DESIGN OF LINEAR-PHASE LATTICE WAVE DIGITAL FILTERS

Transcription

1 DESIGN OF LINEAR-PHASE LAICE WAVE DIGIAL FILERS HŒkan Johansson and Lars Wanhammar Department of Electrical Engineering, Linkšping University S Linkšping, Sweden ABSRAC Lattice wave digital filters can be realized to have approximately linear phase in the passband by letting one of their allpass branches be a pure delay. In this paper, an algorithm for designing these filters is described. Several design examples using this algorithm are also given.. INRODUCION A major advantage of wave digital filters (WDFs) over most other recursive filters is that they can maintain stability under finite-arithmetic conditions. A particularly favourable type of wave digital filter is the lattice wave digital filter. Using lattice WDFs, highly modular and parallel filter algorithms can be obtained. his is advantageous from an implementation point of view. o each wave digital filter there is a corresponding filter in a reference domain. he design of wave digital filters can therefore be carried out in the analog domain using classical filter approximations, whereupon a transformation from the analog domain into the digital domain, applying certain transformation rules, can be performed. For lattice wave digital filters, it is possible to use explicit formulas to directly compute the adaptor coefficients, as given in [Gazs-85]. However, for filters satisfying both magnitude and phase requirements, there exist no closed form solutions. For these cases numerical optimization techniques must be adopted [Kuno- 88, Abo-Z-95, Leeb-9]. In this paper we describe an algorithm for design of linear-phase lattice WDFs. he algorithm was introduced by Renfors and SaramŠki [Renf-86] for design of filters composed of two allpass filters in parallel, of which the lattice WDF is a special case. Approximately linear phase is obtained by

2 letting one of the allpass branches be a pure delay. We show how the adaptor coefficients can be computed from the result of the algorithm, for some different WDF realizations of the allpass branches. Several design examples using the described algorithm are given. LAICE WAVE DIGIAL FILERS A lattice wave digital filter is a two-branch structure where each branch realizes an allpass filter [Fett-74]. hese allpass filters can be realized in several ways [Fett-86]. One approach that yields parallel and modular filter algorithms is to use cascaded first- and second-order sections, as shown in Fig... he first- and second-order sections are here realized using symmetric two-port adaptors. hese can easily be replaced by twoport series- or parallel adaptors using certain equivalence transformations [Fett-86, Gazs-85]. he second-order sections can also be realized using three-port series- or parallel adaptors [Ande-95]. Another approach is to realize the allpass filters using Richards' structures [Fett-86]. a 4 a 8 a 0 a 3 a 7 x(n) / y(n) a a 5 a 9 a a 6 a 0 Figure.. An th-order lattice wave digital filter.

3 3 he transfer function of a lattice WDF can be written as H(z) = ( H 0 (z) + H (z)) (.) where H 0 (z) and H (z) are allpass filters he overall frequency response can therefore be written as H(e jw ) = ( e jf 0 (w) + e jf (w) ) (.) where F 0 (w) and F (w) are the phase responses of H 0 (z) and H (z), respectively. he magnitude of the overall filter is thus limited by H(e jw ) he transfer function of a lattice WDF and its complementary transfer function are power complementary, i.e., H(e jw ) + H c (e jw ) = (.3) where Hc( z) = ( H0() z - H () z ) (.4) his means that, if H(z), for example, is a lowpass filter, then a highpass filter is obtained by simply changing the sign of the allpass filter H (z). An attenuation zero corresponds to an angle w 0 at which the magnitude function reaches its maximum value. For lattice WDFs this occur when He j w ( 0 ) = A transmission zero corresponds to an angle w at which the magnitude function is zero, i.e. when He j w ( ) = 0 At an attenuation zero, the phase responses of the branches must have the same value, i.e.

4 4 F 0 (w 0 ) = F (w 0 ) (.5) Hence, in the passband of the filter the phase responses must be approximately equal. At a transmission zero the difference in phase between the two branches must be F 0 (w ) - F (w ) =±p (.6) hus, the difference in phase between the two branches must approximate ±p in the stopband of the filter. o make sure that only one passband and one stopband occur, the orders of H 0 (z) and H (z) must differ by one... LINEAR-PHASE LAICE WDFS It is possible to obtain a lattice WDF (and more generally, a filter composed of two allpass filters in parallel) with approximately linear phase by letting one of the branches consist of pure delays [Kuno-88, Renf-86]. A linearphase lattice WDF is shown in Fig... he transfer function of a linearphase lattice WDF is H(z) = ( H 0 (z) + z -M ) (.7) he transfer function H 0 (z) corresponds to a general allpass function and can consequently be written as H 0 N å bz i i ()= z i= 0 N bzn i -i å i= 0 (.8) For lowpass and highpass filters N and M must be selected such that N= M ±. he selection N = M + gives the best result in most cases. his will be illustrated in section 4. he overall frequency response can be expressed as H( e j w ) = e j ( w) + e j Mw ( F 0 - ) (.9) In the passband, the phase response of branch zero, F 0 (w), must approximate the phase response of the other branch, which in this case is linear. his forces the overall phase response to be approximately linear in

