Multistage, Multirate FIR Filter Structures for Narrow Transition-Band Filters

Transcription

1 Multistage, Multirate FIR Filter Structures for Narrow Transition-Band Filters Tor A. Ramstad' Scool of Electrical Engineering Georgia Institute of Tecnology Atlanta, GA Tapio Saramaki Signal Processing Laboratory Tampere University of Tecnology P. 0. Box 527, SF Tampere, Finland Abtract - Tis paper presents an FIR fllter system capable of implementing virtually any practical lowpass and igpass filter wit as little as multiplications per sample. Te metod b based on multirate tecniques and complementary Alters. We first give a brief presentation of te principle and te necessary building blocks. Ten, practical design metods for optimitation are presented togeter wit design examples and plots of te multiplication rates. Finally, aliasing and flnite wordlengt effecb are discussed. Te resulting system, wile aving a remarkably low aritmetic complexity, will usually require somewat more memory tan a conventional optimum direct-form filter and a sligt increaee in te internal signal representations. I. INTRODUCTION Narrow transition-band FIR filters often require forbidding filter lengts for practical implementations. Even toug IIR filters can offer significantly lower order, tey suffer from sortcomings like nonlinear pase, instabilities and very large wordlengt requirementa for te same filter specifications, and it is terefore desirable to find dternative FIR structures tat lower te processing load. Several metods ave been deviced to acieve tis goal, most notable among tese are te IFIR [I] and te multirate tecniques [2] wic can bot be applied only to narrow- or broadband filters, and te structure suggested by Jing and Fam [3] and Lim [4] wic can cope wit any bandwidt. We ave earlier presented a metod wic combines te Jing-Fam metod wit multirate tecniques in suc a way tat te overall processing will be less tan for IIR filters wen strict transtion-band requirements apply, and we ave sown tat te multiplication rate is limited to about per sample for any reasonable filter specification [5]. In tis paper we again present te basic principle of te metod and go into finer details about te implementation strategies and optimizations. We also discuss te aliasing noise problem resulting from te multirate operations and present models and results for finite word-lengt effects. 11. BASIC STRUCTURES To simplify te processing in narrow transtion-band low-pass or ig-paw filters we use te following strategy: Assume tat te filter H (z) sown in Fig. 1(a) satisfies te given specifications. We now try to replace tis filter by a cascade of a two-port and a filter Hl(r) M given in Fig. l(b). It is not necessary for te new system exactly to mimic te original filter, but it must satisfy te same filter requirements. Te question is now: can we find two-ports combined wit te filter Hl(z) wic will reduce te multiplication and addition rates wile limiting te necessary filter memory? Before we explain te basic building blocks, we will adopt te terminology narrowband and broadband for filters satisfying fb < f,/4 and fb > f,/4, respectively, were fb is te filter bandwidt and f. is te sampling frequency. Two fundamental building blocks will be sufficient to acieve our goal. One block transforms te problem from a narrowband to a broadband filter, wile te oter one does te opposite. If we manage to reduce te overall processing in going from te original 'On leave from Department of Electrical and Computer Engineering, Te Norwegian InntitUte of Tecnology, N.7034 Trondeim-NTH, Norway Fig. 1. forms Implementation of an FIR filter using two-port trans- Fig. 2. Two basic building blocks for syntesizing a computational& efficient FIR filter. I - (8) (b) filter to te system in Fig. l(b), we can repeat te process on Hl(z) leaving te system in Fig. l(c) wit terminating filter Hz(z). Again a broadband filter Hl(z) is turned into a narrowband Hz(z) or vice versa. Te process can be repeated until te processing left in te terminating filter Hi(.) is negligible or te total processing in te system is minimized. Wat simple two-ports can implement te low-pass to ig-pass and ig-pass to low-pass operations wile restricting te system to remain linear pase? A