This letter proposes an efficient implementation of the Farrow structure using sum-of-powers-of-two (SOPOT) coefficients and multiplier-block (MB). In particular, a novel algorithm for designing the Farrow coefficients in SOPOT form is detailed. Using the SOPOT coefficient representation, coefficient multiplication can be implemented with limited number of shifts and additions. Using MB, the redundancy between multipliers can be fully exploited through the reuse of the intermediate results generated. Design examples show that the proposed method can greatly reduce the complexity of the Farrow structure while providing comparable phase and amplitude responses.published_or_final_versio

Chan, SC

Ho, KL

Pun, CKS

Wu, YC

IEEE Signal Processing Letters

C.K.S. Pun

Y.C. Wu

S.C. Chan

K.L. Ho

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On the design and efficient implementation of the Farrow structure

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Title On the design and efficient implementation of the FarrowstructureAuthor(s) Pun, CKS; Wu, YC; Chan, SC; Ho, KLCitation Ieee Signal Processing Letters, 2003, v. 10 n. 7, p. 189-192Issued Date 2003URL http://hdl.handle.net/10722/42956Rights Creative Commons: Attribution 3.0 Hong Kong LicenseIEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 7, JULY 2003 189On the Design and Efficient Implementationof the Farrow StructureCarson K. S. Pun, Student Member, IEEE, Y. C. Wu, Student Member, IEEE, S. C. Chan, Member, IEEE, andK. L. Ho, Member, IEEEAbstract—This letter proposes an efficient implementation ofthe Farrow structure using sum-of-powers-of-two (SOPOT) coeffi-cients and multiplier-block (MB). In particular, a novel algorithmfor designing the Farrow coefficients in SOPOT form is detailed.Using the SOPOT coefficient representation, coefficient multipli-cation can be implemented with limited number of shifts and addi-tions. Using MB, the redundancy between multipliers can be fullyexploited through the reuse of the intermediate results generated.Design examples show that the proposed method can greatly re-duce the complexity of the Farrow structure while providing com-parable phase and amplitude responses.Index Terms—Farrow structure, fractional-delay digital filters,multiplier-block (MB), sum-of-powers-of-two (SOPOT).I. INTRODUCTIONFRACTIONAL-DELAY digital filters (FD-DF) are veryuseful in providing fractional delay of digital signals.It finds important applications in software radio [1], digitalmodems [2], arbitrary sampling rate conversion, time-delayestimation [3], etc. In general, the fractional-delay digital filterscan be implemented either as finite-duration impulse response(FIR) or infinite-duration impulse response (IIR) digital filters[7]. FIR-based FD-DFs are usually implemented by the Farrowstructure [4] because it can provide variable signal delay,making high-speed online tuning feasible. The implementationof IIR-based FD-DFs is, however, rather complicated due tothe stability constraints and transient problem. Also, to avoidthe implementation of polynomial fractions, approximaterealization such as the gathering structure [12] has to be used.Because of these reasons, we only consider FIR-based FD-DFs.In this letter, an efficient implementation of the Farrowstructure using sum-of-powers-of-two (SOPOT) coefficientsand multiplier-block (MB) is proposed. Furthermore, a novelalgorithm for determining the Farrow coefficients with SOPOTrepresentation is also presented. More precisely, the coeffi-cients of the subfilters in the Farrow structure are representedas SOPOT coefficient or canonical signed digits (CSDs). TheSOPOT coefficient is attractive for very large scale integra-tion or hardware implementation because multiplication ofSOPOT coefficients can be implemented efficiently usingManuscript received August 16, 2001; revised October 4, 2002. The work wassupported by a grant from the Research Grants Council of the Hong Kong SAR.The associate editor coordinating the review of this manuscript and approvingit for publication was Dr. Elias S. Manolakos.The authors are with the Department of Electrical and Electronic EngineeringThe University of Hong Kong (e-mail: carson@ieee.org; ycwu@eee.hku.hk; sc-chan@eee.hku.hk; klho@eee.hku.hk).Digital Object Identifier 10.1109/LSP.2003.813681limited number of hard-wired shifters and adders only (i.e.,multiplierless). To further reduce the number of adders requiredin