3.4 Design Methods for Fractional Delay Allpass Filters

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1 Chater 3. Fractional Delay Filters Design Methods for Fractional Delay Allass Filters Above we have studied the design of FIR filters for fractional delay aroximation. ow we show how recursive or IIR filters can be used for the same roblem. Since the magnitude resonse of an ideal FD element is erfectly flat, we consider only the socalled allass filters. Their magnitude resonse is always flat irresective of the filter coefficients. Allass filters are tyically used for hase equalization and other signal rocessing tasks where the hase characteristics are of greatest concern. Also the FD aroximation is essentially a hase aroximation roblem and thus the allass filter is articularly well suited to this task Proerties of the Discrete-Time Allass Filter A discrete-time allass filter has the transfer function A(z) = z D(z 1 ) D(z) = a + a 1 z 1 +K+a 1 z ( 1) + z 1 + a 1 z 1 +K+a 1 z ( 1) + a z (3.117) where is the order of the filter, D(z) is the denominator olynomial, and the filter coefficients a k (k = 1,,..., ) are real. The block diagram of the allass filter is deicted in Fig The numerator of the allass transfer function is a mirrored version of the denominator olynomial. The oles (i.e., roots of the denominator) of a stable allass filter are located inside the unit circle in the comlex lane and as a result its zeros (i.e., roots of the numerator) are located outside the unit circle so that their angle is the same but the radius is the inverse of the corresonding ole. For this reason the magnitude resonse of an allass filter is flat. It is exressed as A(e jω ) = e jω D(e jω ) D(e jω ) 1 (3.118) Obviously, the name Ôallass filterõ comes from the above roerty that this filter asses signal comonents of all frequencies without attenuating or boosting them. The frequency resonse of the allass filter can be exressed in the form A(e jω ) = e jθ A (ω ) (3.119) This form stresses the fact that the main feature of an allass filter is its hase resonse Θ A (ω). It can be written as Θ A (ω) = arg{ A(e jω )}= ω + Θ D (ω) (3.1) where Θ D (ω) is the hase resonse of 1 D(e jω ), that is The term Ôallass filterõ is sometimes used to denote any discrete-time system (FIR or IIR) whose magnitude resonse is almost flat at all frequencies ( H(e jw )» 1). Throughout this text this term is used only for recursive discrete-time filters which have the exact allass roerty, that is, H(e jw ) º 1.

2 16 Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters ì ì 1 ü Q D (w) = argí î D(e jw ý ) þ = arctan ï í ï î ï å k= å k= ü a k sin(kw) ï ì ý = arctan at s ü í a k cos(kw) ï a T ý î c þ þ ï (3.11a) with a = 1 and the vectors a, s, and c are defined by a = [ 1 a 1 a L a ] T (3.11b) s = [ sin w sin(w) L sin(w) ] T (3.11c) c = [ 1 cosw cos(w) L cos(w) ] T (3.11d) The hase delay t,a (w) of an allass filter can be exressed with the hel of Eq. (3.1) as t,a (w) =- Q A (w) w = - t,d (w) (3.1) where t,d (w) is the hase delay of 1 D(e jw ). The corresonding grou delay is given by t g,a (w) =- dq A (w) dw where t g,d (w) is the grou delay of 1 D(e jw ) Design of Fractional Delay Allass Filters = - t g,d (w) (3.13) There is a noteworthy difference between the design of FIR and allass filters: the coefficients of an FIR filter are easily obtained by the inverse discrete-time Fourier transform of the frequency-domain secifications, since the coefficients of an FIR filter are equal to the samles of its imulse resonse. However, the relationshi between the transfer function coefficients and the imulse resonse of an allass filter is not this simle. Hence, most of the design techniques for allass filters are iterative. Many of the design methods for FD allass filters are counterarts of FIR design methods discussed in Section 3.. The desired or ideal hase resonse in the fractional delay aroximation roblem is Q id (w) =-Dw (3.14) as may be seen from the ideal frequency resonse given by Eq. (3.8). The hase error of an allass filter can be defined as the deviation from the desired hase function Q id (w) as ìï DQ(w) =Q id (w) -Q A (w) = arctan at s b üï í îï a T ý c b þï (3.15) where a is the coefficient vector given by Eq. (3.11b), s b and c b are defined as