narrowband-to-broadband two-port wic uses standard multirate tecniques is depicted in Fig. 2(a). Te decimation ratio r = M/N must be selected suc tat it leaves te terminating filter as a broadband filter. In Fig. 3 we first sow te frequency response of H(z) and a ratio 2 decimation filter (i. e. N = 1 and M = 2) [Fig. 3(a)] and ten te filter requirements for Hi(%) [Fig. 3(b)]. Comparing H(z) and W ~(Z) in te example above, we realize tat te sampling rate in Hl(z) is lowered by T and te relative transition bandwidt is increased by te same factor. Assuming tat te filter lengt is inversely proportional to te relative transition bandwidt, we conclude tat te processing in Hl(z) is lowered by a factor (M/N)' as compared to H(z). Te brodband-to-narrowband transformation can be implemented using a complementation tecnique as sown in Fig. 2(b) wen te internal delay is exactly equal to te delay of te filter Hl(z). Notice tat tis operation also performs a lowpass-to-igpass trans-

2 I f '42 f Fig. 5. Commutative structures for FIR alf-band filters. (a) Decimator. (b) Interpolator. Note tat in te interpolation cme te filter output must be multiplied by two to preserve te signal energy. Fig. S. Design of a narrow-band filter H(z) using te building block offig. 2(a) witb N = 1 and M = 2 and a broadband filter Hi(z). O"o O.M ai s Fig. 4. Frequency response of te filter Hl(z) and te complementary response obtained using te system in Fig. J(b) wit te structure in Fig. 2(b) as a two-port. form, or vice versa. See Fig. 4. Te given block requires only one addition per sample and leaves te sampling frequency uncanged. Wit tese building blocks our main concern in system optimization is to select good substructures for te decimation/interpolation transformations to avoid tat te processing in tese structures surpasses te processing in te original filter. After studying various possible substructures, we ave come to te conclusion tat we only need two different types of decimation/interpolation two-ports, one using decimation by 2 and te oter decimation by 213, bot constructed using alf-band filters as discussed in te next section HALF-BAND FILTERS For constructing te decimation filter HD(z) and te interpol& tion filter H,(z), alf-band filters [6] are particularly efficient. Te transfer function of tese filters can be written as 1 *a, 1-6, 6, -8, If~(z) = I2 z-~ +G(zz), (1) were K is an odd integer and te order of G(z2) is K in z2 or 2K in z. Wen it is nsed for decimation or interpolation, tis filter can be implemented using te commutative structures [7] sown in Fig. 5. Te delay branc z-(~-~)/' can be sared wit G(z). In te decimation case, tis can be done using te transposed directform realization exploiting te coefficient symmetry of G(z). In te interpolation case, te directfrom realization is used. Wen te symmetry in te coefficients of G(z) is exploited, we need only (' + 1)/2 multipliers plus a trivial multiplication by 0.5 in te decimation case. _Since G(z) is working in bot cases at te lower sampling rate of fb = f8/2, te implementation of bot HD(z) and S,(z) requires only fs(k + 1)/2 multiplications per second. Te alf-band filters are caracterized by te facts tat teir passband 2018 and stopband ripples are equal and te passband and stopband edges are related via fat = fs/2 - fp. IV. DESIGN EQUATIONS In tis section, we give te basic design equations for syntesizing te proposed filters. All wat is needed is to determine te conditions under wic a filter H (z) can be syntesized in terms of te building bloyks of Section I1 and a termination filter wic we ere denote by H(z). After knowing tese conditions, we are able te repeat te overall syntesis procedure. First, we concentrate on te required passband and stopband edges of te decimator and interpolator filters and tose of te new termination. Let H(z) be a lowpass filter wit passband and stopband edges of fc * A and sampling rate of fa. Tere are te following tree cases wic require different constructions for H(z): Case A: fc < fa/4 and fc + A is not close to fa/4. Case B: fc > fs/4 and fc - A is not close to fs/4. Case C: fa/4 is close to or inside te interval [fc - A, fc + A]. In Case A, H(z) is a narrow-band lowpass filter and we use te building block of Fig. 