This letter proposes a new multiplierless approximation of the discrete Fourier transform (DFT) called the multiplierless fast Fourier transform-like (ML-FFT) transformation. It makes use of a novel factorization to parameterize the twiddle factors in the conventional radix-2 n or split-radix FFT algorithms as certain rotation-like matrices and approximates the associated parameters using the sum-of-powers-of-two (SOPOT) or canonical signed digits (CSD) representations. The ML-FFT converges to the DFT when the number of SOPOT terms used increases and has an arithmetic complexity of O(N log 2 N) additions, where N = 2 m is the transform length. Design results show that the ML-FFT offers flexible tradeoff between arithmetic complexity and numerical accuracy in approximating the DFT.published_or_final_versio

Chan, SC

Yiu, PM

IEEE Signal Processing Letters

©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE

S. C. Chan

P. M. Yiu

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An efficient multiplierless approximation of the fast Fourier transform using sum-of-powers-oftwo (SOPOT) coefficients

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Title An efficient multiplierless approximation of the fast Fouriertransform using sum-of-powers-of-two (SOPOT) coefficientsAuthor(s) Chan, SC; Yiu, PMCitation Ieee Signal Processing Letters, 2002, v. 9 n. 10, p. 322-325Issued Date 2002URL http://hdl.handle.net/10722/42941Rights©2002 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.322 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 10, OCTOBER 2002An Efficient Multiplierless Approximation of the FastFourier Transform Using Sum-of-Powers-of-Two(SOPOT) CoefficientsS. C. Chan and P. M. YiuAbstract—This letter proposes a new multiplierless approx-imation of the discrete Fourier transform (DFT) called themultiplierless fast Fourier transform-like (ML-FFT) transforma-tion. It makes use of a novel factorization to parameterize thetwiddle factors in the conventional radix-2 or split-radix FFTalgorithms as certain rotation-like matrices and approximates theassociated parameters using the sum-of-powers-of-two (SOPOT)or canonical signed digits (CSD) representations. The ML-FFTconverges to the DFT when the number of SOPOT terms usedincreases and has an arithmetic complexity of ( log2)additions, where = 2 is the transform length. Design resultsshow that the ML-FFT offers flexible tradeoff between arithmeticcomplexity and numerical accuracy in approximating the DFT.Index Terms—Discrete Fourier transform (DFT), fast Fouriertransform (FFT), radix-2 decimation-in-frequency (DIF),sum-of-powers-of-two (SOPOT).I. INTRODUCTIONTHE DISCRETE Fourier transform (DFT) is an importanttool in digital signal processing [1]. A treasure of fast algo-rithms such as the Cooley–Tukey fast Fourier transform (FFT),the prime factor algorithm (PFA) FFT, and the Winogard FFT isavailable to compute efficiently DFT of different lengths. In thisletter, a new multiplierless approximation of the DFT, calledthe multiplierless FFT-like transformation (ML-FFT), usingthe radix- FFT algorithm and the sum-of-powers-of-two(SOPOT) or canonical signed digits (CSD) representation,is proposed. Generalization to the split-radix FFT (SR-FFT)algorithm is also possible. The present work is motivated bya recent interest in the efficient realization of fast transformalgorithms using either integer [6]–[8] or SOPOT [3]–[5]representations. The main objective is to provide efficientmultiplierless approximation to sinusoidal transforms suchas the DFT, which can be implemented efficiently either insoftware (say in microcontrollers with no on-chip multipliers,etc.), hardware, or very large scale integrated circuits (VLSI).Integer FFT [6] and sinusoidal transforms [7], [8] are attractivewhen the transform sizes are small, because the wordlengthgrowth of the intermediate data can still be accommodated in16–32 bits without rounding. As a result, high accuracy canManuscript received August 16, 2001; revised June 10, 2002. