Efficient Filter Banks And Interpolators A. G. DEMPSTER AND N. P. MURPHY Departent of Electronic Systes University of Westinster 115 New Cavendish St, London W1M 8JS United Kingdo Abstract: - Graphical design techniques have been shown to reduce the coplexity of individual ultipliers or structures that use several products of a single ultiplicand, such as FIR filters. The latter uses ultiplier blocks, which in this paper are shown to further reduce the coplexity of filter banks. We design an interpolating filter using the Farrow structure, and also use ultiplier blocks in the design of the extra delay circuitry that this structure requires. Applying ultiplier blocks to whole filter banks proves ore efficient than using separate filters, and applying ultiplier blocks to the Farrow structure akes it ore efficient than using the ore recognised Lagrange interpolator filters for low orders of Lagrange interpolation. Key Words: - Multiplication, Multiplier blocks, Interpolation, Lagrange interpolation, Farrow structure 1. Introduction 1.1 Multiplier Blocks The proble of reducing the nuber of adders used in a shift-and-add ultiplication process has been widely exained. For a single ultiplication, we have introduced a ethod that guarantees the iniu nuber of adders for short wordlengths [1] and algoriths that are the ost efficient available for longer wordlengths [2, 3]. In any structure where several products of a single ultiplicand are required, such as the transposed direct for FIR filter, ultiplier blocks can be utilised to exploit redundancy between the different ultipliers, resulting in efficient designs [4, 5, 6] requiring far fewer adders than other leading techniques [7]. Several IIR structures are also suitable for ultiplier block use and significant savings can be ade [8, 9, 1, 11]. In a recent paper, we exained the savings that can be ade if banks of FIR filters use ultiplier blocks to perfor their coefficient ultiplication operations [12]. Here we extend this work to deterine when the Farrow structure is best. 1.2 Filter Banks and the Farrow Structure Filter banks (parallel connections of digital filters) are used in any signal processing applications including design of analysis and synthesis filters for ultirate signal processing, tie-frequency analysis, wavelets and for fractional delay filter design. This latter application uses the Farrow structure [13], which can be explained as follows. To design a fractional delay filter (i.e. a filter which delays an incoing sapled signal by a non-integral nuber of saples, say D) is to produce a filter with frequency response jω jωd H( e ) = e (1) or ipulse response h( n) = sinc( n D) for all n (2) which is non-causal and so ust approxiated. A popular and effective ethod of approxiating this ipulse response is to use an FIR filter perforing Lagrange interpolation, which has coefficients [14]: N D k h( n) = for n =, 1, 2, N (3) n k k= k n This works well for constant delay D, but the coefficients ust be recalculated each tie D changes.
The ai of Farrow's ethod [13] is to avoid this recalculation. If d is the fractional part of D (ie D = l d for integer l, and d < 1), then a polynoial of order P in d is used to approxiate each coefficient in equation: h ( n) = c ( n) d d P = (4) which leads to transfer function N N P P n n H d ( z) = hd ( n) z = c ( n) d z = C ( z) d (5) n= n= = = where C ( z) = c ( n) z N n= n. This structure can be ipleented as in Figure 1, i.e. a filter bank. Iportantly, the coefficients of the filters are constant, regardless of the delay d. Figure 1, the constant ultipliers and saple delay registers, needs to be replicated M ties, once for each delay value. However, only one Farrow filter bank is required. Such a structure is shown in Figure 2. Although the d i delays are fixed, there are several of the, and the extra coputations required by the Farrow structure are effectively aortised across the range of delays. C P (z) C P-1 (z) C 1 (z) C (z) d d d y (n) C P(z) C P-1(z) C 1(z) C (z) d 1 d 1 d 1 y 1 (n) d d d y(n) d M d M d M y M (n) Figure 1 Farrow structure for polynoial approxiation of filter coefficients The Farrow structure can be used to replace singlefilter interpolators such as the Lagrange interpolator. In fact, because the Lagrange interpolator itself uses a polynoial ethod of approxiation, the Farrow structure can exactly replicate the single Lagrange interpolator filter with N = P [15]. Typically, the Farrow structure, which is a filter bank, is only a useful replaceent for a single filter where the fractional delay is different each tie an output saple is calculated [14], because its fixed coefficients save coputation coplexity. Obviously, for a constant delay, one filter requires less effort than P filters, so the Farrow structure is less attractive. However, another useful application of the Farrow structure that we propose here is that of a regular interpolator, i.e. a structure that produces interpolated outputs at several (fixed) delay intervals. In ultirate signal processing applications, where a signal is being upsapled by a factor M, a regular interpolator or expander is used to reconstruct signals. In such a structure, the lower part of Figure 2 A Farrow structure used to generate several constant delays, as in a regular upsapler or expander. 