Vibrations Fundamentals and Practice Appd

Vibrations Fundamentals and Practice Appd Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Appendix D” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 Appendix D Digital Fourier Analysis and FFT In the frequency domain, vibration analysis can be carried out using Fourier transform techniques Three versions of the Fourier transform are available The frequency content of a periodic signal is conveniently represented by its Fourier Series Expansion (FSE) For nonperiodic (or transient) signals, Fourier-Integral-Transform (FIT) is used For a discrete sequence of points in a signal (i.e., a set of sampled data), a sequence of discrete data in the frequency domain is obtained using Discrete Fourier Transform (DFT), or “digital” Fourier transform This can be interpreted as a discrete-data approximation to FIT Similarly, FSE can be expressed as a special case of FIT In this sense, the three versions of Fourier transform — FSE, FIT, and DFT — are interrelated It should be clear that the DFT is the appropriate version for digital analysis of data, using a computer The direct use of DFT relations, however, is not computationally efficient because it needs a very large number of operations and liberal use of computer memory For this reason, Fourier analysis using a digital computer was not considered feasible until 1965 That year, the Fast Fourier Transform (FFT) algorithm was published by Cooley and Tukey This revolutionized the field of data Fourier analysis by reducing the number of arithmetic operations required for the discrete Fourier transformation of an N-point data sequence by a factor of nearly 2N/ln2N Prior to this, the only economical way to perform Fourier analysis of complex time histories was by analog means, where narrow-band analog filters (circuits) were used to extract the frequency components in various frequency bands of interest Early applications of FFT were limited to offline computations in a batch mode using software in a large mainframe computer It was only after the development of large-scale integration (LSI) and the associated microprocessor technology that software-based and dedicated hardware FFT analyzers became cost effective for general applications The hardware FFT analyzers are particularly suitable in real-time applications Several stand-alone FFT analyzers were marketed in the late 1970s Practically unlimited options for frequency (spectral) analysis are available today, through these dedicated analyzers as well as desktop computers D.1 UNIFICATION OF THE THREE FOURIER TRANSFORM TYPES By discrete Fourier transformation of a set of sampled data from a signal, one cannot expect to generate an exact set of points in the analytical Fourier spectrum of the signal Because of sampling of the signal, some information will be lost Clearly, one should be able to reduce the error in the computed Fourier spectrum by decreasing the size of the data sample step (∆T) Similarly, one does not expect to get the exact Fourier series coefficients by discrete Fourier transformation of sampled data from a periodic function It is very important to study the nature of these errors, which are commonly known as aliasing distortions D.1.1 RELATIONSHIP BETWEEN DFT AND FIT A fundamental result relating DFT and FIT is established in this section In view of the FIT relation, the frequency-spectrum values Xm = X(m · ∆F), m = 0, ±1, ±2, …, sampled at the discrete frequency points of sample step ∆F are given by ©2000 CRC Press ∞ Xm = ∫ x(t) exp(− j2πm∆Ft)dt −∞ = ∞ ( k +1)T k =−∞ kT ∑∫ (D.1) x (t ) exp( − j πmt T )dt where T = 1/∆F On implementing the change of variable t → t + kT (i.e., let t′ = t – kT and then drop the prime) and interchanging the summation and the integration operations, one obtains T ∫ x˜ (t) exp(− j2πmt T )dt Xm = (D.2) where x˜ (t ) = ∞ ∑ x(t + kT ) (D.3) k =−∞ The fact that exp(–j2πmk) = for integers m, k was used in obtaining equation (D.2) Since x˜ (t) is periodic, having the period