These results can be used to design filters with constrained phase as follows: (a) Constrain R(a)) to be 1 cos (OD1) at the start of the transition band (<I>D1 is the required phase deviation there) and 0 at the end of the transition band. Using optimisation on eqn. 6, calculate xf(j) and hence calculate the achieved R(<D). 03 with constraints (b) From the achieved R(co), calculate the required values of Q{(Oi) to get the required values of phase at the frequencies where phase is to be constrained. -10 normalised frequency, aiT I8377H Fig. 2 Phase response (c) Use optimisation with eqn. 7 to determine y/(j). — O — Constrained point Design examples: The optimisation method used to produce these results is the same as that described in Reference 3. Two filters are shown: one has constrained phase in the transition band, the other unconstrained. Both filters have the gain constrained to be 1 and 0 at the edges of the transition band. These points (at normalised frequencies of 0-2 and 0-3, respectively) are marked by circles on Fig. 1, and, between these frequencies, desired values cannot be specified for the optimisation program. The required phase deviation is sin (2coT), and it can be seen from Fig. 2 that the phase of the unconstrained filter deviates rapidly from the desired value in the unspecified transition region. However, the constrained filter (constrained points normalised 0-r0 -50 | 01 Conclusions: In this letter it has been shown that FIR digital filters with nonlinear phase characteristics can be designed to specific phase requirements in the transition region, even though the gain is not specified in this region. T. F. LIAU M. A. RAZZAK L. G. CUTHBERT 0-2 References 1 -100-1 with constrained phase 2 I -200-] CUTHBERT, L. G.: 'Optimizing non-recursive digital filters to nonlinear phase characteristics', Radio & Electron. Eng., 1974, 44, pp. 645-651 HOLT, A. c , ATTiKiouzEL, J., and BENNETT, R.: 'Iterative technique for designing non-recursive digital filters to non-linear phase characteristics', ibid., 1976, 46, pp. 589-592 i 3 £ -250 \ A-RAZZAK, M., and CUTHBERT, L. G.: 'Performance comparisons for nonrecursive digital filters with nonlinear phase characteristics', Electron. Lett., 1978, 14, (12), pp. 370-372 -300-1 -350 J 12th October 1981 Department of Electrical & Electronic Engineering Queen Mary College, University of London Mile End Road, London El 4NS, England frequency, CD I «/ -150 4 XI en being marked with circles) gives a good fit to the required response but with the penalty that there is a deterioration in amplitude response (Fig. 1). The amount of deterioration in amplitude response depends, naturally, on the particular example, especially on the number of constrained points being used. 4 without constrained phase -400 5 A-RAZZAK, M., and CUTHBERT, L. G.: 'Two-dimensional non- recursive digital filters designed to arbitrary phase specifications', IEE J. Electron. Circuits & Syst., 1979, 3, pp. 219-224 STEIGLITZ, K.: 'Design of FIR digital phase networks', IEEE Trans., 1981, ASSP-29, pp. 171-176 0013-5194/81/240910-02$!. 50/0 o- 05 10 -1 -2-3 1-5 xlO" EQUIVALENT-TIME SIGNAL ACQUISITION AND PROCESSING 20 Indexing terms: Signal processing, Noise, Optimal filtering normalised frequency ,uuT 1837/11 Equivalent-time signal acquisition in the presence of stationary additive noise is considered and contrasted with 'realtime' transient recording. It is shown how the substantial noise decorrelation encountered with equivalent-time methods can markedly influence the form of an optimum post-processor and the signal/noise ratio attainable. -4- Introduction: Equivalent-time signal acquisition, widely employed in digital waveform-processing systems, makes use of the sampling oscilloscope principle. Hence signal samples which are adjacent in memory—closely spaced in 'equivalent time'—derive from different realisations of the signal appearing widely separated in 'real time'. Formally, consider a periodic signal x{t), related aperiodic signal a{t) and a stationary additive noise process n(t), all strictly bandlimited to less than 1/2A with normalised frequency x 10 ,aiT x(t) = x(t+T) Fig. 1 a Amplitude response No. of coefficients = 80 No. of constrained points for constrained-phase design = 17 b Enlargement of passband for filter without constrained phase c Enlargement of passband for filter with constrained phase ELECTRONICS LETTERS 26th November 1981 Vol.17 = Yj a(t + iT) We acquire a representation of a(t) corrupted by noise by sampling x(t) + n(t). The sampled equivalent-time signal is given by ye(iA) = x{i{k, T + A)} + n[i(k, T + A)} k t e Z+ = ae{iA) + ne(iA) No. 24 911 In the absence of noise, ye{iA) - ae(iA) = a(iA). The ne(iA), however, correspond to widely spaced samples of n(t), ne(iA) £ n(iA), and are thus uncorrelated if (/c,T + A)2 $> m2, the normalised second temporal moment of the noise autocorrelation function. This renders ne a white-noise process. In contrast, 'real-time' transient signal acquisition schemes derive samples from a