For original paper see Barkat and Boashash (IEEE Trans. Signal Processing, vol.49, p.2232-9, 2001). It is shown that the time-frequency distribution (TFD) proposed in the above paper is not well defined in the ordinary sense for power signals, including the single-tone sinusoid, and it needs the introduction of generalized functions and transforms. It is also shown that the proposed TFD does not satisfy the conditions cited by the authors of the paper to justify the claim that it has the instantaneous frequency property

Hussain, Z

RMIT Research Repository

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004 287
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10-1.7
10-1.6
10-1.5
10-1.4
10-1.3
β
rm
s
SME
AML1
Subdiag.
Proposed
CRB
Fig. 4. RMSE of the DOA estimates versus .m = 8,N = 64, and q = 0
dB.
0 10 20 30 40 50 60 70 80 9010
-2
10-1
100
101
θ0
rm
s
SME
AML1
Subdiag.
Proposed
CRB
Fig. 5. RMSE of the DOA estimates versus  . m = 8, N = 64, q = 0
dB, and  = 0:25.
of [1] provides erroneous estimates, which clearly demonstrates the
ambiguity problem. In contrast, our method does not suffer from this
problem.
V. CONCLUSIONS
In this correspondence, we consider the direction-finding problem
for an extended target whose power spatial density is not necessarily
symmetric with respect to its mass center. Two computationally simple
algorithms were proposed. One is based on the spectral moments of the
target, which are easily related to its DOA. The second borrows ideas
from [1] and extends the range of DOAs that can be estimated unam-
biguously. Both methods provide robust, simple, yet accurate DOA es-
timates.
REFERENCES
[1] O. Besson, P. Stoica, and A. Gershman, “Simple and accurate direction
of arrival estimator in the presence of imperfect spatial coherence,” IEEE
Trans. Signal Processing, vol. 49, pp. 730–737, Apr. 2001.
[2] A. Monakov, “Observation of extended targets with antenna arrays,”
IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp. 297–302, Jan. 2000.
[3] M. Zatman, “How narrowband is narrowband?,” Proc. Inst. Elect.
Eng.—Radar, Sonar Navigat., vol. 145, no. 2, pp. 85–91, 1998.
[4] K. Pedersen, P. Mogensen, and B. Fleury, “A stochastic model of the
temporal and azimuthal dispersion seen at the base station in outdoor
propagation environments,” IEEE Trans. Veh. Technol., vol. 49, pp.
437–447, Mar. 2000.
[5] A. Paulraj and T. Kailath, “Direction of arrival estimation by eigenstruc-
ture methods with imperfect spatial coherence of wavefronts,” J. Acoust.
Soc. Amer., vol. 83, pp. 1034–1040, Mar. 1988.
[6] A. Gershman, C. Mecklenbräuker, and J. Böhme, “Matrix fitting ap-
proach to direction of arrival estimation with imperfect coherence of
wavefronts,” IEEE Trans. Signal Processing, vol. 45, pp. 1894–1899,
July 1997.
[7] R. Raich, J. Goldberg, and H. Messer, “Bearing estimation for a dis-
tributed source via the conventional beamformer,” in Proc. SSAP Work-
shop, Portland, OR, Sept. 1998, pp. 5–8.
[8] R. Moses, T. Söderström, and J. Sorelius, “Effects of multipath-induced
angular spread on direction of arrival estimators in array signal pro-
cessing,” in Proc. IEEE/IEE Workshop Signal Process. Methods Mul-
tipath Environments, Glasgow, U.K., Apr. 20–21, 1995, pp. 6–15.
[9] D. Astély and B. Ottersten, “The effects of local scattering on direction
of arrival estimation with MUSIC,” IEEE Trans. Signal Processing, vol.
47, pp. 3220–3234, Dec. 1999.
[10] M. Bengtsson and B. Ottersten, “Low-complexity estimators for
distributed sources,” IEEE Trans. Signal Processing, vol. 48, pp.
2185–2194, Aug. 2000.
[11] , “A generalization of weighted subspace fitting to full-rank
models,” IEEE Trans. Signal Processing, vol. 49, pp. 1002–1012, May
2001.
[12] O. Besson and P. Stoica, “Decoupled estimation of DOA and angular
spread for a spatially distributed source,” IEEE Trans. Signal Pro-
cessing, vol. 48, pp. 1872–1882, July 2000.
[13] O. Besson, F. Vincent, P. Stoica, and A. Gershman, “Approximate max-
imum likelihood estimators for array processing in multiplicative noise
environments,” IEEE Trans. Signal Processing, vol. 48, pp. 2506–2518,
Sept. 2000.
Comments on “A High-Resolution
Quadratic Time-Frequency Distribution for
Multicomponent Signals Analysis”
Zahir M. Hussain
Abstract—It is shown that the time-frequency distribution (TFD) pro-
posed in the above paper is not well defined in the ordinary sense for power
signals, including the single-tone sinusoid, and it needs the introduction of
generalized functions and transforms. It is also shown that the proposed
TFD does not satisfy the conditions cited by the authors of the paper to jus-
tify the claim that it has the instantaneous frequency property.
Index Terms—Generalized functions, instantaneous frequency, mul-
ticomponent signals, reduced interference distributions, time-frequency
analysis.
Recently, a time-frequency distribution (TFD) of Cohen’s Class,
which is known as the B-distribution (BD), was proposed and claimed
Manuscript received July 26, 2002; revised Autgust 8, 2003. The associate
editor coordinating the review of this paper and approving it for publication
was Dr. Ta-Hsin Li.
The author is with the School of Electrical and Computer Engineering, RMIT
University, Melbourne, Victoria 3000, Australia (e-mail: zmhussain@ieee.org).
Digital Object Identifier 10.1109/TSP.2003.820057
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288 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004
to satisfy the instantaneous frequency (IF) property and have better
time-frequency resolution under noise-free and noisy conditions than
the spectrogram and Wigner–Ville distribution (WVD) [1]–[3]. The
following are four comments about the BD.
I. COMMENT 1
The time-lag kernel of the BD is given by [1]
G(t;  ) =
j j
cosh2(t)

