DIGITAL FILTER FOR QUADRATURE SUB-SAMPLE DELAY ESTIMATION

Transcription

1 DIGITAL FILTER FOR QUADRATURE SUB-SAMPLE DELAY ESTIMATIO Ewa Hermanowic Multimeia Systems Department, Faculty of Electronics, Telecommunications an Informatics Gans University of Technology, Gans, Polan ABSTRACT In this paper we present a novel igital filter intene for quarature sub-sample elay estimation base on the Hilbert (complex filtering an fractional elaying of an incoming realvalue signal. It is a Hilbert Transform Filter (HTF having variable fractional elay (VFD. The filter is base on a pair of rotate VFD filters in the Farrow structure. It is capable of performing the Hilbertian as well as VFD filtering of the incoming iscrete-time signal at the same time. Thus one can substitute a hitherto use cascae of the HTF an the VFD filters with an aggregate HTF-VFD filter propose here. The technique is simple to implement. The avantages lie in lower total elay introuce by the compoun filter an in moular structure, which is compose of the same linear-phase rotate VFD Farrow subfilters. The overall rotate VFD filters in the pair iffer only in the value of one parameter the VFD. The propose filter can be applie to aaptive quarature sub-sample estimation of elay. I. ITRODUCTIO Fractional elay filters (FDFs fin a variety of applications in, e.g., soun synthesis, timing ajustment in igital receivers, image ooming an many others, which have been wiely escribe so far see, e.g., []. Recently a FDF having variable fractional elay (VFD has been applie to aaptive sub-sample elay estimation using a quarature etector base on the Hilbert (complex filtering an fractional elaying of an incoming real-value signal [], [], [3]. In the abovementione references both of these operations have been realie by two inepenent filters, namely the Hilbert transform filter (HTF an the VFD filter. In this paper we propose to substitute these filters with a single aggregate one, further calle the HTF-VFD, having the same functionality, but reuce transport elay. For this aim we use polynomial interpolation. The polynomial interpolation can be implemente efficiently using the iea base on the Farrow structure [4], [5]. This structure is very attractive because it can provie variable signal elay by changing only one parameter. This maes on-line elaying straightforwar an feasible. The signal passes through FIR Farrow subfilters either symmetric or anti-symmetric see [4], hence their phase response linearity, an is multiplie by appropriate powers of elay to prouce the output. Organiation of the paper is the following. Section II provies inispensable bacgroun information about the fractional elay filter an the Farrow structure. Section III formulates the proceure of esigning the HTF having fractional elay. Section IV provies illustrative esign examples. Section V summaries the results. Fractional Delay Filter II. BACKGROUD The frequency response of an ieal FDF is efine as D ( e exp( τ, ω π ( τ < where ω πft is the normalie angular frequency in ra per sample, F stans for the physical frequency in H an T stans for the sample interval in secons. The impulse response of this filter is π n τ ( ω π Dτ e e π ( sinπ ( n τ sinc( n τ π ( n τ In the case of an FIR approximation of length, τ is efine as a sum of a transport elay of the system an introuce fractional elay τ ( / + (3 where the values of are restricte to the interval.5.5 in orer to eal with central interpolation the most accurate one. The transfer function of an FIR FDF in a irect form is efine as D n where [, n,, stans for the impulse response of the FIR filter. This filter, in case of variable value of, is calle the VFD. Farrow Structure High efficiency of a VFD can be achieve by applying the Farrow structure [4]. The main avantages of this structure is that the coefficients of its subfilters are either symmetrical or anti-symmetrical an fixe, inepenent of the value of introuce fractional elay see [4]. It means that we can change the fractional elay without reesigning the filter. n (4 7 EURASIP 53

