IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 6, JUNE 2000
TABLE II
COMPARISON BETWEEN BURG'S, RER AND SPAC METHODS; m
= 2000.
n
1803
= 105
;
a very good approximation of the SPAC one. Table II indicates that
the four methods are equivalent in (1; 1) estimation. Otherwise, the
Burg-type extensions give very bad results for the other parameters.
This seems to be a consequence of the presence of constraints in the recursive filters construction. This clearly appears in the 12 case, where
the bias corresponds to overestimation of residual variances of each period. It can be expected that this failure has an increased effect when
the model order is large.
[9] W. A. Gardner, Ed., Cyclostationarity in Communications and Signal
Processing. New York: IEEE, 1994.
[10] E. G. Gladysev, “Periodically random sequences,” Sov. Math., vol. 2, pp.
385–388, 1961.
[11] S. M. Kay, “Recursive maximum likelihood estimation of autoregressive processes,” IEEE Trans. Signal Processing, vol. 41, pp. 56–65, Jan.
1993.
[12] S. Lambert-Lacroix, “Fonction d'autocorrélation partielle des processus
à temps discret non stationnaires et applications,” Ph.D. dissertation, J.
Fourier Univ., Grenoble, France, 1998.
[13] M. Morf, A. Vieira, D. Lee, and T. Kailath, “Recursive multichannel
maximum entropy spectral estimation,” IEEE Trans. Geosci. Electron.,
vol. GE–16, pp. 85–94, 1978.
[14] A. H. Nuttall, “Positive definite spectral estimate and stable correlation
recursion for multivariate linear predictive spectral analysis,” Naval Underwater Syst. Cent., New London, CT, NUSC Tech. Doc. 5729, Nov.
14, 1976.
[15] M. Pagano, “On periodic and multiple autoregressions,” Ann. Stat., vol.
6, no. 6, pp. 1310–1317, 1978.
[16] D. T. Pham, “Maximum likelihood estimation of autoregressive model
by relaxation on the reflection coefficients,” IEEE Trans. Acoust.,
Speech, Signal Processing, vol. ASSP–38, pp. 175–177, Jan. 1988.
, Fast scalar algorithms for multivariate partial correlations and
[17]
their empirical counterparts, Grenoble, France, Rap. Recherche, LMC
(URA 397), RR 903-M, 1992.
[18] H. Sakai, “Circular lattice filtering using Pagano's method,” IEEE Trans.
Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 279–287, Feb.
1982.
[19] O. N. Strand, “Multichannel complex maximum entropy (autoregressive) analysis,” IEEE Trans. Automat. Contr., vol. AC-22, pp. 634–640,
July 1977.
Nonuniform
-Band Wavepackets for Transient Signal
Detection
Seema Kulkarni, V. M. Gadre, and Sudhindra V. Bellary
VII. CONCLUSION
The SPAC and RER methods are extended to the periodically correlated processes case. They are compared with the Yule-Walker method
and two extensions of Burg's method. Simulation results show that they
eliminate some shortcomings of the other methods. On the other hand,
we also consider the relationship between these different approaches
and those related to the stationary multivariate process. The advantage
of scalar approaches is to avoid use of matrices and to allow estimation
of autoregressive models of any order.
Abstract—In this paper, we present a scheme to detect significantly overlapping transients buried in white Gaussian noise. A nonuniform
-band
wavepacket decomposition algorithm using
-band, translation-invariant
wavelet transform (NMTI) is developed, and its application to transient
signal detection is discussed. The robustness of the NMTI-based detector
is illustrated.
REFERENCES
I. INTRODUCTION
[1] G. Boshnakov, “Periodically correlated sequences: Some properties
and recursions,” Luleå Univ., Luleå, Sweden, Res. Rep. 1, Div. Quality
Technol. Stat., 1994.
[2] J. P. Burg, “Maximum Entropy Spectral Analysis,” Ph.D. dissertation,
Dept. Geophys.. Standford Univ., Stanford, CA, 1975.
[3] S. Dégerine, “Canonical partial autocorrelation function of a multivariate time series,” Ann. Statist., vol. 18, no. 2, pp. 961–971, 1990.
