IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 6, JUNE 1996
Instantaneous Frequency Estimation of Quadratic and Cubic FM Signals Using the Cross Polynomial Wigner-Ville Distribution
1549
that is
.w7gl,(t,f,
Wazl+b22,Y(t, (k) f )= -1 bwg!,(t> f). The selection of coefficients c, is based on the following two requirements [4], [5].
Branko Ristic and Boualem Boashash Abstract-The cross polynomial Wigner-Ville distribution (X-PWVD) is applied to instantaneous frequency (IF) estimation of polynomial FM signals. The X-PWVD is used in a simple iterative algorithm, similar to the one recently proposed for the cross Wigner-Ville distribution (X-WVD) 121. The advantage of the X-PWVD over the polynomial Wigner-Ville distribution [3] is that the former is linear with respect to the observed signal and as a result it attains the Cramer-Rao bound at lower signalto-noise ratios (SNR's).
1) The PWVD must be real valued. This condition amounts to the order k to be even, and 02% = -czZ--l for i = 1,.. . , k / 2 . 2) The PWVD of a polynomial FM signal s ( t ) = 4 exp {j27r Cy=.=,a , t z } must yield ii delta function along the signal IF, i.e., W!(i?(t:f)= i l k 6 ( f - E.=, i u , t " ' ) . These two conditions when combined result in the set of nonlinear equations (for details see [4] and [ 5 ] )as follows: k/2
c2z
11. THECROSSPOLYNOMIAL WIGNER-VILLE DISTRIBUTION We adopt the definition of the X-PWVD of two complex signals ~ ( tand ) y ( t ) [5],[9], as follows:
n J: x ( t + C k T )
n
k-1
Pt[y(t
+
CtT)]?-J2"fr
( t .f ) E R2
Xc.2 = O
(for odd values of nr;3
5
m
(1)
where ct. 7 E R, PL[.] is the complex-conjugate operator if i is odd, ) the kernel of the Xand identity operator otherwise; I i $ ; $ ( t , ~is PWVD. If z (tj = 'y ( t ) then (1) becomes the definition of the (auto) PWVD [4], as follows: ~
q 2 (t.f )
(3)
pj.
2=1
Solving (2) and (3) for polynomial order p := 4 (which is of interest in this paper), we found that the coefficient!; of the PWVD of order k = 6 [4], [5] are as follows:'
where c6 > 112 for c2 and c4 to exist and to be real. The remaining coefficients are found using the fact that c2, = - c ~ ~ - I . The IF Estimation Algorithm: The X-PWVD defined by (1) requires, in general, knowledge of both the observed signal z ( f ) and the reference signal y ( t ) . However, in situations where the reference signal y ( t ) is unknown, it is possible to estimate the IF using an iterative algorithm similar to the one recently proposed in the context of the X-WVD [ 2 ] . The IF estimation algorithm based on the XPWVD is summarized in Table I. The input parameter t in Table I controls the tradeoff between the accuracy and the required number of iterations.
111. CONVERGENCE ISSUES IN THE ABSENCE OF
dT
2=1
=FT-I{IC:Fi(t.T)}
(2)
kI2
The polynomial Wigner-Ville distribution (PWVD) has been recently defined as a method for time-frequency analysis and instantaneous frequency (IF) estimation of nonlinear polynomial FM signals [3],[5].A serious limitation of the PWVD is its poor performance in the presence of additive noise [5].In order to overcome this problem, which is mainly due to the nonlinear nature of the PWVD, we propose to use the X-PWVD, defined here as a linear transform with respect to the observed signal. As a result, an IF estimator of quadratic and cubic FM signals is constructed, characterized by an improved performance at low SNR. For other types of FM signals, the performance of the estimator would degrade. The correspondence is organized as follows. Section I1 defines the X-PWVD and describes the iterative IF estimation algorithm. Section I11 shows analytically the convergence of the algorithm in the case of quadratic FM signals, unaffected by noise. The performance of the IF estimator in the noisy case has been studied by simulations in Section IV.