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Giovanni Ramponi.The authors are with the Department of Electrical and Electronic Engi-neering, The University of Hong Kong, Pokfulam Road, Hong Kong (e-mail:scchan@eee.hku.hk; pmyiu@eee.hku.hk).Digital Object Identifier 10.1109/LSP.2002.803408.be achieved with simple integer arithmetic. However, it is verydifficult to design such integer matrices when the transformlength increases. Up to now, only order-8 integer FFT [8] andorder-16 integer DCT [7] have been reported. The proposedmultiplierless FFT-like (ML-FFT) transformation is basedon the SOPOT representation of the twiddle factors in theconventional FFT algorithms using a special parameterizationof the rotation-like matrices [3]. This parameterization allowsus to implement both the forward and inverse transforms withthe same set of SOPOT coefficients. Furthermore, by using theperiodic properties of the rotation-like matrix and the twiddlefactors, the number of parameters to be optimized is reducedby a factor of four and the dynamic range of the parameters islimited between zero and one, greatly improving the numericalproperty. Unlike integer-based transformations, multiplierlesstransformations with length up to 1024 and higher can beobtained without much difficulty, and they converge to theirideal counterparts when the SOPOT terms used increases. Theproposed approach differs from the multiplierless DCT of [5] inthe parameterization used, which has the advantages of limiteddynamic range and reduced number of parameters mentionedearlier. Design results show that the proposed ML-FFT offers aflexible tradeoff between arithmetic complexity and accuracyin approximating the DFT. Another point worth mentioning isthat the proposed approach is also applicable to approximateother sinusoidal transforms such as the four types of DCTsand DSTs [4] and the discrete Hartley transform (DHT) [9].This letter is organized as follows, Section II is devoted tote proposed ML-FFT algorithm, where its principle and theparameterization of the rotation-like matrices into SOPOTcoefficients are described. Section III gives the design methodand several examples demonstrating its usefulness. Finally,conclusions are drawn in Section IV.II. ML-FFT ALGORITHMThe DFT of a finite-length sequence of lengthisfor (2.1a)where and . Its inverse, called theinverse DFT (IDFT), is given byfor (2.1b)1070-9908/02$17.00 © 2002 IEEECHAN AND YIU: AN EFFICIENT MULTIPLIERLESS APPROXIMATION OF THE FAST FOURIER TRANSFORM 323If is a power-of-two number, the DFT can be computed by theradix or the split-radix FFT. Without lossof generality, let us consider the decimation-in-frequency (DIF)FFT algorithm. From (2.1a), it can be shown that the even-num-bered frequency samples can be computed via an -pointDFT of the sequence as follows:(2.2)Similarly, the odd-numbered frequency samples can be com-puted as an -point DFT of the sequenceas shown in the following:(2.3)is called the twiddle factor. If is a a power-of-twonumber, then the DFT can be reduced successively using theabove decomposition to eventually two-point DFTs.The arithmetic complexity is complex additions andcomplex multiplications (including possibletrivial multiplications). Fig. 1 shows the flow graph of theeight-point DFT computed using the DIF FFT algorithm. Theproposed ML-FFT algorithm approximates the twiddle factormultiplication, which can be implemented as rotations, usingthe SOPOT representation and a certain parameterization ofthe rotation-like matrix. More precisely, letbe a complex number. The multiplication of with ,, can be written as(2.4)whereFig. 1. Flow graph for computing an eight-point DFT using the DIF radix-2FFT algorithm (W = e ).is the rotation-like matrix that we are interested in and. The main difficulty in constructing a multiplierlesstransformation using the SOPOT representation of is that thecoefficients of the matrix transformation and its inverse cannot,in general, be expressed in terms of SOPOT coefficients. Ifand in (2.4) are expressed directly in terms of SOPOT co-efficients, say and , then the inverse ofisAs and are SOPOT coefficients, the termcannot, in general, be expressed as SOPOT coefficients. Thebasic idea of the proposed ML-FFT is to parameterize andits inverse as follows:(2.5)Since these factorizations of and involve the same set ofcoefficients, i.e., and , they can be directly quan-tized to SOPOT coefficients to form(2.6)where and are, respectively, SOPOT approximations toand having the form , where, ; is the rangeof the coefficients, and is the number of terms used in eachcoefficient. In practice, it is usually limited to