[4]
, “Sample partial autocorrelation function,” IEEE Trans. Signal
Processing, vol. 41, pp. 403–407, Jan. 1993.
[5]
, “Sample partial autocorrelation function of a multivariate time
series,” J. Multivariate Anal., vol. 50, no. 2, pp. 294–313, 1994.
[6]
, Partial autocorrelation function of a nonstationary time series with
application to periodically correlated processes, preprint, 1998.
[7] B. W. Dickinson, “Autoregressive estimation using residual energies ratios,” IEEE Trans. Inform. Theory, vol. IT–24, pp. 503–506, Apr. 1978.
[8]
, “Estimation of partial correlation matrices using Cholesky decomposition,” IEEE Trans. Automat. Contr., vol. AC–24, pp. 302–305, Apr.
1979.
Index Terms—
wavepackets.
-band, receiver operating characteristics, transient,
The transient signal detection problem has been studied using various techniques with different assumptions regarding a priori information about the signal like time of arrival, duration, time-bandwidth
product and relative bandwidth, signal model, etc. [1]–[3]. In case of no
prior knowledge, wavepackets and their variations have shown better
performance due to their property to capture the transient signal in a
Manuscript received January 5, 1999; revised September 23, 1999. This work
was supported in part by the Naval Research Board of India, CDAC Pune, and
the All-India Council of Technical Education. The associate editor coordinating
the review of this paper and approving it for publication was Dr. Lal C. Godara.
S. Kulkarni is with the Centre for Development of Advanced Computing
(CDAC), Pune, India.
V. M. Gadre and S. V. Bellary are with the Department of Electrical Engineering, Indian Institute of Technology, Powai, Mumbai, India.
Publisher Item Identifier S 1053-587X(00)04048-4.
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fewer number of coefficients [4], [5]. The problem of matched subspace
detection has also been addressed [10] for detection of one signal. In
the case of multiple transients, however, the problem becomes more
difficult, and it becomes necessary to develop a transform that is sufficiently generic.
In this paper, we have extended the transient signal detection algorithm based on UMTI and UETI1 [4],[5] for multiple transients overlapping in either of the time or frequency domains. No a priori information is assumed. We also have developed a new transform, a nonuniform
M -band wavepacket decomposition algorithm using the translation invariant wavelet transform (NMTI), which is an extension of UMTI, and
discussed its application to the detection of single as well as multiple
transients. Detectors based on UMTI and NMTI are compared. The
latter is seen to be more robust.
II. NONUNIFORM M -BAND TRANSLATION-INVARIANT WAVELET
TRANSFORM (NMTI)
In M -band wavepacket decomposition, each node of the tree is decomposed using the same set of filters into M bands [4]. Here, we decompose a node in either P or Q bands, depending on which decomposition minimizes the cost. The cost function used in the algorithm,
as well as elsewhere, is an additive and shift-invariant cost function
C (x) =
0
jx j
i
2
j j
log xi
2
(1)
i
which, when minimized, minimizes the Shannon entropy function [7]
E (x) =
0
jx j2
kxk2
i
i
log
jx j2 :
kxk2
i
(2)
Consider for any M 2 Z , M QMF’s ai (n) for i = 0; 1;
satisfying the following equations:
111; M 0 1
ai (n) = M i; 0
n
ai (n + kM )aj (n + lM ) = M i; j k; l :
(3)
NMTI
define the NMTI Fpq
as
Fp (x);
Fp S (x);
if C [Fp (x)] = C min
if C [Fp S (x)] = C min
..
.
NMTI
Fpq
(x) =
Fp S P 01 (x);
Fq (x);
Fq S (x);
..
.
Fq S Q01 (x);
if C Fp S P 01 (x) = C min
if C [Fq (x)] = C min
if C [Fq S (x)] = C min
(6)
if C Fq S Q01 (x) = C min.
The whole wavepacket decomposition is performed using
NMTI
Fpq
(x). After the entire decomposition is performed, the best
basis is searched using bottom up search algorithm [8]. The result is a
nonuniform frequency decomposition.