2(a) wit N = 1-and M = 2. Wen tis block is cascaded wit te termination H(z), te relation between te z-transforms of te input signal ~(n) and te output signal y(n) becomes Here, Fl(z) is a conventional transfer function from te input to te output. Tis transfer function must satisfy te conditions stated for H(z). F2(z)X(-z) is te aliased term due to te sampling rate alteration and te response of F~(z) must be small in te frequency range [0, fs/2]. Te desired result is obtened by selecting te edges of H(z) worliing at te sampling_rate of f3 = fa/2 to be tose of te overall filter, i.e., fc zk A wit fc = fc. Because of te periodicity of If(.'), it as an extra passband [fa/? - ( fc - A), fa/2] and an extra transition band [fa/2- ( fc +A), f8/2 -.( fc - A)] (see Fig. 6). Te required passband and stopband edges for HD(z) and H,(r) are fid) = fc - A, fat = fa/2 - (fc +A). (3) Te resulting HD(z) and Hi(z) preserve te first passband region of G(z2) and-attenuate te extra transition band and te extra passband of H(z2), giving te desired response for F~(z). In te case of F2(2), Hl(z) attenuates te second transition band and te second passband of H(z2) and HD(--z) takes care of te lower ones, resulting in a small aliased term Fz(z)X(-z). It as been observed experimentally tat te required stopband edge of HD(z) and Hl(z) can be selected to be

3 f C fs/2-fc fs/2 f fc fj2-fc fj2 f Fig. 6. Transfer functions for te aiiased and undiased components in Case A. (a) Terms in Fl(z). (b) Terms in Fz(z). Tis is because te termination filter i(z2) provides some attenuation in te transition bands near te stopband edges. In Cue E, H(z) is a wideband design and we first use te building block of Fig. 2(b) to convert te problem to te design of narrowband igpass filter. Te paasband edge is fc + A and te stopband edge is f c - A. Tis filter can be constructed aa in Case A using te building block of Fig. 2(a) wit N = 1 and M = 2. Te desired result is obtained by selecting te edges of a lowpass H(z) to be fc * A, were = j6/2 - fc. (5) Te required HD(z) and HI(z) are igpass filters wit edges ffd' = fc + A, fi:) = f8/2 - j c + Af3. (6) See [5] for detaila. Using alf-band filters wit properties aa discussed above, in Case A, te passband edge as to be selected as SiD) = jc + A/3 (7) to give te desired stopband edge. In Case B, te alf-band filter is a igpaas design wit te stopband edge f,':' as given in Eq. (6). In Caae C, fi/4 is eiter in te transition band of H(z) or te passband or te stopband edge of H(r) is close to f8/4. In te former case, we cannot use N = 1 and M = 2 at all. In te latter c w, te transition bands of HD(z) and H,(z) become narrow, resulting in ig filter orders. To avoid tis problem, one alternative is to use te building block of Fig. 2(a) wit N = 3 and M = 2. In tis case, te input-output relation is werefl(z) and Fz(z) aregiven by Eqs. (2b) and (2c), respectively. In tbcase, te relation between te input of HD(z) and te output of H,(z) is te same as in Case A. Te basic difference is tat te input sampling rate of HD(z) is now f: = 3f, and te input of HD(z) contains one and a alf periods of te original input signal spectrum [see Fig. 7(a)]. Te specifications for te overall filter consisting of te termination filter, te decimator, and te interpolator are te same except tat te sampling rate is now f: = 3f. [see Fig. 7(b)]. Te second basic difference compared to Case A is tat tere are also aliased terms wen finally decimating b. tree, as sown by Eq. (ab). However, because of filtering, te components aliased from te region [f8/2, 3 f8/2] are very small [see Fig. 7(c)]. A Te sampling frequency of te center filter H(z) is in tis case = 3/2fs, i.e., 3/2 times tat of te overall filter H(z). Te &\antage of using r = 2/3 lies, owever, in te following facts. First. te edges for te alf-band designs of HD(z) and H,(z) are ji,"' = f c + A/3 and f,:' = jl/2 - (fc + A/3). Terefore, te relative transition bandwidt is very wide, resulting in very low filter ordys. Second, te passband and stopband edges of g(z) are close to fi/3 80 tat te design procedure can be easily repeated using te building block of Fig.?