w::;(t> f )
= 112
Z=l
I. INTRODUCTION
I\iOISE
In this section, we examine analytically the underlying mechanism of the iterative algorithm in the case of a quadratic FM signal, unaffected by noise. Suppose the observed signal is
.(t) = b o e 3 [ L " ( S O t + " t 2 + f ~ t 3 ) + ~ O l with the IF given by fz,(t) = fo reference signal be y(o) (1) = , ~ ~ ~ [ ( f 0 + A
05t 5
T
(6)
+ 2at + 3 d t 2 . L,et the initial S j t + ( ~ + h ~ ) t 2 + ( ~ + ~ ~ j ~ ~ I
O l t l T
(7)
where -If,Am and Ad represent arbitrary errors in the initial IF coefficients. Next, we apply the X-PWVD of order k = 6 in the For k = 2 and CL = -c-2 = 112, (I) amounts to the conventional iterative scheme described in Table I. The X-PWVD o €signals r ( t ) X-WVD [6].Note that the X-PWVD is linear with respect to ~ ( t ) ;and y(f), given by (1) and for k = 6. can be expressed as Manuscript received February 24, 1994; revised November 14, 1995. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Patrick Flandrin. The authors are with the Signal Processing Research Centre, School of Electrical and Electronic Systems Engineering, University of Technology, Brisbane, Australia (e-mail:
[email protected]). Publisher Item Identifier S 1053-587X(96)03041-3.
W:'J(t..f) =
1
Z(f
.y*(t
-k C 6 T ) l / * ( t
- CzT)Y(t
+
CLT)
- r q T ) y ( t + L ~ T T )(~t * rGT)e
JPnfr
dr (8)
' For p = 4 and k = 4, (2) and (3) have no real-valued solutions
1053-587X/96$05.00 0 1996 IEEE
I ssn
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 6, JUNE 1996
TABLE I PSEUOOCOOE OF AN ITERATIVE ALGORITHM BASEDON THE X-PWVD FOR TIME-FREQUENCY REPRESENTATION A~DIOR INSTANTANEOUS
FREQUENCY ESTIMATION OF POLYNOMIAL FM
1200 -
SIGNALS 1000 -
z ( t )- observed signal (non-linear FM signal in noise e);
Input:
t-
800 -
acceptable error; n 3 z t
2
600-
2
Algorithm: 400 -
Create initial reference signal y(O)(t); m = 0; Repeat
oi
1. Calculate X-PWVD, W:lk)lml(t,f )
2 . Estimate the IF from the X-PWVD's peak as follows: (vt) i:m+l)(t)
= argmaxf
lwJ$ml( t ,f ) I
Fig 1
3. Synthesise reference signal ~ ( ~ + ' = ) (expb2n t)
until J, If,(m+"(t) - f,'"(t)12cit
Output:
+
-
f:m+')(t)
+
+.
+
cz~)]*y(O'(to
( 2 ~ )
+
(6) hz y(o) ( t o , T)
- 327r(Afto
+
+
{ J ~ T [ ~ " T2 0 7 t o
(AfT
+ 3 3rt; + (1
+
-
C6)A$]f;
+
-
2(Am
f'""(f)
Consider now the magnitude of the X-PWVD (at t = t o ) l J ~ - ( r ) ( o j ( ~ o . f ) l = IS7-f{",(6)y ( 0 ) ( f O T))l. J-Y
(10)
(11)
T
- C6)A.f)
+ s(fi+ (1
+ 3 ( 3 + (1 - e6)Ao)t:
i ~ n ~ ) t
+ 3cgABT)t'
+ (1 c6)"'Af + ($'"A0T] + 2[a+ (1 c ~ ) ' ~+A3ci"A/?T]t N + 3[/3+ (1 ~ ) ~ A p ] t ' .