a small number sothat the multiplication can be implemented with limited numberof additions and shifts. Since the coefficient multiplications arenow replaced by a limited number of additions, the transforma-tion can be realized efficiently without multipliers. Interestedreaders are referred to [3]–[5], [10] for more references on ef-ficient realization of digital filters using SOPOT coefficients.Direct application of (2.6) might lead to large dynamic rangefor when is close to . Although the factorization ofin (2.5) is unique for a given value of , the periodic property324 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 10, OCTOBER 2002of can be utilized to construct slight variation of this factor-ization so as to reduce the dynamic range of the coefficients, aswell as the number of parameters to be optimized. More pre-cisely, it can be shown that the quantities , , ,, and can be expressed completely in terms of anyone of the following quantities:for (2.7a )for (2.7b)for (2.7c )for (2.7d )Using these relationships, it is possible to limit within therange . Thus, the dynamic range of the parameters to beapproximated by the SOPOT representation is always limited tothe range , and the number of parameters to be optimizedcan be reduced by a factor of four. The former allows us to limitthe dynamic range of the parameters even if the transform lengthis very large, resulting in better numerical property. The approx-imations , , , and of their respectiverotation-like matrices can be generated from by replacingwith in (2.7). As all the twiddle factors in the radix-2 DIF FFTalgorithm are just a subset of those in the first stage, there arealtogether twiddle factors, including . Using the pro-posed parameterization, which has two parameters per rotation,there are approximately parameters in an -pointML-FFT. The SOPOT twiddle factors for the radix-4, radix-8,and split-radix ML-FFTs can be generated similarly.III. DESIGN METHOD AND EXAMPLESThe random search algorithm proposed in [4] is used toperform the discrete optimization. More precisely, a randomvector with all its elements bounded by 1 is first multipliedby a scaling factor , and is added to the parameter vectorcontaining the real-valued and . It is then quantized tothe nearest SOPOT coefficients. The objective function is thenevaluated for this SOPOT candidate. The one with the bestperformance at a given number of additions is recorded. Thesearch continues until the maximum allowed number of trials isexceeded. The scale factor controls the size of the neighborhoodto be searched. A number of solutions with different tradeoffsbetween implementation complexity and performance are thenobtained. Without loss of generality, the Frobenius norm of thederivation of the ML-FFT from its ideal counterpart is used asthe objective function(3.1)where and are the th entries of the ML-FFTmatrix and its ideal counterpart, respectively. Table I showsTABLE IFROBENIUS NORM OF THE ERROR MATRIX IN APPROXIMATING THE DFTBY THE ML-FFT USING DIRECT QUANTIZATIONTABLE IITHE OBJECTIVE FUNCTION, THE FROBENIUS NORM, THE AVERAGE NUMBEROF SOPOT TERMS, AND THE COMPUTATIONAL COMPLEXITY FOR8-, 16-, 32-, 64-, AND 128-CHANNEL REAL-VALUED FFT ANDML-FFT (MULT: MULTIPLICATIONS; ADD.: ADDITIONS)(in decibels) obtained by directly rounding the coeffi-cients to the SOPOT representations without performing anyoptimization. It can be seen that the precision of the ML-FFTis directly related to the number of SOPOT terms used, whichin turn affects its arithmetic complexity. Table II shows ,the average SOPOT terms, and the computational complexitiesfor the 8-, 16-, 32-, 64-, and 128-channel ML-FFT obtainedusing the random search algorithm. The numbers of SOPOTterms are limited to five to simplify hardware implementation,though it is not necessary, say, in software implementation.The arithmetic complexities of the radix-4 FFT and ML-FFTare also shown as a comparison. It can be seen that the numberof additions required for the ML-FFT is approximately twotimes that of the radix-2 FFT algorithm for an averaged numberof SOPOT terms per coefficient of five. In general, if isthe average number of the SOPOT terms per coefficient, thenfrom (2.6), the arithmetic complexity of the radix- ML-FFTalgorithm derived from a radix- FFT algorithm with an arith-metic complexity of complex