2 In orer to apply the Farrow structure we have to express the equation (4 in a form of a polynomial of fractional elay. The coefficients of the filter can be rewritten as M c, n,, (5 where M+ stans for a number of subfilters in the Farrow structure. From the above equations we have where M M n D c[ C n C c n { n} n (6,,, M (7 are the transfer functions of subfilters, which in the original Farrow wor [4] are implemente in irect form. This structure gives us a flexibility to change the elay by manipulating only one coefficient, the aforementione fractional elay (FD. x[ C M ( C M Fig.. VFD in the Farrow structure; y x( n ( / an is given by (b. The subfilters are linear-phase [4]. III. FILTER DESIG AD IMPLEMETATIO Hilbert Transform Filter C ( C The frequency response of an ieal Hilbert transform filter having the so-calle generalie linear phase-response is efine [6] as exp( j( ω π / β, ω (, π H β ( e (8, ω ( π, where β stans for the elay introuce by the filter. We can write the impulse response of this filter using a pair of VFDs [6] n / ( [ n / ], n, ±, ± 4, K β / hβ (9 n / ( ( β / [( n / ], n ±, ± 3, K Therefore the esign of an FIR approximation of length for this ieal HTF requires two VFDs of length each. A general scheme of implementing this HTF having variable fractional elay as an FIR filter in the Farrow structure is presente in Fig.. The total elay of this filter expresse in terms of is β + y[ (a an the fractional elay to be varie is / (b In Fig. in the lower branch we nee to introuce an aitional elay necessary to interlace the coefficients of the filter. In case of Lagrangian polynomial approximation the coefficients of a maximally flat VFDcan be obtaine using the following expression n n ( for n,,, where ( β + / from (a. x[n ] VFDR Fig.. Bloc scheme for implementing the HTF-VFD in one stage with variable fractional elay / (b. The proceure of esigning the filter in Fig. is as follows. From equation ( we compute the coefficients of the VFD in irect form an then, using equations (4, (6 an (7, we compute the coefficients { c [ ]} n. The next step is multiplication of these coefficients by appropriate powers of (, as given by { ~ n c [ ]} {( [ ] n n c n } n ( in orer to get their rotate version (cf. (9. The resulting filter is calle in Fig. the VFDR variable fractional elayer rotate. oteworthy is that we introuce the elay into the upper branch in Fig. an / into the lower branch. The output of the HTF-VFD is a complex signal, y[, escribe as y Re y[ + j Im y[ (3 The transfer function of the resulting HTF-VFD having variable fractional elay is ~ ~ H HTF -VFD C ( + j C ( ( IV. DESIG EXAMPLES In this Section we present some examples of esigne HTF-VFD filters. A. Maximally flat HTF-VFD / VFDR Re y[ n ] Im y[ n ] Example. The coefficients of Farrow subfilters with a maximally flat VFDR of length for the HTF-VFD from Fig. of length 4, compute in accorance with the escription as above are the following 7 EURASIP 54

3 { ~ } { /, / } an { c ~ } {, c }. n n Consequently ~ ~ C / / an C(, an H HTF-VFD / + j ~ ~ C ( + j C ( ( (/ j 3 + ( The impulse response an the coefficients of H HTF -VFD for an arbitrary value of for this aggregate filter an also for the corresponing cascae can be reaily obtaine using the symbolic function HTF-VFD_symb( written in MATLAB an given below, where small plays the role of in (4. An exemplary isplay of results for is also provie. function [htr,htrcascae]htf-vfd_symb(; % % program function for the generation of the coefficients of % FIR fractional elaying Hilbert transform filter % HTF-VFD maximally flat aroun omegapi/ % using symbolic expressions for the impulse response % coefficients htr(n % as a function of FD ; it is assume that -.5<<.5 % input % - impulse response length of the VFD solely % output % htr the aggregate HTF-VFD coefficients vector of % htrcascae - the HTF an VFD cascae imp. resp. of % use [htr,htrcascae] HTF-VFD_symb(; to start with syms h h h h htr hfs hfs hfs htotal syms htrafo hfs sum h g % HTF-VFD aggregate hffsymb(; hffsymb_shifte(; for n:; h(n((-^(n-*h(n; h(nj*((-^(n- *h(n; en htr(:*; for n:; htr(*n-h(n; htr(*nh(n; en %Cascae of linear-phase HTF an VFD %Linear-phase HTF huppervf(,.5; hlowervf(,; for n:; hupper(n((-^(n-*hupper(n; hlower(nj*((-^(n-*hlower(n; en h(:*; for n:; h(*n-hupper(n; h(*nhlower(n; en htrafo(:*-h(:*; hfsffsymb(; %VFD for n:*-; hfs(n; en; for n:; hfs(*n-hfs(n; en for n::*-; hfs(nimag(((-^(n/*hfs(n; en; %Cascae % convolution of htrafo an hfs is compute symbolically x*-; hhtrafo; ghfs; sum(h(*g(; for n:x; sum(nh(*g(n; for :n; sum(nh(*g(n-++sum(n; en; en sum(*x-h(x*g(x; for n*x-:-:x+; p; sum(nh(x*g(x-p; for x:-:n-(x-; sum(nh(*g(n-++sum(n; en; pp+; en; htrcascaesum; HTF-VFD_imp_respconj(htr' HTF-VFD_cascae_imp_respconj(htrcascae' function hffsymb( syms h h; tr(-/; for n:; h(n; for :; if ~n h(nh(n*(tr+-(-/(n-; en; en; en function hffsymb_shifte( syms h h; tr(-/;%transport elay for n:; h(n; for :; if ~n h(nh(n*(tr+-/-(-/(n-; en; en; en function hvf(,e taug(-/+e; for n:; h(n; for :; if ~n h(nh(n*(taug-(-/(n-; en; en; en The isplay of results for HTF-VFD_imp_resp [ /-]; [ i*(-+]; [ -/-]; [ -i*]; HTF-VFD_cascae_imp_resp [ /*i*(/-]; [ -/+]; [ -/*i*(/-+/*i*(-/-]; [ /+]; [ -/*i*(-/-]; Fig. 3. Continue on the next page. Fig. 3 presents the performance of the esigne narrowban aggregate filter, where is efine in (b. 7 EURASIP 55

4 means that at the front en of processing a linear-phase half-ban filter (HBF prouces two-times over-sample signal which is passe through a half-ban VFD. The result is finally own-sample bac to the base-ban frequency. In this arrangement the requirements for the VFD are relaxe relative to one-stage approach allowing for a smaller number of taps in the Farrow structure. In [7] it was shown that the two-rate structure coul be realie efficiently as two-stage one-rate with two parallel branches. Fig. 3. The performance of an aggregate HTF-VFD of length 4, maximally flat at f π /, from Example. For phase an group elay responses in Fig. 3 the transport elay having the value of.5 sample intervals is remove for better visualiation of the FD. The FD values are set here from the region <,. 5 > for the upper branch of Fig. an / <.5, > for the lower branch, with an increment equal to.. It finally gives the VFD value / <.5,.5 > as epicte in Fig. 3. It is worth noticing that magnitue responses are isplaye in the range f < /, /, where f ω /( π, on the normalie frequency axis an other responses are isplaye within the range f <, /. The latter interval covers the pass ban of the filter, where these characteristics are the object of our interests an neglects the stop ban, where only the attenuation is important. For the sae of comparison Fig. 4 presents the performance of the corresponing cascae filter see [], [] or [3], for the same FD elay values as in Example. The target filter length is 5. The transport elay introuce by the cascae is sample intervals, thus is greater than in Example. Also the with of pass ban is unfavorably narrower than for the aggregate esign propose here. Thus in this example we have shown that the novel HTF- VFD filter from Fig. 3 performs better than its hitherto cascae counterpart from Fig. 4, both base on the same VFD moule. This hols inepenently of the VFD moule length. B. Minimax HTF-VFD optimie in two-stage esign Finally, in Fig. 5, we present the performance of a wieban HTF-VFD with a pair of minimax VFDs optimie in two-stage esign [7], [8]. Two-stage esign Fig. 4. The performance of a cascae of maximally flat HTF an VFD. The specification on VFDs given in [7] is the following. The banwith is.45 expresse in terms of the normalie frequency f. The maximal absolute value of the complex approximation error (CAE magnitue efine as the ifference between the ieal frequency response ( an its minimax approximate is.4 which is equivalent to 47.5 B. The coefficients of Farrow VFD subfilters roune to four ecimal igits are given in Figure 4 in [7]. The number of these subfilters as well as their length is jointly optimie [7]. The HBF is a yquist filter [] of orer 58, optimal in the Chebyshev sense, with pass ban an stop ban ege normalie frequencies:.45 an.55, respectively, requiring 5 taps of ifferent values. The overall VFD nees 4 taps in parallel branches. The transport elay introuce by this wieban filter is 6.5 sample intervals. 7 EURASIP 56

