the strip width approaches that of the dielectric support, the effective permittivity decreases because the edge field lines will be concentrated in the air. 78, I Hilbert transformer, the IIR filter becomes a stable Hilbert transformer with linear phase characteristic. Secondly, an allpass network for the IIR Hilbert transformer is proposed to make the magnitude response constant by averaging the coefficients of numerator and denominator. Design method of IIR Hilbert transformers: The ideal frequency response of the Hilbert transformer is defined as The impulse response is then given by 03 04 0 5 06 0 7 w/d 0 8 0 9 1 0 Fig. 4 Effective dielectric constants us function of strip width a = I O m m , b = 100mm. c = Imm, d = 3 3mm, t = 3 5 p , E, = 9 8, frequency = IOGHz 0 IEE 1993 Electronics Letters Online No: 19931350 Because the impulse response for n < 0 cannot be realised and the number of filter coefficients must be finite, it is impossible to obtain an ideal frequency response using eqn. 2. Therefore, in designing a reference FIR Hilbert transformer, the Remez algorithm has been proposed to obtain a satisfactory frequency response [I]. 20 September 1993 M. Aubrion, H. Aubert, M. Ahmadpanah and H. Baudrand (Luboratoire d’elertronique, ENSEEIHT, 2 rue C. Camirhel 31071 Toulouse cedex, France) References I 2 3 4 5 K., and VAMASHITA, E.: ‘Transmission line aspects of the design of broad-band electrooptic traveling-wave modulators’, J. Lighrwave Technol., 1987, LT-55,pp. 316-319 YOUNG, B., and ITOH. T.:‘Analysis and design of microslab waveguide’, IEEE Trans., 1987, MTT-35, pp. 85Ck857 VAMASHITA, E., OHASHI, H., and ATSUKI. K.: ’Characterization of microstrip lines near a substrate edge and design formulas foe edge compensated microstrip lines’, ZEEE Trans., 1989, MlT-37, pp. 89&896 THORBURN. M., AGOSTON, A., and TRIPATHI. v.K.: ’Computation of frequency-dependent propagation characteristics of microstriplike propagation structures with discontinuous layers’, IEEE Trans., 1990, MlT-38, pp. 148-153 AUBERT, H., s o w , E.,and BAUDRAND, H.: ‘Origin and avoidance of spurious solutions in the transverse resonance method’, IEEE Trans., 1993, M T T 4 , pp. 45&456 Kalman ATSUKI. Fig. 1 Configurationfor designing IIR Hilberr transformer The transfer function of an IIR filter is written as where a, = 1. As shown in Fig. I , the IIR filter coefficients are controlled by a Kalman filter to minimise the output error e(t) between the output of the IIR filter and that of the reference FIR Hilbert transformer. If the output error is sufficiently small, then the IIR filter becomes an IIR Hilbert transformer with the same frequency characteristics as a reference FIR Hilbert transformer. The vectors of coefficients and signals are defined by 0 = [ai.az,.. .U,. bo, b i , . . . ,bnIT t-n+l), s(t + l ) ,. . . .s(t - n + 1)]T (4) z ( t + I ) = [ - y ( t ) ;...-y( Design of IIR allpass networks for Hilbert transformers S. Shimizu and T. Yahagi Indexing r e m : Digitalfilters, Allparsfiliers (5) where x ( t ) is an input signal which is a unit impulse sequence or Gaussian noise and y ( t ) is an output signal. The relationship between the input and output can then be expressed as + e(t A new method for designing an IIR allpass network for a Hilbert transformer is proposed. The coefficients obtained by a Kalman filter are used for an allpass network for an 1IR Hilbert transformer with a flat magnitude response. ELECTRONICS LETTERS 25th November 7993 + 1) = z T ( t + 1)8(t)- y ( t + 1) + 1) = E[{e(t+ 1) - P ( t + I)}’] R(t)z(t+ 1) kit + 1) = a;(t + 1)+ z T ( t + l ) R ( t ) z ( t + 1) u:(t Introduction: FIR filters are popular in the design of Hilbert trans- formers and in other applications [I]. However, their orders must be very high to achieve the required level of performance. To overcome this problem, some methods for designing IIR Hilbert transformers by using half-band filters have been proposed [2, 31. In this Letter, we propose a new approach to designing an IIR allpass network for a Hilbert transformer. First, the IIR filter coefficients are obtained by a Kalman filter as in [4] to have the same output characteristics as that of the reference FIR Hilbert transformer. If the difference can be neglected between the output of an IIR filter and that of the reference FIR + y ( t 1) = z T ( t l)O (6) We can obtain the unknown vector 0 by a Kalman filter. The computational procedures of a Kalman filter are as follows: d(t R(t + 1) = 8 ( t ) - e(t + l)k(t + 1) + 1) = [I- k(t + l ) z T ( t + I)]R(t) (7) (8) (10) (11) where the initial conditions are O(0)= 0 R(0) = y I y>0 A design example of a 16th-order IIR Hilbert transformer is shown in Fig. 2. The magnitude responses of IIR Hilbert transformer lH(z)l, its numerator [&)I, and denominator (I/A(z)l are Vol. 29 No. 24 2087 method would be available not only to a Hilbert transformer but also to general phase shifters. 