Adjustable Fractional-Delay Filters Utilizing the Farrow Structure and Multirate Techniques

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1 Ajustable Fractional-Delay Filters Utilizing the Farrow Structure an Multirate Techniques Håkan Johansson an Ewa Hermanowicz *Division of Electronics Systems, Department of Electrical Engineering, SE Linköping University, Sween **Multimeia Systems Department, Faculty of Electronics, Telecommunication an Informatics, Gansk University of Technology, Gansk, Polan ABSTRACT The Farrow structure can be use for efficient realization of ajustable fractional-elay FIR filters, but espite its efficiency compare to other approaches, its implementation complexity grows rapily as the banwith approaches π. To reuce the complexity, a multirate approach has been propose. In this approach, the input signal is first interpolate by a factor of two via the use of a fixe half-ban linear-phase FIR filter. Then, the actual fractional-elay filtering takes place. Finally, the so generate signal is ownsample to retain the original input/output sampling rate. In this way, the banwith of the fractional-elay filter use is halve compare to the overall banwith. Because the complexity of half-ban linear-phase FIR filter interpolators is low, the overall complexity can be reuce. In this paper, we give further etails of the multirate approach that have not been publishe before. In aition, we introuce the use of an approximately linear-phase IIR filter instea of a linear-phase FIR filter in the interpolation process in orer to reuce the complexity even further. Design examples are inclue emonstrating this point. E L (z) E (z). INTRODUCTION Many applications require the use of igital filters having ajustable (variable) frequency responses []. This paper eals with ajustable fractional elay (FD) filters which are require in several contexts [] [4]. One example is sampling rate conversion by arbitrary conversion factors where the traitional interpolators an ecimators use for integeran rational-factor conversion fail or imply very high interpolation an ecimation factors [3]. An efficient structure for ajustable FD filtering is the so calle Farrow structure [5] [7] which makes use of a number of fixe FIR subfilters an only one variable parameter as seen in Fig.. The implementation complexity is therefore much lower for Farrow-base filtering methos than for methos base on either on-line esign or storage of a (large) amount of ifferent impulse responses. However, even if the complexity of the Farrow structure is relatively low, it grows rapily as the banwith of the FD filter approaches π [6], [7]. To reuce the complexity, a multirate approach was propose in [8] []. In this approach, the input signal is first interpolate by a factor of two via the use of a half-ban linear-phase FIR filter. Then, the actual FD filtering takes place. Finally, the so generate signal is ownsample to retain the original input/output sampling rate. In this way, the banwith of the FD filter use is halve compare to the overall banwith. Because the complexity of half-ban linear-phase FIR filter interpolators is low, the overall complexity can be reuce. For a banwith of.9π, savings of some 3-5% were reporte in []. In this paper, we give further etails of the multirate approach that have not been publishe before. Furthermore, we introuce the use of an approximately linear-phase IIR filter instea of a linear-phase FIR filter in the interpolation process. It is emonstrate through esign examples that this alternative can reuce the complexity further by some 5%. Following this introuction, Section recapitulates the basics of Farrow-base FD FIR filters. Section 3 consiers half-ban linear-phase FIR an approximately linear-phase IIR filters. This section inclues a comparison between the two filter classes that have not been publishe before. Section 4 escribes the multirate approach. Section 5 iscusses the filter esign an provies a esign example. Finally, Section 6 conclues the paper.. ADJUSTABLE FRACTIONAL-DELAY FIR FILTERS Let the esire frequency response H es ( e jωt ) of an ajustable FD filter be H es ( e jωt ) = e jd ( + )ωt () = e jdωt e jωt, ωt ω T < π c where D an are fixe an ajustable real-value constants, respectively. In this paper, it is assume that D is either an integer, or an integer plus a half, whereas takes on values in the interval [, ]. In this way, a whole sampling interval is covere by, an the fractional elay equals (+.5) when D is an integer (an integer plus a half). The ieal filter response in () can be approximate using the Farrow structure epicte in Fig.. The transfer function can then be expresse as where E k ( z) are FIR subfilters. Further, it is possible an efficient (ue to the coefficient symmetry) to impose linear- E (z) Figure. Farrow structure. L Gz ( ) = k E k ( z) k = E (z) ()

