data Points from the DFT Coef cients The signi cance of the DFT coef cients is that they can be used to reconstruct the time series at the sampling instants tnAt From the coef cients X0 through XNg we form the trigonometric series A 1 W4 2 1 xt X0 Z l Xk l cos27239fkt k XN2 cos27rfNZt N m N N f kT k arngkl At the sampling instants tnAt the trigonometric series and data will exactly match 01m xnAt Between the sampling instants t nAt the reconstruction few will not be equal to the actual data xt unless special precautions are taken The trigonometric series contains N2 sinusoids ranging in frequency from f11T Hz to fN2N2T Hz The rst frequency f1 is the fundamental frequency of the set while the last and highest frequency fN2 is known as the Nyquist frequency denoted here as fquot The Nyquist frequency turns out to be one half the sampling frequency ie fN2 qN2TN2NAFl2lAtl2 Parseval s Theorem for DFT In general for a time signal xt the relation between the standard deViation of sampled data points xquot in time and the DFT coef cients Xk in the frequency domain is NV N271 f 2 l l 2 2 l 2 F E xnx23 E Xk 37Xm quot0 k1 is known as Parseval39s theorem This theorem says that the RMS value of a time signal can be computed either from its sampled points in the time domain xn or from its DFT coef cients in the frequency domain Xk This theorem is extremely useful because it allows one to modify or weight the frequency components in a signal based upon the frequency Random and Deterministic Signals Acoustic noise like many signals in nature may be classi ed on a scale ranging from random to deterministic Consider the following two signals x 8n and y cos272394348t 018 where gquot is uniformly distributed random noise selected from the range 01 ObViously the signal xquot is purely random while yquot is predominantly deterministic When randomness is present in noise signals the DFT coef cients obtained in a frequency decomposition must be averaged overM time records of N samples in order to determine the spectrum Consider the amplitude spectra of the process xquot shown in the figure below A total of 1600 points xquot were generated This data was subdivided into M50 segments of data Current Averaged O N A O N A O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 Frequency Hz O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 O N A O 1000 2000 3000 4000 5000 Frequency Hz Current and averaged spectra of random process xquot The vertical aXis for the quotCurrentquot column is le 1 and the vertical aXis for the quotAveragedquot column is ARk records N32 points long A DFT decomposition of each data record in k02N2l and il250 was computed with an FFT subroutine In this notation the k index corresponds to frequency gkNAFkT and the 139 index identi es the data record number from 1 to M50 The DFT amplitudes were averaged over R records at each frequency using the formula R l 2 E IX 1m 1 From the gure in the column labeled quotCurrentquot it is evident that there is considerable variation in DFT amplitudes from record to record If the amplitudes are averaged as shown in quotAveragedquotcolumn the average amplitude spectrum converges For the case of random noise it can be theoretically shown that the spectrum of a random process is constant and equal amounts of each frequency are present in the signal This is the situation observed for M20 and M50 data records a A An equivalent analysis of the deterministic signal yn is shown in the Figure in the following page Since the random component of the signal yn is small there is little variation in each amplitude spectrum as observed in the quotCurrentquot column Likewise the quotAveragedquot column shows the same spectrum Because the deterministic signal does not vary from record to record the spectrum likewise is stable Current Averaged 20 20 10 10 M1 M1 O O 200 1000 2000 3000 4000 5000 200 1000 2000 3000 4000 5000 10 10 M5 M5 0 0 200 1000 2000 3000 4000 5000 200 1000 2000 3000 4000 5000 10 10 M20 M20 0 200 1000 2000 3000 4000 5000 200 1000 2000 3000 4000 5000 10 10 M50 M50 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Frequency Hz Frequency Hz Current and averaged spectra for yquot The vertical axis for the quotCurrentquot column is Km and the vertical axis for the quotAveragedquot column is ARk
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