(1)
where  is a real positive number less than one. Hence, the BD for any
time signal z(t) = a(t) exp[j(t)] is given by the general formula for
Cohen’s Class [4]–[8]
z(t; !) =
1
 1
1
 1
G(t  u;  )Kz(u;  )e
 j! dud
= F
!!
fpz(t;  )g (2)
where F represents the Fourier transformation, ! = 2f is the radian
frequency, Kz(u;  ) = z(u + =2)z(u   =2) is the instantaneous
autocorrelation product, pz(t;  ) = G(t;  ) (t) Kz(t;  ), and (t)
stands for time convolution. Note that G(u  t;  ) = G(t   u;  ). In
fact, the structure of the BD is equivalent to a smoothed pseudo-Wigner
distribution. The notion of such a distribution was first time defined by
Flandrin as detailed in [6, p. 254].
Due to the diverging factor j j in the integrand, the above integral
does not exist in the ordinary sense unless z(t) is an energy signal with
either
1) finite length;
2) infinite length such that there exists an ordinary (nonsingular)
function r() for which the convolution r() () pz(t;  ) is
defined 8 and limj j!1[r() () pz(t;  )] = 0 [11].
For example, let us consider the sinusoidal signal z(t) = exp(j!ot).
This power signal is of fundamental importance in signal analysis.
TheWVD for this signal is given by
W (t; !) = 2(!   !o) (3)
where  is the Dirac delta function. This means that theWVD gives the
best possible representation of this signal since it gives ideal concen-
tration around the IF of the signal, which is !o. The BD for this signal
is given by
z(t; !) = k
1
 1
j j exp[ j(!   !o) ] d (4)
where k = (1=2; );  is the beta function [9]. Note that k is time
independent. The above integral, which is simply the Fourier transform
of j j (with respect to the frequency variable F = ! !o) scaled by
k, is undefined in the ordinary sense for all > 0 at every point (t; !)
in the time-frequency plane since the integrand diverges indefinitely
when j j increases. To further clarify this point, note that the above
integral exists if and only if the integral
I(; F ) =
1
0
 cos(F )d (5)
exists for 0 <  < 1 and F > 0. This is so because the Fourier trans-
form of a real and even function is real and even. Using the technique
of integration by parts and the tables in [9], it follows that
I(; F ) = 
sin(F )
F
1
=0
 