A. Computational Complexity
Calculating level i NMTI requires first calculating the single level
M -band wavelet transforms for P and Q circulant signal shifts. For a
length-m signal, each transform requires O(m) operations. This gives
O(m) operations for the transform of all (P + Q) shifts. If NMTI is
performed to give a full wavepacket decomposition, then there can be
a maximum of a total of logM m levels, where M is the smaller of the
P and Q. Hence, the total work of the NMTI is O(m logM m), where
M is the smaller of the P and Q.
III. DETECTION ALGORITHM FOR MULTIPLE TRANSIENTS
The detection problem consists of choosing between the signal absent hypothesis H0 and signal present hypotheses H1 1 1 1 Hk , where k
is the number of transients. The idea is to detect the first transient by
treating the overall waveform as comprised being of transient signal
plus remainder signal and noise. At every step, the previously detected
transient are zeroed out, and a new transient is identified. This could
be expressed by the following steps:
n
The filter a0 (n) represents the lowpass filter, aM 01 (n) represents
the highpass filter, and the remaining filters represent the bandpass filters. Let the operators Fi; p be defined for i = 0; 1; 1 1 1 ; P 0 1 and
Fi; q be defined for i = 0; 1; 1 1 1 ; Q 0 1 on a signal s, p = (P )N ,
q = (Q)N for P; Q; N > 0, P; Q; N 2 Z by periodic convolutions
Fi; p (s) =
s(n)ai (n
n
Fi; q (s) =
s(n)ai (n
0 P k)
(4)
0 Qk):
(5)
The operators correspond to filtering the input signal with the
filters fai g followed by a decimation by a factor of M . Let S
represent the periodic shift operator applied to sequences of
length n. Then, component-wise S [x(l)] = x(l 0 1) on the
time index, for addition, performed mod n. Let x represent a
subband sequence at level m to be decomposed by construction
of the full wavepacket decomposition. Then, on x, the NMTI calculates
111; F
p
S P 01 (x); Fq (x); Fq S (x);
and chooses that coefficient
function. That is, for
Cmin = min
set
that
q
q
1 1 1 ; F S 01 (x)
q
minimizes
111; C
C [F (x)]; C [F S (x)]; 1 1 1 ; C
C [Fp (x)]; C [Fp S (x)];
Fp S
Fq S
P
Q
the
cost
01 (x)
01 (x)
Q
For brevity, we refer to M -band wavepacket decomposition based on
M -band translation invariant wavelet transform [TI] as UMTI and that based on
M -band extended translation invariant wavelet transform as UETI throughout
1
this correspondence.
H 1: x(m) = s1 (m) + s(m) + n(m):
H 2: x(m) = s1 (m) + s2 (m) + s(m) + n(m)
..
.
k
Hk : x(m) =
si (m) + n(m):
(7)
i=1
n
Fp (x); Fp S (x);
H 0: x(m) = n(m):
The detector follows the steps described as follows.
1) Full wavepacket decomposition is performed using UETI,
UMTI, or NMTI.
2) Best basis is selected by minimizing the cost function as in (1)
over bases [8].
3) For J -level deep decomposition, a template at level L, block
B , and index I consists of coefficients from (I 0 (J 0 L))
to (I + (J 0 L)). The best basis is searched for the template
having the maximum energy. The maximum energy serves as the
detection statistic.
4) The detection statistic is compared with a threshold to choose
between hypotheses.
5) The maximum energy template is then forced to zero.
6) Steps 3)–5) are repeated k times, where k is the maximum possible number of transients in a signal.
A. Performance Measures
ROC and tiling pictures are used to evaluate the performance of the
detectors. ROC is a plot of probability of detection versus the proba-
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Fig. 1. ROC’s for detectors based on different transforms showing superiority
of NMTI over rest of the transforms. Upper-most ROC corresponds to the
detector based on matched filter. Below that one is ROC for NMTI. The
detectors based on other transforms, namely, two-band MTI, two-band ETI,
three-band MTI, and three-band ETI have more or less similar performance in
this case.
bility of false alarm for various values of the threshold. Tiling picture is
a time-scale representation of a signal. It is plotted with low frequency
bands below and higher frequency bands toward the top. The coefficients are indexed from left to right in order. Gray-scale represents
the magnitude of the coefficient, where more darkness corresponds to
higher magnitude.
