Adaptive Subsample Time Delay Estimation for Narrowband Signal Synchronisation Saul R. Dooley and Asoke K. Nandi Department of Electronic and Electrical Engineering, University of Strathclyde, 204 George Street, Glasgow G1 1XW, UK. E-mail:
[email protected] Abstract This paper addresses the problem of subsample or fractional sample time delay estimation of narrowband signals of known centre frequency, in order to synchronise the signals. The Lagrange interpolator filter is incorporated into the explicit time delay estimator (ETDE) method. In order to achieve this, a new form of the Lagrange interpolator is presented in this paper. Further modifications are made to the ETDE algorithm. Simulations show delay estimation bias using the new algorithm is orders of magnitude smaller than conventional “sinc”-based ETDE.
1 Introduction The problem of estimating the (possibly fractional sample) delay D accurately between two discrete narrowband signals is of particular importance in many areas, including synchronisation in digital communications systems and narrowband interference rejection. Adaptive techniques are widely used due to their ability to track a nonstationary delay. One such method is the explicit time delay estimator (ETDE) [1]. This technique is ^ is explicitly updated in the adaptive process, thus alleattractive as the delay estimate D viating the need for interpolation of filter weights which can be one source of estimation bias. The adaptive configuration (effectively the system identification setup), is shown in figure 1, with input signals
x(k) = s(k) + (k) (1) y(k) = s(k ? D) + (k) (2) where s(k ) is a narrowband signal of centre frequency !c (with bandwidth W Q k=0 l=p?k
mi N ? m j i ci;j (p) = pi P ?p jp (?Q)?p
bi;j (m) = (?1)i+j P i Qj
h
(11) (12)
0 0 0 where * denotes a discrete convolution and x is defined as x = x; x > 0; x = 0; x < 0. The operator b is defined as the Stirling number of the first kind with argument (a; b) to convert between binomials and polynomials, which are positive integers tabulated in, e.g. [6] and in mathematical tables, and are defined as the number of permutations of a symbols having exactly b cycles. Taking the definition as in [6], the gamma function can be expressed in terms of Stirling numbers of the first kind through this formula, for any c 2 R: a hai X ?(c + 1) b (13) = (?1)a?b b c: ?(c ? a + 1) b=0 The proofs of (9) and (10) are straightforward but lengthy and thus are omitted here. It should also be noted the coefficients ap;m need computing only once for a set fractional delay filter order. The filter can then be made maximally flat at ! = !c by modulating ^ (k )) = ej!c (m?D^ (k))h0 (n; D^ (k )): thus: h(n; D
3 Modified algorithm The ETDE can be realised with the Lagrange interpolator instead of the sinc filter. With h(m; D^ k ) in the form of equation 8, its derivative, f (m; D^ k ), can be derived exactly. This is one advantage of the filter form of equation 8. ^ (k )) and the unmodulated part of f (m; D^ (k )) can A second advantage is that h0 (m; D be implemented exactly using the fast Farrow structure [3]. The ETDE algorithm can now be implemented using a complex-valued form of equation 3. Additionally, it can be shown ^ (k ) = D (restricting the search space that the minimum of the performance surface is at D ^ (0)) < 1 as in [1]) and that the estimated gradient is unbiased. to ?1 < (D ? D
4 Simulations The simulations were performed with a noise-free sinusoidal input signal, for a variety of sinusoidal frequencies and for a given delay of D = 0:3. The stepsize was chosen to be = 1=(5000!c2 ) to ensure convergence to within 2% of the final value in 5000 iterations. ^ (0) was set to zero. The initial estimate D 3
Figure 2 illustrates the difference between the proposed modulated Lagrange and unmodulated (conventional) sinc ETDE for each tested input signal frequency (denoted as normalised frequency fc = !c = ). What is shown is the magnitude of the estimation bias, which is defined as the difference between the final value of the estimate and the true value D. It is easily seen that the bias is much lower for Lagrange ETDE estimation than sinc ETDE estimation for values of L = 4 and L = 10. Additionally, it should be noted that even the Lagrange ETDE with L = 4 significantly outperforms the sinc ETDE with L = 10 in terms of estimation bias. This gives some indication of the benefits of using fractional delay filters different to the sinc in the ETDE method. Next, we study the effects of the modulation and signal-to-noise ratio (SNR). We use the following four L = 4 fractional delay filters—two versions of the sinc and Lagrange interpolator, both modulated to !c and both unmodulated. The delay estimates were ob^ (k ) after convergence; was set to 0.005. tained by averaging the last 2000 values of D At first, the simulated signal was a single sinusoid of frequency !c = 0:4 . Figure 3(a) illustrates the mean squared estimation error (MSEE) ensemble averaged over 20 runs for SNRs in the range 40 dB to 60 dB. The MSEE is defined as
^) = MSEE(D
20 1 X (D^ (i) ? D)2 20 i=1
(14)
^ (i) is the estimated delay for each simulation run. It is evident that the where each D unmodulated sinc ETDE yields the worse MSEE followed by the unmodulated Lagrange interpolator. The flat MSEE is due to the fact both these filters yield biased estimates. Modulating the filters appears to give a significant improvement in terms of MSEE, with the modulated Lagrange being the better by around 1.5 dB. Figure 3(b) demonstrates a very similar scenario where the simulated signal is now taken as a combination of three equal-amplitude sinusoids of frequencies 0:3 , 0:4 and 0:5 , scaled so that the signal power equals that of the signal in the first example. This means that our signal has finite bandwidth. Again, the MSEE is plotted for all four filters and remarkably, the sinc, Lagrange and modulated Lagrange fractional delay filters are largely unaffected - only a matter of up to around 1 dB different. The significant change is in the performance of the modulated sinc filter, whose MSEE has shot up dramatically. The modulated Lagrange, again, can be seen to be the best filter for the situations presented. The reason why both Lagrange filters outperform their sinc counterparts i s because of the significantly smaller fractional delay approximation error for frequencies near the maximally flat frequency as noted in [4] [7].
5 Conclusions Simulations show that the proposed Lagrange ETDE method significantly outperforms the sinc ETDE method in terms of estimation bias using noise-free sinusoidal inputs of known frequency. The proposed form of (8) allows for the exact derivative of the Lagrange filter coefficients improving implementation accuracy and lowering computational complexity. The authors wish to thank the Engineering and Physical Sciences Research Council for funding the research on this project. 4
References [1] So H.C., Ching P.C. and Chan Y.T. A new algorithm for explicit adaptation of time delay. IEEE Trans. Signal Processing, 42(7):1816–1820, July 1994. [2] Widrow B. and Stearns S.D. Adaptive signal processing. Prentice-Hall, 1985. [3] Laakso T.I., V¨alim¨aki V., Karjalainen M. and Laine U.K. Splitting the unit delay. Signal Processing magazine, 13(1):30–60, Jan. 1996. [4] Dooley S.R. and Nandi A.K. Adaptive subsample time delay estimation using Lagrange interpolators. Submitted to IEEE Signal Processing Letters, 1998. [5] Kootsookos P.J. and Williamson R.C. FIR approximation of fractional delay systems. IEEE Trans. Circuits & Systems II, 43:269–271, March 1996. [6] Knuth D.E. The art of computer programming, volume 1: Fundamental algorithms. Addison-Wesley, 1997. [7] Cain G.D., Murphy N.P. and Tarczynski A. Evaluation of several variable FIR fractional-sample delay filters. ICASSP, pages 621–624, 1994. 0
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