TIME-DELAY ESTIMATION USING FARROW-BASED FRACTIONAL-DELAY FIR FILTERS: FILTER APPROXIMATION VS. ESTIMATION ERRORS

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1 TIME-DEAY ESTIMATION USING FARROW-BASED FRACTIONA-DEAY FIR FITERS: FITER APPROXIMATION VS. ESTIMATION ERRORS Mattias Olsson, Håkan Johansson, an Per öwenborg Div. of Electronic Systems, Dept. of Electrical Engineering, inköpings universitet Campus Valla, SE-58 8 inköping, Sween phone: , {matol, hakanj, perl}@isy.liu.se web: ABSTRACT This paper provies error analysis regaring filter approximation errors versus estimation errors when utilizing Farrowbase fractional-elay filters for time-elay estimation. Further, a new technique is introuce which works on batches of samples an utilizes the Newton-Raphson technique for fining the minimum of the corresponing cost function.. INTRODUCTION The nee for estimating time-elays between two signals arises in many ifferent fiels, incluing biomeicine, communications, geophysics, raar, an ultrasonics. In [], a technique utilizing Farrow-base igital fractional-elay (FD) filters [] was introuce for this purpose. The use of Farrow-base FD filters has two major avantages over other elay estimation techniques working in the igital omain. First, it is eminently suitable to hanle elays that are fractions of the sampling interval. This is in contrast to cross correlation-base methos that require aitional interpolation []. Secon, it can hanle general banlimite signals. This is in contrast to techniques that assume a known input signal, like a sinusoial signal [4]. In [], the iea of using Farrow-base FD filters for elay-estimation was propose. However, no analysis was provie as to filter approximation error versus estimation error. Such an analysis will be provie in this paper. Furthermore, a new technique is introuce that works on batches of samples an utilizes the Newton-Raphson technique for fining the minimum of the corresponing cost function. Since the fractional elay of Farrow-base FD filters is governe by only one parameter, analytical erivatives can be erive for this purpose. Thereby, the problems associate with the use of numerical erivatives are avoie. Following this introuction, Section will provie a short introuction to time-elay estimation using FD filters, followe by a presentation of the elay-estimation technique in Section an an error analysis in Section 4. In Section 5 we write FD filter esign an in Section 6 we verify the error analysis. Finally some conclusions are rawn.. TIME-DEAY ESTIMATION USING FD FITERS Two (or more) iscrete-time signals, originally coming from one source, might experience ifferent elays. We moel this as x(n) = x a (nt)+e () v(n) = x a (nt T)+e () F The cost function F for =.5 an w =π/ Figure : Example of the cost function F for a sinus input with =.5. where is the unknown ifference in elay between the signals, T is the sampling perio an e an e are uncorrelate aitive noise. We assume that the elay is a fraction of T an that any integer sample elay has alreay been taken care of in a proper manner. Now, let v(n) act as a reference signal an let the other signal x(n) pass through a FD filter generating the output y(n, ) [see (7)]. The average square ifference function (ASDF) F() for a certain elay over a batch of N samples can then be written as F() = N n +N (y(n,) v(n)). () n=n An estimate of the unknown fractional elay in the reference signal can then be compute by minimizing F as = argmin F(). (4) An example of this cost function F for a sinusoial input can be seen in Fig.. Ieally, the function woul be equal to sin ( ω( ) ), which is approximately square for small, however, if noise, magnitue errors, elay errors, etc, are present, it will eviate from the square shape. Due to lack of space, the effect of noise is not inclue in this paper. It will not affect the filter approximation error vs. estimation error which is in focus in this paper. The noise can be reuce to a level that is negliable compare to the filter approximation error by increasing the batch length N.