this structure, the subfilters in the Farrow structure areimplemented in transposed form, which allows us to implementall the SOPOT multiplications with a single MB [5], [6]. MB isan efficient technique for reducing the number of additions inmultiplying a variable input with a fixed set of coefficients inbinary or CSD representations. The basic principle of the MBis to reuse the intermediate results generated in realizing someof the coefficients in order to reduce the adders or additionsrequired for the other coefficients. For example, if there aretwo SOPOT coefficients “7” and “21” to be multiplied bythe input signal , the direct implementation will requirethree adders, since “ ” equals “ ,” and “ ” equals“ ” in SOPOT form. However, if the multiplicationsare implemented together in an MB , the number “21” will berewritten as “ .” Therefore, the result of canbe realized by first computing the result “ ” and thenmultiplying it with “ .” By so doing, only two addersare required in the MB as a result of reusing the intermediatevalue . MB has been proposed to implement interpolatedfilter and filter banks [5]. Unfortunately, detailed design ofsuch multiplierless Farrow structure was not addressed. Thismotivates the proposed design and implementation methods ofthe Farrow structure.The proposed algorithm consists of two different steps: anFD-DF in Farrow structure with real-valued coefficients is firstdesigned using any conventional methods [4], [7], [10]; a flex-ible and efficient “random search” algorithm is then employedto search for the SOPOT coefficients around the real-valued co-efficients while minimizing some criteria such as the number ofSOPOT terms used subject to a given frequency specification.Design results show that significant saving in hardware com-plexity can be obtained by the proposed method.II. EFFICIENT FARROW STRUCTURE WITH SOPOTCOEFFICIENTS AND MBThe output of the FD-DF is given by(1)where is the input signal sampled at a period ;is the impulse response of the FD-DF with a system delay of, where is an integer constant, and is the fractional1070-9908/03$17.00 © 2003 IEEE190 IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 7, JULY 2003Fig. 1. Original Farrow structure.delay parameter between zero and one. is the length of .To avoid the implementation of a large number of filters withdifferent delays, Farrow [4] proposed to approximate each im-pulse response with the following th-order polynomialin delay value such that the delay control is independent ofthe filter coefficients(2)Substituting (2) into (1) gives(3)Fig. 1 shows the Farrow structure for implementing (3), wherethe input signal is passed through a number of subfilters ,, and is multiplied by the appropriate powers ofto produce the output.Though the Farrow structure is very useful in providing a con-tinuous value of signal delay for digital signals by changing thevalue of , it still requires large number of multiplications forthe subfilter implementation, especially when and are largeto provide very precise control of the frequency characteristicsof the FD-DF. One method to avoid the expensive multipliers isto convert the Farrow coefficients , which is fixed, into thefollowing SOPOT representation:(4)with and ,where is a positive integer, and its value determines the rangeof the coefficients. is the number of terms used in the coef-ficient approximation and is usually limited to a small number.The coefficient multiplications can therefore be implemented aslimited shifts and additions, resulting in a significant reductionin implementation complexity. Very often, there is also signifi-cant redundancy in these SOPOT coefficients, which appears ascommon subexpressions among different SOPOT coefficients.Due to the operator, it is somewhat difficult to remove thesesubexpressions. Fortunately, thanks to the transposed form ofthe subfilters, the Farrow structure can be rewritten as in Fig. 2.In this new structure, the input is multiplied to a number of con-stant coefficients. Hence, the common subexpressions withinthe SOPOT coefficients can be eliminated [8], [9] using a singleMB, which further reduces the complexity of the Farrow struc-ture. Next, we shall consider the problem of determining theFarrow coefficients in SOPOT form.III. SOPOT COEFFICIENTS DESIGN AND MB GENERATIONLet and be the frequency re-sponses of the ideal fractional delay filter and that of the Farrowstructure with SOPOT