F
1
0
 1 sin(F )d
= lim
!1

sin(F )
F
 
 (+ 1)
F+1
sin

2
: (6)
The above limit does not exist; hence, the BD is not well defined in
the ordinary sense for single-tone sinusoidal signals. However, the BD
of a sinusoid can be understood in terms of generalized functions (or
functionals) defined on the class S of “good” functions, i.e., functions
that are infinitely differentiable and tend to zero as ! !1 faster than
the reciprocal of any polynomial function [11, Sec. 7.1]. Only in this
sense, it can be shown that [11, pp. 138–139] for 0 <  < 1
g() = j j , G(!) =
2 (+ 1) cos 
2
(+ 1)
j!j+1
: (7)
Note that the right-hand side of (7) is not an ordinary function. Rather,
1=j!j+1 in (7) is a functional defined on the class S in the sense that
1=j!j+1 =  
1

1
j!j
sgn(!)
0
(8)
where the derivative 0 is the generalized derivative [11, p. 130]. In
this case, the BD is meaningful only in conjunction with the integral
1
 1
z(t; !)'(!)d! for a “good” frequency function '(!) [11].
II. COMMENT 2
The authors of [1] state that the BD has better concentration (reso-
lution) than the WVD for stepped FM (a combination of finite-length
sinusoids) and linear FM signals (see [1, Figs. 4 and 7]). A more gen-
eral statement in the Conclusion of [1] confirms that “The proposed
distribution outperforms the WVD and the spectrogram in terms of
time-frequency resolution and cross term suppression.” No reasoning
or analysis is provided in [1] to support this claim, knowing that the
WVD has the best possible concentration for infinite-length sinusoids
(see Comment 1 above) and linear FM signals [4], where it gives the
delta function around the IF law in both cases.
III. COMMENT 3
It is claimed in [1] that the first moment of the BD yields the instan-
taneous frequency of the signal, that is
!i =
1
 1
!z(t; !)d!
1
 1
z(t; !)d!
(9)
because its Doppler-lag kernel g(;  ) satisfies the following condi-
tions:
@g(;  )
@ (0;0)
=
@g(;  )
@ (0;0)
= 0
g(; 0) = constant 8: (10)
First, it must be pointed out that the correct conditions for the IF
property are [4]–[8], [10]
@g(;  )
@
=0
= 0 8
g(; 0) = 1 8: (11)
The condition (@g(;  ))=(@)j(0;0) = 0 (or its correct version
(@g(;  ))=(@)j=0 = 0 8 ) is inappropriate here as it is related
to the time delay property and not the IF property [4]–[8], [10].
Second, I would like to point out that the Doppler-lag kernel of the
BD satisfies neither (10) nor (11), as shown below.
Proof: According to [1, (7)], we have
g(;  ) = j j
22 1
 (2)
 (+ j) (  j)
= g1()g2() (12)
where g1() = j j, and  (z) is the gamma function defined by
 (z) =
1
0
e ttz 1 dt (13)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004 289
Fig. 1. Doppler-lag kernel g(; ) of the BD for different values of .
Contrary to all known quadratic TFDs, the BD kernel has a highpass shape in
the lag direction with a minimum (zero) at the origin.
and has simple poles in the complex z-plane at z = 0; 1; 2; . . ., etc.
[9]. For the range of  specified in [1], i.e., 0 <   1, the derivative
(@g(;  ))=(@) = g2()dj j
=d does not exist at  = 0 since
lim
!0
dj j
d
=
+1  ! 0+; 0 <  < 1
 1  ! 0 ; 0 <  < 1
+1  ! 0+;  = 1
 1  ! 0 ;  = 1
(14)
while g2() = (22 1)=( (2))j (+j)j2 is always positive and
finite for all  when 0 <   1. Therefore, it cannot be concluded that
the BD has the IF property unless a valid proof is given.
IV. COMMENT 4
The authors of [1] state that the kernel of the BD is designed intu-
itively to retain the auto-terms and suppress the cross-terms. However,
in order to suppress the cross-terms [that accumulate far away from the
origin of the Doppler-lag (a.k.a. ambiguity) domain] while keeping the
auto-terms (that accumulate around the origin), it is well known that the