2. Filter Bank Structures Using Multiplier Blocks Multiplier blocks are useful in structures where several products of a single ultiplicand are required. In [12], we show how network anipulations can be used so that, starting with a bank of parallel FIR filters, all of the filter coefficients could be incorporated into a single block. This structure has NP delays (for P order N filters), but another structure can be defined which is canonic in delays (ie N delays only), but has a separate ultiplier block for each individual filter in the bank. 3. Farrow Structures Using Multiplier Blocks Typically the Farrow structure is drawn as in
Figure 1, i.e. a filter bank with separate identical ultipliers separated by delays. The ai of the work presented here is to iniise the nuber of nuerical operations perfored in such structures, and the ethod in the previous section can be used to iniise the operations in the filter bank section of the structure. Previous authors have used siilar siplifications for specific siple exaples of Farrow filter banks [15, 16]. If the Farrow structure is used to ipleent a regular grid of delay points, a structure such as that in Figure 2 is ore appropriate. It can be seen fro that figure that a ultiplier block could replace the M products of the output of C P (z). However, even greater siplifications can be ade with soe network anipulation. In [12], we describe how the Farrow interpolator bank can be configured as in Figure 3, which we call Farrow block I and Figure 4 which we call Farrow block II. C P(z) C P-1(z) C 1(z) C (z) y (n) y 1(n) y (n) Figure 4 The Farrow structure of Figure 2 odified by transposing each single delay filter, ipleenting the coefficients in a ultiplier block, then retransposing the whole filter bank (Farrow block II) C P (z) C P-1 (z) C 1 (z) C (z) d M P P P d 1 d d 1 P-1 d M P-1 d P-1 d M d 1 d Figure 3 The Farrow structure for several constant delay outputs. Note the outputs of each bank filter are then ultiplied by several coefficients, which can be gathered into ultiplier blocks. The highest-order coefficient block is shown dotted. (Farrow block I) y (n) y 1 (n) y M (n 4. Varying Design Paraeters Our previous work [12] showed that ipleentations with all coefficients in the bank in a single block are less costly. For both the Lagrange and Farrow structures, gains ade by including all coefficients in a single block outweigh the penalty of using a few ore delay eleents, consistent with earlier results [5, 7, 8, 9, 1]. In that paper, the exaple Lagrange bank filter was the cheapest. However, despite the extra circuitry required by the Farrow structure, it is not far behind. There is reason to believe that the Farrow structure ay in soe circustances be better. 4.1 Varying the Lagrange order The above experient was repeated for wordlength w =12, upsapling rate M = 16 and Lagrange orders between 2 and 1. For each of these specifications, four different filter structures were designed: i) individual Lagrange filters, each designed as in [5] (see also [17]), ii) a Lagrange bank (all coefficients in a single block, as discussed in section 2 and Figure 3c of [12]), iii) a Farrow structure with individual filters blocked (the canonic design discussed in section 2 and Figure 4 of [12]), and iv) a Farrow block (either Figure 3 or Figure 4b, whichever was ore efficient). The results are shown in Figure 5. For Lagrange filters, as in [12], it is ore efficient to block all of the coefficients in the
bank. For the Farrow structure, due to the extra structural coponents, this advantage is not established. In fact, for soe orders, there is a very slight advantage if the individual filters are used. The interesting point to note fro these results, however, is that for orders 4 and below, the Farrow structure gives the ost efficient solution, and for orders 5 and above, the Lagrange filters give the ost efficient solution. precision (a siilar phenoenon was noted in [5] where as the nuber of coefficients increased, total cost saturated as no new coefficients were introduced). The individual filter ipleentation of Lagrange quickly becoes expensive. The Farrow structure is ore efficient than Lagrange for high wordlengths. 4 35 3 35 3 25 Lagrange ind Farrow ind Lagrange bank Farrow bank adder cost 25 2 15 adder cost 2 15 1 1 5 5 Lagrange ind Farrow ind Lagrange bank Farrow bank 5 1 15 2 25 3 35 upsapling rate 2 3 4 5 6 7 8 9 1 Lagrange order Figure 5 Costs of the different filter configurations for different Lagrange orders 4.2 Varying the upsapling rate A siilar experient was run, with Lagrange order N = 5, wordlength w = 12, with upsapling rate varying fro 2 to 32. The results are shown in Figure 6. For both the Farrow and Lagrange structures, the bank ethod is ost efficient throughout. The nononotonic behaviour for low upsapling rates is due to the rate of 4 producing cheap powers-of-two type coefficients, aking it ore efficient than a rate of 3. Note that for higher upsapling rates (above 18), the Farrow structure is superior. 