coefficients. Our objective is to deter-mine the SOPOT coefficients such that the distortion fromthe ideal response is minimal. Commonly used distortion mea-sures such as the least squares and the minimax criterion can beused. Without loss of generality, the minimax criterion is em-ployed and the problem can be stated as follows.Given the maximum number of terms in each coefficient,the dynamic range of the coefficients and the bandwidth ofinterest , determine the SOPOT coefficients suchthat the maximum phase response error is minimized subjectto a constraint that the peak amplitude error , where(5)Let be the vector containing the real-valued coefficients ofthe Farrow structure designed by using any conventionalFD-DF design method and interpolation. The principle of thePUN et al.: DESIGN AND EFFICIENT IMPLEMENTATION OF THE FARROW STRUCTURE 191Fig. 2. Proposed implementation of the Farrow structure.proposed random search algorithm is to generate random can-didate SOPOT coefficients in the neighborhood of so as tosearch for the optimal discrete solution. More precisely, a newcoefficient vector is generated by adding a random vectorto the original coefficient vector as follows:(6)where is a scale factor that controls the size of neighborhoodto be searched; is a vector with its elements being randomnumbers in the range [ 1, 1]; and is the roundingoperation that convert its argument to the nearest SOPOT coeffi-cients with maximum number of terms in each coefficient beingand dynamic range . The performance measures and ofthe new coefficients are then calculated. The set that yields theminimum phase error with smaller than a given peak am-plitude error is the optimum solution under the constraints ofand . As this is a random search algorithm, the longer thesearching time, the higher the chance of finding the optimal so-lution.After the SOPOT representation is obtained, the minimumnumber of adders needed in the MB is determined using thealgorithm proposed in [6]. Due to space limitation, the detailis not included here. Interested readers can find more detail in[6].This random search algorithm is similar to the mutation ofgenetic algorithm and the random walk in stimulated annealing[13]. The main difference here is that we have limited its searchspace to a small neighborhood of the real-valued solution. Thisgreatly shortens the search time to a few minutes. In fact, forthe problem considered here, the overall design time only takesless than 5 min to complete on a typical Pentium-400 PC usingMatlab 5.3., including both the design of SOPOT coefficientsand the MB design. Another advantage of this algorithm is thatit can also be used to minimize directly the hardware cost suchas adder cells of the filters, taking into account round-off andoverflow noise [11]. Moreover, a set of possible solutions rep-resenting different tradeoffs between computational complexityTABLE IOPTIMIZED SOPOT COEFFICIENTS IN EXAMPLE 1and performance will be generated during the search. Therefore,it helps one to achieve an appropriate tradeoff for a given appli-cation. It is also possible to combine the two stages together toimprove the performance, but the computational time will begreatly increased. Finally, it should be noted that the proposeddesign and realization methods are also applicable to the multi-plierless realization of the gathering structure recently proposedin [12] for implementing IIR FD-DFs.IV. EXAMPLESExample 1: Let us consider the coefficients provided byFarrow in [4]. The bandwidth under consideration for this filteris from 0 to . is set to 0.006. The interpolation order isthree, and the length of each FIR filters used in the Farrowstructure is eight. The value of “ ” in (4) is chosen to be ten.The optimized SOPOT coefficients of the Farrow structure aregiven in Table I. For the Farrow structure with real-valued coef-ficients, the peak ripple error and maximum phase delay errordeviated from the ideal frequency response are 0.006 271 and0.0032, respectively; whereas for the SOPOT Farrow structure,and are found to be 0.005 371 and 0.0046, respectively.This shows that a comparable performance can be achievedusing SOPOT coefficients. After common-subexpressionselimination, the MB requires only 13 adders, which comparesfavorably with 32 real multiplications in the original Farrow192 IEEE SIGNAL PROCESSING LETTERS, VOL. 10, NO. 7, JULY 2003Fig. 3. Comparison of amplitude and phase responses between the