2 upate equation is here n+ = n + F ( n ) F ( n ) (9) Figure : The moifie Farrow-base FD Filter where G k (z) are linear-phase FIR filters. In an implementation elays must ae to make sure the elay is equal for all the subfilters.. PROPOSED TIME-DEAY ESTIMATOR. Farrow-Base FD Filters The esire frequency response of an FD filter is H es (e jωt,) = e j(d+)ωt, ωt < ω c T < π (5) where D is an integer elay an is a subsample (fractional) elay. In practice an approximation of (5), esigne for frequencies up to ω c T, is use. The FD filters in this paper make use of the moifie Farrow structure shown in Fig. [5], [6] which is a moification of the original Farrow structure [] in that the subfilters G k (z) are linear-phase FIR filters. The avantage of using linear-phase filters is that their impulse responses are symmetrical an can be implemente with fewer multiplications. The overall transfer function can thus be written as H(z) = k= k G k (z) (6) where G k (z) are linear-phase FIR filters of either o or even orer, say M k, an with symmetric (anti-symmetric) impulse responses g k (n) for k even (k o), i.e., g k (n) = g k (M k n) [g k (n) = g k (M k n)] for k even (k o) [7]. When is fixe, the overall filter approximates an allpass filter with the fractional elay, provie that the subfilters have been esigne in a proper manner [5], [6]. The output y(n,) from the FD filter in Fig. can be written as where y(n,) = k= k y k (n). (7) y k (n) = g k (n) x(n). (8) In this paper, it is assume that the orer M = max{m k } is even. The reason is that the elay of H(z) is then an integer, D = M/, when =. This is suitable for the timeelay estimation problem. The filter is normally optimize for.5 to cover one sampling interval.. The Estimator To fin the minimum of F with respect to, the well-known Newton Raphson (NR) algorithm is use. The algorithm is iterative an tens towars the closest zero of a oneimensional function, in this case the erivative of F. The where F () an F () are the erivatives of F(). The principle of the estimator is epicte in Fig.. Compare to the common east-mean-square (MS) algorithm we o not nee to specify a step length since it is compute explicitly as /F ( n ). For a perfectly quaratic function this step length is optimal an only one iteration is neee. However, in a real situation, a few more iterations are neee. Each new iteration can use a new batch of samples N or the same batch, epening on the amount of memory at han an if the estimation is run on-line or off-line. To be able to calculate the next iterative estimate in (9), the first an secon erivatives of F(n,) with respect to are neee, which can be calculate as an F () = N n +N (y(n,) v(n))y (n,) () n=n F () = n +N N (y(n,) v(n))y (n,)+(y (n,)). n=n () When a Farrow-base FD-filter is use, the erivatives of (7) can be calculate analytically as an y (n,) = y (n,) = k= k k y k (n) () k(k ) k y k (n), () respectively. In Fig. 4, a straightforwar implementation of the erivatives () an () can be seen. Note that the subfilters G k (z) only have to be use once. Since a Farrow-base FD filer is an approximation of the ieal response in (5), the approximation errors will affect the estimator performance. In the next section we will investigate the performance of the algorithm when the FD filter is not ieal. 4. APPROXIMATION ERROR VS. ESTIMATION ERROR We moel the non-ieal FD filter as H(e jω,) = (+δ(ω,))e jω(d++ (ω,)) (4) where δ(,ω) enotes the magnitue error an (,ω) enotes the elay error of the FD filter for a certain fractional elay an a certain angular frequency ω. D is the integer elay of the FD-filter. This is a formulation which is general for all types of FD filters, not just the Farrow-base FD filter use in this paper, an the resulting expressions in this section is hence inepenent of the FD filter type. In Section 5, expressions specific for Farrow-base FD filters are erive.