T, it has an FSE given by x˜ (t ) = T ∞ ∑ X exp( j2πnt T ) n (D.4) n =−∞ which follows from the FSE equation The sampled values x˜ m = x˜ (m · ∆T), m = 0, ±1, ±2, …, at sample steps of ∆T are given by x˜ m = = T T ∞ ∑ X exp( j2πnm N ) n n =−∞ ∞ ( k +1) N −1 k =−∞ n = kN ∑∑ (D.5) Xn exp( j πnm N ) where ∆T = T/N = 1/(N · ∆F) In a manner analogous to the procedure for obtaining equation (D.2), the change of variable n → n + kN (i.e., let n′ = n – kN and then drop the prime) is implemented, and the summation operations are interchanged This results in x˜ m = T N −1 ∑ X˜ exp( j2πnm N ) n (D.6) n=0 where X˜ ( f ) = ∞ ∑ X( f + kF) k =−∞ ©2000 CRC Press (D.7) Note that X˜ n = X˜ (n · ∆F), n = 0, ±1, ±2, …, are the sampled values of the periodic function X˜ (f), having the period F The frequency parameter F = N · ∆F = 1/∆T = N/ T represents the number of samples in a record of unity time duration It is not possible to extract any information about the frequency spectrum for frequencies f > F/2 = fc from time-response data sampled at steps of ∆T The parameter fc = 1/(2∆T) is known as Nyquist frequency It is evident that by comparing equation (D.6) with the inverse DFT relation that the sequence { X˜ n} = [ X˜ 0, X˜ 1, …, X˜ N–1] represents the DFT of the sequence { x˜ m} = [ x˜ 0, x1, …, xN–1] The forward transform is given by N −1 ∑ x˜ X˜ n = F m exp( − j πmn N ) (D.8) m=0 In summary, if X(f) is the FIT of x(t), then the N-element sequence { X˜ n} is the DFT of the N-element sequence { x˜ m} The periodic functions x˜ (t) and X˜ (f) are related to x(t) and X(f), respectively, through equations (D.3) and (D.7); { x˜ m} and { X˜ n} being their individual sampled data D.1.2 RELATIONSHIP BETWEEN DFT AND FSE A fundamental result relating DFT and FSE will be established in this section From the FSE equation, it follows that, for a periodic signal x(t) of period T, the sampled values xm = x(m · ∆T), m = 0, ±1, ±2, …, are given by xm = T = T ∞ ∑ A exp( j2πnm∆T T ) n n =−∞ ∞ ( k +1) N −1 k =−∞ n = kN (D.9) ∑ ∑ A exp( j2πnm N ) n By definition, the sequence {xm} is periodic with N-element periodicity where N = T/∆T The procedure for obtaining equation (D.6) is now adopted to obtain xm = T N −1 ∑ A˜ exp( j2πnm N ) n (D.10) n=0 where ∞ A˜ n = ∑A n + kN (D.11) k =−∞ The sequence { A˜ n} is periodic, with N-element periodicity By comparing equation (D.10) with the inverse DFT equation, it becomes clear that the N-element sequence { A˜ n} = [ A˜ 0, A˜ 1, …, A˜ N–1] is the DFT of the N-element sequence {xm} = [x0, x1, …, xN–1] The forward transform is given by N −1 A˜ n = ∆T ∑x m=0 ©2000 CRC Press m exp( − j πrm N ) (D.12) In summary, if {An} are the coefficients of the FSE of a periodic signal x(t), then the N-element sequence { A˜ n} is the DFT of the N-element sequence {xm}, where xm = x(m · ∆T) and { A˜ m} is given by equation (D.11) D.2 FAST FOURIER TRANSFORM (FFT) The direct computation of the discrete Fourier transform (DFT) is not recommended, particularly in real-time applications, because of the inefficiency of this procedure For a sequence of N sampled data points, N2 complex multiplications and N/(N – 1) complex additions are necessary in the direct evaluation of the DFT, assuming that the complex exponential factors exp(–j2πmn/N) are already computed Many of these arithmetic operations are redundant, however The Cooley and Tukey algorithm, commonly known as the radix-two fast Fourier transform (FFT) algorithm, is an efficient procedure for computing DFT Efficiency of the algorithm is achieved by dividing the numerical procedure into several stages so that redundant computations are avoided D.2.1 DEVELOPMENT OF THE RADIX-TWO FFT ALGORITHM The multiplicative constant ∆T in the DFT equation is a parameter that was introduced to maintain the consistency with the conventional FIT equation This constant can be treated as a scaling factor for the final results; or, equivalently, it could be combined with the input data sequence {xn} In any event, the primary