Comments, with reply, on 'positive time-frequency distributions' by l. Cohen

Comments, with reply, on 'positive time-frequency

distributions' by l. Cohen

Citation for published version (APA):

Janssen, A. J. E. M. (1987). Comments, with reply, on 'positive time-frequency distributions' by l. Cohen. IEEE

Transactions on Acoustics, Speech, and Signal Processing, ASSP-35(5), 701-705.

https://doi.org/10.1109/TASSP.1987.1165192

DOI:

10.1109/TASSP.1987.1165192

Document status and date:

Published: 01/05/1987

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T-ASSP/35/5/13502

73•

A Note on “Positive Time-Frequency Distributions”

A. J. E. M.

Janssen

Reprinted from: IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,

Vol. ASSP-35, No. 5, MAY 1987.

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING, VOL. ASSP-35. NO. 5. MAY 1987 701

A Note on “Positive Time-Frequency Distributions”

A. J. E. M. JANSSEN

Abstract—We discuss the performance of certain recently proposed positive time-frequency distribution functions with correct marginals. It turns out that for signals of the FM type, satisfactory results can only be obtained with a strongly signal dependent choice of the param eters in these distribution functions.

I. INTRODUCTION

For some time there has been a discussion in the signal analysis community on how to deal with the occurrence of negative values in certain popular time-frequency distributions. This discussion was recently revived by the introduction, by Cohen and Zaparovanny [l}, of a class of distributions with many desirable properties such as positivity and the correct marginals property. In j2[ and the above paper,’ these distributions are recommended as an alternative to, e.g., the Wigner distribution, since the latter one cannot be inter preted as a true probability density. There is agreement that the Wigner distribution can he of great help in displaying time-fre quency characteristics of time-varying signals and signals with transient behavior (see, e.g..

[3[.

(4J), but the application of this distribution requires some skill. This has to do with the fact that the Wigner distribution involves the signal bilinearly, so that for multicomponent signals, disturbing crossterms leading to negative values must he expected [6[. In this correspondence we raise the question of whether the positive distributions proposed in [2[ and the above paper’ can be equally successful in displaying signal characteristics of time-varying signals. It turns out this is not so for signals of the FM type.

II. BILINEAR DISTRtBUTIONS AND Pos!TtvE DtsTRtauTtoNs The Wigner distribution is a member of a large class of distri butions which can be parameterized by means of a function 4) of two real variables.

C~’(t.

~)

= exp (—2~i(Ot + rw — Cu)) 4)(O. r)

X

.1(0

+ T)

I

*(u — r)dO dr dii (II.

wheref is the signal under study, and 4) is independent off. The distributions so obtained have been studied extensively, especially with respect to the point how various “natural” requirements are reflected in restrictions on 4). The Wigner distribution results on taking 4) (0. r) = I in (11.1). and it satisfies a long list of desirable

mathematical properties [61.

Among these properties are the following:

a) correct marginals. so that integration overtand u, yields the power spectral density and the instantaneous power,

b) time-frequency translations of the signal are reflected by cor responding translations of the distribution. and

c) vanishing of power spectral density or instantaneous power outside intervals implies vanishing of the distribution outside the corresponding strips.

The class described by (11.1) is a subclass of Cohen’s class in troduced in

[71.

where the kernel 4) may depend on

I.

To obtain everywhere positive distributions satisfying a), one must take 4)’s

Manuscript received September 4. 1985; revised November 6. 1986. The author is with the Philips Research Laboratories. 5600 JA Eind hoven. The Netherland,,.

IEEE Log Number 8613502.

‘L. Cohen and T. E. Posch, iEEE Trans. .4 roust., Speech, Signal Pro cessing. vol. ASSP-33. pp. 3l—38. Feb. t985.

702 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING. VOL. ASSP.35. NO. 5. MAY 1987

depending on the signal since by Wigner’s theorem and extensions of it [5], [8], for any choice of 4’ such that a) holds, there is an abundance of signalsf for which C~ takes negative values.

The positive distributions introduced by Cohen and Zaparo vanny in [ii can be generated according to the formula

Cf(t, w) f(t) j F(c~) (1 + cp(x(t). v(~))).

where,

withE

SIf(r)I2dr,

x(t) ~ f(r)j dr.

(~)

~ F(X)~ dX.

The function p and the constant c act as parameters in (11.3). They must be chosen such that a) is satisfied and that I + cp(x, y) ≥ 0

for all x, y E [0, 1], but are otherwise unrestricted. The general

form of p and the manner in which p and the 4’ of (11.1) are related are given in [1], [2], and the above paper.’ The distributions (11.3), with p. c chosen as stated above, satisfy conditions a), b), and c); in addition, they satisfy d) positivity.

III. POSITIVE TIME-FREQUENCY DISTRIBUTIONS FOR FM

SIGNALS

Although the mathematical properties of Section II are impor tant, the signal analysts want something else. They have quite often an intuitive idea or advance knowledge where the signal energy of a particular signal should be concentrated. When dealing with an application yielding signals of a certain type, they need a distri

bution that exhibits the expected behavior for signals on which they can check, so that they can expect meaningful results for signals that are only known to be of that particular type. To achieve this end, they are willing to sacrifice quite a few of the mathematical properties. It has been demonstrated in various applications that the Wigner distribution is an excellent distribution to start from. It can be modified and enhanced so that the information one looks for can be better obtained from it. The enhancement techniques needed may depend on the particular application, but remain essentially fixed for all the signals encountered in one application.