single realisation of a(t) + n(t) to give Results are presented in Table 1 for four illustrative examples: (a) b a n d l i m i t e d w h i t e noise, N(f) = n/2, \f\ coincide for A = A m a x < B. T h e results (b) bandlimited white signal, \A(f)\ = a, | / | < B yt{iA) = x(iA) + «(/A) (d) as (b) with = a(iA) + nt(iA) If n(t) is a coloured-noise process, n, is similarly coloured and adjacent samples are correlated; while lst-order statistics for n, ne, n, coincide, higher-order statistics for ne can differ markedly from those for n, nt. The purpose of this letter is to illustrate quantitatively how this can significantly influence the form of subsequent signal processing appropriate to ye compared with yt. By way of example, we consider optimal detection filtering to achieve maximum signal/noise ratio. In the interests of clarity, we neglect finite word and record length effects and assume ideal interpolation (zonal lowpass filtering) applied to the sampled signals ye, yt. This enables us to isolate effects intrinsic to the signal acquisition processes and facilitates an analysis in terms of equivalent ideal analogue processors. Noise spectral density: Without loss of generality, we normalise the problem such that N(f) = X(f) = 0, | / | > l / 2 , giving A < A ^ = 1. The input noise power spectral density N(f) is preserved by sampling and ideal interpolation in real-time recording, since I N(f) | = 0, | / 1 > 1/2A, so that Nt(f) = N(f), \f\< 1/2A. For equivalent-time systems, however, n(t) is generally grossly undersampled. Multiple aliasing occurs to yield a substantially white spectrum. This insensitivity of the spectrum of undersampled noise to the spectrum of the input process is well known and has been studied in detail in another context.1"3 The variance of the sample weights, and hence of the reconstructed process, corresponds to the input noise variance N(f)df= - 1/2A \f\<B- Concluding remarks: The analysis and examples presented demonstrate that widely different results may be obtained by processing optimally single representatives of a signal-plusnoise waveform acquired using 'equivalent-time' and 'realtime' recording methods. We note that, for equivalent-time systems, the attainable signal/noise ratio is proportional to sample density, while, for real time systems, it is insensitive to sample density within the limits of the sampling theorem. Consequently, it may often be the case that a higher signal/noise Table 1 COMPARISON OF SIGNAL/NOISE RATIO EXPRESSIONS FOR FOUR ILLUSTRATIVE EXAMPLES SNRf SNR( IE 1 A r, ".I mdf EC E N(f) df (c) 2E/M 2E /£ 1 \B 2A (d) Hence ajL 1 2^B^ 1 /i 1 2E1/A r\ \BJ 1 ™, IE A r, (c) with/, <g B 1/2A (ti 1+ 1 2£ 1 tan" | 1+ T n \BJ 2 3\/J Ji 00 el2 =A f N(f)df (d) with/, <? B Optimal filtering: It follows that optimum processors—in the sense of maximising the signal/noise ratio—for the 'equivalenttime' and 'real-time' acquired signals are, respectively, a matched filter and a whitened matched filter.4 The maximum signal/noise ratios attainable are thus A(f)A*(f) df SNR F T = —5 A [ N(f) df * SNRBT= ^ 2£ B 2 A r] /, n 2 £ 1 IB] 7M7W ratio is obtained by processing an equivalent-time signal rather than a real-time signal with the same number of samples. However, this is rather dependent on the signal and noise spectra. As an illustration, 'white' signals with coloured noise concentrated at high frequencies [case (c)] and concentrated at low frequencies [case (d)] have been examined. In both cases, we find that SNR £ r < SNR ? r by approximately 2(fi/B) if A = Amax. We note in passing that the addition of signal averaging modifies the detail of the analysis but does not alter the general tenet. 00 - v J. J. O'REILLY 12th October 1981 Department of Electrical Engineering Science University of Essex Colchester CO4 3SQ, Essex, England ^ . where A{f) is the Fourier spectrum of a(t) with complex conjugate A*(f). Note that increasing the sample density above the minimum required by the sampling theorem has no effect with real-time recording, while, for equivalent-time systems, the signal/noise ratio attainable is proportional to sample density I/A. Illustrative examples: Consider A(f), N(f), bandlimited such that | A{f) | = | N(f) | = 0 , | / 1 > B. This determines a maximum A, A^^ = 1/2B. Also, we define signal energy References 1 CATTERMOLE, K. w.: 'Principles of pulse code modulation' (Iliffe, 1969), pp. 156-159 2 LEVER, K. v., and CATTERMOLE, K. W.: 'Quantising noise spectra', Proc. IEE, 1974, 121, (9), pp. 945-954 3 LEVER, K. v.: 'Quantising noise spectra'. Digest of colloquium: Mathematical topics in telecommunications—nonlinear operations on stochastic processes, University of Essex, Feb. 10th, 1981, pp. 54-70 4 O'REILLY, J. J.: 'Matched filter receivers'. Digest of colloquium: Mathematical topics in telecommunications—calculus of variations, University of Essex, Nov. 25th, 1980, pp. 39-46 E= 0013-5194/81/240911-02$l .50/0 912 ELECTRONICS LETTERS 26th November 1981 Vol.17 No. 24

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