4.3 Varying the wordlength Wordlength w was varied fro 4 to 32 bits for the case Lagrange order N = 5, upsapling rate M = 16. These results are shown in Figure 7. Each of the cost curves saturates because, due to the way the coefficients are generated, we find that extra wordlength gives no ore Figure 6 Costs of the different filter configurations for different upsapling rates adder cost 35 3 25 2 15 1 Lagrange ind Farrow ind Lagrange bank Farrow bank 5 5 1 15 2 25 3 35 wordlength Figure 7 Costs of the different filter configurations for different wordlengths 4.4 When to use the Farrow structure For three wordlengths, w = 12, 16, 24, the order was varied fro 2 to 1 in search of the lowest order where
the Farrow structure was not superior to the standard Lagrange ethod, i.e. the transition fro where a Farrow design is prefered to Lagrange. These results are shown in Figure 8, and again a nuber of interesting observations can be ade: Certain upsapling rates (e.g. 15 and 25) have a low transition order, indicating that the Lagrange ethod is ore preferable than for neighbouring values of upsapling rate. Siilarly, there are rates (e.g. 16, 24 and 3) where the Farrow structure is ore preferable. At wordlength w = 12 in particular, for upsapling rates of 24 and 3, the Farrow structure is prefered for all orders up to 15. 1) As upsapling rate increases, the transition order increases. In other words, the Farrow structure becoes ore attractive, i.e. it is ore efficient for ore Lagrange orders. This growth in the transition order is greater for the shorter wordlengths. 2) In general, wordlength w = 16 sees to have a lower transition order than either w = 12 or 24, although it could also be said that there is not a great variation in the results with wordlength. transition order 22 2 18 16 14 12 1 8 6 4 2 w=12 w=16 w=24 5 1 15 2 25 3 upsapling rate Figure 8 The iniu order at which the cost of the Lagrange ethod is less than that using the Farrow structure (both with all coefficients in the filter bank in a single block) 5. Conclusions It has been shown that there are significant savings to be ade if ultiplier blocks are used for all the coefficients in a filter bank. In the case of upsapling interpolators, the Farrow structure becoes cost copetitive with low-order standard Lagrange designs if ultiplier blocks are used. References [1] A G Depster and M D Macleod, Constant integer ultiplication using iniu adders, IEE Proc E: CD&S, vol 141, no 5, pp47-413, Oct 1994 [2] A G Depster and M D Macleod, General algoriths for reduced-adder integer ultiplier design, Electronics Letters, vol 31, no 21, pp18-182, Oct 1995 [3] A G Depster and M D Macleod, Coents on Miniu nuber of adders for ipleenting a ultiplier and its application to the design of ultiplierless digital filters, IEEE Trans C&S II, vol. 45, no. 2, pp. 242-243, Feb 1998 [4] D R Bull and D H Horrocks, Priitive operator digital filters, IEE Proc. G, vol 138, no 3, pp41-412, Jun 1991 [5] A G Depster and M D Macleod, Use of iniu-adder ultiplier blocks in FIR digital filters IEEE Trans C&S II, vol 42, no 9, pp569-577, Sept 1995 [6] A G Depster and M D Macleod, Use of ultiplier blocks to reduce filter coplexity, ISCAS94, London [7] A G Depster and M D Macleod, Coparison of fixed point FIR filter design techniques, IEEE Trans C&S II, vol. 44, no. 7, pp. 591-593, July 1997 [8] A G Depster and M D Macleod, Multiplier blocks and the coplexity of IIR structures, Electronics Letters, vol 3, no 22, pp1841-1842, Oct 1994 [9] A G Depster and M D Macleod, Coparison of IIR filter structure coplexities using ultiplier blocks, Proc ISCAS95, vol 2, pp858-861, IEEE, Apr-May 1995 [1] A G Depster, The Cost of Liit-Cycle Eliination in IIR Digital Filters Using Multiplier Blocks ISCAS97, vol 4, pp 224-227, June 1997 [11] A G Depster and M D Macleod, IIR Digital Filter Design Using Miniu-adder Multiplier
Blocks, IEEE Trans C&S II, vol. 45, no. 6, pp. 761-763, June 1998. [12] A G Depster and N P Murphy, Efficient Interpolators and Filter Banks using Multiplier Blocks, accepted for publication in IEEE Trans Signal Processing [13] C W Farrow, A Continuously Variable Digital Delay Eleent, ISCAS88, pp2641-2645 [14] T I Laakso et al, Splitting the Unit Delay, IEEE Signal Processing Magazine, vol 13 no 1, pp3-6, Jan 1996 [15] V Valiaki, A New Filter Ipleentation Strategy for Lagrange Interpolation, Proc International Syposiu on Circuits and Systes (ISCAS95), pp361-364 [16] Ewa Heranowicz, Weighted Lagrangian Interpolating FIR Filter, Signal Processing VIII, Proceedings of EUSIPCO-96, pp123-126, 1996 [17] A G Depster and N P Murphy, Lagrange Interpolators and Binoial Windows, Signal Processing, vol. 76, no. 1, pp. 81-91, July, 1999