Fig. 2. Tiling picture for the signal whose parameters are given in Section IV
using two-band UETI and three-band UETI.
IV. SIMULATION RESULTS
The received signal was simulated as a superposition of exponentially damped sinusoids buried in white Gaussian noise
k
0A [sin(Bi )]Ci + n(m)
exp
(8)
i=1
x(m) = s(m) + n(m), where
Ai
=i (m 0 0i );
= sin[!i (m 0 0i ) + i ];
Bi
Ci
=U (m 0 0i ) + n(m);
0i arrival time of the ith transient;
U (n) unit step function;
k
number of transients;
n(m) added Gaussian noise.
A large set of signals with wide ranges of , ! , and was generated,
with k ranging from one to five. The SNR is defined as the average
x(m) =
signal energy to mean-square noise ratio. All the simulations were carried out using NMTI (P = 2; Q = 3), UMTI (M = 2 or 3), or UETI
(M = 2 or 3). Standard Daubechies and Coifflet filters were used
for the two-band filter bank. For the three-band filter bank, the coefficients were derived using the scheme given in [6] and are given in the
Appendix. ROC’s, tiling pictures, and locations of detected transients
were found.
1) Fig. 1 shows ROC’s for the detectors based on different transforms in comparison with the matched filter. The signal with
= 1=16, ! = =2, and k = 1 with noise of variance of 0.1
was used. NMTI is clearly seen to perform better than the rest
of the transforms. The ROC corresponding to NMTI is closer to
that for the matched filter, as compared with other transform.
2) The second simulation illustrates the robustness of NMTI to resolve the transients overlapping in time domain. The signal parameters with length = 216 and k = 3 in (8) are 01 = 02 =
Fig. 3. Tiling picture for the signal whose parameters are given in Section IV
2, Q = 3. Here, labels A, B , and C correspond to
using NMTI with P
different transients.
=
03 = 50, 1 = 2 = 3 = 3=64, 1 = 2 = 3 = ,
!1 = =6, !2 = =2, and !3 = 5=6. The maximum number
of levels is J = 3 for all the transforms. Tiling pictures for UMTI
and UETI are similar and are shown in Fig. 2. The tiling picture
for the NMTI is shown in Fig. 3. For this particular example,
only NMTI could resolve all the transients.
3) Finally, we have compared different transforms based on the correctness of the time-frequency location of the detected transient.
The signal parameters with k = 5 are given in Table I. SNR
= 0 1:23 dB. Figs. 4 and 5, respectively, show original and noisy
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TABLE II
STATISTIC, TIME OF ARRIVAL, AND FREQUENCY OF THE TRANSIENTS
DETECTED USING 2-BAND UETI WITH LEVELS = 3 FOR THE SIGNAL
TABLE III
STATISTIC, TIME OF ARRIVAL, AND FREQUENCY OF THE TRANSIENTS DETECTED
USING 3-BAND UETI WITH LEVELS = 3, FOR THE SIGNAL SHOWN IN FIG. 5
Fig. 4. Signal containing five transients whose parameters are given in Table I.
TABLE IV
STATISTIC, TIME OF ARRIVAL, AND FREQUENCY OF THE TRANSIENTS
DETECTED USING NMTI WITH P = 2 AND Q = 3 FOR LEVELS = 3, FOR THE
SIGNAL SHOWN IN FIG. 5
APPENDIX
THREE-BAND FILTER BANK
Fig. 5. Signal containing five transients plus Gaussian noise of var
The parameters for the original signal are given in Table I.
= 0 05.
