efficient direct manual computation of discrete fourier transform in ...

Sci.Int (Lahore), 25(4), 719 - 723, 2013 ISSN 1013-5316; CODEN: SINTE 8 719EFFICIENT DIRECT MANUAL COMPUTATION OFDISCRETE FOURIER TRANSFORM IN TIME DOMAINEjaz A. Ansari 1 , Saleem Akhtar 2 , Muhammad Nadeem 3Department of Electrical Engineering, COMSATS Institute of Information Technology1.5 km Defense Road, off Raiwind Road, 53700, Lahore, Pakistan{dransari 1 , sakhtar 2 , munadeem 3 {@ciitlahore.edu.pk}Corresponding Author’s email: {dransari@ciitlahore.edu.pk}ABSTRACT: Digital Signal Processing (DSP) is an important and integral component of WirelessCommunications, Speech, Image and Video Processing in particular and of Electrical Engineering ingeneral. With the advancement in integrated technology, sophisticated algorithms have been designed andimplemented on the machines to achieve the results of many complex problems related to real worldphenomena. In this paper, we thus propose an algorithm which facilitates the instructors teaching DSPcourse in class to do manual computation of N-point discrete Fourier transform (DFT) and inverse discreteFourier transform (IDFT) of a complex data signal, x[n] with ease in time domain. We show in ouranalysis and through numerical results that our proposed algorithm is more efficient than the directcomputation of N-point DFT and IDFT of x[n]. It further provides 50 percent saving in terms of realadditions (RAs), real multiplications (RMs) and real arithmetic operations (RAops) using our proposedalgorithm in comparison with the direct computation of N-point DFT and IDFT of x[n] for large Npreferably (N ≥ 1000). Moreover, for real data, our percentage saving in the form of real arithmeticoperations remains unchanged even if we compute its N-point DFT and IDFT using the proposedalgorithm.Key Words: Computation, Discrete Fourier transform, Efficient Algorithm, Inverse discrete Fourier transform, saving1 INTRODUCTIONDigital signal processing (DSP) is concerned with therepresentation of signals in digital form and with theirprocessing along with the information which they carry. Thefield of DSP has been at explosive growth during the pastfive decades as phenomenal advances both in research andapplications have been made. Fueling this growth have beendue to the advances in digital computer technology andsoftware development. With a tremendously rapid increasein the speed of DSP processors along with correspondingincrease in their sophistication and computational power,DSP algorithms are extensively used for modeling discretetimesignals, estimating the channel parameters and powerspectrum of a random process [1-5].DSP course is taught at undergraduate level of electrical andcomputer engineering programs in all universities of theworld. Instructors teaching this course delineate theimportance of DFT and IDFT understandable to the class byperforming their computation for small values of N = 2, 4, 6manually using the direct approach [1-2]. The manualcomputation of DFT and IDFT of x[n] becomes quitedifficult to perform in class when N exceeds the value 6.Thus, in this paper, we present a simple and efficientalgorithm to perform the manual computation of N-pointDFT and IDFT of a complex or a real data sequence, x[n] intime domain. For this purpose, we exploit the properties oflinear transformation matrix, W to improve the efficiency ofour proposed algorithm. Due to unitary (orthogonal)transformation of the matrix W [6], we succeed in achieving50 percent saving in real arithmetic operations (RAops)using the proposed algorithm in comparison with the directcomputation of N-point DFT and IDFT of x[n]. We shallmake use of matrix theory [6] to keep the analysis simple.The