2 Magnitue Response Error Magnitue Response π.4π.6π.8π π.π.4π.6π.8π.9π Figure. Illustration of magnitue response an magnitue response error for an ajustable Farrow-base FD filter taken from Example 4 in [7], with o-orer subfilters for =.5 (soli),.375 (ashe),.5 (ashot),.5 (otte). phase constraints on the subfilters [5] [7]. Hence, we assume that each E k ( z) is an N G th-orer linear-phase FIR filter with either symmetric (for k even) or anti-symmetric (for k o) impulse response e k ( n), i.e., e k ( n) = e k ( N G n) an e k ( n) = e k ( N G n), n =,,..., N G, for even an o, respectively. Furthermore, it has been emonstrate that it is beneficial, in terms of implementation complexity, to make use of subfilters of unequal orers instea of equal orers [7]. To keep the notation simple, this is here taken care of by assuming that N = max{n k }, N k enoting the subfilter orer, an introucing aitional elays to make all branches have the same elay. Introucing aitional elays in an FIR filter correspons to appening zeros in the beginning an the en of the impulse response. The filter with a transfer function in the form of () can approximate the ieal response () as close as esire by choosing L an esigning the filter appropriately [6], [7]. Figures an 3 show the responses for Example 4 in [7]. 3. HALF-BAND FILTERS The transfer function F(z) of the half-ban filters ealt with here can be written as [], [] Az ( Fz ( ) ) + z = K (3) where K is an o integer. It is note that (3) is a special case of a general polyphase ecomposition [] Fz ( ) = F ( z ) + z F ( z ) for F ( z) = z ( K ) an F ( z) = Az ( ). The function Az ( ) is chosen as explaine below for the FIR an IIR filter cases. For convenience later on, we also efine a complementary transfer function F c ( z) accoring to Az ( F c ( z) ) z = K. (4) It follows from (3) an (4) that F c ( z) = F( z) = [ z K Fz ( )] (5) Phase Delay Phase Delay Error π.4π.6π.8π π which, in the frequency omain, correspons to a frequency shift of π ra, i.e., F c ( e jωt ) = Fe ( j( ωt π) ). (6) Further, every secon impulse response value is zero except for the centre tap that equals /, i.e., fn ( ) x -3 -.π.4π.6π.8π.9π Figure 3. Illustration of phase elay an phase elay error for an ajustable Farrow-base FD filter taken from Example 4 in [7], with o-orer subfilters for =.5 (soli),.375 (ashe),.5 (ashot),.5 (otte). =, n = N F = K, n = 3,,, N F, n N F. (7) This can, e.g., be utilize to eliminate intersymbol interference in communication systems. A. Half-Ban Linear-Phase FIR Filters In this case, Az ( ) is the transfer function of a Type II linearphase FIR filter of o orer K []. The impulse response is therefore symmetric accoring to an ( ) = ak ( n) for n =,,, K. This implies that the overall filter Fz ( ) becomes a Type I linear-phase FIR filter of even orer N F = K having a symmetric impulse response accoring to fn ( ) = fn ( F n) for n =,,, N F. A typical feature of half-ban linear-phase FIR filters is the following. If the filter approximates unity in the passban region ωt [,, ω c T < π, with the tolerance δ c, then it approximates zero in the stopban region ωt [ ω s T, π], ω c T + ω s T = π, with the same tolerance. In other wors, the stopban ripple δ s equals the passban ripple δ c. This can easily be shown by noting [from (5)] that the magnitue of the complementary transfer function F c ( z) equals the magnitue of the function z K Fz ( ). i.e., F c ( e jωt ) = e jkωt Fe ( jωt ). (8) Since Fz ( ) is a Type I linear-phase FIR filter of elay K (orer N F = K), the passban ripple (stopban ripple) of the function z K Fz ( ) equals the stopban ripple (passban ripple) of Fz ( ). Further, F c ( z) is a frequency shifte version of Fz ( ) which implies that these two filters have the same passban an stopban ripples. Finally, since F c ( z)

3 Magnitue Resp [B] Magnitue Resp [B] π.4π.6π.8π π 4 x -3.π.π.3π.4π Figure 4. Magnitue response of a half-ban linear-phase FIR filter. coincie with z K Fz ( ) (except for the minus sign), this means that the passban an stopban ripples of Fz ( ) must be equal. B. Half-Ban Approximately Linear-Phase IIR Filters In this case, Az ( ) is the transfer function of an allpass filter of orer K+. This means that the overall filter will be a (K+)-orer IIR filter. A typical feature of this class of IIR filters is that the passban ripple will be extremely small for practical stopban attenuations. Another feature is that the phase error, i.e., the ifference between the filter s phase response an the esire linear-phase response KωT roughly equals the stopban ripple. These two features can be shown as follows. It is first note that the transfer function pair in (3) an (4) here form a power complementary filter pair. This is easily shown by noting that the frequency responses can be written as Fe jωt e jφ A ( ωt) + ( ) = e jkωt e j.5( Φ A ( ωt) KωT) Φ A ( ωt) + KωT = cos (9) an F c e jωt e jφ A ( ωt) e ( ) = jkωt je j.5( Φ A ( ωt) KωT) Φ A ( ωt) + KωT = sin () respectively, where Φ A ( ωt) enotes the phase response of Az ( ). From (9) an () it follows immeiately that the filters are power complementary, i.e., Fe ( jωt ) + F c ( e jωt ) =. () Since F c ( z) is a frequency shifte version of Fz ( ), it follows from () that the passban an stopban ripples of Magnitue Resp [B] Magnitue Resp [B] Phase Error Fz ( ) satisfy which implies.