in Proc. IEEE 32nd Asilomar Conf.Signals, Systems, Computers
Pacific Grove, CA, pp. 282–286, Nov. 1998
Copyright 1998 IEEE
TIME-VARYING SPECTRA FOR UNDERSPREAD AND OVERSPREAD
NONSTATIONARY PROCESSES*
Gerald Matz and Franz Hlawatsch
Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology
Gusshausstrasse 251389, A-1040 Wien, Austria
phone: +43 1 58801 38916, fax: +43 1 5870583, email: [email protected]; [email protected]
web: http://www.nt.tuwien.ac.at/dspgroup/time.html
The generalized W i g n e r - Ville spectrum (G WVS) [4],
wp(t,f) 4 LZJ((t,f) .
ABSTRACT
We introduce an extended concept of underspread and overspread nonstationary random processes and show its importance for time-varying spectral analysis. We consider two
classes of time-varying power spectra that comprise most
existing spectra, including Wigner-Ville and evolutionary
spectra. For underspread processes, all spectra are shown
to yield similar results. For overspread processes, the spectra may yield very different results and contain oscillatory
“cross-terms” indicating time-frequency correlations. These
cross-terms can be attenuated via smoothing techniques.
1
Under mild conditions [4],the GWVS is the expectation of the generalized Wigner distribution [5,6]. The
Wigner- Ville spectrum [4,7] and the Rihaczek spectrum
181 are obtained for a = 0 and Q = 112, respectively.
The generalized evolutionary spectrum (GES) [3],
f)I2,
G W , f) 4 I@(t,
(1)
with H an innovations system of ~ ( t )For
. Q = 112,
ct = 0, and a = -112, one obtains the euolutzonary
spectrum [9], the W e y l spectrum [3], and the transztory
evolutionary spectrum [3, IO], respectively.
INTRODUCTION
This paper discusses the time-varying spectral analysis of
nonstationary random processes. In Section 2, we introduce
an extended notion of underspread and overspread processes,
and we show that in the case of overspread processes two fundamental families of time-varying power spectra contain oscillatory cross-terms that indicate time-frequency (TF) correlations. In Section 3, we define two generalized classes
of time-varying power spectra that allow an attenuation of
cross-terms by means of a TF smoothing. Section 4 provides bounds which show that the spectra of underspread
processes are approximately equivalent and satisfy other desirable properties. Finally, simulation results are presented
in Section 5 .
Much of our theory will be based on the generalized W e y l
symbol [l] and the generalzzed spreading f u n c t i o n [l] of a
linear operator H that are defined as
UNDERSPREAD AND OVERSPREAD
PROCESSES
Whereas stationary processes feature only temporal correlations and (nonstationary) white processes feature only
spectral correlations, general nonstationary processes feature
both temporal and spectral correlations. These “TF correlations” can be described by the generalized expected ambiguity
f u n c t i o n (GEAF) [ll,121 defined as
2
A p ( 7 , v) e Sk“,’(T,v ) .
The GEAF magnitude is independent of a , and hence we
write IA&a,”)(7,v)I
= IA,(.r,v)I. The value of lA,(r,v)l at
a specific point ( 7 , ~is) a global measure of the statistical
correlation of process components separated in time by 7
and in frequency by v [ll,121. The GEAF is related to the
GWVS by a 2-D (symplectic) Fourier transform,
Jr Jv
respectively, where H ( t 1 , t z ) is the kernel of H and Q is a
, are related
real-valued parameter. L $ ) ( t , f) and S g ) ( r v)
by a 2-D (symplectic) Fourier transform.
Two Families of Time-varying Power Spectra. The
second-order statistics of a nonstationary process z ( t ) are
characterized by the 2-D correlation function R,(tl, t z ) =
E(z(t1) z * ( t z ) } or the correlation operator R, (whose kernel
is R,(tl, t z ) ) . Alternatively, we may consider an innovations
system H of z ( t )which is obtained as a (non-unique) solution
to HH’ = R, [2,3], with H+ the adjoint of H.