[fo
- C6)Ao")tll
(13)
-
(14)
Provided that 0 < CG < 1: (14) in the limit tends to the IF of ~ ( t i.e. ): liiii f " " ' ( t ) = fi,l.(t) = f o
n1-x
+ 2at + 3 / ~ t . ~
(15)
Recall that CG must be greater than 1 / 2 ; hence, coefficient CG must be in the range (0.5, 1). The freedom in choosing CG may help to speed up the rate by which the unwanted terms in (14) approach zero. We choose the value of Cg to be the one for which all unwanted terms in (14) diminish by the same rate: that is, when (1 - r e ) = 62.After solving this quadratic equation and rejecting the solution that is less than 112, we obtain the following for C G :
vanable,
function / M ( t o , f ) 1 is symmetncally spread about the central frequency as follows
+ (1
c6)Aa
+c
~
A typical waveform of ~ K ' ~ 6 ~ l , o , is (to shown . f ) ~ in Fig 1 Since
:6ico)
- c6)ilf
-
+ 3A3to)c;r
A:6i(o) ( t o . T ) is effectively a linear FM with respect to
-
+ (1
+ (;j+ (1- ~ A , w ~ I } .
+ 2AOTt" + 3 A j T t ; ) l )
+ + R[$ + (1
(f0
+ (a+ (1
- c6)
The signal described by the time-lag product ( t o . T ) has a linear frequency vanafion (with respect to T vanable) given by 1 (1 ___ drg { ~ ~ ~ ~ ( o )T)} ( ~ O . 2n dT = [fo (1 - C 6 ) A f l 401 (1 - f 6 IAnIto
FO
= exp { j a r [ ( f o
By repeating the described procedure, the IF of the reconstructed reference signal after m iterations becomes
+ Ant; + 1j t i ) }
exp { - ~ 2 7 r ( ~ a 3 ~ d t o ) c ~ ~ ~ ) pxp
+ 3APto)c; . T
where T is the duration of the signal. By locating the peak of the magnitude of the X-PWVD as t = t o ,one extracts, at the worst case, the value of frequency at AA f ( l ) ( t o )= Fo -. (12) 2 Assuming that the sign in (12) is always the same (which is again the worst case), say the reconstructed reference signal (step 3 in the repeat-until loop in Table I) is given by
. [ + O ) ( t o - c l T ) ] * y ( o ) ( t oc s T ) [ y ( o ) (-t oL 6 T ) ] * (9) For x ( t )and y ( " ( t ) expressed by (6) and (7), respectively, using (2) and (3), the expression in (9) yields
= exp {JdO
Typical waveform of i W ~ 6 ~( (t oo, f, ) l
A,4 = 2 ( A n
The kernel of the X-PWVD at t = t o then reads
(to, T ) r ( t o c b r ) [ y(0)( t o
5
Ok5
with a spread measured as
or the time-frequency representation W ~ ~ $ l m( +t ,fl l)
-
0'4
J f:m+l)(u) i du]
< t; m = m + I;
Instantaneous frequency estimate
FO
NORM 0'2 FREQUENCY 0 2 5 h - O b 5
0 , i A d O \ 5
(C6LPt
=
~
&2-
- 0.61803.
21n general, the optimal value of Cg for fast convergence of the algorithm, depends on the initial errors Af and Au and A,3. Since these errors are unknown, we treat the unwanted terms in (14) as if they have the same weight.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 6, JUNE 1996
1551
IF estimation, followed by a signal reconstruction algorithm), since the spectrogram is known to have a low SNR threshold [l].In order to illustrate this point, consider an FM signal z ( t ) = ho rxp { j h ( t ) } with the cubic instantaneous frequency law given (in Hertz) by
f z ( t ) = 20
+ 1031.252
-
4394.5t2
+ 4577.6t3.
Fig. 3 displays (a) the spectrogram and (b) the PWVD of signal ~ ( taffected ) by additive white Gaussian noise (WGN), with SNR = 25 dB. The PWVD produces a much sharper t-f representation,
initial IF >
I
TIME (a) 55 TRUE VALUE
- x-
50 X-PWVD
451
40
w
x
,XI
>