computational effort in the DFT equation is directed toward computing the sequence [A(0), A(1), …, A(N – 1)] from the data sequence [a(0), a(1), …, a(N – 1)], using the relationship A( m) = N −1 ∑ a(n)W mn for m = 0, 1, K, N − (D.13) n=0 where W = exp( − j 2π N ) (D.14) The FFT algorithm requires that N be highly composite (i.e., factorizable into many non-unity integers) In particular, for the radix-two algorithm, it is required that N = 2r, where r is a positive integer If the given data sequence does not satisfy this condition, it must be augmented by a sufficient number of trailing zeros A systematic development of the radix-two FFT algorithm is presented now The integers m and n are expressed in the binary-number-system expansion Recalling that ≤ m ≤ N – and ≤ n ≤ N – 1, one can write m = mr −1 r −1 + mr −2 r −2 + K + m0 n = nr −1 r −1 + nr −2 r −2 + K + n0 (D.15) in which mi and nj take values or Equivalently, m = Binary( mr-1 , mr-2 , K, m0 ) n = Binary(nr-1 , nr-2 , K, n0 ) ©2000 CRC Press (D.16) Next, the indices of the elements A(·) and a(·) in equation (D.13) are expressed by their binary counterparts: A( mr −1 , K, m0 ) = 1 ∑ ∑ ∑ a(n L n0 = n1 = r −1 , K, n0 )W mn (D.17) nr −1 = where W mn = W =W ( mr −1 2r −1 +K+m0 )( nr −1 2r −1 +K+n0 ) nr −1 r −1 m0 W nr − 2 r − ( m1 + m0 ) W ( nr − r − 2 m2 + m1 + m0 ) K W n0 ( mr −1 2r −1 +K+m0 ) (D.18) r The fact that W = WN = has been used in the foregoing expansion By defining the intermediate set of sequences {A1(·)}, {A2(·)}, {Ar(·)}, equation (D.17) can be expressed as the set of equations: A1 ( m0 , nr −2 , K, n0 ) = ∑ a(n r −1 , K, n0 )W nr −1 r −1 m0 nr −1 = A2 ( m0 , m1 , nr −3 , K, n0 ) = ∑ A (m , n r −2 , K, n0 )W nr − 2 r − ( m1 + m0 ) (D.19) nr − = M Ar ( m0 , m1 , K, mr −1 ) = ∑A r −1 ( m0 , K, m r-2 , n0 )W ( n0 mr −1 r −1 +K+ m0 ) n0 = with A( mr −1 , K, m0 ) = Ar ( m0 , K, mr −1 ) (D.20) Consequently, the single set of computations given by equation (D.13) for the N-element sequence {A(·)} has been replaced by r stages of computations In each stage, an N-element sequence {Ai(·)} must be computed from the immediately preceding N-element sequence {Ai–1(·)} It soon will be apparent that, as a result of this r-stage factorization, the number of arithmetic operations required has been considerably reduced It should be noted that each relationship given in equation (D.19) corresponds to a set of N separate relationships, because the index within the parenthesis of {Ai(·)} runs from to N – In order to observe some important characteristics of the FFT algorithm, the ith relationship of equation (D.19) is examined Specifically, Ai ( m0 , m1 , K, mi −1 , nr −i −1 , K, n0 ) = ∑ nr −1 = ©2000 CRC Press Ai −1 ( m0 , m1 , K, mi −2 , nr −i , K, n0 )W ( nr − i r − i 2i −1 mi −1 +L+ m0 ) i = 1, K, r (D.21) where A0(·) = a(·) The summation on the right-hand side of equation (D.21) is expanded as Ai ( m0 , m1 , K, mi −1 , nr −i −1 , K, n0 ) = Ai −1 ( m0 , K, mi −2 , 0, nr −i −1 , K, n0 ) + Ai −1 ( m0 , K, mi −2 , 1, nr −i −1 , K, n0 )W (D.22) ( r −1 2i −1 mi −1 +L+ m0 ) which involves only one complex multiplication and one complex addition, assuming that the complex exponential terms Wp are precomputed It is noted that the variable mi–1, which is the binary coefficient of 2r–i in the index of the left-hand-side term Ai(·), does not appear in the binary index of the right-hand-side terms Ai–1(·) Consequently, the values of the Ai–1(·) terms on the righthand side remain unchanged as mi–1 switches from to in the left-hand-side index This switch corresponds to a jump in the index of A1(·) through a value of 2r–i Accordingly, the computation of, for example, the kth term Ai(k) and the (k + 2r–1)th term Ai(k + 2r–i) of the sequence {A1(·)} in the ith stage involves the same two terms Ai–1(k) and Ai–1(k + 2r–i) of the previous (i – 1)th sequence Ai–1(·) It follows that equation (D.22) takes the more familiar decimal format: ( ) Ai (k ) = Ai −1 (k ) + Ai −1 k + r −i W p ( ) ( ) Ai k + r −i = Ai −1 (k ) + Ai −1 k + r −i W p˜ ( p = r −i i −2 mi −2 + L + m0 ) ( (D.23) (D.24) p˜ = r −i i −1 + i −2 mi −2 + L + m0 ) (D.25) Equation (D.24) results when mi–1 = 0, and equation (D.25) results when mi–1 = On closer examination, it is evident that p˜ = N/2 + p, which follows from 2r–i2i–1 = 2r–1 = 2r/2 Hence, W p˜ = W N W p (D.26) From the definition of W [equation (D.14)], however, WN/2 = –1 Consequently, W p˜ = −W p (D.27) By substituting equation (D.27) in equation (D.23), one obtains ( ) Ai (k ) = Ai −1 (k ) + Ai −1 k + r −i W p ( ) ( ) Ai k + r −1 = Ai −1 (k ) − Ai −1 k + r −i W p (D.28) for i = 1, …, r and k = 0, 1, …, 2r–i – 1, 2r–i+1, …, where p is given by equation (D.24) The in-place simultaneous computation of the so-called dual terms Ai(k) and Ai(k + 2r–i) in the N-term sequence {A·(·)} involves just one complex multiplication and two complex additions As a result, the number of multiplications required has been further reduced by a factor of two At each stage, the computation of the N-term sequence requires N/2 complex multiplications and ©2000 CRC Press N complex additions Because there are r stages, the radix-two FFT requires a total of rN/2 complex multiplications and rN complex additions, in which r = ln2 N In other words, the number of multiplications required has been reduced by a factor of (2N/ln2 N), and the number of additions has been reduced by a factor of (N/ln2 N) For large N, these ratios correspond to a sizable reduction in the computer time required for a DFT This is a significant breakthrough in real-time digital Fourier analysis An insignificant shortcoming of the Cooley-Tukey FFT procedure is evident from equation (D.20) The final sequence {Ar(·)} is a scrambled version of the desired transform {A(·)} To unscramble the result, it is merely required to interchange the term in the binary location (m0, …, mr–1) with that in the binary location (mr–1, …, m0) It should be remembered not to duplicate any interchanges while proceeding down the array during the unscrambling procedure Because an in-place interchange of the elements is performed, there is no necessity for defining a new array There is an associated saving in computer memory requirements D.2.2 THE RADIX-TWO FFT PROCEDURE The basic steps of the radix-two FFT algorithm are as follows: N = 2r elements of the data sequence {A(·)} are available Step 1: Initialize variables Stage number i = Sequence element number k = Step 2: Determine p as follows: From equation (D.24), p = binary (mi–1, …, m0, 0, …, 0)Nbits From equation (D.21), k = binary (m0, m1, …, mi–1, nr–i–1, …, n0) Shift k register through (r – i) bits to the right, and augment the vacancies by leading zeros This gives binary (0, …, 0, m0, …, mi–1)Nbits Reverse the bits to obtain p Step 3: Compute in place, the dual terms Ai(k) and Ai(k + 2r–i), using equation (D.28) Note: Since Ai–1(k) and Ai–1(k + 2r–i) are not needed in the subsequent computations, they are destroyed by storing Ai(k) and Ai(k + 2r–i) in those locations As a result, only one array of N elements is needed in the computer memory Step 4: Increment k = k + If an already-computed dual element is encountered, skip k through 2r–i (i.e., k = k + 2r–i) If k ≥ N, increment i = i + If i > r, go to step Otherwise, go to step Step 5: Unscramble the sequence, using equation (D.20), and stop D.2.3 ILLUSTRATIVE EXAMPLE Consider the data sequence [a(0), a(1), a(2), a(3)] of block size N = Its DFT sequence [A(0), A(1), A(2), A(3)] is obtained as follows Note that r = 2, and the required complex exponents Wp are available as tabulated data Stage (i = 1): From equation (D.28), the matrix form of the equations with the indices expressed as binary numbers is  A1 (0, 0) 1     A1 (0, 1)  = 0  A1 (1, 0)  1     A1 (1, 1)  0 1 W0 −W 0  a(0, 0)   W   a(0, 1)    a(1, 0)    −W   a(1, 1)  Note that the dual jump 2r–i = 22–1 = for this stage Furthermore, for k = = binary (0,0), p = binary (0,0); and for k = = binary (0,1), p = binary (0,0) = ©2000 CRC Press Stage (i = 2):  A2 (0, 0) 1     