:
TABLE I
PARAMETERS FOR THE SIGNAL SHOWN IN FIG. 4
The full wavelet matrix A, with M = 3, number of vanishing moments N = 2, and the rank-3 DCT for its characteristic Haar matrix
[6] is given as
0:58
0:91
1:25 0:41 0:08
:
1:39 0:27
:
0:00 0:00
00 17 00 27 00 37
0 70 01 41
0 70
:
:
:
:
00 25
00 85
00 00
:
:
:
:
REFERENCES
signals. Tables II–IV give the mean time of arrival, mean frequency, and energy statistic for the two-band UETI, three-band
UETI, and NMTI, respectively. Results for UETI and UMTI
were more or less similar. It is seen that NMTI is able to resolve
and detect all the transients. The results obtained using NMTI
are comparable with those obtained using UETI.
V. CONCLUSION
In this correspondence, we have extended the scheme of single transient detection based on [4] for multiple transients. NMTI is developed,
and its application to transient signal detection is discussed. NMTI is
seen to outperform UMTI over a wide range of signals. It provides more
flexibility and, hence, adaptability to the nature of the signal.
[1] M. Frisch and H. Messer, “Transient signal detection using prior information in the likelihood ratio test,” IEEE Trans. Signal Processing, vol.
41, pp. 2177–2191, June 1993.
[2]
, “The use of the wavelet transform in the detection of an unknown
transient signal,” IEEE Trans. Inform. Theory, vol. 38, pp. 892–897,
Mar. 1992.
[3] B. Friedlander and B. Porat, “Performance analysis of transient detectors
based on a class of linear data transform,” IEEE Trans. Inform. Theory,
vol. 38, pp. 665–673, Mar. 1992.
[4] S. Marco and J. Weiss, “M -band wavepacket-based transient signal detector using a translation-invariant wavelet transform,” Opt. Eng., vol.
33, pp. 2175–2182, July 1994.
[5]
, “Improved transient signal detection using a wavepacket-based
detector with an extended translation-invariant wavelet transform,”
IEEE Trans. Signal Processing, vol. 45, pp. 841–850, Apr. 1997.
[6] P. N. Heller, “Rank M wavelets with N vanishing moments,” SIAM J.
Matrix Anal., Apr. 1995.
[7] R. R. Coifman and M. V. Wickerhauser, “Entropy-based algorithms for
best basis selection,” IEEE Trans. Inform. Theory, vol. 38, pp. 713–718,
Mar. 1992.
[8] C. Taswell, “Satisficing search algorithms for selecting near-best bases
in adaptive tree-structured wavelet transforms,” IEEE Trans. Signal Processing, vol. 44, pp. 2423–2438, Oct. 1996.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 6, JUNE 2000
[9] H. L. Van-Trees, Detection and Estimation Theory. New York: Wiley,
1971.
[10] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE
Trans. Signal Processing, vol. 42, pp. 2146–2157, Aug. 1994.
An Aperiodic Phenomenon of the Extended Kalman Filter
in Filtering Noisy Chaotic Signals
Henry Leung, Zhiwen Zhu, and Zhen Ding
Abstract—In this correspondence, we report an interesting behavior of
the extended Kalman filter (EKF) when it is used to filter a chaotic system.
We show both theoretically and experimentally that the gain of the EKF
may not converge or diverge but oscillates aperiodically. More precisely,
when a nonlinear system is periodic, the Kalman gain and error covariance
of the EKF converge to zero. However, when the system is chaotic, they either converge to a fixed point with magnitude larger than zero or oscillate
aperiodically. Our theoretical analyses are verified using Monte Carlo simulations based on some popular chaotic systems.
Index Terms—Chaos, extended Kalman filter, Lyapunov exponent, stability.
I. INTRODUCTION
Recently, there has been a great deal of interest in applying chaos
to engineering problems such as control [1], signal processing [2], [3]
and communications [4]. Not only have many real systems and signals
been shown to be chaotic but simulated chaotic signals can also be used
in various applications such as image encryption [5], secure communication [6], and multiple access [7]. Since chaotic signals are neither
entirely deterministic or stochastic, new signal processing techniques
are under developement to process them efficiently.