5 5 the passband. here exist no closed form solutions for design of linearphase lattice WDFs. herefore, numerical optimization algorithms have to be used. a 3 a a x(n) / y(n) 5 Figure.. An th-order linear-phase lattice WDF 3. ALGORIHM In this section we describe in detail an algorithm for design of linear-phase lattice WDFs. It was introduced by Renfors and SaramŠki [Renf-86], for design of approximately linear-phase filters composed of two allpass filters in parallel, of which the linear-phase lattice WDF is a special case. Here, we consider design of lowpass filters. Let w c and w s denote the passband and stopband edges respectively, and let the magnitude specification be - d c (w) H(e jw ), w Î[ 0, w c ] (3.a) He ( jw ) ds( w), w ws, p (3.b) Î[ ] he overall frequency response can be written as H( ejw ) = ej e jm ( F0( w ) + - w ) = = ej( F0( w) - Mw) / ej( ( ) M ) / e j( ( ) M ) ( F0 w + w + - F0 w + w / ) M e j( ( w ( ) ) Mw) / æ w + w ö = F0 - F0 cos è ø (3.) he magnitude function is consequently given by

6 6 H(e jw æ ) = cos F 0 (w) + Mw ö è ø (3.3) he design of the overall filter can therefore be performed with the aid of the phase response F 0 (w). o obtain a lowpass filter the difference between F 0 (w) and ÐMw must approximate zero in the passband and Ðp (N = M + ) in the stopband. Using Eq.(3.3) we see that the specification of Eq.(3.) is met if where L(w) F 0 (w) U (w), w Î[ 0, w c ]È [ w s, p ] (3.4a) ì ï L(w) = í îï ì ï U (w) = í îï - Mw - cos - (- d c (w)), w Î[ 0, w c ] - Mw - p - sin - (d s (w)), w Î[ w s, p ] - Mw + cos - (- d c (w)), w Î[ 0, w c ] - Mw - p + sin - (d s (w)), w Î[ w s, p ] (3.4b) (3.4c) Next, an error function that is to be minimized is defined as where E(w) = W (w)[ F 0 (w) - D(w) ] (3.5a) ì cos - (- d c (w)), w Î 0, w c ï W (w) = í ï sin - (d s (w)), w Î [ w s, p ] î ï [ ] (3.5b) and ì ï D(w) = í îï - Mw, w Î[ 0, w c ] - Mw - p, w Î[ w s, p ] (3.5c) he algorithm works in a way that is similar to Remez multiple exchange algorithm for polynomial approximation problems. his algorithm is often used in, e.g., design of linear-phase FIR filters [Oppe-89]. In the first step of the algorithm, N + extremal points are selected on the union of the passband and stopband regions. Next, the coefficients b i of

7 7 H 0 (z), as given by Eq.(.8), and d are computed such that the error function E(w) alternatingly equals ±d at these extremal points. hen, N + new extremal points are determined by finding the N + angles at which E(w) has its local extrema. he process is then repeated until the new extremal points are the real extremal points of E(w). Finally the adaptor coefficients are computed. he algorithm is thus performed in the following five steps: ) Select N + initial extremal points W = [ w w wn + ] w Î[ 0 wc]è [ ws p],,,,, (3.6) ) Solve the system of N + equations E( wi) = W( wi) [ F 0 ( wi) - D( wi) ]= (- ) id, i =, L, N + for d and the filter coefficients b i of H 0 (z). (3.7) 3) Find the N + local extrema of E(w) on [ 0, w c ]È [ w s, p ] with the condition that the maxima and minima alternate. Let the angles at which these extrema occur be W = w, w, w [ N + ] 4) If w - w e, for i =, L, N + then go to step 5. Otherwise, set W i i = W ' and go to step. 5) Compute the adaptor coefficients. he specification of Eq.(3.) is satisfied if d. he final problem is consequently to find the minimum N such that d. In the first step of the algorithm N + initial extremal points must be selected. he locations of these extremal points are not as crucial for the convergence of the algorithm as is the distribution of these to the passband and stopband. A good starting point is to select the number of points in the passband as w c (N + )/(w c + p Ð w s ) rounded to the nearest integer [Renf-86]. he extremal points in the passband and stopband can then be distributed equidistantly in each respective band. In step two, Eq.(3.7) is to be solved. For each value of d the coefficients b i can be computed as functions of d. his can be done conveniently in the Y- domain using simple recurrence formulas [Henk-8, Renf-87, Joha-96a]. We first rewrite Eq.(3.7) as