π.4π.6π.8π π x -6.π.π.3π.4π x -4 5.π.π.3π.4π Figure 5. Magnitue response an phase error of a half-ban approximately linear-phase IIR filter. ( δ c ) + δ s = δ c.5δ s (). (3) Hence, the passban ripple will be very small in practice. Further, utilizing (9), it follows that the phase error, efine as, Φ err ( ωt) = Φ( ωt) + KωT, ωt [,, (4) with Φ( ωt) enoting the phase response of F(z), becomes Φ A ( ωt) + KωT Φ err ( ωt) = (5) In the passban region of F(z), ωt [,, which correspons to the stopban region of F c (z), we know from () that Φ A ( ωt) + KωT sin δ. (6) s For small phase errors, it then follows from the Taylor series expansion of the sine function that Φ err ( ωt) < δ s. (7) C. Comparison Figures 4 an 5 show typical frequency responses for linear-phase FIR an approximately linear-phase IIR filters,

4 Filter Orer /Δ Figure 6. Filter orer of half-ban FIR (ashe) an half-ban IIR (soli) versus the inverse of the transition banwith (Δ = (ω s T ω c T)/π). H(z) H (z) Figure 8. Multirate (two-rate) FD filtering approach. F(z) G(z) Multiplication ratio Stopban attenuation [B] Figure 7. Complexity comparison (multiplication ratio) between the half-ban IIR an FIR filters. respectively. The FIR filter has been esigne with the ai of remez.m in MATLAB which implements the fast McClellan-Parks-Rabiner algorithm [3]. The IIR filter has been esigne using the algorithm propose in [4] for N =. Figure 6 an 7 illustrate the filter orers require for the two options. It is seen that the approximately linear-phase IIR filters have lower complexity in terms of number of arithmetic operations require to implement the filter. 4. MULTIRATE APPROACH In the multirate approach for realizing fractional-elay filters, the input signal is first upsample by a factor of two, then filtere through a cascae connection of a half-ban an fractional-elay filter, an finally ownsample by a factor of two. The esire overall fractional elay D+ is obtaine by letting the elay of the filter between the upsampler an ownsampler be (D+). This is because the filter works at twice the input/output rate. However, in the final realization, all filtering takes place at the lower sampling rate. In aition, the arrangement in Fig. 8 is in fact a linear an time-invariant system with the transfer function being the th polyphase component of H(z) [], i.e., H (z) in the polyphase representation Hz ( ) = H ( z ) + z H ( z ). (8) This is because the overall transfer function of the structure in Fig. 8 can be written as.5[ Hz ( ) + H( z )] which equals H (z). Here, H(z) is basically a cascae of the FD an HB filters consiere in Sections an 3. However, we may also a an aitional elay in orer to avoi half-integer elays. This leas us to the following three ifferent cases an structures. Structure : In this case, H(z) is given by. The multiplication by two require to retain the energy in the interpolation process it inclue in Fig. but omitte elsewhere for simplicity. z F (z) F (z) G (z) G (z) Figure 9. Structure in the multirate FD filtering approach. A(z) z (K+)/ E L (z) E (z) Figure. Structure in etail. Figure. Structure in the multirate FD filtering approach. Figure. Structure 3 in the multirate FD filtering approach. Hz ( ) = Fz ( )Gz ( ) (9) where G(z) an F(z) are transfer functions of the FD an HB filters consiere in Section an 3, respectively. Utilizing polyphase representation an the equivalence in Fig. 9, we obtain the polyphase component H (z) as E (z) E (z) E L (z) E (z) E (z) E (z) z F(z) G(z) z F (z) F (z) G (z) G (z) z F(z) G(z) F (z) F (z) G (z) G (z) H ( z) = F ( z)g ( z) + z F ( z)g ( z) ()

5 The elay of H (z) is D+ = (D H + H )/ where D H + H is the elay of H(z). The fixe part D H is given by D H = N G /+K. With G(z) an F(z) in the form of () an (3), Structure can be implemente as shown in Fig. where E k ( z) an E k ( z) are the polyphase components of E k ( z). Structure : In this case, we introuce a elay which gives us the transfer function H(z) accoring to Hz ( ) = z Fz ( )Gz ( ) () Utilizing the equivalence in Fig., we now obtain the polyphase component H (z) as H ( z) = z [ F ( z)g ( z) + F ( z)g ( z) ] () The elay of H (z) is now D+ = (D H ++ H )/. Structure 3: In this case we introuce a negative elay which gives us the transfer function H(z) accoring to Hz ( ) = zf( z)gz ( ) (3) Here, utilizing the equivalence in Fig., we