(a) wit ' = 1 and M = \ I a * fsn f'sn f Fig. 7. Filtering in Case C. (a) Periodic input signal spectrum of HD(z). (b) Transfer function from teinput of HD(z) to te output ofh,(z). (c) Output signalspectrum ofh,(z) beforesampling rate reduction by 3. It can be sown tat by ignoring te aliased terms, te passband and stopband ripples of te overall design are at most were dk) is te ripple of te kt decimator and interpolator and S is te set of stages in front of wic terz isan even number of building blocks of Fig. 2(b) or no blocks. 6 (6) is te passband (stopband) ripple of te termination filter if tere is an even number of delay blocks in front of it. Oterwise, it is te stopband (passband) ripple. Te simplest way to determine te required ripples is to make all te terms in te summations equal. V. DESIGN EXAMPLES As an example we consider te design of a lowpass filter wit passband and stopband edges of 0.2s. and 0.201f6 [( f0.0005)f8] and passband and stopband ripples (60-dB attenuation). Te minimum order of a conventional direct-form design to meet tese criteria is 3256, requiring 1629 multiplications per input sample. Using te syntesis procedure described in te previous sections, te first building block is selected to be te one in Fig. 2(a) wit N = 1 and M = 2. Te edges of te termination filter are, in terms of its sampling rate f,(') = f6/2, (0.401 ~O.OOI)f,('). Te passband edge for bot HD(z) and H,(z) is ( /3)f8. Te termination filter is now wideband. Terefore, we use te building blocks of Figs. 2(b) and Z(a). Te edges of te termination are (0.198 f 0.002)f.(2) were = f8/4 is its sampling rate. HD(z) and H,(Z) are igpass alf-band filters wit stopband edge of ( O.O01/3)f,('). Proceeding in te same manner, te sampling rates for te tird, fourt, fift and sixt terminations become f;") = f,/2k for k = 3,4,5,6. For te tird and fift stages, we use te building block of Fig. 2(a) wit N = 1 and M = 2 and te passband edges of te subfilters are ( O.O02/3)f$') and ( O.O08/3)f$'), respectively. For te fourt and sixt stages, we use te building blocks of Figs. 2(b) and 2(a) wit N = 1 and M = 2. Te subfilters are igpass alf-band filters wit stopband edges of ( /3)f,(3) and ( O.O16/3)f$'), respectively. If four or six stages are used, te edges of te remaining terminations are (0.208t0.008)f,(4) and (0.168 * 0.032)fi6), respectively. In te case of four stages, te ripples for all te stages become wen determined from Eq. (9) in suc a way tat all te

4 terms in te summations become equal. In te case of six stages, te ripples are 0.001/7. Filter Complezity Wit four stages te number of multipliers (te filter orders) for te decimation and interpolation stages are 11 (42), 4 (14), 11 (42), and 4 (14). Te implementation of all te decimators and interpolators requires multiplications per input sample. Te minimum order of a direct-form FIR filter to meet te criteria of te termination is 264. Tis filter requires 133 multipliers. Since it is working at te sampling rate of fn/16, it requires multiplications per input sample. Te overall multiplication rate is tus For te first building block of Fig. 2(b), te delay term is z-1210 and, for te second block, z-"*. Te number of delay elements required in implementing te overall filter is 1864, wic is lower tan tat for te direct-form design (3256). Te delay for te unaliased component is 2462, wereas te delay of te direct-form equivalent is Te multiplication rate can be reduced by designing te termination as an IFIR filter in te form F(z2)C(r) [l]. Te required orders of F(z) and G(z) are 136 and 12, respectively. In tis case, te overall multiplication rate is 21. Wen six stages are used, ten te number of multipliers (orders) for te decimators and interpolators become 12 (46), 4 (14), 11 (42), 4 (14), 15 (58), and 4 (14). Te order of te termination is 70. Te overall multiplication rate is in tis case Te price paid for te reduction in te multiplication rate is te increased overall delay (3970) and te increased number of delay elements (2700). Aliasing Noise Wen four stages are used, ten te overall output can be written in te form % I i -100 p -120 I I I I I I I I I 0 LIN. AMP r j R 0.4~ 0.6~ 0.8~ X FREQUENCY -40!