Splitting the Unit Delay - IEEE Signal Processing ... - IEEE Xplore

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implying thatOetken also observed that the amplitude ripple is not thesame for all the polyphase branches (in our case: for allNh(n)LJnRk = e-jakNI2, k = 1,2 ,..., Kfractions of the delay). Instead, it depends on the fractionaln=O (47) part d in the following manner [go]:where NI2 is the delay of the filter. Assuming that the zerosremain the same for noninteger values of D as well, the filtercoefficients can be solved from Eq. 47 for a chosen total delayD, which is close to Nl2. This can be expressed in matrix formaswhereisaKx(N+ 1)matrixand(494where 61,2 is the maximum amplitude ripple of the linearphaseprototype filter with the delay D = N/2 (= Int(D) + 112)and Sd is the ripple of a filter approximating the (noninteger)delay D with the fractional part, d. Note that the amplituderipple is largest in the linear-phase case (D = NI2 or d = 0.5)and reduces to zero when the fraction approaches an integervalue (d = 0 or d = l), which corresponds to the case that theimpulse response reduces to a unit pulse. The almost-equirippleapproximation computed using the Oetken method isillustrated in Fig. A6. The 4-tap FIR filter does not have manyripples, but the 10-tap filter responses are seen to be veryclose to equiripple.Relation to Interpolation/Decimation FIR Filters-jDQl ,-jDQ, ,-jDOk T As already discussed above, the polyphase structure of decieD=[e ...l(49b) mationlinterpolation filters can be utilized for fractional delayimplementation with fixed steps ([5, 9, 23, 24, 741). ForEquation 48 is a set of K complex equations with L = 2Kexample, in order to break the unit delay into Q steps, one canwhich can be expressed as asetdesign a Qth-band lowpass filter with the normalized pass-Of equations by equating the and parts band width of Q and form the Q-branch polyphaseof both sides asby picking up every Qth sample to one branch. It can beshown that each band approximates a fractional delay of the(50a) valuewithandPa =[;IPn = [ 3where the matrices and vectors contain appropriate cosineand sine elements such that Ea = Cn -j&, and e D= c D-jsD.Note that the design scheme can be interpreted as a complexversion of the frequency sampling technique [84] where thefrequency samples are unequally spaced.Hence, one first has to design the linear-phase prototypefilter, to find its zero frequencies Qk and then to invert thecosine-sine matrix of Eq. 50b. Since the zeroes of the phaseerror function need not be known with high precision, simplenoniterative search on the employed frequency grid is sufficientin general. After that, the coefficients of a new filterapproximating any given delay are readily obtained via asingle matrix multiplication. Note that matrix Eq. 50b isindependent of the delay D and only needs to be inverted onceoff-line so that the approach is suitable for real-time coefficientupdate.D=e-k, k=0,1,2 ,..., Q-IQThe accuracy of approximation depends naturally on thelength of the prototype Qth-band lowpass filter. In order toachieve comparable frequency response for each branch, thelength of the prototype filter should be a multiple of Q. Thelowpass filter should be linear-phase, but it can be designedwith any method. However, the optimality of the prototypefilter is not shared with the branch filters; e.g., equiripplemagnitude characteristic will be lost.The multirate approach is straightforward and well suitedfor table look-up applications. If one is satisfied for example,with Q = 50-step division of the unit delay, one simplydesigns a length-501 filter and uses the desired length4 filterof the set.However, small delay steps and strict specifications for theapproximation error may result in an FIR filter with hundredsof taps, which makes it impossible to use the Parks-McClellanalgorithm for the prototype design. In that case, windowingmethods may be used for which there is practically no limitfor the filter order. However, as there are several other simplemethods that provide smaller error, the multirate approachnow appears somewhat outdated for this application.For reference, we chose Q = 10 to design prototype filtersof lengths 40 and 100 to provide a set of length-4 andJANUARY 1996 IEEE SIGNAL PROCESSING MAGAZINE 43
