close

Вход

Забыли?

вход по аккаунту

?

el%3A19810635

код для вставкиСкачать
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
Документ
Категория
Без категории
Просмотров
2
Размер файла
256 Кб
Теги
3a19810635
1/--страниц
Пожаловаться на содержимое документа