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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
_-
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