multiplicationsand complex additions is approximatelyreal additions. For simplicity, we have ignored thosetrivial multiplications in the real-valued algorithm. Asan example, the radix-2 FFT algorithm has an arith-metic complexity ofand . Then, the corre-sponding complexity of the ML-FFT algorithm isreal additions. Furthersaving can be obtained by using the radix-4 (as illustrated inCHAN AND YIU: AN EFFICIENT MULTIPLIERLESS APPROXIMATION OF THE FAST FOURIER TRANSFORM 325TABLE IIITHE SOPOT COEFFICENTS OF THE 32-CHANNEL ML-FFTTABLE IVTHE SOPOT COEFFICENTS OF THE 128-CHANNEL ML-FFTTable II) and the SR-FFT algorithms. The details are omitteddue to page length limitations. For illustration purposes, theparameters of the proposed 32- and 128-channel radix-2 DIFML-FFT are also shown in Tables III and IV, respectively.Finally, it is noted from Table II that fewer than five averagenumber of SOPOT terms per coefficient are needed to give abelow 40 dB. By increasing the number of SOPOTterms, the error between the ML-FFT and the ideal DFT can bereduced to an arbitrary small value. This allows one to choosean appropriate tradeoff between implementation complexityand numerical accuracy required in a given application. Forexample, in DFT filter bank, only the frequency characteristicsof the final filter banks are of major concern. Therefore, alower average SOPOT term of say two to three will sufficefor a variety of applications [3], [4]. As an illustration, thefrequency response of the 32-channel ML-FFT mentionedearlier in Table III is shown in Fig. 2. Its frequency response isalmost visually identical to the ideal DFT. On the other hand,if the ML-FFT is used to compute circular convolutions withhigh accuracy, then a larger number of SOPOT terms is needed.Fig. 2. Frequency response of the 32-channel ML-FFT using radix 2 DIF FFTalgorithm.IV. CONCLUSIONA new multiplierless approximation of the FFT called themultiplierless FFT-like transformation is proposed. It param-eterizes the twiddle factors in the conventional radix- orsplit-radix FFT algorithms as certain rotation-like matrices andapproximates the associated parameters using the SOPOT orcanonical signed digits representations. The ML-FFT convergesto the FFT when the number of SOPOT terms used increasesand has an arithmetic complexity of additions,where is the transform length. Design results showthat the ML-FFT offers flexible tradeoff between arithmeticcomplexity and numerical accuracy in approximating the FFT.REFERENCES[1] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Pro-cessing. Englewood Cliffs, NJ: Prentice-Hall, 1989.[2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. EnglewoodCliffs, NJ: Prentice-Hall, 1993.[3] S. C. Chan, W. Liu, and K. L. Ho, “Perfect reconstruction modulatedfilter banks with sum-of-powers-of-two coefficients,” in Proc. ISCAS,Geneva, Switzerland, May 28–31, 2000.[4] S. C. Chan and P. M. Yiu, “Multiplier-less discrete sinusoidal and lappedtransforms using sum-of-powers-of-two (SOPOT) coefficients,” in Proc.ISCAS, Sydney, Australia, May 6–9, 2001.[5] T. D. Tran, “A fast multiplier-less block transform for image and videocompression,” in Proc. ICIP, vol. 3, 1999, pp. 822–826.[6] S. C. Pei and J. J. Ding, “The integer transforms analogous to discretetrigonometric transforms,” IEEE Trans. Signal Processing, vol. 48, pp.3345–3364, Dec. 2000.[7] W. K. Cham and Y. T. Chan, “An order-16 integer cosine transform,”IEEE Trans. Signal Processing, vol. 39, pp. 1205–1208, May 1991.[8] A. Nallanathan, S. C. Chan, and T. S. Ng, “Lapped orthogonal transformwith integer coefficients,” in Proc. 42nd Midwest Symp. Circuits andSystems, vol. 2, 2000, pp. 1065–1068.[9] S. C. Chan, “Multiplier-less approximation of DFT and DHT,” Dept.Elect. Electron. Eng., The Univ. Hong Kong, Int. Rep., May 2000.[10] Y. C. Lim and S. R. Parker, “FIR filter design over a discretepower-of-two coefficient space,” IEEE Trans. Acoust. Speech, SignalProcessing, vol. ASSP-31, pp. 583–591, June 1983.

An efficient multiplierless approximation of the fast Fourier transform using sum-of-powers-of-two (SOPOT) coefficients

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An efficient multiplierless approximation of the fast Fourier transform using sum-of-powers-of-two (SOPOT) coefficients

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