8 8 i ( ) d F 0 ( wi) = - + D( wi), i =, L, N + (3.8) W( w ) i he problem now is to determine the coefficients of the allpass filter H 0 (z) of order N such that Eq.(3.8) is satisfied, i.e. design H 0 (z) such that ( ) = = arg H ( ejw 0 i F 0( wi), i, L, N (3.9) he corresponding allpass transfer function in the Y-domain can be expressed as H0( Y ) = PN ( Y ) P (-Y ) N (3.0) he numerator and denominator polynomials of this function contribute with the same amount to the phase response. It is therefore sufficient to find a numerator polynomial P N (Y) having half the desired phase response values at the extremal points. hus, using Richards' variable as given by Y = z - z + (3.) the equations that must be satisfied are F ( ) arg P ( j 0 wi ( N Wi) ) =, i =, L, N (3.a) where W i æ w = i ö tan, i =, L, N (3.b) è ø he polynomial P N (Y) is generated using recurrence formulas. First, the parameters b, b, L, b N are computed using the following recursive continued fraction formula:

9 9 b b b ( W ) = - æf0( w) ö tan è ø = W b W æf0( w) ö tan è ø M b i- Wi - Wi- bi Wi - Wi - M - W i( Wi - Wi-) = - ( ) ( ) - -3 ( ) b Wi - W - bwi - æf ( tan 0 wi) ö è ø, < i N (3.3) he polynomial P N (Y) can then be derived using the recurrence formula P ( Y ) = 0 P ( Y) = + b Y ( ) Pi ( Y) = P i-( Y) + b i Y + Wi- Pi -( Y), i N (3.4) he coefficients b i in Eq.(.8) can be computed from P N (Y) via the substitution of Eq.(3.). he remaining equation of Eq.(3.8), after rewriting it, can now be expressed as f( d, b ( d), b ( d), L, b ( d)) = f( d) = (3.5) N 0 Equation (3.5) can be solved using standard methods for solving f(x) = COMPUAION OF ADAPOR COEFFICIENS he allpass function H 0 (z) can be realized in several ways. For the approach of cascaded first- and second-order sections using symmetric twoport adaptors, according to Fig. 3., the transfer functions are, for the firstorder section -a 0 z + z - a 0 (3.6) and for the second-order section

10 0 -a z - a (- a )z + z - a (- a )z - a (3.7) he adaptor coefficients can be derived either from the transfer function H 0 (z) as given by Eq.(.8), or from the numerator polynomial P N (Y). Here, we use the latter approach. he numerator polynomial is then first factored into first- and second-order factors of the form -Y + a 0i and Y - a j Y + a 0 j where the indices indicate the first-order factor i and second-order factor j, respectively. he coefficients for the first- and second-order sections can now be computed as a 0i = - a 0i + a 0i (3.8) and a -a - j 0j a j = aj a0j a a j = - + a j 0 j (3.9a) (3.9b) respectively. a a a x(n) y(n) x(n) y(n) a) b) Figure 3.. Realizations of first- and second-order allpass sections using symmetric twoport adaptors. a) First-order section. b) Second-order section.

11 he second-order sections can also be realized with three-port series (or parallel) adaptors as shown in Fig. 3.. In this case the transfer function is ( g+ g - ) z + ( g - g) z+ z + ( g - g ) z+ g + g - (3.0) he coefficients of second-order section j can be computed as + + g j = aj a0j a 0 j g j = aj a0j + + (3.a) (3.b) he coefficients can also be computed using the following relations: gj = ( a a - j)( + j) (3.a) gj = ( a a - j)( - j) (3.b) An alternative to using cascaded first- and second-order sections is the Richards' structure shown in Fig he adaptor coefficients of this structure can be computed from the port resistances of the two-port adaptors, which can be derived iteratively from P N (Y) by using Richards theorem [Erik-78], or by using Schur parametrization [Neir-79]. Here, we use the algorithm in Box which is based on the Schur parametrization [Joha-96b]. he algorithm derives the coefficients iteratively given the coefficients of the polynomial P N (Y), where P N ( Y) N = + a i å iy i= (3.3) -g -g - - x(n) y(n) Figure 3.. A second-order allpass section using a three-port series adaptor.