A2 (0, 1)  = 1  A2 (1, 0)  0     A2 (1, 1)  0 W0 −W 0 0 1   A1 (0, 0)     A1 (0, 1)  W   A1 (1, 0)    −W   A1 (1, 1)  The dual jump for this stage is 2r–i = 22–2 = Also, for k = 0, p = as before Now, one must shift k through the dual jump This gives k = = binary (1,0) Shift this though r – i = – = ⇒ no shifts, and bit reverse to get p = binary (0,1) = Unscrambling: In binary index form, this amounts to a simple bit reversal  A(0, 0)  A2 (0, 0)      A(0, 1)  =  A2 (1, 0)   A(1, 0)   A2 (0, 1)       A(1, 1)   A2 (1, 1)  The corresponding decimal assignments are  A(0)  A2 (0)      A(1)  =  A2 (2)  A(2)  A2 (1)       A(3)   A2 (3)  When real x(t) sequence is used, note that half of the X(f) sequence (N/2 points) is wasted because X * N −n = Xn for real x m Hence, one can make some gains in computational effort by converting a real-time sequence to an equivalent complex sequence prior to DFT D.3 DISCRETE CORRELATION AND CONVOLUTION D.3.1 DISCRETE CORRELATION The sampled data are formed according to x m = x ( m ⋅ ∆T ) =0 yk = y(k ⋅ ∆T ) =0 ©2000 CRC Press for m = 0, K, M − otherwise for k = 0, K, K − otherwise (D.29) Using the trapezoidal rule, the sequence {zn} that approximates the sampled values z(n · ∆T) of the correlation function of x and y can be computed, using z(n ⋅ ∆T ) ≅ zn = N −1 ∑x y N (D.30) r r +n r =0 in which N > max(M, K) It is noted that, in the summation, an upper limit greater than min(M – 1, K – – n) is redundant Because the divisor is the constant value N rather than the actual number of terms in the summation, equation (D.30) represents a biased estimate of the mean lagged product Nevertheless, it is convenient to use equation (D.30) in this analysis Discrete Correlation Theorem A DFT result for discrete correlation is now established for discrete data The inverse DFT equation is used in equation (D.30) in conjunction with the fact that xm = [xm]* for real xm: zn = N N −1 ∑ r =0 = N ∆T N∆T N −1 ∑[X ] * m exp( − j πmr N ) m=0 N −1 N −1 N∆T N −1 ∑ Y exp[ j2πk(n + r) N ] k k =0 N −1 (D.31) ∑ ∑ [ X ] Y exp( j2πkn N )∑ exp[ j2πr(k − m) N ] * m k m=0 k =0 r =0 The orthogonality condition is used in the last summation term of equation (D.31) Consequently, Tzn = N∆T N −1 ∑[X ] Y * m m exp( j πmn N ) (D.32) m=0 where T = N · ∆T It follows that T{zn} is the inverse of DFT of {[Xm]*Ym} Equation (D.32) is the discrete correlation theorem Discrete Parseval’s theorem is given by N −1 ∆T ∑ m=0 N −1 ym2 = ∆F ∑Y n (D.33) n=0 Discrete Convolution Theorem The convolution theorem equation of two signals u(t) and h(t), defined over the finite durations (0,T1) and (0,T2), respectively, can be computed, using a digital processor, to obtain y(t) First, the sample step ∆T is chosen and the two sequences {um} and {hk} of sampled data are formed according to um = u( m ⋅ ∆T ) =0 hk = h(k ⋅ ∆T ) =0 for m = 0, K, M − otherwise for k = 0, K, K − (D.34) otherwise in which M = integer (T1/∆T) and K = integer (T2/∆T) In order to eliminate the wraparound error, it is required that the number of samples of y(t) be N = M + K – The direct digital computation of convolution can be performed using the trapezoidal rule: ©2000 CRC Press y(n ⋅ ∆T ) = yn = ∆T N −1 N −1 ∑ um hn− m = ∆T ∑u for n = 0, 1, K, N − h n−m m (D.35) m=0 m=0 In view of the zero terms in the two sequences {um} and {hk} as given by equation (D.34), it is equally correct to make the lower and the upper limits of the first summation be max(0, n – k + 1) and min(n, M – 1), respectively Similarly, the two limits in the second summation could be max(0, n – M + 1) and min(n, K – 1) In any event, by direct counting through summation of series, it can be shown that the computation of equation (D.35) needs KM real applications and KM – N real additions Alternatively, the discrete convolution result that is analogous to the continuous counterpart in the frequency domain can be used to evaluate equation (D.35) indirectly By substituting the inverse DFT equation in equation (D.35), one obtains N −1 yn = ∆T ∑ m=0 = N ∆T N∆T N −1 ∑ Ur exp( j πrm N ) r =0 N −1 N −1 N∆T N −1 ∑ H exp[ j2πk(n − m) N ] k k =0 N −1 (D.36) ∑ ∑ U H exp( j2πkn N )∑ exp[ j2π(r − k )m N ] r k r =0 k =0 m=0 The orthogonality condition is used in the last summation Consequently, yn = N ⋅ ∆T N −1 ∑ U H exp( j2πrn N ) r r (D.37) r =0 D.4 DIGITAL FOURIER ANALYSIS PROCEDURES Proper interpretation of the DFT results is extremely important in digital Fourier analysis For example, only the first N/2 + points of the DFT array approximate the Fourier transform of the data signal The remaining N/2 – points correspond to the negative frequency spectrum and should be interpreted accordingly The error caused by interpreting all N points in the DFT array as the positive frequency spectrum corresponding to the data signal is so great that the analysis would become worthless In this section, some useful DFT procedures are outlined Emphasis is placed on correct interpretation of the results Some ways to reduce computation time and memory requirements in real-time applications are described D.4.1 FOURIER TRANSFORM USING DFT Given an analog signal (continuous time) x(t), the major steps for obtaining a suitable approximation to its Fourier transform X(f), using digital Fourier analysis, are as follows: Pick the sample step ∆T Theoretically, ∆T = 1/(2 × highest frequency of interest) This value should be sufficiently small in order to reduce the aliasing distortion in the frequency domain Sample the signal up to time T, where T = N·∆T and N = 2r The duration [0, T] of the sampled record must be sufficiently long in order to reduce the truncation error (leakage) Obtain the DFT { X˜ n} of the sampled data sequence {xm} using FFT A discrete approximation to the Fourier transform X(f) is constructed from { X˜ n} according to: ©2000 CRC Press X (n ⋅ ∆F ) ≅ Xn for n = 0, 1, K, N , and X ( − n ⋅ ∆F ) ≅ X N −n for n = 1, K, N − 1, where ∆F = 1/ T D.4.2 INVERSE DFT USING DFT The inverse DFT can be written as [ ] * xn = N∆T N −1 ∑ [ X ] exp(− j2πmn N ) * m (D.38) m=0 where [ ]* denotes the complex conjugation operation It is observed that equation (D.38) is identical to the forward DFT equation except for a scaling factor Consequently, the forward DFT algorithm can be used in the computation of the inverse DFT The sampled data should be reorganized and complex-conjugated, however, before using DFT Finally, the scaling factor should be accounted for so that the final results have the proper units Given the complex spectrum X(f), which is the FIT of a real signal x(t) with x(t) = for t < 0, the main steps of determining a good approximation to the original signal using digital Fourier analysis are as follows: Let F be the highest frequency of interest in X(f), and let [0, T] be the interval over which real signal x(t) is required The sample step ∆F = 1/T It is required that ∆F be sufficiently small (T sufficiently large) to reduce aliasing distortion in the time domain Also, F should be sufficiently large to reduce truncation error Furthermore, the number of samples F/∆F = N = 2r, if radix-two FFT is used Sample X(f) at intervals ∆F over the frequency interval [–F/2, F/2] according to Xn = X(n · ∆F) for n = –N/2, …, 0, …, N/2 and properly scale the data Form the sequence { X˜ n} according to: X˜ n = Xn for n = 0, 1, K, N = Xn− N for n = N + 1, K, N -1 Form the complex conjugate sequence {[ X˜ n]*} Obtain the DFT of {[ X˜ n]*} using FFT This results in {[ x˜ m]*}, which has complex elements with negligible imaginary parts Construct: [ ] x ( m ⋅ ∆T ) ≅ real x˜ m D.4.3 SIMULTANEOUS DFT OF * for m = 0, 1, K, N -1 TWO REAL DATA RECORDS Considerable computational advantages can be realized when the DFTs {Ym} and {Zm} of two real sequences {yn} and {zn} are required simultaneously The procedure given in this section achieves this using only a single DFT rather than two separate DFTs It is recalled that {xn} is generally a complex sequence When a real sequence is used, half the storage requirement is wasted Instead, the DFT of the