The following two families of time-varying power spectra
of nonstationary processes have been defined previously:
An Extended Concept of Underspread Processes.
In [11,12], a process has been called underspread if its GEAF
has compact support with area 5 1. This implies a sharp
limitation of Ohe process’ TF correlations. Since in practice
GEAFs are almost never compactly supported, we now formulate an extended concept of underspread processes which
merely requires the GEAF to be concentrated about the origin of the (T,v)-plane. We propose t o measure the GEAF’s
concentration via the weighted integrals
*Funding by FWF grant P11904-OPY.
0-7803-5148-7/98/$10.0001998 IEEE
282
(4
ative components. Since (3) implies that the innovations
systems of overspread processes are overspread, the GES
contains oscillatory components too; these are nonnegative
and thus more difficult to identify than in the GWVS case.
These oscillatory GWVS and GES components are related
to cross-terms that indicate the strong T F correlations existing in overspread processes (such “statistical cross-terms”
have previously been reported for a special case in [15]).
In order to illustrate the mechanism and geometry of
statistical cross-terms, we consider the two-component process z ( t ) = z l ( t ) z 2 ( t ) with z l ( t ) = alzo(t - tl)e32Kf1t
~,
z:o(t) is a proand z z ( t ) = a2zo(t - t ~ ) e ” ~ f 2where
cess whose GWVS is localized about the origin of the T F
plane and a l , a2 are random factors uncorrelated with zo(t).
Hence, the components z l ( t ) and m(t) are localized about
the T F points (tl,fi) and (t2,f2), respectively, and they
are correlated if a1 and a2 are correlated. In the overall process z ( t ) , the T F points (t1,fl) and ( t 2 , f i ) will
then be correlated. The GWVS of z ( t ) consists of two
“auto-terms,” $ : ) ( t , f ) = E{la1I2} $r’(t - t1,f - f l )
(b)
+
Figure 1. Contour-line plots of the GEAF magnitude of (a) an
underspread process and (b) an overspread process. The small
rectangles shown have area 1 and thus permit to assess the respective process ’ underspread or overspread properly. The lowest
contour line is 20 d B below the mazimum value, IAl(O, 0)l.
Here, $(r,v) with d(7,v) 2 +(O, 0) = 0 is a weighting function penalizing GEAF contributions away from the origin.
and Mi’) are the GEAF moments
Special cases of
and Mi”‘“) (with k , l E NO)which are defined using the weighting function d ( ~v), = 171k 1 ~ 1 ’ . We note that
mi’)
mik’‘)
-4
F
and Mi““) 5 M : ~ ~ , O ) M ( O , ~ ~ ) .
The above defined weighted GEAF integrals/moments
characterize the T F correlations of nonstationary processes
in a flexible manner. In particular, a process will be called
underspread if the weighted integrals/moments are small;
otherwise, it will be called overspread. Fig. 1 contrasts the
GEAFs of an underspread and an overspread process.
The innovations system H provides an alternative characterization of T F correlations. H is called underspread
if S ~ ’ ( T , Vis) concentrated about the origin, which indicates that H introduces only small T F displacements
[12-141. The concentration of S$’(T, v) can be measured by
weighted integrals mg),M$) and moments mg”),M:’” of
S~)(T
v), whose definitions are analogous to the GEAF integrals/moments [14]. The relation R, = HH+ implies [12,14]
mik,l)
5
IAz(7,.)I
5
.)I ** I s H ( - T ,
wir)(t
and $ z ) ( t , f ) = E{la2I2}
- t 2 , f - f i ) , that are
correctly localized about ( t l ,11) and ( t 2 , f2), respectively,
and two oscillatory “cross-terms” which are localized about
the T F points
- a(t1- t 2 ) ,
a(f1-fz))
and
a(t1 - t Z ) ,
- a ( f 1 - f 2 ) ) (these cross-terms
can be calculated using results from [IS]). In particular, for
a = 0 the two cross-terms collapse into a single real-valued
cross-term given by c(t fwith
(F
+
w+
(9
w
F,
c(t,f) = 217-1 Tvi:)(t,
w)
f) cos [ 2 7 4 f 1 - f 2 ) t - ( t l 4 2 ) f )
+ 91 ,
+
where cp = 27r(f1 - f Z ) ( t i - t 2 )
arg{r}, with T =
E{ala;}. This cross-term is located about the center point
and is oscillatory with significant negative
components. From this cross-term (whose geometry equals
that of the Wigner distribution [16]), it is even possible to infer the correlated T F locations (tl,f i ) and ( t z , f 2 ) . One can
similarly show the existence of oscillatory (but nonnegative)
cross-terms also for the GES.