12 x(n) y(n) a N a a Figure 3.3. An Nth-order Richards' allpass section. c, c, = -a 0 = + a for i = to N cii, = ai+ ci-, i- ci, = (- ) ia i + ci-, 0 0 for j = i -downto i! cij, = i ja i ci, j ci, j j!( i j)! ( - - ) end end cn, j bn, j =, j =,, L, N N + m å (-) a m m= for i = N downto if i > for j = i -downto ëi / û+ bi, j- bii, bii, - j bi-, j = - b ii, bii, - j- bii, bi, j bi-, i- j = - b ii, end end if i even end end a bi-, i/ bii, / = + bii, j =- bj, j, j =,, L, N Box. Algorithm for adaptor coefficients of Richards' structures.

13 3 4. EXAMPLES In this section we design some linear-phase lattice WDFs using the algorithm described in section 3. he allpass filter H 0 (z) is realized using cascaded first- and second-order sections according to Fig. 3.. Example Consider the following specification: w c = 8, w s = 36, and A min = 40 db. For the ripple in the passband we use three different specifications, A max = 0. db, 0.00 db, and db. he orders of the filters meeting these requirements become 5, 5, and 33. As a comparison, a linearphase FIR filter, using equiripple approximation, requires a filter order of 44 for A max = 0. db. he adaptor coefficients for the three different filters are compiled in able. he magnitude and group delay for the filters are shown in Figs As can be seen, the passband ripple must be very small in order to obtain a small group delay variation. he required coefficient word length after rounding is for all three filters 9. his is under the assumption that the stopband attenuation is still larger than 35 db. he passband ripples for the three different filters, using quantized coefficients, then become db, db, and db, respectively. A max a i a j, a j 0. Ð Ð , Ð Ð , Ð Ð , Ð0.853, Ð , Ð Ð0.4574, Ð Ð , Ð Ð , Ð , Ð , Ð Ð0.558, Ð Ð0.5307, Ð Ð , Ð Ð , Ð Ð , Ð , Ð , Ð , able. Adaptor coefficients for the filters in example.

14 4 Figure 4.. Magnitude and group delay for a 5th-order linear-phase lattice WDF. Figure 4.. Magnitude and group delay for a 5th-order linear-phase lattice WDF. Figure 4.3. Magnitude and group delay for a 33rd-order linear-phase lattice WDF. Example he condition for approximately linear-phase IIR filters is similar to that of FIR filters. hat is, for the cases in which the transition bands are very narrow, the filter orders will be very high, even if they generally will be

15 5 lower than the orders of the corresponding FIR filters. For example, reducing the stopband edge from 36 to 7 in the example above (A max = 0. db) increases the filter order from 5 to 3. he magnitude and group delay of this filter are shown in Fig 4.4. For a linear-phase FIR filter, using equiripple approximation, the required order is 86. Figure 4.4. Magnitude and group delay for a 3st-order linear-phase lattice WDF. Example 3 We have so far in this paper only considered the case where the order of the filter H 0 (z) larger than the order of the delay branch (N = M + ). he reason for this is that the overall filter order in this case is much less than for the case where the order of H 0 (z) is lower than the delay branch (N = M Ð ). o illustrate this, we have designed these two different types of filters for the following specification: w c = 36, w s = 90, A max = 0.00 db, and A min = 40 db. he orders of the filters become 9 for the case where N = M +, and for the case where N = M Ð. he required order for the latter filter type is thus more than two times larger than for the former filter type. he magnitude and group delay for the two filters are shown in Figs. 4.5 and 4.6, respectively. From the figures we can see that the group delay variation is better for the former filter type. Figure 4.5. Magnitude and group delay for a 9th-order linear-phase lattice WDF, N = M +.