length-10 FIR filters for 10-step increments of the fractionaldelay. The results are shown in Fig. A7. The startling featureis that the magnitude approximation error is large for the casethat the delay is zero. The reason is that the impulse responsedoes not have a single nonzero value as it would have in theideal case. This can be alleviated by using the “TRICK’,which forces exact zeroes in the impulse response [ 1171.However, rather than for approximation, the multirateapproach can be used for implementation of high-qualitywideband FD filters, as proposed in [75]. Using a polyphaseimplementation with two branches the accuracy of approximationcan be increased without excessive total delay.Controlling the Delay of Arbitrary FIR FiltersAbove, we have assumed that the fractional delay filter isdesigned exclusively for producing the desired delay. However,in time-critical applications this may be impossiblesince an additional FIR filter always introduces some netdelay, as discussed earlier. Instead, it may be more advantageousto control the delay of an FIR filter that is alreadyincluded in the system. As the FIR filter may be adaptive orvariable or for other reasons its coefficients may not beknown, it is sometimes essential that the delay control algorithmbe independent from the filter coefficients.1) Fourier Transform Based MethodsWe first present a Fourier transform based delay controlalgorithm which is related to ideal bandlimited interpolationdiscussed in [81]. Let us assume an Nth-order (length L = N+ 1) prototype filter whose frequency response isH,(e”)N= xhp(k)e-jkwk=O (53)If the filter is linear-phase, it has constant group delay N/2.Assume that it is desired to change the delay by a fractionfrom this nominal value. This is equal to multiplying thefrequency response by e-1adw orThis is thus the desired frequency response. Taking thesymmetric inverse discrete-time Fourier transform (Eq. 1 1)of Eq. 54, we obtain the real-coefficient FIR filter as. EN= hp (k)sinc(n- k - Ad)k=O(55)which is seen to be a weighted sum of sinc functions (Eq. 13)with an infinitely long impulse response and consequentlywith truncation problems. This can be alleviated in the usualmanner by employing a suitable time-domain window totruncate h(n). When the lengths of h(n) and h,(n) are chosento be the same L, the solution (Eq. 55) can be given in explicitmatrix form ash= WMSMh,(56)where h and hp are the new and the prototype FIR coefficientvectors, respectively, and the elements of the L x L squarematrix Sad areSM,k,l = sinc(k-E - Ad) k,l= 1,2 ,._. L(574and Wad is a diagonal matrix with the value of the length-Lwindow function as its elements, orNote that the length of the new h vector can also be chosenlonger or shorter than L by simply adding rows to or cancelingthem from the S4d matrix. Furthermore, if a fixed step Ad issufficient, one can compute Sad and multiply it by the Wadmatrix in advance, which considerably reduces the computations.A similar delay control method employing two discreteFourier transforms (DFT) was presented by Adams in [2]. Itis in fact closely related since, by adapting to our notation,the two complex DFTs can be packed into a single real-coefficientmatrix Shof the form (Eq. 57a) with the elementsA sin[n(k -1 - Ad)]k,l=1,2, ... LS‘’k’ = Lsin[.rc(k - I - Ad) / L]The Adams method is thus seen to be a discrete-frequencycounterpart of the first method, as the essential difference isthat the sinc function is replaced by the periodic samplingfunction, also known as the Dirichlet kernel [11]. The differencein performance of Eq. 57a as compared to Eq. 58 is thusexpected to be small for large L. However, both formulas areclearly more advantageous for practical use than the twoexplicit complex DFTs as proposed in [2] (even if the FFTalgorithm is employed), since here only real-valued multiplicationsare needed. Furthermore, the matrix (Eq. 58)-as wellas Eq. 57a-has a symmetric Toeplitz structure, which meansthat only L different coefficients are required.For illustration, we approximate the equiripple filters ofFig. A6 using this method. The prototype filter was chosen inthe middle (d = 0.2) and shifted Dolph-Chebyshev windowswith 40 dB ripple were used. From the resulting Fig. A8 it isobserved that the magnitude and phase delay responses arepoorly preserved for the 4-tap filter but fairly good for the10-tap filter.2) Matrix Transform Method Basedon Zeros of the Error FunctionIt is natural to assume that the zeros of the error functionremain the same not only independently of the approximation44 IEEE SIGNAL PROCESSING MAGAZINEJANUARY 1996
Page 9: constant with high precision when t
Page 12 and 13: where N is the1w0=I 0.8tZcl2 0.60.4
Page 16 and 17: method but also irrespective of the
Page 18 and 19: Another major disadvantage is the p
Page 20 and 21: y angular frequency (Eq. lo), the L
Page 22 and 23: 2.62.43 2.2WnIi2
Page 24 and 25: 19. T. Chen, H. P. Graf, and K. Wan
Page 26 and 27: ~ dAppendix A. Examples of FD FIR F
Page 28 and 29: 11.61.40.83 130.6t ; 1.25 I 0.4 2 l
Page 30 and 31: Appendix B. Examples of FD Allpass

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