paper is organized as follows. In section 2, developmentof the proposed algorithm is presented. Section 3 describesits various associated steps. Section 4 discusses its timecomplexity. Section 5 describes the way how to computeinverse discrete Fourier transform of x[n] using the proposedalgorithm. Comparison of our proposed algorithm with thedirect its computation is carried out in section 6. Finally, wediscuss our numerical results in section 7 and conclusionsare drawn in Section 8.2 DEVELOPMENT OF THE PROPOSEDALGORITHMConsider x[n] = a[n] + j b[n] be a given complex signalwhich is defined for 0 ≤ n ≤ N – 1; where a[n] and b[n] aretwo real constants. We are interested in the manualcomputation of N-point DFT of x[n] easily but efficiently.This can be achieved by exploiting the properties of theunitary transformation matrix, W obtained from the twiddlefactors, W N = exp(-j2π/N) [5] and expressing the N- pointDFT into some convenient form. The N- point DiscreteFourier Transform (DFT) of x[n] may be computed using itsdefinition as [7 9]N−1 N−1 N−1kn ⎛ kn ⎞ ⎛ kn ⎞Xk [ ] = ∑xnW [ ] N= ⎜∑anW [ ] N ⎟+j⎜∑bnW[ ] N ⎟n= 0 ⎝n= 0 ⎠ ⎝n=01442443 1442443 ⎠[ ] = [ ] + [ ] ( 1)Xk Ak jBkWhere 0 ≤ k ≤ N – 1; and W N = exp(-j2π / N) also known astwiddle factor [2]. We first compute A[k] for all possiblevalues of ‘k’ by expressing it into matrix form; computationof B[k] will also be carried out in the similar way.[ ]Ak[ ]Bk

720[ 0] = [ 0] + [ 1] + K + [ −1]1 N−1[] 1 = [ 0] +N [] 1+ K +N [ −1]22( N−1) [ 2] = [ 0] + [ 1] + K + [ −1]A a a a NA a Wa W a NA a WNa WNa NM M M M( N−1) ( N−1)( N−1) [ − 1] = [ 0] + [] 1+ K + [ −1] ( 2)AN a W a W aNNIn matrix form, eq.(2) may be written as [6], R = Wa,where R and a are two column vectors each of size N x 1which are defined as R = ⎡A[ 0] A[ 1] A [ N −1]⎤t⎣Kand⎦[ ] [ ] K [ ]a = ⎡a 0 a 1 a N −1 ⎤t⎣⎦. The unitary matrix, W as [6]t[ r r r r r ]W = K(3)0 1 2 N−2 N−1is a square matrix of size N x N respectively in which r i ’srepresent its corresponding ‘N’ rows for different values ofi = 1, 2...,N – 1 and r 0 = (1 1 1….1) t is the first row of thematrix, W and ‘t’ denote the matrix transpose operationrespectively [6]. From the structure of the transformationmatrix, W it is self evident that r 1= r ∗ N − 1; r2 = r ∗ N −2and soon, where * denotes the complex conjugate of the respectiverow. This argument can easily be proved from the rows ofthe unitary matrix, W asN( ) ( ) ( ) {N −1 −1 −1 1∗N N N N N;= 12⎛ ⎞2( N −1)N−2 −2 2∗N= {N × ( N ) =N= ( N ) ;⎜ ⎟⎝ = 1 ⎠N −1⎛ ⎞( N −1)( N −1)N−( N −1)−( N −1) N −1∗N= {N × ( N ) =N= ( N⎜ ⎟)⎝ = 1 ⎠∗1 2 ( N −1)∗N −1 = ⎡1⎤N N N=1W = W × W = W = WNISSN 1013-5316; CODEN: SINTE 8 Sci.Int(Lahore),25(4),719-723,2013N − 1t[ 0 ] =0. = ∑ [ ] ⇐ ( − 1 )A r a a n N RAsn = 0N −1⎛ 2 N RM s ⎞tnA[] 1 = r1. a = ∑ WNa[ n]⇐ ⎜ ⎟n = 0⎝ 2( N − 1)RAs ⎠N − 1⎛ 2 N RM s ⎞t2 nA[ 2 ] = r2. a = ∑ WNa[ n]⇐ ⎜ ⎟n = 0⎝ 2( N − 1)RAs ⎠M M Mt∗t∗[ − 2 ] = ( N − 2) . = ( 2 ) . = ( [ 2]) ⇐ ( )t∗t∗[ − 1 ] = ( N − 1) . = ( 1 ) . = ( [] 1 ) ⇐ ( ) ( 5)A N r a r a A NilA N r a r a A N ilwhere N, RAs, and RMs denote the total real values in thecolumn vector a, real additions and real multiplications ineq. (5) respectively. Two separate cases need to beconsidered according to the integer values of N.2.1 Case no. 1 (when N is even)Under this situation, we come to know that there is definitelya middle row in the matrix, W after the first row (r 0 ) andcomplete grouping of rows is performed around this middlerow. This middle row is located in the matrix W at the indexvalue (i= N/2) excluding r 0 (the first row). We need only tocompute the first half values of A[n] after r 0 till r N/2 (one rowbefore the middle row) for i = 1, 2, …N/2 – 1 values fromthe illustrated dot products in eq. (5) and the remaining halfvalues of A[n] for i = N/2 + 1, …., N – 1 are automaticallycomputed by taking the conjugates of first half computedvalues of A[i]. Here, A[0] is computed by just taking the sumof ‘N’ real values of a and the middle row A[N/2] is alsocomputed in the similar manner as illustrated in eq.