3 Figure : The principle of the propose estimator. Accumulator F () x(n) G (z) G (z) G (z) G (z) y (n) y (n) y (n) y (n) v(n) y(n,) Aer Aer (.) Aer xx ( ) Accumulator F () G (z) y (n) Figure 4: A straightforwar implementation of the erivatives. Now, if we assume that x(n) is sinusoial (for the choice of signal see the conclusions), the output from the FD filter at a frequency ω is y(n,) = (+δ)sin(ω (n )) (5) an the reference signal v(n) is v(n) = sin(ω (n )). (6) Henceforth, the epenence of δ an δ on an ω will be omitte in the notation for the sake of clarity. After inserting (5) an (6) into () an some simplifications we arrive at F (ω,) = δ [(+δ) cos(ω( + ))]+ +(+δ)ω(+ )sin(ω( + ))+ [ (+δ)ω (+ )sin(ω ( ( + + )+ϕ) N (+δ) ω (+ )sin(ω ( ( + )+ϕ)+ + δ cos(ω ( ( + + )+ϕ) (+δ)δ cos(ω ( (+ )+ϕ) ] sin(nω ) sin(ω ) = F (ω,)+f N(ω,) (7) where F oes not epen on the batch length N an F N contains the terms that o. At the minimum of F, as we will see later, + will be close to. Using this fact we can rewrite F N as F N(ω,) + = = δ sin(nω ) N sin(ω ) [ (+δ)ω (+ ) sin(ω ( + N ))+δ cos(ω ( + N )) ]. (8) Since δ an δ usually are very small, (8) will be small even for a small N. However, when ω tens towars, (8) will ten towars δ δ, inepenently of N. The iterative Newton-Raphson algorithm in (9) will, if the function is behaving well, converge towars the zero of F. If we assume that N is so large that F N can be approximate as zero for all an ω, it can be seen from (7) that if δ an are zero, F will be zero when =, irrespective of δ. On the other han, if δ is small, but not zero, the effect of a constant magnitue error δ will affect the estimator. The effect of a elay error erivative is normally neglible since it tens to be small compare to. Unfortunately, it is impossible to irectly from (7) analytically fin the that makes F zero. It can be one numerically an to fin an approximation of the estimation error we o a first-orer Taylor expansion of the sine an cosine in F an write it as F δ (δ + ω ( + ) ) (+δ)(+ )ω ( + ) (9)

4 If + an +δ are approximate by, we can solve for err an get err δ + ω δ ω (δ ) δ. () This expression can be use to preict the final estimation error. The first part becomes small when ω is large compare to (δ ) δ an the error will then be ominate by. For small ω, if δ or δ are not zero, the error eviates more an more from. If = the error becomes err =. 5. FITER DESIGN As was seen in (7) the main error source is the elay error which irectly affects the estimate. However, when the erivative of the magnitue error, δ, is nonzero the magnitue error δ an the frequency ω will come into effect. To fin an FD filter which is optimize for small estimation errors, the expression in () can be use as cost function in an optimization problem. The optimization problem can for example be formulate as a minimax problem Simulate error x 6. Simulate estimator error, N= ω x 5. Simulate estimator error, N= Figure 5: The simulate estimation error for N = an N = samples.. < ω < π/. Simulate error ω. minimize ε subject to err (ω,) < ε () which is then optimize over a range of angular frequencies ω k an elays k. Thefminimax-routine in MATAB efficiently implements a sequential quaratic programming metho that is capable of solving the minimax constraint problem in (). However, the solution is not guarantee to be the global minimum as the problem is nonlinear. The erivative of δ, which is neee to calculate the expecte estimation error, can be foun analytically by noting that the magnitue of the Farrow-base FD filter can be written as H(,e jωt ) = / k= k G (k)r (ωt)+ (+)/ + j k G (k )R (ωt) = A()+ jb() = +δ () where G (k)r an G (k )R are the zero phase frequency responses for the even-orer linear-phase FIR filters with impulse responses g k (n). Since the erivative of the magnitue is equal to the erivative of the magnitue error we can calculate δ as δ = A()A ()+B()B () A() + B() () where A () an B () can be calculate as an / A () = (+)/ B () = k k G (k)r (ωt) (4) (k ) k G (k )R (ωt). (5) Figure 6: The expecte error..π < ω < π/. 6. PERFORMANCE OF THE ESTIMATION ERROR PREDICTION To verify the error analysis, a number of simulations were performe. An FD filter with = 7 subfilters was optimize for minimal error up to the frequency ω = π/. The resulting even-orer subfilters ha orers M k ranging from 8 to 8. The maximum magnitue error was δ max =.8 4, the maximum elay error was max =.68 5 an the maximum erivative of the elay error was max =. 4. In Fig. 5 the simulate estimation error for N = an N = samples can be seen. As preicte by () the estimation error increases for ecreasing frequencies when there is a magnitue error. The reason for the lower performance for N = samples is that N must be larger for the variance of F N to become small, especially when δ an δ are relatively large, which they ten to be for small ω an large. In Fig. 6 the expecte error err can be seen. Compare to the simulate error in Fig. 5 a smaller frequency range has been use to enhance the etails.