The set of distributions generated by formula (11.3) could otter an alternative to this procedure. It is therefore interesting to see whether one can make choices for the parameters c and p such that

~‘_~ all signals of a particular type are accommodated by the distribu

tion (11.3) with this c and this p. We discuss here the performance for signals exp (27ri~s(1)) of the FM type.

According to signal analysts’ intuition, the signal energy of exp (2iri~(fl) should be concentrated around the curve (, ~‘(.‘)).

Consider the chirplike signal

f(t) = (2~)’ exp (~~u2 + ~j~j2) (111.1)

where > 0. Now

I 4 . I 2

/

7I~W

F(~e) = (2e) (e — ‘L3) exp ~ + ~ — ~ +

It follows from the definition of x( r) and v(si) in (2.3) that x(r) = +

(111.2)

(~)

= + 1(~[2f/(~2 +

~

)]‘

~)

. (111.3)

where 1(a) = ~ exp (—iru2 ) do.

When~3 is not too small, compared to~, both f~)j and

F(w)I

are very flat functions of t and o,. respectively. Therefore.

C’1(t, w) can only exhibit the desired behavior when one succeeds in choosing the parameters c and p such that I + cp(.~(r). s (

is large near the straight line ( . (31), 1 C R and small away from this line.

Assume

(3

> 0. When > 0 is small and w is close to (3t, we

have that y(w) is close to + 1(t~~/~J). Hence, when C~(~’. c~) exhibits the desired behavior. I + cp(x, v) must be large only in

a small neighborhood of a curve that passes through

(L

~) and that coincides with the diagonal

{

(x, x)jx E [0. I}

}

when ~ ~ 0. (A

good choice in this case forp would be p(x, v) = 5(x — y) — I,

(11.2) C = I; this p can be derived in the manner of (2.2) in the paper’

from h(x, y) = 5(x — y).

Assume j3 < 0. When > 0 is small and w is close to

~3r,

we

have that y(c,) is close to — 1(t~J~). Now I + cp(x, v) must

(11.3) be large only in a small neighborhood of a curve that passes through ()~, ~) and that coincides with the cross diagonal

{

(x. I — x)

I

xC

[0, lfl when~ ~ 0. (Nowp(x. y) = h(x + s — I) — I, c =

would be a good choice.)

The two conditions on C,p thus obtained are clearly incompat

ible so that one cannot accommodate chirps

f

with positive and negative sweep rate i3 with a single choice of the parameters c, p.

More dramatically, when one chooses C and p such that the

Cj(r, w) forchirpsfwith positive sweep rate are satisfactory, then for any other signal g one will get a distribution that looks a bit like the distribution of that chirp. This is so sinceCg(t, ~) is large

near the set ((1,

+)Ix(r)

= y(~e)}, and both x(r) and y(ø) in

crease in r and w, respectively, whence Cg(t, c~) exhibits a ridge

extending from the third quadrant into the firstquadrant of the time-frequency plane.

We have included the pictures of the distributions of

f,

andf2

obtained from (111.1) by taking e = 1, f3 = I and E 1, j3 0, -~ ;___‘~

(a)

(b)

Fig. I. Cohen—Posch—Zaparovanny distribution for thefof(111.1) with (a)

= I, (3 = I and (b) = l,(3 = 0. respectively, and with hand c as

IEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING. VOL. ASSP.35. No. 5. MAY 987 703

respectively. To accommodatef,. we take a p that is derived from

(b(l —

a’lv

— p(x)~). v — ≤ a

h(x.

v)

= (111.4)

(~O. — i’(.v)I > ~‘

in the manner of (2.2) in the paper.’ Here p (xl = + ‘J~(x —

a = , b is such that h is properly normalized, and c is such

that mm 1 + cp(x, v) = 0. The choice of h is such that near

(t, w) = (0, 0), the distribution off, exhibits a ridge along the linet = w. and the resulting picture is quite satisfactory. However, the picture that results forf, exhibits a similar ridge along the line

=

u/J~,

which should definitely not be the case.

REFERENCES

II] L. Cohen and Y. 1. Zaparovanny, i. Math. Phvs.. vol. 21, pp. 794-796, 1980.

[2] L. Cohen, in Proc. ICASSP. 1984. p. 41B.l.

131

C. P. Janse and A. J. M. Kaizer. i. Audio Eng.Soc..vol.31 pp. 198— 223. 1983.

[41

N. M. Marinovich and G. Eichmann. inProc. 1CASSP. 1985. p. 27.3.

[51

A. J. E. M. Janssen. J. Math. Phvs,. vol. 26. pp. 1986-1994. 1985. [6] L. Cohen, J. Math. Phvs., vol. 7. pp. 781-786, 1966.

171

T. A. C. M. Claasen and W. F. G. Mecklenhräuker,PhilipsJ. Rex..

vol. 35, pp. 372—389, 1980.

[81 E. Wigner. in Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, Eds. New York: Dover. 1979, ch. 4.