-60 L -80 a : a R ~ 0.8~ R FREQUENCY (b) k=o so tat tere is an unaliased component and 15 aliased components. Figure 8(a) gives, in te case were te termination is syntesized in te form F(z2)C(z), lho(d")l in db (amplitude response for te unaliased component). Figure 8(b), in turn, gives lh6(e'")/ (solid line), lh8(dw)i (dased line), and IH4(eJw)I (dot-dased line). Te minimum attenuations for tese responses are 74 db, 74dB, and 83 db, respectively. 'For te oter responses, te minimum attenuation is more tan 86 db. Anoter way of assessing te noise contribution due to aliasing is by simulation wit a stocastic input. Wen te same stocastic signal is input to a multirate filter and to te corresponding timeinvariant filter (filter giving only te unaliased output), te aliasing noise can be found aa te difference between te two output signals. Tis simulation wit a wite input signal, yields for te 4-stage system using an IFIR terminating filter a signal-to-aliasing-noise ratio of 73.1 db referenced to te 0-dB output level of te signal passband. Te non-wite noise spectrum is sown in Fig. 9. Wen comparing tis figure wit Fig. 8(b), it is seen tat te spectrum of Fig. 9 ie basically due to te components sown in Fig. 8(b). Multtpltcatton Rate a# a Function of Cut08 Frequency In order to examine ow te multiplication rate varies as a function of te center of te transition band, Fig. 10 is provided. It plots a case were te widt of te transition band is O.OOlfs and te passband and stopband ripples are, as in te previous example, In contructing tis plot, te upper allowable number of stages as been fixed to be 8 and te upper limit for te overall delay as been two and a alf times tat of te direct-form equivalent (4070). As seen from te figure, te maximum number of multiplications per input sample is in tis case 40. VI. FINITE WORDLENGTH EFFECTS Two effects of finite register lengt8 ave to be considered in FIR systems: te effect of coefficient rounding on te frequency reponse of te system and noise generation due to internal signal rounding. Fig. 8. Responses for te proposed multirate filter wit four decimation and interpolation stages. (a) Response for te unaliased component. (b) Responses for some aliased components. -60,,, I I I I I I -65,, RELATIVE FREQUENCY Fig. 8. DFT spectrum of te alias output signal component from te 4-stage system wit IFlR terminal filter based on 5000 samples. Input signal: wite noise, 0 db. Coeficient Sensitivity Wen te coefficients in our system are rounded, it affects eac individual subfilter. If te subfilter responses are preserved witin our specification, so will te overall response. In order to compare our filters wit direct-form filters, we estimate te necessary number of bits in te two systems using te statistical approac given in [8]. From equations (40) and (41) in tat paper we find tat te frequency response error is wit ig probability less tan U e- 2 -(y@tij7t, (10) were t is te total number of bits and N is te filter order. We now solve for te numer of bits and apply te equation to bot filters 2020

5 , I,,....,,L I..,....,... i j....,....j,.,...,!!.I..,...,...; p , -95 f ' " ' 1 ' 1 ' 1 ' TRANSITION BAND CENTER Fig. 10. Multiplication rate versus te center of te transition band for filters m't transition band widt of0.001f8 and passband and stopband ripples of under investigation, inserting indeces d and m for te multirate and direct-form filters, respectively. Assuming tat te relative allowable error in te stopband due to coefficient quatization is te same for bot filters (ne = a6,a < 1) we obtain, after some manipulation, for te difference in te number of coefficient bits, fd - 1, = 0.51Ogz(Nd/Nm) - log2(6d/6m), (11) were 6d and 6, are te stopband ripples. To simplify te equation we ave ere assumed tat N is large for bot filters. Applying Eq. (11) to compare te direct-form filter wit 6d = order of 3256 wit te terminating filter of te 6-stage design aving 6, = 0.001/7 and order 70, one finds tat te two filters require approximately te same number of bits. One may expect tis result to be quite general, because on one and, te multirate system as stricter inband tolerances, wile on te oter and, te filter order of te direct-form filter is iger, eac pulling in different directions. Quantization Noiae Space limitations do not allow a detailed noise analysis. We will terefore rater present a qualitative discusussion and give a SNR for our example compared wit a