(F,
v)
(3)
where ISH(T,
v)[ denotes the (a-independent) magnitude of
S $ ) ( T , ~and
) ** denotes 2-D convolution. Hence, if H is
3 SMOOTHED TIME-VARYING SPECTRA
v)I will be concentrated and thus z ( t ) The “statistical cross-terms” discussed above may be useful
underspread, ~A,(T,
will be underspread too. Conversely, if z ( t ) is underspread,
as indicators of TF correlations. On the other hand, they
one can always find an underspread innovations system H.
tend to mask the “auto-terms” that characterize the mean
Since in the underspread case Ap’(7,v) and S ~ ) ( T , V )T F energy distribution of the process and thus indicate the
T F locations of the process’ components. An attenuation or
are concentrated about the origin and since A ~ ) ( T , vf))
suppression of cross-terms can be achieved by a TF smoothWL”’(t,f ) and S$)(T,v) U L g ) ( t ,f) are Fourier transform
ing, as explained in the following.
pairs, the GWVS and GES of an underspread process must
Type I Spectra. It can be shown [4] that all T F shiftbe smooth functions. In fact, for any process z ( t )the partial
covariant spectra with linear dependence on the correlation
derivatives of the GWVS and GES are bounded as
’function R x ( t l ,t 2 ) can be written as
IsH(7,
(’
’
(k-i,l-j)
(;.)m&’)m,
Here, H is the innovations system used in G p ) ( t , f).
Overspread Processes and Statistical Cross-Terms.
If a process is overspread, i.e., if its GEAF has significant
components far away from the origin, then it follows from
( 2 ) that the GWVS must contain oscillatory and partly neg-
with
PH
(%T)”+’IIsHII;
C ( t ,f) 4 $(t,f) ** i 8 ” ( t , f) ,
(4)
where $(t,f ) is some function that does not depend on
Rx(tl,t 2 ) . Here, without loss of generality, we have used the
GWVS with (Y = 0 (using a different a value in (4) would
result in the same class of spectra). The 2-D Fourier transform of cl(t,
f ) is given by *(T,v)ALo)(7,
v) with *(T,v) =
$(t,f ) e j 2 R ( T f - u t )d t d f . Hence, if * ( T , v) is concentrated about the origin of the (T,v)-plane, which implies that
$(t,f) is a smooth function, then components of AC’(7, v)
located away from the origin and, in turn, oscillatory cross-
s, &
283
wLo)(t,
based on the same innovations system H is bounded as
terms in
f ) will be attenuated. Note that the convolution in (4) here becomes a smoothing: We shall assume
q ( 0 , O ) = 1 which guarantees that j , jfC, ( t ,f) d t d f yields
the mean signal energy E, = Rz(t,t)dt.
The class of time-varying spectra defined by (4), hereafter called type I spectra, has previously been considered
in [4,17,18]. Under mild conditions [4], the type I spectra
equal the expectation of TF signal representations of Cohen’s
class [19,20]. The GWVS with Q # 0, Page’s instantaneous
power spectrum [21], Levin’s spectrum [22], and the physical
spectrum [23] are obtained for specific choices of Q(T, v).