complex sequence {x } = {y } + j{z } n ©2000 CRC Press n n (D.39) is obtained using FFT This results in {Xm} It is evident from the DFT equation that N −1 X N − m = ∆T ∑x n exp( j πmn N ) (D.40) n=0 recalling that exp(–j2πn) = Consequently, [X ] N −1 * N −m = ∆T ∑ [ X ] exp(− j2πmn N ) * (D.41) n n=0 Since [xn]* = yn – jzn, it is straightforward to observe from the equation and (D.41) that ( ) (D.42) ]) (D.43) Ym = * X + {X N −m } m Zm = X − X N −m 2j m and ( [ * From the complex sequence {Xm}, the required complex sequences {Ym} and {Zm} are constructed according to equations (D.42) and (D.43) D.4.4 REDUCTION OF COMPUTATION TIME FOR A REAL DATA RECORD The DFT of a 2N-element real sequence [x0, x1, …, x2N–1] can be accomplished by means of a single DFT of an N-element complex sequence, using the concept discussed in the preceding section From the DFT equation, one has N −1 Xm = ∆T ∑x n exp[− j πmn (2 N )] n=0 N −1 = ∆T ∑x 2n exp[− j πm(2n) (2 N )] (D.44) n=0 N −1 + ∆T ∑x n +1 exp[− j πm(2n + 1) (2 N )] n=0 Consequently, N −1 Xm = ∆T ∑ n=0 x n exp( − j πmn N ) + exp( − jπm N )∆T N −1 ∑x n +1 exp( − j πmn N ) (D.45) n=0 Two real sequences, each having N elements, are defined by separating the even and the odd terms of the given sequence {xn} according to: ©2000 CRC Press yn = x n zn = x n+1 n = 0, 1, K, N − (D.46) The DFT sequences {Ym} and {Zm} of the two real sequences {yn} and {zn} are obtained using the procedure given in the preceding section Finally, the required DFT sequence is obtained using equation (D.45): Xm = Ym + exp( − jπm N ) Zm for m = 0, 1, K, N − (D.47) It should be noted that only the first N terms of the transformed sequence are obtained by this method This is not a drawback, however, because it is clear that due to aliasing distortion in the frequency domain, the remaining terms correspond to the negative frequencies of X(f) D.4.5 CONVOLUTION OF FINITE DURATION SIGNALS USING DFT Direct computation of the convolution is possible using the trapezoidal rule Also, from equation (D.37), it is clear that the required sequence {yn} is the inverse of DFT of {UrHr}, in which {Ur} and {Hr} are the DFTs of the N-point sequences {ur} and {hr}, respectively This gives rise to the following procedure for evaluating the convolution: Determine {Ur} and {Hr} by the DFT of the N-point sequences {ur} and {hr}, respectively Evaluate {yn} from the inverse DFT of {HrUr} If the slow DFT is used, the foregoing procedure requires 3N2 + N complex multiplications and 3N(N – 1) complex additions If the FFT is employed, however, only 1.5Nln2 N + N complex multiplications and 3Nln2N complex additions are necessary For large N, this can amount to a considerable reduction in computer time It can be shown that the trapezoidal rule is the most economical method for N < 200 (approximately) For larger values of N, the FFT method is recommended Wraparound Error A direct consequence of the definition of the DFT equation is the N-term periodicity of the sequence {Xm}: Xm = Xm +iN for i = ±1, ± 2, K (D.48) Similarly, from the inverse DFT equation, it follows that the sequence {xn} has the N-term periodicity x n = x n+iN for i = ±1, ± 2, K (D.49) Accordingly, whenever a particular problem allows variation of the indices of Xm or xn beyond their fundamental period (0, N – 1), the periodicity of the sequences should be properly accounted for, and the indiscriminate use of DFT should be avoided under such circumstances An example for such a situation is the evaluation of the discrete convolution equation (D.35) using DFT The direct evaluation of equation (D.35) using the trapezoidal rule does not cause any discrepancy because the correct values as given by equation (D.34) are used in this case When the DFT method is used, however, the N-term periodicity is assumed for the sequences {um} and {hk} Since this is not true according to equation (D.34), the use of DFT can introduce a technical error into compu- ©2000 CRC Press tation It can be shown that, unless N ≥ M + K – 