One problem in chaotic signal processing that has drawn a lot of
attention is noise filtering. In a recent workshop on complexity and
chaos [8], developing a “nonlinear Kalman filter” has been identified
as an important research topic for future directions. While most techniques developed for chaotic modeling and synchronization in the literature assume a noise-free environment, measurement noise, which is
unavoidable in real-life engineering problems, must be handled carefully for these chaotic techniques to be of practical use. Some noise reduction methods have been proposed to smooth out noisy chaotic time
series [9], [10], but these methods usually require a long data sequence,
and the smoothing is carried out in an off-line fashion. Their applicability to real-time engineering problems such as signal processing and
communications is therefore limited.
A natural choice to filter a chaotic system on-line is through a nonlinear filter developed using statistical estimation theory [11], [12]. Although a chaotic system is not stochastic, it fits into the stochastic state
1807
space representation used in the filtering theory with a zero system
driven noise. In fact, the most widely used nonlinear filter, namely,
the extended Kalman filter (EKF), has recently been applied to chaotic
signal processing [13], chaotic control [14], and chaotic communications [15], [16]. It is hoped that the EKF will have an optimal or near
optimal performance in processing noisy chaotic signals. In [15], we
apply the EKF to demodulate a chaotic communication system. This
chaotic modulation scheme has been shown to have great potential in
spread spectrum communication and multiple access because it has a
high capacity and does not require any code synchronization. While
the concept of this novel chaotic communication scheme has been validated in a noise-free environment, the main problem for putting this
system into practice is the lack of an efficient demodulator when noise
exists. We therefore propose using an adaptive filter such as the least
mean square (LMS) algorithm and the EKF in the receiver to demodulate the noisy signal. Although the EKF demodulator is based on a correct state-space representation and the LMS method only uses a simple
approximated model that treats the measurement noise as the system
driven noise, the performance of the EKF is found to be even worse
than that of the LMS demodulator in some cases.
This unusual behavior and the potential of chaotic communication
have motivated us to investigate why the EKF performs unexpectedly
poorly on the simple logistic map used in chaotic modulation system.
Here, we use the one-dimensional (1-D) discrete time dynamical systems to investigate the EKF performance in filtering chaotic systems
because they almost include all of the behaviors and characteristics
in the multidimensional systems, and they are most widely used for
chaotic communication and signal processing. We show in this correspondence that when the EKF is applied to certain types of chaotic
maps, the Kalman gain does not necessarily converge to zero but oscillates aperiodically. To the best of our knowledge, this strange oscillatory behavior has never been reported. Since the Kalman gain does
not converge to zero, the performance of the EKF in noise filtering and
chaotic demodulation is therefore rather poor.
It is also of interest to note that our EKF implementation follows
the standard approach used in the tracking literature. When the EKF is
applied to radar tracking, the EKF may produce large tracking error.
It is because many tracking systems to date use a threshold on the
Kalman gain to prevent filter divergence. More precisely, if the gain
exceeds the threshold, then the filter is considered to be divergent and
will be restarted. Apparently, this divergent test is not effective in detecting the strange oscillatory behavior of the EKF when the underlying state-space model reaches a chaotic state. This new aperiodic behavior of the EKF is not only useful in designing an effective receiver
for chaotic communications, but it also provide some new insights on
target tracking and nonlinear filtering.
II. THEORETICAL ANALYSIS
Consider a nonlinear dynamical system
xn = f (xn01 )
Manuscript received July 8, 1998; revised November 23, 1999. The associate
editor coordinating the review of this paper and approving it for publication was
Dr. Ali H. Sayed.
H. Leung is with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alta., T2N 1N4, Canada (e-mail:
[email protected]).
Z. Zhu and Z. Ding are with the Communications Research Laboratory,
McMaster University, Hamilton, Ont., L8S 4K1, Canada (e-mail: [email protected]; [email protected]).
Publisher Item Identifier S 1053-587X(00)04066-6.
(1)
R ! R1 is a smooth function, and the measurement
where f : 1
equation is given by
yn = xn + wn
(2)
where wn is a zero mean white Gaussian noise process with variance
w2 . When the global Lyapunov exponent of (1) is larger than zero, we
consider (1) to be chaotic. Otherwise, the system is nonchaotic.
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