3 Chater 3. Fractional Delay Filters 17 and b(w) as [ { } sin{ b(w) - w} L sin{ b(w) - w} ] T (3.16a) s b = sin b(w) [ { } cos{ b(w) - w} L cos{ b(w) - w} ] T (3.16b) c b = cos b(w) b(w) = 1 [ Q id (w) + w] (3.16c) When aroximating a fractional delay D = + d, or the ideal hase function Q id (w) = =-Dw =-( + d)w, the last form reduces to b(w) =- wd (3.17) The weighted least-squares hase error that is to be minimized is defined as E LS = 1 ò W(w) DQ(w) dw (3.18) where W(w) is a nonnegative weighting function. Lang and Laakso (1994) have introduced an iterative algorithm for the design of the coefficient vector a. The algorithm tyically converges to the desired solution, but this cannot be guaranteed. The LS hase delay error can be defined as E LS = 1 ò W(w) Dt (w) dw = 1 = 1 ò W(w) w DQ(w) dw ò W(w) DQ(w) w dw (3.19) In other words, the hase delay error solution is obtained by introducing an additional weighting function W (w) = 1 w to the hase error, Eq. (3.18). Allass filters may be designed using the iterative algorithm modified from that mentioned above. There exists a large variety of algorithms for equirile or minimax allass filter design in terms of hase, grou delay, or hase delay (see Laakso et al., 1994 for references). ew iterative techniques for equirile hase error and hase delay error design are described in Laakso et al. (1994). It aears that, just as with the FD FIR filters, a well suited digital allass filter for FD aroximation in audio signal rocessing is obtained by using the maximally flat design at w =. This technique deserves more attention than the others Maximally Flat Grou Delay Design of FD Allass Filters The maximally flat grou delay design is the only known FD allass filter design technique that has a closed-form solution (Laakso et al., 199, 1994). Here we discuss this solution. Thiran (1971) roosed an analytic solution for the coefficients of an all-ole lowass filter with a maximally flat grou delay resonse at the zero frequency

4 18 Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters with the binomial coefficient a k = (-1) k æ ö d + n ç è k Õ for k =,1,,K, (3.13) ø d + k + n n= æ ö ç = è k ø! k!( - k)! (3.131) It has been shown by Thiran (1971) that when d > the denominator olynomial obtained by the above method will have its roots within the unit circle in the comlex lane, i.e., the filter will be stable. A drawback of ThiranÕs design technique is that the magnitude resonse of the all-ole lowass filter cannot be controlled. In the allass design, however, this kind of roblem does not exist. Thus it seems that the result of Thiran is better suited to the design of allass than all-ole filters. As can be seen from Eq. (3.13) the grou delay of an allass filter is twice that of its all-ole counterart. Thus ThiranÕs solution may easily be alied to the design of allass filters by making the substitution dõ = d/. The solution for the coefficients of a maximally flat (MF) allass filter is thus (Laakso et al., 199) a k = (-1) k æ ö d + n ç è k Õ for k =,1,,K, (3.13) ø d + k + n n= ote that a = 1 always and thus the coefficient vector a need not be scaled. This closed-form solution to the allass filter aroximation is called the Thiran allass filter. The allass filters designed using Eq. (3.13) aroximate a grou delay of + d samles. This kind of arametrization is rather imractical. For this reason we substitute d = D Ð into Eq. (3.13). As a result we obtain the following formula: a k = (-1) k æ ö D - + n ç è k Õ for k =,1,,K, (3.133) ø D - + k + n n= ow the delay arameter D refers to the actual delay rather than the offset from samles. In Table 3.1 we give the coefficients for the Thiran allass filters with = 1,, and 3. Table 3.1 Coefficients of the Thiran allass filters of order = 1,, and 3. = 1 - D - 1 D + 1 a 1 a a 3 = - D - D + 1 = 3-3 D - 3 D + 1 (D - 1)(D - ) (D + 1)(D + ) (D - )(D - 3) 3 (D + 1)(D + ) (D - 1)(D - )(D - 3) - (D + 1)(D + )(D + 3)

5 Chater 3. Fractional Delay Filters 19 Phase Delay in Samles D = 3 D =.75 D =.5 D =.5 D = D = 1.75 D = 1.5 D = 1.5 D = 1 D =.75 D =.5 D =.5 D = ormalized Frequency Fig. 3. The hase delay of a first-order ( = 1) Thiran allass filter. The dashed lines indicate the ideal hase delay which is constant. ThiranÕs roof of stability imlies that this allass filter will be stable when D >. We have exerimentally observed that the Thiran allass filter is also stable when the delay is Ð 1 < D <. At D = Ð 1, one of the oles (as well as one zero) of the Thiran allass filter will be on the unit circle making the filter asymtotically unstable. When D < Ð 1, one or more oles will be outside the unit circle and the filter will be unstable. In Figs. 3., 3.1, and 3., the hase delay resonses of first, second, and thirdorder MF allass interolators are resented for several values of the delay arameter D starting at D = Ð 1. The ideal hase delay (constant at all frequencies) is illustrated with a dotted line in each figure. It is seen that the Thiran allass filter is most accurate at low frequencies. This is an exected result since Thiran designed the denominator olynomial so that the grou delay and its derivatives evaluated at the zero frequency match the desired result. As the filter order is increased, the hase delay curves remain nearly constant at higher frequencies. The closed-form design of the first-order allass filter for varying the length of a delay line has been discussed in aers by Jaffe and Smith (1983) and Smith and Friedlander (1985). They obtained the solution by first exressing the hase resonse of the first-order allass filter by its Taylor series with the exansion oint z = and then truncating this series after the first term. The exression for the filter coefficient a 1 of the first-order ( = 1) allass filter is the same as that given in Table 3.1.

