Effect of Sampling Errors On

Charles R. Baker Department of Statistics University of North Carolina Chapel Hill, North Carolina 27514 Louis R. Chow Department of Computer Science Tamkang College Taipei, Taiwan The "Bryn Processor" is a well-known array processor; its design depends on spectral densities of the signal and noise processes. Array output signal-to-noise ratio can be decreased because of errors in sampling times: (1) When array design parameters are estimated from sampled data; (2) when operating on sampled input data. We model errors in sampling times as a discrete-parameter random process. Expressions for array output SNR of the Bryn processor are determined for the two situations listed above. Numerical results are presented for some specific assumptions on signal, noise, and sampling error. Introduction This paper contains an analysis of the effect of sampling time jitter on the array output signalto-noise ratio and array gain of the "Bryn processor". This is a well-known processor [1], [2] for the detection of a stationary band-limited Gaussian signal in independent stationary band-limited Gaussian noise. The design of the processor is data-dependent, varying with the spectral properties of signal and noise. Errors in sampling time can arise in either the evaluation data (data being tested for signal) or in the design data (the data used to determine the components of the processor). Thus, one could have sampling jitter in the evaluation data and jitter in the design data, or jitter in evaluation data and no jitter in design data, or jitter in design data and no jitter in the evaluation data. Each of these combinations is considered here. We proceed as follows. First, the digitized Bryn processor is derived, assuming no jitter. The processor is then implemented with an array followed by a spectral filter. Assuming this implementation, we compute array output SNR for the various combinations of jitter in design data and evaluation data. The resulting expressions for array SNR are then analyzed to determine the effect of jitter. Finally, we present curves of array output SNR as a function of frequency, under two sets of assumptions on signal, noise, and jitter characteristics. Our results are stated in terms of array output SNR. Array gain is Usually defined as the ratio of output SNR to input SNR. Thus, the comparisons presented here for output SNR can be interpreted as comparisons of array gain. CH 1285—6/78/0000—0 143 SOD. m@1D7SrEEE 143 Derivation and Implementation of the Bryn Processor The basic philosophy of the Bryn processor is given in [1], and its implementation (for continuous-time operation) under the assumptions to be used here is given in [2]. We give the derivation of the digitized processor, and its implementation; this will also help to fix our definitions and framework. A set of K sensors is assumed; the output of the ith sensor is i(t), 0 < t < T0. L(t) is the K-component vector with i(t) as 1th component. If noise alone is present, ijt) = n1(t); if signalplus-noise is present, i1(t) = s1(t) + n(t). The following assumptions will be made. (1) Signal and noise are segments of independent and stationary zero-mean Gaussian processes, with spectral densities S(w) (for signal) and N(w) (for noise), the spectral density being the same for each sensor. (2) Signal is present at one sensor only if signal is present at all sensors. (3) The signal and noise spectral densities are essentially band-limited, with the frequency band having positive frequencies in the region (4) In implementing the processor, the signal will be assumed to be a plane wave, so that s(t) = s(t ri), where T1 is a time delay measured from some reference. (5) The vector of sensor outputs, L(t), can be reresented by — fu 0 jkGt < ') L_f T ' — — U uO where 0 = 2Tr/T0, and Z(kG) is a K-component column vector with th component Z1(kQ). In order for equation (1) to accurately represent L(t), one must have either f -'or T0 ÷ . We assume here (as in [1], [2]) that T0 is sufficiently large, so that equation (1) adequately represents L(t) Further, since Z(kD) is a Fourier coefficient, Z (kG) = Z(-kQ). (6) Z.(kG) Y.(kG) when noise alone is present; Z(kQ) X(kQ) + Y(kQ) when signal-plusnoise is present. We assume that and Y. can be adequately approximated by a discrete Fourier transform. Thus X.