5 aroun the normalie frequency.5. ote that in case of, the target HTF-VFD filter performance can be assesse in a way shown in [9], [] or [] on the basis of the performance of the VFD use in Fig. as the VFDR. V. COCLUSIOS A novel efficient esign of a igital filter for quarature sub-sample elay estimation was propose an escribe in this paper. The coefficients of this filter are obtaine from the coefficients of a pair of ientical VFDs appropriately rotate an interlace. Maximally flat (Lagrangian [6] approximation as well as minimax approximation [7], [8] of VFDs optimie in two-stage esign was employe to compute coefficients of the esigne HTF-VFD filter. Using in this esign the structure invente by Farrow [4] gives a possibility of changing the value of introuce elay in real-time, without reesigning the coefficients. The original Farrow structure [4] is compose of linear-phase subfilters an as such can be use straightforwarly without any nee for moifications. (This seems unnotice by many authors. It was shown that the propose aggregate HTF- VFD exhibits wier banwith an introuces smaller integer elay in comparison with a hitherto applie cascae esign base on a series connection of two inepenent filters: the linear-phase HTF an the VFD. REFERECES Fig. 5. The performance of an aggregate minimax HTF-VFD of length 4 with VFDs optimie in two-stage esign. The maximal value of the CAE magnitue of the resulting HTF-VFD (efine as the absolute value of the ifference between the ieal frequency response (8 an its approximate arrange as shown in Fig. is approximately 5.5 B in the banwith of.5 locate symmetrically [] D.L. Masell an G.S. Woos, Aaptive subsample elay estimation using a moifie quarature phase etector, IEEE Trans. Circuits Syst. II, vol. 5, o., Oct. 5, pp [] D.L. Masell, G.S. Woos, an A. Kerans, A harware efficient implementation of an aaptive subsample elay estimator, in Proc. ISCAS 4, vol. III, pp [3] D.L. Masell an G.S. Woos, Aaptive subsample elay estimator using quarature estimator, Electr. Letters, vol. 4, o. 5, 4 th March 4, pp [4] C.W. Farrow, A continuously variable elay element, in Proc. IEEE Int. Symp. Circuits Syst., vol. 3, Espoo, Finlan, June 7-9, 988, pp [5] S. Samai, A. M. Omair an M..S. Swamy, Results on maximally flat fractional-elay systems, IEEE Trans. Circuits Syst. I, vol. 5, o., ov. 4, pp [6] E. Hermanowic, Special Discrete-Time Filters an Applications. EXIT, Warsaw, Polan, 5, Chapter 4. [7] E. Hermanowic an H. Johansson, On esigning minimax ajustable wieban fractional elay FIR filters using two-rate approach, in Proc. European Conference on Circuit Theory an Design, ECCTD 5, Cor, Irelan, 9 August September 5, vol. I, pp [8] H. Johansson an E. Hermanowic, Ajustable fractional-elay filters utiliing the Farrow structure an multirate techniques, in Proc. of The Sixth International Worshop on Spectral Methos an Multirate Signal Processing, SMMSP6, Florence, Italy, -3 September 6. [9] P.P. Vaiyanathan an T.Q. guyen, A tric for the esign of FIR half-ban filters, IEEE Trans.Circuits Syst. vol. CAS-34, Mar. 987, pp [] P.P. Vaiyanathan, Design an implementation of igital FIR filters, in Hanboo on Digital Signal Processing, Eite by D.F. Elliott, Acaemic Press Inc., 987, pp [] P.P. Vaiyanathan, Multirate Systems an Filter Bans, Prentice Hall, 993, pp [] T. I. Laaso, V. Valimai, M. Karjalainen an U. K. Laine: Splitting the unit elay. Tools for fractional elay filter esign. IEEE Signal Processing Magaine, January 996, pp EURASIP 57