Type I1 Spectra. We define the novel class of type II
spectra as the GES (again with Q = 0, without loss of generality) augmented by a convolution that is implemented prior
I
to taking the squared magnitude in (l), i.e.,
IG?’(t,f) - G?)(t,f)l 5 2 l
s,
mg)
@ ) ( t ,f) M
We will now show that for underspread processes, different
type I and type I1 spectra are approximately equal and approximately satisfy some desirable properties. As a mathematical underpinning of these approximations, we present
bounds on the approximation errors that are formulated in
terms of the weighted integrals/moments from Section 2.
Equivalence of Type I Spectra. It can be shown that
the difference between any two type I spectra cil)(t,
f) and
cp)((-t,
f) is bounded as
p%,f) - W ( t ,f)l 5 IIAxII1 d’), (5)
1pp cpq25 IIAzl12Mi’),
with +(T,v) = I ~ ( ’ ) ( T , v )
- qC2)(~,v)1.
Thus, if z ( t ) is underspread with small mi’) and Mi’), then we have
@ ) ( t ,f ) M ep’(t,f) .
-
As a special case, the difference between two GWVS with
different a parameters is bounded as
lw;orl)(t,f)-wx
- ( W ) (t,f)I I 27rIa1-azI
~ ~ w ? ~ ) - Iv: : y ~ 2
27rlCr1-
a21
I G ~ ’ ) ( t , f ) - G ~ ’ ’ ( t , f )5
I 4 ~ 1 ~ x 1Q- ~ ~ ~ ~ S H ~ ~ ~ V Z ~ ’
(This bound extends a result in [3].) Hence, for an underwe have
spread innovations system H with small &”),
G:”(t,
(These bounds extend a result in [ll].) Hence, for an underspread process z ( t ) with small m?”) and MA’”), we have
-
w:*”(t,f) M w;-)(t,
f) .
Note that small mc’l) and Mi”’) imply that A%’(T,v)is
concentrated along the T axis and/or the v axis.
Equivalence of Type I1 Spectra. The difference between two type I1 spectra z?’(t,f ) and G?’(t, f) that are
f)
M
G p 2 ) ( tf)
, .
Equivalence of Type I and Type I1 Spectra. Extending a result in [11,12],the difference between the GWVS and
the GES with the same a can be shown to be bounded as
lw$or)(t~
f) - G?)(t,f)l
5 271. l l s H l l l &(a)
(7)
with
where H is the innovations system used in G P ) ( t ,f). Hence,
for an underspread process with small moments of HI
-
Wi”’(t,f)
G p ) ( t ,f) .
M
Note that ,$$’in (8) is smallest for a = 0. Here, the bound
(7) can be tightened by using the metaplectic covariance of
the generalized Weyl symbol with Q = 0 [l,14,241:
Iwio’(t,
f) -Gio)(t,f ) l
5
hlls~lltir&f{mg$,+mg&+},
(9)
where the infimum is over all unitary metaplectic operators
U (corresponding to linear, area-preserving TF coordinate
transforms) [l,6,14,24].
By combining the bounds in (5), (6), and (9) using the
triangle inequality, we finally obtain the following bound on
the difference between any type I and type I1 spectrum,
IW(t,f)-m t , f ) l
5
I I S H I I[mi’’)
~
+ 2mg2) + 2rinf{mgil+nlC:l+>]
U
,
with c # q ( ~ , v) = Il-q(‘)(~,
.)I and + z ( T , v ) = ll-q(’)(~,
v)l.
Hence, if the process z ( t )and the innovations system H are
underspread such that the respective weighted integrals and
moments are small, we have
llAxlllmcJ),
llAzllzM p )
Gi2’(t,f) .
In particular, the difference between two GES with different
Q parameters is bounded as
where $ ( t , f ) is some function that does not depend on
the innovations system H. The 2-D Fourier transform of
$ ( t ,f ) ** L g ) ( t ,f) is * ( T , v ) S g ’ ( ~v), with * ( T , v) the 2-D
Fourier transform of $(t,f). Hence, if *(T,V) is concentrated about the origin of the ( T , v)-plane, i.e., $(t,f) is a
smooth function, then components of S$(T, v) located away
from the origin and, equivalently, oscillatory cross-terms in
@ ( t , f) (and, hence, in Gio’(t,f)) will be attenuated. We
note that the GES is obtained with ~ ( T , v )= e - j Z x o r v r .