(6).kN−1 N 1N ⎛ ⎞−⎢ N 22 ⎥ ⎜ ⎟k= 0=−1k=0⎡ ⎤ N 2{ [ ] ( ) kA = r . a = ∑ ⎜ W [ ] ( )( )N ⎟ ak = ∑ − 1 ak ⇐ N − 1 real + ' s 6⎣ ⎦ ⎝ ⎠W W W W W2.2 Case no. 2 (when N is odd)We come to know under this situation that there will NOTbe any middle row in the matrix W after the first row (r 0 )W W W W W ; and complete grouping of rows is present there in the matrixW. We thus need only to compute the first half values of A[i]rfor i = 1, 2, …(N – 1)/2 values from the illustrated dot⎣W W K W⎦( r ) ( 4 ) products in eq. (5) and the remaining half values of A[i] fori = (N + 1)/2, ….., N – 1 are automatically computed bytaking the conjugates of first half computed values of A[n]Similarly, we can show that rN2r ∗−=2and so on. Hence, as shown in eq. (4). Complex conjugation is usuallywe may express the matrix, W given in eq. (3) with this implemented in hardware just by interchanging the real andtsubstitution as∗ ∗imaginary part values of the result [5]. Thus, it requires NOW = ⎡ ⎣ r0 r1 r2 K r2 r ⎤1 ⎦ . Thus, wearithmetic operations (Aops) for its implementation.conclude that 2 nd row (r 1 ) of the matrix W is a complex However. A[0] is again computed by just taking the sum ofconjugate of its last row, (r N-1 ). Similarly, 3 rd row (r 2 ) of W ‘N’ real constant values of a.is a complex conjugate of its second last row (r N-2 ) and so on 3 STEPS OF THE PROPOSED ALGORITHMtill we reach the middle row of the matrix W (when N is We describe our proposed algorithm discretely in theeven). The same is true with the columns of the unitary following six simple steps:matrix W. We thus require the following operations and 1. Compute the twiddle factors, W N for n = 0,1,… , N – 1complexity (i.e., in computing the dot products of r i ’s witha) in finding the entries of the column vector R as(assume N to be even).2. Compute the first half values of A[n] by finding the dotproduct of rows of W with real part of x[n]3. Compute the first and the middle row of A[n] usingeqs. (5) and (6)

Sci.Int (Lahore), 25(4), 719 - 723, 2013 ISSN 1013-5316; CODEN: SINTE 8 7214. Compute the remaining half values of A[n] by simplycomplex conjugating the first half values of A[n].5. Repeat steps 2 – 4 by considering the imaginary partsof x[n]This will provide us N-point DFT of the complex datasignal, x[n] which is X[k] for k = 0,1,2,…,N – 1.4 DETERMINING TIME COMPLEXITY OF THEPROPOSED ALGORITHMCase # 01 and eq. (5) show that in order to compute first halfsamples of A[n], we therefore require (N – 2)/2 x 2N =N(N – 2) real multiplications and (N – 2)/2 x 2(N – 1) =(N – 1)(N – 2) real additions. 2 x (N – 1) more real additionsare required to compute the first and the middle values of R,i.e., (A[0] and A[N/2]. Thus, to compute A[n] completely forall even integer values of N, we require N(N – 2) realmultiplications and {(N – 1)(N – 2) + 2 x (N – 1)} =N(N – 1) real additions respectively.Case # 02 and eq. (5) show that in order to compute onesample of A[n] for n ≠ 0; we require 2N real multiplicationsand 2(N – 1) real additions. To compute first half samples ofA[n], we therefore, require ( N - 1) 2× ( 2N ) = N ( N - 1 )real multiplications and ( N - 1) 2× 2( N - 1 ) = ( N - 1 ) 2real additions. To compute A[0], we also require (N – 1)more real additions. Thus, to compute A[n] completely forall odd integer values of N, we require N(N – 1) realmultiplications and ( N - 1) 2+ ( N − 1) = N( N - 1)realadditions