5 Difference x Difference between expecte an simulate error, N=.. Figure 7: The ifference between the expecte estimation error an the simulate error, N =..π < ω < π/. Difference x Figure 8: The ifference between the expecte estimation error an the simulate error, N =..π < ω < π/. When err was erive we omitte F N, assuming that N is infinite or at least large enough. If the batch size N is ecrease the expecte estimation error will increase, which can be seen in Fig. 7 an 8. As expecte, the ifference between the estimate error an the simulate error is increase when N becomes smaller so that F N no longer can be approximate to be zero. For N = samples the estimator still performs quite well, see Fig. 5, but the expecte error err no longer works as a goo estimation of the actual error since the expecte error is almost as large as the actual ifference between the simulate an expecte error. This is ω. ω.... because for small N noise an the variance of F N will ominate, while for large N the filter approximation errors will ominate. 7. CONCUSIONS We have presente a novel metho to estimate the elay error between two sets of samples using a FD filter. The iea is to use an iterative, Newton-Raphson-base, estimator to minimize the mean-square-ifference between a reference signal an a signal with an unknown elay. The effects of a nonieal FD filter with magnitue an elay errors have been stuie theoretically an in simulations. An expression of the expecte estimation error err was erive, which can be use to optimize the FD filter in the estimator for a minumum estimation error. The expecte estimation error err is a bias, or in other wors, the best one we can achieve without noise, i.e when N. In the analysis we have assume a single sinusoi, but the time elay estimator can be use for more general banlimite signals. However, in this case, the estimation error will iffer from the case where a sinusoial is use. The analysis one in this paper is still useful since it gives an insight into errors cause by the nonieal interpolation performe by the fractional-elay filters. Aitionally, in some applications where the training sequence may be chosen freely we can actually choose a sinusoi an achieve the limits compute in this paper. The time-elay estimation metho can easily be extene to more sets of samples, using one set as the reference, which coul be useful e.g. in the calibration of time-interleave analog-to-igital-converters (TIADCs). REFERENCES [] S. R. Dooley an A. K. Nani, On explicit time elay estimation using the Farrow structure, Signal Processing, vol. 7, pp. 5 57, 999. [] C. W. Farrow, A continously variable igital elay element, in Proc. IEEE Int. Symp. Circuits & Syst., June , vol., pp [] J. O. Smith an B. Frielaner, Aaptive interpolate time-elay estimation, IEEE Trans. Aerospace, Electronic Syst., vol. AES-, no., pp. 8 99, Mar 985. [4] D.. Maskell an G. S. Woos, The iscrete-time quarature subsample estimation of elay, IEEE Trans. Instrum. Meas., vol. 5, no., pp. 7, Feb. [5] J. Vesma an T. Saramäki, Optimization an efficient implementation of FIR filters with ajustable fractional elay, in Proc. IEEE Int. Symp. Circuits Syst., June 9 997, vol. IV, pp [6] H. Johansson an P. öwenborg, On the esign of ajustable fractional elay FIR filters, IEEE Trans. on Circuits an Syst. II: Analog an Digital Signal Processing, vol. 5, no. 4, pp , Apr.