Hereafter, we assume I ~ ( T , v)l 5 @ ( O , O ) = 1.
UNDERSPREAD APPROXIMATIONS
(6)
with +(r,v)= I * ( ’ ) ( T , V )
- ~ ( ‘ ) ( T , v ) [ .Hence, if H is unis small (which is only possible for
derspread such that
z ( t ) underspread), then
U f ) LL [ W , f )**LJIO)(t,f)I2,
4
l s H l / ~ ~ ~ ) ,
CL1)(t,f)
M
@’(t, f) .
Properties of Type I Spectra. Next, we discuss some
desirable properties of type I spectra.
Real-valuedness. A type I spectrum
f) is real-valued
if and only if q * ( - ~ , - v )= ! P ( T , ~ ) . In particular, the
GWVS is real-valued only for cy = 0. In general, the imaginary part of C,(t, f) satisfies
cz(t,
IIm{Cx(t,f))l I
;llAzIllmi’)l
I~(T,v)
p l { c z } ~ ~=2 ~llAzllzMiO’
-.)I
with c$(T,v) =
-q*(-~,
Hence, for an underspread process with small
and Mi’),
( t ,f ) will be
284
mi’)
ex
st
nearly real-valued. For the GWVS with a # 0, there also exist simpler but coarser bounds since here m:’)
47rla1~n:”’)
and Mi’)
4~rIalMi”’).
Marginal properties. A type I spectrum C,(t, f ) satisfies
the marginal properties
fies Jf e,(t,f ) d t d f = E , if and only if I@(T, v)I 1. This
is satisfied by all GES but not necessarily by other type
I1 spectra. However, it can be shown that the difference
A4
G,(t, f) dtdf - E, is bounded as
<
<
J; C X ( tf), df = Rz(t,t ),
st sJ
II~SHII;
1
CZ(tIf) d t = R x ( f ,f) ,
with R x ( f , f ) = E{lX(f)lz}), if and only if @(O,v)= 1
and @ ( T , 0) = 1, respectively. In the general case, the error
A,(t) Jf C x ( t ,f ) d f - R,(t, t ) can be shown to satisfy
IAl(t)l 5
Ax,
llAlll2
=
1
where H is the innovations system underlying E, (t,f ) and
+(T,v) =
Thus, as long as l@(r,
v)I M 1on
d
@ ( 0 , v ) 1 2 1 ~ , ( 0 , v ) 1 2 d v . Similarly, A2(f) 4 J, C , ( ( t , f ) d t Rx (f, f ) satisfies
11A2112 = 0, I
lA2(f)l 5 0, >
where Ox = J
, 11 - @ ( r0)l
, /A,(T,0)l d r and 0, = 11 @ ( T ,O ) l z l A x ( ~
0 ),l Z d T . (Note that the quantities A,, 0, and
A, , 0, can formally be viewed as degenerate weighted inteand Mi’), respectively.) Thus, for underspread
grals
processes where these quantities are small, the marginal
properties will be satisfied at least approximately.
Moyal-type relation. The Moyal-type relation
.
the effective support of ~SH(T,
v)I, the mean energy property
will be approximately satisfied.
Marginal properties. Bounds for the errors Jf &(t, f) df -
s,
G,(t, f ) d t - R x ( f , f) can be derived but they
are somewhat involved in general. Therefore, we shall only
consider the special case of the GES G e ) ( t , f ) . Extending previous bounds in [3], it can be shown that the differences A,(t)
SfG?)(t, f ) d f - R,(t,t) and A,(f)
G?)(t, f ) d t - R x ( f ,f) are bounded as
R,(t, t ) and
where A, = J
, 11 - 9(O,v)I IA,(O,v)ldv and A, = J, 11 -
m
s,
s,
l A ~ ( t ) lI 27r11 -
mi’)
l&(f)l
5
m p {J,
2 r l l -t 2 4 m p { s ,
ls~(7,v)I
d v } h%lll m:”),
lsH(7,V)I
d T } llSHlll m:”).