6 11 Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters Phase Delay (samles) D = 4 D = 3.75 D = 3.5 D = 3.5 D = 3 D =.75 D =.5 D =.5 D = D = 1.75 D = 1.5 D = D = ormalized Frequency Fig. 3.1 The hase delay of a second-order ( = ) Thiran allass filter Choice of Otimal Range for D in the Thiran Interolator In the case of FIR FD filters we noticed that it is a good choice to use values ( Ð 1)/ D < ( + 1)/, since then the mean squared error is the smallest in all cases of d. In the case of recursive filters this rule does not aly. In the following we analyze the otimal range for the delay arameter in the case of the Thiran allass filter. Let us examine the mean squared error function of the Thiran allass filter given by E S = 1 ò E(e jw ) dw = 1 ò A(e jw ) - H id (e jw ) dw = 1 e jq A ò (w ) - e - jwd dw = 1 e jq A ò (w ) - e - jwd [ ][ e - jq(w ) - e jwd ] dw (3.134) = ì ï 1 - cos Q üï í ò [ A (w) + wd]dw ý îï þï Integration of the last form is difficult and we thus use numerical methods for investigating the aroximation error. We use a numerical integration technique to evaluate the error for Thiran allass filters of different order. Figure 3.3 gives the error E S as a

7 Chater 3. Fractional Delay Filters 111 Phase Delay (samles) D = 5 D = 4.75 D = 4.5 D = 4.5 D = 4 D = 3.75 D = 3.5 D = 3.5 D = 3 D =.75 D =.5 D =.5 D = ormalized Frequency Fig. 3. The hase delay of a third-order ( = 3) Thiran allass filter. function of the delay arameter D for Thiran allass filters of orders 1 to 6. It is seen that the error is zero for integral delay values Ð D = Ð1 or. This imlies that the Thiran allass filter yields the original signal samles in these cases. As discussed before, the case Ð D = Ð1 yields an asymtotically unstable filter. It is suggested not to use this kind of filter in ractical work. The Thiran allass filter can thus only be used for delay aroximations such that D > Ð 1. In the case D = the Thiran filter reduces to a delay line because all the coefficients of the numerator olynomial are zero. This is easily shown by noticing that each coefficient comuted according to (3.133) has a factor D Ð in its numerator. A fractional delay filter can usually be used for aroximating a delay D for which it holds D D < D + 1 (3.135) To solve for the otimal choice for D we may evaluate the average error E ave written as E ave (D ) = D +1 E S D ò (D)dD (3.136) We can evaluate this integral numerically. Figure 3.4 resents the average error as a function of D for some low-order Thiran allass filters. The minimum of each curve is indicated by a circle in Fig These oints corresond to the otimal D for each

8 11 Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters.6.5 Squared Integral Error D - Fig. 3.3 Squared integral error of Thiran allass filters of orders = 1 to 6 (to to bottom). filter. The values of D and the corresonding average error E ave are resented in Table 3.. Some general observations can be made from Fig Aarently, the aroximation error of the Thiran allass filter is quite small for delay values Ð 1 < D < reaching a minimum at the value given in Table 3.. For much larger values of D, the aroximation error increases considerably. It is seen in Fig. 3.4 that the average error does not increase very raidly in the neighborhood of the minimum. Thus, if the best ossible accuracy is not essential, one may use a simle rule of thumb to choose the Table 3. The otimal values for D and minimum average errors for low-order Thiran allass filters. E ave ( -.5) is the average error when D = Ð.5. D, ot E ave (D, ot ) E ave ( -.5)

9 Chater 3. Fractional Delay Filters Average Error D - Fig. 3.4 The average error of low-order Thiran allass filters as a function of D. The filter orders from to to bottom are = 1,, 3, 4, 5, and 6. The minimum of each error function is indicated by a circle. range for D: for the Thiran allass filter, a close-to-minimum error is obtained when D = Ð.5, which means that D is -.5 D < +.5 (3.137) A similar statement as in the case of FIR FD filters can be made: the total delay of the Thiran allass filters increases linearly with order Discussion This section has dealt with allass filter aroximation of fractional delay. Digital allass filters are a good choice for this task since their magnitude resonse is exactly flat and the design can concentrate on the hase roerties. The Thiran interolator was discussed in detail since it is easy to design with closed-form formulas and it is very accurate at low frequencies. The design of general IIR filters (with different numerator and denominator olynomials) for FD aroximation has not been much discussed in the DSP literature. One examle is the aer by Tarczynski and Cain (1994) where reduced-bandwidth IIR filters are otimized iteratively. The design of otimal IIR filters is a difficult roblem but new techniques are certainly worth trying since generally seaking recursive filters are more owerful than FIR filters. This is a fascinating toic for future research.

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