16 6 Figure 4.6. Magnitude and group delay for a st-order linear-phase lattice WDF, N = M Ð. Example 4 From Figs , we see that the group delay has a small variation for low frequencies and deteriorates at the passband edge. he variation is largely dependent on the passband ripple. One way to increase the stopband attenuation for a given filter order, without increasing the group delay variation, is therefore to allow a larger ripple for low frequencies. A simple way to do this is to let d c (w) decrease linearly in the passband from Kd c to d c, i.e., d ( w) = Kd + ( - K) d w / w (3.4) c c c c for some constant K. Consider the following specification: w c = 36, w s = 7, A max = db, and A min = 40 db. he specification is met by a 7th-order filter. We use K = and K = 5. he magnitude and group delay for the two cases are shown in Figs. 4.7 and 4.8. he group delay variations are in both cases 0.6 in the passband. he stopband attenuations are 40 db for K =, and almost 44 db for K = 5. Figure 4.7. Magnitude and group delay for the 7th-order linear-phase lattice WDF when K =.

17 7 Figure 4.8. Magnitude and group delay for the 7th-order linear-phase lattice WDF, when K = CONCLUSIONS In this paper an algorithm for designing linear-phase lattice WDFs has been described in detail. he filters consist of a parallel connection of two allpass filters of which one corresponds to a pure delay. Several design examples using the algorithm were given. It was observed that in order to obtain a filter with a small group delay variation in the passband, the passband ripple must be very small. Further, for a narrow transition band specification, the filter order becomes high. It was also demonstrated that the overall filter order is minimized if the order of the filter that corresponds to a pure delay is less than the order of the other allpass filter. We also suggested a simple way to increase the stopband attenuation for a given filter order, without increasing the group delay variation in the passband. REFERENCES [Abo-Z-95] Abo-Zahhad M., Yaseen M., and Henk.: Design of Lattice Wave Digital Filters with Prescribed Loss and Phase Specifications, European Conf. on Circuit heory & Design, pp , Istanbul, urkey, 995. [Ande-95] Anderson M. S., Summerfield S., and Lawson S.S.: Realization of Lattice Wave Digital Filters Using hree-port Adaptors, Electronic Letters, Vol.3, No. 8, pp , April 995. [Erik-78] Eriksson S. and Wanhammar L.: idsdiskreta filter, Vol., , Linkšping University. [Fett-74] Fettweis A., Levin H., and Sedlmeyer A.: Wave Digital Lattice Filters, Intern. J. on Circuit heory and Applications, Vol., pp. 03-, June, 974. [Fett-86] Fettweis A.: Wave Digital Filters: heory and Practice, Proc. IEEE, Vol. 74, No., pp , Feb [Gazs-85] Gazsi L.: Explicit Formulas for Lattice Wave Digital Filters, IEEE rans. on Circuits and Systems, Vol. CAS-3, No., pp , Jan. 985.

18 8 [Henk-8] [Joha-96a] [Joha-96b] [Kuno-88] [Leeb-9] [Neir-79] [Renf-86] Henk.: he Generation of Arbitrary-Phase Polynomials by Recurrence Formulae, Intern. J. Circuit heory and Applications, Vol. 9, pp , 98. Johansson H. and Wanhammar L.: Design of Bireciprocal Linear-Phase Lattice Wave Digital Filters, Report LiH-ISY-R-877, Linkšping 996. Johansson H. and Wanhammar L.: An Algorithm for the Adaptor Coefficients of Richards' Structures, IEEE Nordic Signal Processing Symp., pp , Espoo, Finland, Sept Kunold I.: Linear Phase Realization of Wave Digital Filters, IEEE Intern. Conf. on Acoustics, Speech, and Signal Processing, pp , New York 988. Leeb F.: Lattice Wave Digital Filters with Simultaneous Conditions on Amplitude and Phase, IEEE Intern. Conf. on Acoustics, Speech, and Signal Processing, pp , oronto 99. Neirynck J. and Vinckenbosch C.: Design and Properties of Canonic Symmetric Digital Filters by Schur Parametrisation, Proc. 979 IEEE Int. Symp. on Circuits and Systems, pp , okyo, Japan, July 979. Renfors M. and SaramŠki.: A Class of Approximately Linear Phase Digital Filters Composed of Allpass Subfilters, IEEE Intern. Symp. on Circuits and Systems, pp , 986. [Renf-87] Renfors M. and SaramŠki.: Recursive Nth-Band Digital Filters-Part I: Design and Properties, IEEE rans. on Circuits and Systems, Vol. CAS-34 No., pp. 4-39, Jan [Oppe-89] Oppenheim A.V. and Schafer R.W.: Discrete-ime Signal Processing, Prentice Hall, 989.