proposedstructure (dotted line) and the original Farrow structure with real-valuedcoefficients (solid line) in Example 2.structure. The resultant Farrow structure filter has a much lowercomplexity than the real-valued Farrow structure but providingnearly the same phase delay and amplitude response.Example 2: This example will be on a Farrow structure withhigher polynomial order. Although there are many methods fordesigning FD-DF with real-valued coefficients [4], [7], [10],the prototype fractional delay filters in this example are de-signed using complex Chebyshev approximation, because it isreadily available in MATLAB. They are then interpolated bya fifth-order polynomial to obtain the real-valued Farrow coef-ficients. The length of each FIR filter in the Farrow structureis again chosen to be eight, and the value of “ ” is ten. Thebandwidth under consideration is from 0 to . is set to0.03. The peak ripple error and maximum phase delay error ofthe original Farrow structure are 0.023 256 and 0.0054, respec-tively. After SOPOT coefficients searching and common-ex-pression elimination, the subfilters require only 18 adders toachieve and . In Fig. 3, the fre-quency responses of the proposed structure (dotted lines) and itsreal-values counterpart (solid lines) are compared for differentvalues of . It can be seen that they are very close to each other.The performance and arithmetic complexity of the two exam-ples are summarized in Table II. Again, significant reduction inhardware complexity by the proposed method is observed.TABLE IISUMMARY OF DESIGN EXAMPLESV. CONCLUSIONA new method for the design and efficient implementationof Farrow-based FD-DF using SOPOT coefficients and MB ispresented. Considerable saving in hardware complexity can beachieved by the proposed method with a reasonable fast designtime and a frequency response comparable to their real-valuedcounterpart. The usefulness of the method is demonstrated withseveral design examples.REFERENCES[1] T. Hentschel and G. Fettweis, “Sample rate conversion for softwareradio,” IEEE Commun. Mag., vol. 38, pp. 142–150, Aug. 2000.[2] L. Erup, F. M. Gardner, and R. A. Harris, “Interpolation in digitalmodems. II. Implementation and performance,” IEEE Trans. Commun.,vol. 41, pp. 998–1008, June 1993.[3] H. C. So, P. C. Ching, and Y. T. Chan, “A new algorithm for explicitadaptation of time delays,” IEEE Trans. Signal Processing, vol. 42, pp.1816–1820, July 1994.[4] C. W. Farrow, “A continuously variable digital delay element,” in Proc.ISCAS, 1988, pp. 2641–2645.[5] A. G. Dempster and N. P. Murphy, “Efficient interpolators and filterbanks using multiplier blocks,” IEEE Trans. Signal Processing, vol. 48,pp. 257–261, Jan. 2000.[6] A. G. Dempster and M. D. Macleod, “Use of minimum-adder multiplierblocks in FIR digital filters,” IEEE Trans. Circuits Syst. II, vol. 42, pp.569–577, Sept. 1995.[7] T. I. Laakso, V. Valimaki, M. Karjalainen, and U. K. Laine, “Splittingthe unit delay, tools for fractional delay filter design,” IEEE Signal Pro-cessing Mag., pp. 30–60, Jan. 1996.[8] M. Potkonjak, M. B. Srivastava, and A. P. Chandrakasan, “Multiple con-stant multiplications: Efficient and versatile framework and algorithmsfor exploring common subexpression elimination,” IEEE Trans. Com-puter-Aided Design, vol. 15, pp. 151–165, Feb. 1996.[9] R. Pasko, P. Schaumont, V. Derudder, and D. Durackova, “Optimiza-tion method for broadband modem FIR filter design using commonsubexpression elimination,” in Proc. Symp. System Synthesis, 1997, pp.100–106.[10] J. Vesma and T. Saramaki, “Design and properties of polynomial-basedfractional delay filters,” in Proc. ISCAS, 2000, pp. 104–107.[11] C. K. S. Pun, S. C. Chan, and K. L. Ho, “Efficient design of a classof multiplier-less perfect reconstruction two-channel filter banks andwavelets with prescribed output accuracy,” in Proc. 11th IEEE Work-shop Statistical Signal Processing, Singapore, Aug. 2001, pp. 599–602.[12] M. Makundi, T. I. Laakso, and V. Valimaki, “Efficient tunable IIR andallpass filter structures,” Electron. Lett., vol. 37, pp. 344–345, Mar. 2001.[13] D. Karaboga and D. T. Pham, Intelligent Optimization Techniques: Ge-netic Algorithms, Tabu Search, Simulated Annealing and Neural Net-works. Berlin, Germany: Springer-Verlag, 2000.

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On the design and efficient implementation of the Farrow structure

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