respectively.As the formulation of B[k] given in eq. (1) resembles withthat of A[k] except for the multiplication with the operator‘j’, we therefore conclude that the same arithmetic1. Compute the complex conjugate of the matrix, W2. Apply the proposed algorithm on the given sequence, Rusing its stepsoperations will be required to compute B[k] as X = W a + 3. Divide the result obtained above by N in order to get backj W b = W(a + j b) = for 0 ≤ k ≤ N – 1. As A[k] and B[k] the original complex sequence, x[n]comprise of complex values due to the matrix, W, so, weneed to add them up together in order to find out N – point 6 COMPARISON WITH DIRECT COMPUTATION OFDFT of x[n]. For this purpose, we make use of two real N-POINT DFT AND IDFTadditions to compensate for one complex addition. Thus, we To compute one sample of DFT using eq. (1) directly, onerequire additional 2 x (N – 1) / 2 = (N – 1) real additions requires N complex multiplications (CMs) and (N – 1)when N is odd and 2 x (N – 2) / 2 = (N – 2) real additions complex additions (CAs) respectively. To compute allwhen N is even. Thus, we require 2N(N – 1)+ (N – 1) = samples (N) of DFT, N 2 complex multiplications and(N – 1)(2N + 1) and 2N(N – 1)+(N – 2)=(2N + 1)(N – 1) – 1 N(N – 1) complex additions are required [4]. We nowreal additions to compute N – point DFT of x[n] when N is convert these complex arithmetic operations to realodd and even respectively. We determine the complexity of arithmetic operations using the following relations as:our proposed algorithm (for large N) in terms of real 1 CM = 4 RMs and 2 RAs and 1 CA = 2 RAs, N 2 CMs = 4N 2additions (RAs) and real multiplications (RMs) asRMs and 2N 2 RAs and N(N -1) CAs = 2N(N – 1) RAs. Thus,2( N 1)( 2N 1)2 N ; for N 2k12( 2N + 1)( N −1) −1≈ 2 N ; for N = 2k( 7)2⎧⎪2N ( N −1)≈ 2 N ; for N = 2k+ 122N ( N − 2) ≈ 2 N ; for N = 2k( 8)⎧⎪− + ≈ = +RAs = ⎨⎪⎩RMs = ⎨⎪⎩5 COMPUTATION OF IDFT USING THE PROPOSEDALGORITHMMatrix representation of eq.(2) permits us to compute thevalue of the complex signal, x[n] after multiplying it withinverse of the matrix, W as a = W -1 R . But, the definition ofinverse discrete-time Fourier transform (IDFT) [7]-[9] canalso be written into Matrix form as Na = W * R, where *denote conjugate of the matrix W. The comparison of aboveFigure 1 shows the comparison of real additions required bythe computer to compute N-point DFT and IDFT of thecomplex sequence, x[n] using the proposed algorithm and itsdirect computation in time domain. It can easily be seenfrom the graph that computation of N-point DFT and IDFTby direct computation requires twice real additions than theproposed algorithm when N ≥ 1000. Thus, we obtain apercentage saving of 50% in the computation of N-pointDFT and IDFT of x[n] using the proposed algorithm.two representations reveal that W -1 = W * /N. This relationtells us to compute the inverse of the matrix, W by firstconjugating it and then normalizing its entries by N. W -1 canbe expressed ast∗t( ⎡⎣ 0 1 2K⎤2 1 ⎦ ) ⎡⎣ 0 1 2K⎤2 1⎦( 9)= =−1 −1 ∗ ∗ −1∗ ∗W N r r r r r N r r r r rwhere we have interchanged the transpose operation with thecomplex conjugate operation in computing the inverse and r 0remains unchanged as its all entries comprise of real values.It can easily be seen from (9) that same relation existsamong the columns of inverse matrix, W. This also showsthat there will be no effect on the complexity of directmanual computation of IDFT as conjugation involvesexchange of real and imaginary parts only. The followingsteps are essential to perform in order to compute directmanual IDFT of a sequence, R with N complex values:total RAs = 2N 2 + 2N(N – 1) = 2N(2N – 1) ≈ 4N 2 for large N.Similarly, N 2 CMs = 4N 2 RMs. Saving