(kG) a (I/vM)2 s(ma)e3' (2) Authorized licensed use limited to: Tamkang University. Downloaded on March 29,2010 at 21:13:53 EDT from IEEE Xplore. Restrictions apply. Y. (kG) S l/v M-l n. (ma)emma 1 m=O 1 where a<l/2B is the nominal sampling interval and M=T0/a is the total number of samples taken. (7) The Fourier coefficients fiTo < k < fTo} are a set of independent random variables for each i = 1,... ,K. If one had T0 and equation (3) were actually a Fourier transform rather than an approximation by a DFT, this would necessarily follow from the fact that s1(t) and n(t) are Gaussian processes [4]. To optimally detect the signal, we form the likelihood ratio [4]. Since all of the information in the data is contained (under the above assumptions) in the positive-frequency Fourier coefficients, this involves taking the ratio of the joint probability density function of the positivefrequency Fourier coefficients under the hypothesis of signal present to the ratio of the joint probability density function under the hypothesis that signal is absent. Since the Fourier coefficients are assumed independent for different values of kG, and the data is Gaussian, the likelihood ratio becomes f T * A(Z) = y eyp{-l/2 kfT [Z T(kG){(cov[X(kQ) + Y(kQ)]Y1 (cov[Y(kG)]Y1}Z(kG)]) where y is a constant, and cov(.) denotes the covariance matrix. Let Q(kG) be the matrix of normalized correlation coefficients of the vector noise process, with ij element qjj (kG). N(kG)q1 (kG) is approximately (large M) the cross-spectral density for the noise process in the ith and Jth channels; q.. (kG) = E Y.(kG)Y(kG)/N(kG). The covariance matrix of the vector noise process is thus N (kG) Q (kG), while the covariance matrix of the plane wave ector signal process is S(kG)V(kG)V T(kG), where V(kG) is a K-component vector whose th component is j k3rj e (see matrices). frequencies If A(Z) represents the likelihood ratio for the Fourier coefficients {Z(kG), fiTo < k < fTo), then one has [3] -2 lo AZ = fTo S(1)IZ*T(kG)W(kG)I2 5 g ( ) LkfT B(kG) where W(kG) Q(kG)V(kG), and B(kG) N(kQ) [N(kG) + S (kG) V*T (kG) Q (kG)V(kG)]. The processor can be implemented as shown in Figure 1, where the Fast Fourier Transform in the th channel computes Z(kG) l/VM ui(ma)ema, and the spectral filter is F(kG) [S(kG)/B(kG)]. The output of the spectral filter is passed through a square-law device, and then summed over k = f1T0,... fuT0• This sun is -2 logA(Z), under the above assumptions, and is compared with a threshold. (3) We assume hereafter that the processor is implemented as shown in Figure 1; the array is that part of the processor preceding the spectral filter F(kG). Bryn Processor with Sampling Jitter Suppose now that the actual sampling times are (ma + where 5m E s(ma) represents the jitter for mtl sample. m' 0 < m < M-l} is assumed to be a random process independent of both signal and noise. Em is also assumed to be independent of for m n, and the probability distribution function of c is the sane as that of c , with characm juEm n teristic function (w) E e . The output of the FFT for the ith array element at angular frequency kG is then rM-l -jkna [Zi(kG)]jitter = l/VM Lm0 (ma + c)e The covariance matrices of the jittered signal and noise processes are derived in [3]. For the jittered signal one obtains E[X(kG)X*T(kG)] = U + 0(kG)2 S(kG)V(1GV*T(k() where D = [d..], 3w(T. -T.) 13 d. = (l/2ii 1Band e 1 3 [1 j(w) I ]S(w)du. For the jittered noise, the covariance matrix is 2 E[Y(kG)Y (kG)] = U + N(kG)14(kG)I Q(kG), where (4) U = [u] a/2 1Band1 0(w) I2}1 (w)N(w) dLs. The spectral densities for the jittered signal and noise thus become S(kG)= l/2r)fBd [1 I(w)l2]S(w)ds + I0(kG)12 S(kG) 2 N(kG) = "2Band [1 (w) 2]N(w)d + I0(kG) I N(kG). The normalized correlation matrix for the noise is now Q(kG) = (u11 + N(kG)jO(kG)12)-l [U + N(kG)I0(kQ)12Q(kG)]. In computing array output SNR, under the assumption that the design data contains jitter, one has [W(kG)] = Ql(kG)V(kG). Expressions for Array Signal-to-Noise Ratio The array output SNR at angular frequency k is defined as the ratio of the difference in the average power output when signal is present and signal is absent, to the average power output when signal is absent, or E{ Iw*T(kG) [X(kG)+Y(kQ)]2IW*T(kG)Y(kG) 12)