obtain the polyphase component H (z) as H ( z) = F ( z)g ( z) + F ( z)g ( z) (4) The elay of H (z) is now D+ = (D H + H )/. The three structures are use as follows in two ifferent cases. Case A: In this case, either Structure or Structure is use straightforwarly. The overall elay is D+ = (D H + H )/ [(D H + H +)/] for Structure [Structure ]. The avantage of this case is that a fixe structure is use. This makes it suitable for applications where the fractional elay changes from sample to sample (or at least frequently). The (minor) isavantage is that the variable fractional elay at the higher rate, i.e., H, must cover the interval [, ] when the esire overall variable fractional elay H covers [, ]. The wier this range is, the higher will the filter orer of the fractional elay filter G(z) be, but, as we shall see, the overall complexity will espite of this fact be lower than that of a regular Farrow-base FD filter. Case B: In this case, all three structures are utilize in such a way that the interval for H becomes [, ]. The avantage of this approach is that the complexity ue to the fractional-elay filter G(z) thereby can be reuce. The isavantage is that, ue to the switching between the ifferent structures, the implementation becomes more complex if one wants to avoi transient problems. This case is therefore more attractive for applications where the fractional elay oes not change too often. We now explain the basic principle for the case where D H = D (other cases can be explaine similarly) an recall that it is esire to obtain the elay D+. For [ 4, 4], we set D+ = (D H + H )/ an recognize that this elay can be obtaine by using Structure with H =. For ( 4, ), we set D+ = (D H ++ H )/. This elay can be obtaine by using Structure with H =. Likewise, for (, 4), we set D+ = (D H + H )/. This elay can be obtaine by using Structure 3 with H = FILTER DESIGN In this paper, the overall FD filter is esigne by minimizing the maximum of the complex error s moulus. This is one by solving the following optimization problem: Given the number of Farrow subfilters L+, the subfilter s orer N G, an the half-ban FIR filter orer N F = K, or the half-ban IIR filter orer N F = K+, fin the unknown coefficients g k (n), k =,,..., L, n =,,..., (N G )/, a m, m =,,..., K, an δ, to minimize δ subject to H ( e jωt ) H es ( e jωt ) δ, ωt [,. (5) for ωt [, an [,5,,5]. Here, a m represent the coefficients of either the Type II linear-phase FIR filter (in the half-ban FIR case) or the allpass filter (in the approximately linear-phase IIR case). In the FIR (IIR) case, K = K (K = K+). The problem state above is a stanar nonlinear optimization problem for which one can use stanar solvers like fminimax.m in Matlab. It is nonlinear because H(z) is compose of filters in cascae. Consequently, one cannot guarantee that the solution obtaine in the optimization is the global optimum, but only the local optimum. It is therefore important to use a goo initial solution. This can be one by esigning the half-ban filter an the Farrow-base FD filter separately to meet about the same approximation error as the esire overall approximation error. This is because the overall transfer function can be written as H ( z) =,5[ Hz ( ) + H( z )] (6) where Hz ( ) = Fz ( )Gz ( ). The main contribution in the frequency ban of interest emanates from Hz ( ) because H( z ) has its stopban region there. The overall frequency response is therefore roughly etermine by the response of Fz ( )Gz ( ). To minimize the overall complexity, an to make sure that δ δ spec, where δ spec is a specifie acceptable approximation error, one must select L, N G, an N F in a proper way. The optimum values of L, N G, an N F can be foun through an exhaustive search aroun the neighborhoo of the values obtaine through the separate optimization mentione above. In this way, the number of optimization problems that nees to be solve is limite to a rather small number. A. Design Example The specification is ω c T =.9π an δ spec =.4, which has been consiere before in [7], []. To meet this specification, the Farrow structure, with subfilters jointly optimize as outline in [7], requires 45 fixe an 5 variable multipliers. The multirate approach, with FIR filters only, requires 3 (3) fixe an 6 (3) variable multipliers for Case A (Case B) []. For the multirate IIR/FIR approach, the corresponing figures are 6 (9) fixe an 6 (3) variable multipliers for Case A (Case B). Hence, by using approximately linear-phase IIR filters instea of linear-phase FIR filters for the interpolation by two, the complexity of the fixe parts can be reuce by some 5%. Further, the integer elay has been reuce from 6 to for both Case A an Case B. Figures 3 an 4 show for the multirate IIR/FIR approach the complex error magnitue, magnitue response, an phase elay for some values of the fractional elay.