Doppler-lag kernel g(;  ) of any quadratic time-frequency distribu-
tion should have a two–dimensional lowpass shape with a maximum at
the origin (;  ) = (0; 0) (see, for example, [5] and [6]). If the preser-
vation of the signal total energy is considered, this maximum should
be one, i.e., g(0; 0) = 1 (see [5, p. 164] and [6, p. 107]). Contrary to
all known TFDs in the quadratic class (a.k.a. Cohen’s class), the BD
kernel has a highpass shape in the lag direction with a minimum (zero)
at the origin, as shown in Fig. 1. No reasoning is provided in [1] for
this abnormality.
V. CONCLUSION
In this correspondence, it is shown that the B-distribution (BD),
which was recently proposed as a time-frequency distribution
(TFD), is not well-defined in the ordinary sense for power signals,
including the pure sinusoid, which is of fundamental importance in
time-frequency analysis. A correct definition may be introduced using
generalized functions. It is also shown that the BD does not satisfy the
conditions cited by the authors of [1] to justify the claim that it has the
conventional IF property. In addition, it is pointed out that the BD has
an abnormal time-lag kernel with a minimum at the origin, contrary to
the original purpose of designing the BD to attenuate the cross-terms
and pass the auto-terms.
ACKNOWLEDGMENT
The auhtor would like to thank the Editor-in-Chief, Prof. A. H.
Sayed, for his valuable time and sincere efforts in reviewing these
comments. He is deeply grateful to the Associate Editor, Dr. T.-H. Li,
for his insightful comments, fruitful suggestions, and patience with
him throughout this discussion. Many thanks to the reviewers for their
constructive comments and to the IEEE for this unique dedication to
the truth. He would like also to thank the authors of the BD paper for
their reply.
REFERENCES
[1] B. Barkat and B. Boashash, “A high-resolution quadratic time-frequency
distribution for multicomponent signals analysis,” IEEE Trans. Signal
Processing, vol. 49, pp. 2232–2239, Oct. 2001.
[2] B. Barkat, “Design, estimation, and performance of time-frequency dis-
tributions,” Ph.D. dissertation, Queensland Univ. Technol., Brisbane,
Australia, 2000.
[3] B. Barkat and K. Abed-Meraim, “A blind components separation pro-
cedure for FM signal analysis,” Proc. IEEE Int. Conf. Acoust., Speech,
Signal Process., vol. 2, pp. 1425–1428, May 2002.
[4] L. Cohen, “Time-frequency distributions—A review,” Proc. IEEE, vol.
77, pp. 941–981, July 1989.
[5] , Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-Hall,
1995.
[6] P. Flandrin, Time-Frequency/Time-Scale Analysis. Boston, MA: Aca-
demic, 1998.
[7] J. Jeong and W. Williams, “Kernel design for reduced interference dis-
tributions,” IEEE Trans. Signal Processing, vol. 40, pp. 402–412, Feb.
1992.
[8] P. J. Loughlin, J. W. Pitton, and L. E. Atlas, “Bilinear time-frequency
representations: New insights and properties,” IEEE Trans. Signal Pro-
cessing, vol. 41, pp. 750–767, Feb. 1993.
[9] I. S. Gradshtein, Table of Integrals, Series, and Products, 6th
ed. Boston, MA: Academic, 2000.
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290 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004
[10] T. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribu-
tion—A tool for time-frequency signal analysis—Part III: Relationswith
other time-frequency signal transformations,”Philips J. Res., vol. 35, no.
6, pp. 372–389, 1980.
[11] A Handbook of Fourier Theorems, Cambridge Univ. Press, New York,
1987.
Reply to “Comments on ‘A High-Resolution
Quadratic Time-Frequency Distribution for
Multicomponent Signals Analysis’”
B. Barkat and B. Boashash
I. REPLY TO COMMENT 1
The recently proposed quadratic time-frequency distribution