0 IEE 1993 27 September 1993 Electronics Letters Online No:- 19931370 S. Shimizu and T. Yahagi (Department of Information and Computer Sciences, Chiba University. 1-33 Ya.voi-rho Inage-ku Chiba-shi. 263, Japan) References 1 0 1 0 2 03 0 4 normalised frequency 0 0 5 Fig. 2 Magnitude responses of IIR Hilbert transformers __ IIR _________ numerator denominator shown. The magnitude response lH(z)l is tlat only in the area around o = 0.25. However, the reference FIR Hilbert transformer is not good enough to be used for the design of an IIR Hilbert transformer; then, an IIR allpass network can be designed for obtaining an IIR Hilbert transformer with constant magnitude response in the following way. Consider the transfer function where CR(z)= 7"C(7').In this case the magnitude response IH'('(z)I is constant. Fig. 2 shows that the numerator and denominator of the IIR Hilbert transformer compensate their frequency characteristics for each other. Therefore the transfer function of an allpass network may be obtained by averaging the numerator and the denominator as follows: H'(z)= RALIINER. L . R . , and O L D . B.: 'Theory and application of digital signal processing' (Prentice Hall, Englewood Cliffs, NJ, 1975) 2 ANARI. R.: 'IIR discrete-time Hilbert transformers', IEEE Trans., 1987, A S P - 3 5 , pp. 111&1119 3 IKEHARA. M., TANAKA. H., and MATSUO. H : 'Design of IIR Hilbert transformers using Remez algorithm', IEICE Trans., 1991, J75-A. pp. 414-420 4 HENRIQUES. M A.A., and Y A H A G I .T.: 'Time and frequency domain design of approximately linear phase IIR digital filters', IEICE Trans., 1992, E75-A, (lo), pp. 1429-1437 Generalised filters and optimum sampling frequencies for the reconstruction of bandlimited signals from past samples J. Le Bihan and L.R. Watkins Indexing terms: Signal processing, Digitalfilters, Sampled data sysrems + + AR(z) B(z) A(z) B R ( z ) Generalised first- and second-order filters are presented for the reconstruction of bandlimited signals from past samples. In each case the minimum sampling frequency is derived and the corresponding filter coefficients obtained. A tradeoff exists between the rate of convergence of the reconstruction and the sampling frequency. From eqns. 12 and 13, the coefficient c, is given by c,=a,+b,-, i + O , 1 , . . . ,n (14) The frequency response of this IIR allpass network is shown in Fig. 3. We can design an IIR allpass network for a Hilbert transformer very easily by this method. ,o 1-2 -3 0 -60 m * -90 m 06 EO 4 -120 00 % S 02 ~ ," -1 50 a 0 n1o r m a0l i2Os e d frequency 03 0 4 0 -15 l 80 0 Fig. 3 Frequency response of allpass network for a Hilbert trans- Introduction: According to the Nyquist theorem, a bandlimited signal f i t ) can be reconstructed from its samples fikT) provided the sampling frequency I / T is at least twice the signal bandwidth f,. The reconstructed signal is theoretically obtained through an infinite summation of sinc functions weighted by the sample values. Methods have been developed that use past samples alone because in practice, only these are available. For example, Papoulis [I] considered n past samples weighted by the binomial coefficientsand showed that the reconstruction converges tofit) in the mean square sense provided I / T 2 3(2f,). In a recent Letter, F'illai and Elliott [2] introduced a particular second-order windowing function that enabled unconditional reconstruction to be achieved provided I / T 2 2(2f,). In this Letter, we generalise the first-order and second-order cases as considered, respectively, by Papoulis and Pillai and Elliott. In both cases we show that reconstruction can be achieved with lower sampling frequencies than those indicated above and give the minimum theoretical values. l Theory: Letfit) be a bandlimited signal such that F(o)= 0 for IwI >k f,and the approximation fn(t) of f i t ) be given by former ~ F n ( t ) = G , ( t ) F ( t ) = 11 - H " ( z ) ] F ( z ) magnitude P k Conclusion: A new design method for an IIR allpass network for a Hilbert transformer has been proposed. First, the coefficients of an IIR Hilbert transformer are obtained by using a Kalman filter. Stability of the IIR Hilbert transformer is then assured because the reference FIR filter is stable. Secondly, this method has been applied to obtain an allpass network for an IIR Hilbert transformer with a flat magnitude response. The proposed design 2088 (1: where z = exp(iwT) and l / T is the sampling rate to be determined. The approximation error E.(I) is given by En(t) and converges to 0 as n 7 -. = f(t) - fn(t) m len(t)12 dt = 1 2rr -w ELECTRONICS LETTERS (2: in the mean square sense if 7 IEn(w)12 dw -m 25th November 1993 Vol. 29 ~ 1 No. 24 _-

1/--страниц