1, the first M + K – – N terms in the N-point sequence {yn} not represent the correct discrete convolution results In the first relation of equation (D.35), as m varies from to N – 1, the highest value of m for which um ≠ is M – The corresponding index of h is n – M + Because of the N-term periodicity assumed in DFT, the terms in the sequence {hk} with indices ranging from (–N) to (–N + K – 1) are also non-zero; but if they are included in the discrete convolution, they lead to incorrect results because, in the correct sequence [equation (D.34)], these terms are This is known as the wraparound error It follows that, in order to avoid the discrepancy, one must require n – M + > –N + K – In other words, the condition n > M + K – – N must be satisfied to avoid the discrepancy Since n ranges from to N, the condition is satisfied if and only if M + K – – N ≤ –1 Consequently, it is required that N ≥ M + K – in order to avoid the wraparound error Data-Record Sectioning in Convolution The result ∞ ∞ ∫ u(τ + t )h(t − τ + t )dτ = ∫ u(τ′)h(t + t + t −∞ − τ ′ ) dτ (D.50) −∞ is obtained using the change of variable τ′ = τ + t1 In view of the convolution equation, one obtains ∞ ∫ u(τ + t )h(t − τ + t )dτ = y(t + t + t ) 2 (D.51) −∞ From equation (D.51), it follows that, if the two convolving functions are shifted to the left through t1 and t2, the convolution shifts to the left through t1 + t2 Suppose that the time history u(t) is of short duration and that the nonnegligible portion of h(t) represents a relatively long period If proper sampling of h(t) can exceed the available memory of the digital computer, the function h(t) is sectioned into several portions of equal length T2, and the convolution integral is computed for each section Finally, the total convolution integral is obtained using these individual results The concept behind this procedure is as follows: h(t ) = ∑ h (t ) i (D.52) i On substituting in the convolution equation, one obtains ∑ y (t ) (D.53) ∫ u(τ)h (t − τ)dτ (D.54) y(t ) = i i where ∞ yi (t ) = i −∞ ©2000 CRC Press However, hi(t) = over ≤ t < iT2 Because of these trailing zeros, the use of the DFT method becomes extremely inefficient for large i To overcome this, each segment hi(t) is shifted to the left through iT2, which results in a set of modified functions hi(t + iT2) that not contain the trailing zeros The corresponding convolutions, yi (t + iT2 ) = ∞ ∫ u(τ)h (t − τ + iT )dτ i (D.55) −∞ can be evaluated very efficiently using FFT in the usual manner Subsequently, the functions yi(t + iT2) are shifted to the right through iT2 to obtain yi(t) Finally, y(t) is constructed by superposition [equation (D.53)] It should be noted that evaluation of equation (D.55) using DFT or FFT is performed as described earlier The major steps of the procedure are as follows: Choose the sample step ∆T in the usual manner Choose T2 based on computer memory limitations or computational speed requirements Section h(t) at periods of T2 Move each section to the origin and sample each section A separate memory or storage segment can be used to store the sectioned and sampled data sequences {hk}i Sample u(t) at ∆T This results in the sequence {um} Using N = and {hk} (T1 + T2 ) as the period, obtain the discrete convolution {y } of each pair {u } ∆T Shift each sequence {yn}i to the right through iK = overlapping elements) ©2000 CRC Press n i m iT2 elements and superpose (add the ∆T ... x˜ (t) and X˜ (f) are related to x(t) and X(f), respectively, through equations (D.3) and (D.7); { x˜ m} and { X˜ n} being their individual sampled data D.1.2 RELATIONSHIP BETWEEN DFT AND FSE... implementing the change of variable t → t + kT (i.e., let t′ = t – kT and then drop the prime) and interchanging the summation and the integration operations, one obtains T ∫ x˜ (t) exp(− j2πmt... index of the left-hand-side term Ai(·), does not appear in the binary index of the right-hand-side terms Ai–1(·) Consequently, the values of the Ai–1(·) terms on the righthand side remain unchanged