in real arithmeticoperations (both additions and multiplications) = 4N 2 – 2N 2= 2N 2 . Percentage saving in real arithmetic operations (bothadditions and multiplications) = 2N 2 / 4N 2 x 100 = 50%.This shows that our proposed algorithm provides 50%saving in the computation of N-point DFT and IDFT forlarge N in time domain and thus outperforms its directcomputation. There is no effect on the percentage saving, i.e,it remains 50 % when we want to compute N-point DFT of areal data sequence, ( x r [n] = x[n] with imaginary part, b[n] =0) using the proposed algorithm. In this case, step 5) in the

722ISSN 1013-5316; CODEN: SINTE 8 Sci.Int(Lahore),25(4),719-723,2013proposed algorithm will not be performed. Moreover, itenables the instructors teaching DSP course to compute itmanually with comfort and ease in the class.6 NUMERICAL RESULTS AND DISCUSSIONIn this section, we present the numerical results to show theperformance of our proposed algorithm in comparison withthe direct computation of N-point DFT in time domain. Forthis purpose, we have varied the length, N of the complexsequence, x[n] from 2 to 2500 and have computed itscorresponding N-point DFT and IDFT using the proposedalgorithm and direct computation. The results are thusgathered and plotted in Figures 1, 2 and 3 respectively.Comparison of Real Additionssaving in computations while performing N-point DFT andIDFT of x[n] using the proposed algorithm. We also showedthat the percentage saving remains unchanged when we x[n]reduces to a real data sequence. Instructors in class teachingthe course of DSP can also demonstrate the manualcomputation of DFT and IDFT using the steps of ourproposed algorithm with ease and comfort for reasonablylarge values of N without any complication.1614Comparison of Real Multiplicationssaving = 50%DirectProposedA r ith m etic O p e ra tion s161412108618 x 106 Number of points in DFT, Nsaving = 50%DirectproposedA rithmetic O perations18 x 106 Number of points in DFT, N12108642400 500 1000 1500 2000 2500200 500 1000 1500 2000 2500Figure 2: Comparison of real multiplications (RMs) of proposedalgorithm with direct computation of DFT and IDFT of x[n]Figure 1: Comparison of real additions (RAs) of proposedalgorithm with direct computation of DFT and IDFT of x[n]Comparison of Total Arithmetic Operations (+'s plus *'s)Saving = 50 %3.5 x 107 Number of points in DFT, NFigure 2 and 3 describe the comparison of realmultiplications and arithmetic operations (including bothreal additions and multiplications) required by the computerto compute N-point DFT and IDFT of the complexsequence, x[n] using the proposed algorithm and its directcomputation in time domain. It can easily be seen from thegraphs shown in figure 2 and 3 that computation of N-pointDFT and IDFT by direct computation requires twice realmultiplications and real arithmetic operations than theproposed algorithm when N ≥ 1000. Thus, we again obtain apercentage saving of 50% in the computation of N-pointDFT and IDFT of x[n] using the proposed algorithm.7 CONCULUSIONSIn this paper, we proposed an algorithm which computesN-point DFT and IDFT of a complex or real data sequence,x[n] very efficiently and with ease in time domain. For thispurpose, we made use of relationships existed amongvarious rows or (columns) of the linear transformationmatrix, W. We showed analytically and demonstratedthrough numerical results that our proposed algorithmoutperforms the direct computation of N-point DFT andIDFT of x[n]. We also showed that we achieve 50 percentA rithm etic O perations32.521.510.5DirectProposed00 500 1000 1500 2000 2500Figure 3: Comparison of real arithmetic operations (RAops) ofproposed algorithm with direct computation of DFT and IDFT.