6 Complex error magn 5 x π.6π.9π Complex error magn 5 x π.6π.9π Magnitue response π.4π.6π.8π π Magnitue response π.4π.6π.8π π Phase elay π.4π.6π.8π π Figure 3. Multirate IIR/IIR Case A for ifferent values of. Phase elay π.4π.6π.8π π Figure 4. Multirate IIR/IIR Case B for ifferent values of. 6. CONCLUSION This paper has consiere a multirate approach for ajustable fractional-elay filtering. In this approach, the signal is first interpolate by a factor of two before the actual fractional filtering takes place via the Farrow structure. Finally, ownsampling by two takes place to retain the original sampling rate. It was emonstrate through a esign example that this approach can reuce the complexity significantly compare to the traitional single-rate Farrow structure. It was also emonstrate that the complexity can be further reuce by using approximately linear-phase IIR filters instea of linear-phase FIR filters for the interpolation. REFERENCES [] G. Stoyanov an M. Kawamata, Variable igital filters, J. Signal Processing, vol., no. 4, pp , July 997. [] L. Erup, F. M. Garner, an R. A. Harris, Interpolation in igital moems part II: Implementation an performance, IEEE Trans. Commun., vol. 4, no. 6, pp , June 993. [3] J. Vesma, Optimization an Applications of Polynomial-Base Interpolation Filters, Diss. no. 54, Tampere University of Technology, 999. [4] H. Johansson an P. Löwenborg, Reconstruction of nonuniformly sample banlimite signals by means of igital fractional elay filters, IEEE Trans. Signal Processing, vol. 5, no., pp , Nov.. [5] C. W. Farrow, A continuously variable igital elay element, in Proc. IEEE Int. Symp. Circuits Syst., Espoo, Finlan, June 7 9, 988, vol. 3, pp [6] J. Vesma an T. Saramäki, Optimization an efficient implementation of FIR filters with ajustable fractional elay, in Proc. IEEE Int. Symp. Circuits Syst., Hong Kong, June 9, 997, vol. IV, pp [7] H. Johansson an P. Löwenborg, On the esign of ajustable fractional elay FIR filters, IEEE Trans. Circuits Syst. II, vol. 5, no. 4, pp , Apr. 3. [8] N. P. Murphy, A. Krukowski, an I. Kale, Implementation of a wieban integer an fractional elay element, Electron. Lett., vol. 3, no., pp , 994. [9] E. Hermanowicz, On esigning a wieban fractional elay filter using the Farrow approach, in Proc. XII European Signal Processing Conf., Vienna, Austria, Sept. 6, 4, pp [] E. Hermanowicz an H. Johansson, On esigning minimax ajustable wieban fractional elay FIR filters using two-rate approach, in Proc. European Conf. Circuit Theory Design, Cork, Irelan, Aug. 9 Sept., 5. [] T. Saramäki, Finite impulse response filter esign, in Hanbook for Digital Signal Processing, es. S. K. Mitra an J. F. Kaiser, New York: Wiley, 993, ch. 4, pp [] P. P. Vaiyanathan, Multirate Systems an Filter Banks, Englewoo Cliffs, NJ: Prentice-Hall, 993. [3] J. H. McClellan, T. W. Parks, an L. R. Rabiner, A computer program for esigning optimum FIR linear phase igital filters, IEEE Trans. Auio Electroacoust., vol. AU-, pp , Dec [4] M. Renfors an T. Saramäki, Recursive Nth-Ban Digital Filters- Part I: Design an Properties, IEEE Trans. Circuits, Syst., vol. CAS- 34, no., pp. 4-39, Jan. 987.