direct-form filter meeting te same specifications. To evaluate te effects of representing te internal signals by a finite number of bits, we ave employed te following model: We assume tat all adders ave a sufficient number of bits to avoid any significant contribution bot to quantization noise and saturation effects. At te output of all filters and after te additions of te delay Line outputs and te interpolating filter outputs, tere are scang multipliere followed by quantizers. Te scaling multipliers amplify te sign& aa muc aa possible witout bringing it to saturation. Te quantizers round te signal to te required number of bits. It ie not ard to realize tat te noise at te filter oucput will be non-wite due to te number of noise sources influencing te various frequency regions. Te transfer function from te terminating filter and tose close to it will basically ave a bandpass caracter wit te passband at te edge of te overall filter system passband. Terefore, we will expect tat te noise wiil grow towarb te passband edge for lowpass filters. In our simulation experiment we ave used a wite Gaussian zerwmean unit-variance signal source. All te signal levels ave been acded to unit variance were applicable. Te exception is at te external summation points were te signals to be added from te different brances must be scaled by exactly te same total amount. Assuming tat te signals at all quantization points are Gaussian, we fix te saturation levels at six times te signal vari an ce. " RELATIVE PREQuENcr Fig. 11. DFT spectrum of simulated quantization noise in te six-stage structure wit 16 internal bits based on 5000 samples. Input signal: wite noise, 0 db. In our example using te above described sceme and a 16-bit signal representation, one obtains a 78.5 db signal-tsquantization noise ratio. In Fig. 11 te noise spectrum is sown. As expected, te noise level increases towards te passband edge. To reduce tis effect, one migt resort to using more bits in te filters close to te center of te structure as compared to te oter ones. To appreciate tis result, we ave compared it to te noise in a direct-form design using te same input and output scaling. Te resulting signal-to-quantization-noise is 80.4 db wen all additions are performed prior to output scaling and quantization. In oter words, to obtain te same SNR, we need at most one extra bit in our system for internal signal representation. VII. CONCLUSION In tis paper we ave demonstrated tat FIR filters can be implemented wit low computational complexity in a structure based on multirate filters. A comparison wit IIR filters will reveal tat for filters wit tigt filter specifications tese FIR structures will require less computations, but te delay and te memory will, of course, be muc larger. We feel tat tis is a strong demonstration of te efficiency possible in time-varying systems as compared to sift-invariant systems. REFERENCES [I] T. Saramiiki, Y. Neuvo, and S. K. Mitra: "Design of computationally efficient interpolated FIR filters," IEEE Trans. Circuits Syaf., vol. CAS-35, pp , Jan [2] L. R. Flabiner and R. E. Crociere, "A novel implementation for narrow-band FIR digital filters," IEEE Trana. Acoust., Speec, Signal Proceaaing, vol. ASSP-23, pp , Oct [3] 2. Jing and A. T. Fam, "A new structure for narrow transition band, lopaas digital filter design," IEEE Trana. Acoust., Speec, Signal Processing, vol. ASSP-32, pp , April [4] Y. C. Lim, "Frequency-response masking approac for te syntesis of sarp linear pase digital filters,", IEEE Trana. Circuit8 Syst., vol. CAS-33, pp , Apr [5] T. A. Ramstad and T. Saramiiki, "Efficient multirate realization for narrow transition-band FIR filters," in Proc IEEE Int. Symp. Circuits Syal. (Espoo, Finland), pp. ao , June [6] M. G. Bellanger, J. L. Daguet, and G. P. Lepagnol, "Interpolation, extrapolation, and reduction of computation speed in digital filters," IEEE Trana. Acoud., Speec, Signal Proceaaing, vol. ASSP-22, pp , Aug [7] R. E. Crociere and L. R. Rabiner, Multirate Digital Signal Processing. Englewood CWs, NJ: Prentice-Hall, (81 D. S. K. Can, L. R. Rabiner: "Analysis of quantization errors in te direct form for finite impulse response digital filters", IEEE Trana. Audio Electroacoud., vol. AU-21, pp , August