(TFD), referred as BD, handles power signals in the same way
the Wigner–Ville distribution (WVD) or any other time-frequency
distribution (TFD) handles them. These TFDs, being in essence
Fourier transforms (FTs) of the signal or quadratic functions of it,
cannot be evaluated in an ordinary way but have to be evaluated using
functionals. As an illustration, let us reconsider the signal used in the
comments, namely, a sinusoid expressed as z(t) = exp(j!0t). The
WVD of this signal is given by
Wz(t; f) =
1
 1
[z(t+ 0:5 )  z(t  0:5)] exp[ j! ] d: (1)
Because [z(t+0:5)z(t 0:5)] = exp[ j!0 ] is not absolutely
integrable over the considered interval, its FT cannot be obtained by
direct evaluation, and one has to resort to transforms in the limit or
functionals in order to obtain the final result in (3) of the comments. The
same discussion applies to the BD or any other TFD. FTs in the limit
have existed for decades and are not new concepts for the community.
As a consequence, Comment 1 does not provide any new information.
II. REPLY TO COMMENT 2
It was clearly stated that the BD can solve some problems that the
WVD or the spectrogram cannot. It was never claimed that the BD
performs better than the two other distributions at all times and all sit-
uations. To be more specific, in the paper introduction, the following
statement was given: “This comparison is performed with respect to
some criteria detailed later in the paper.” All the criteria pertaining
to the comparisons in the paper were given in detail. In addition, the
paper comparisons were basically numerical and not analytical, and in
a numerical implementation, it is not possible to use an infinite-length
Manuscript received November 14, 2002; revised August 25, 2003. The asso-
ciate editor coordinating the review of this paper and approving it for publication
was Dr. Ta-Hsin Li.
B. Barkat is with the School of Electrical and Electronic Engi-
neering, Nanyang Technological University, Singapore 639798 (e-mail:
ebarkat@ntu.edu.sg).
B. Boashash is with the Signal Processing Research Centre, Queens-
land University of Technology, Brisbane, Qld 4001, Australia (e-mail:
b.boashash@qut.edu.au).
Digital Object Identifier 10.1109/TSP.2003.820059
Fig. 1. Doppler-lag kernels of the cone-shape distribution for a small value
(top plot) and a large value (bottom plot) of its parameter.
signal, and consequently, the theoretical “best possible concentration”
of the WVD is not guaranteed. This point was well illustrated in the
paper where Monte-Carlo analysis as well as several examples, in-
cluding real-life data, were provided to support the claim.
III. REPLY TO COMMENT 3
To start with, let us observe that conditions (11) are just sufficient
conditions, and consequently, there is the following.
i) Other, less restrictive, conditions can also be valid conditions for
expression (9). This means that conditions (10) reported in [2,
p. 14] cannot be excluded without a detailed demonstration to
prove it.
ii) The first moment of a TFD that violates conditions (11) may still
be a good estimator of the signal instantaneous frequency (IF).
In addition, in the paper, expression (9) was not used to estimate
the signal IF. Instead, the peak of the BD was used as an IF estimator.
Therefore, whether the BD verifies or violates conditions (11) does
not have any negative implications on the results of the BD paper or
on the simulations therein. Furthermore, many known TFDs do not
verify (11). One of them is the spectrogram, which is still today one
of the most popular and widely used quadratic TFDs. Since this has
not limited its application in real-life problems, the same statement can
be made about the BD.
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