Thus, if mg”) is small, i.e., ~ S H ( T
v)]
, is concentrated along
the r axis and/or the v axis, the marginal properties will be
approximately satisfied.
1lC,oC;(t,f)dldf,(t,f)C;(t,f)dtdf
=
R,(tl,tz)R,:(tl,tz)dtldtz
is satisfied if and only if l @ ( ~ , v )E
l 1. In the genJJ C , ( t , f ) Cy*(t,f)d t d f eral case, the difference A3 A
RX(tl,t2)
R j ( t l l t 2 ) d t 1 d t 2 is bounded as
s,
s,, s,,
(6)
(4)
IA3I 5 IIA,II2IIAyIIzMz M y
with q 5 ( ~ , v=) 11 - ~ @ ( T , v ) I I .
Hence, for underspread pro) y(t) where Mi4) and M;’) are small, the
cesses ~ ( tand
Moyal-type relation will be approximately satisfied.
Positzvity. For underspread processes, the type I spectra
have been shown t o be approximately equal to the (nonnegative) GES, and hence they are approximately nonnegative. Specifically, we will consider the Wigner-Ville specf ) which satisfies many desirable properties.
trum
Whereas wLo)(tl
f) is not strictly nonnegative, its negative
part, w;”-(t,f)
- w.”(t,f)]/2, can be
shown to be bounded as
7
w~”(t,
e [IwLo)(t,f)(
1wL”’-(4 f)I I min(P1 Pzl ,
I
where
PI
= 47r i n f H { l l S H I I ~ m ~ ’ l ) m ~ ’ o
(with
) } H innova-
tions systems of z ( t ) ) and / 3 ~= ZllAzlll infg{mi’g)}. Here,
~ , ( T , v )= ll-Ap)(T,v)l whereAp)(-r,v) = J t g ( t + $ ) g * ( t e-32“”tdt with llgll = 1. Hence, the Wigner-Ville spectrum of an underspread process is approximately nonnegative, which conforms t o observations made in [4,25].
Properties of Type I1 Spectra. Since type I1 spectra
are always real-valued and nonnegative, we shall only consider the marginal properties and a mean energy property.
Me a n energy property. A type 11 spectrum Gx(t,f) satis-
a)
5 SIMULATION RESULTS
Fig. 2 shows various type I and type I1 spectra, with and
without smoothing, of an underspread process that was generated according to [26] and whose GEAF was shown in Fig.
l(a). All spectra yield practically identical results and correctly characterize the process’ mean TF energy distribution.
Fig. 3 shows the results obtained for the overspread process whose GEAF was shown in Fig. l(b). Some of these
results are quite different. The GWVS and GES in parts
(a), (b), (d), (e) contain oscillatory cross-terms that indicate strong statistical correlations between the ‘T’ and ‘F’
components. In the smoothed spectra in parts (c) and (f),
these cross-terms are effectively suppressed and the process’
mean TF energy distribution is better visible. However, the
smoothed spectra no longer indicate the TF correlations.
6 CONCLUSION
The distinction between underspread and overspread nonstationary processes is of fundamental importance for timevarying spectral analysis. In the underspread case, all timevarying spectra tend to yield similar and valid results. In the
overspread case, different spectra may yield dramatically different results. Here, one can choose between
‘ e non-smoothed spectra containing statistical crossterms, which indicate the strong time-frequency (TF)
correlations inherent in overspread processes but tend
t o obscure the process’ mean TF energy distribution,
e smoothed spectra with attenuated cross-terms, which
better represent the process’ mean TF energy distribution but fail t o indicate TF correlations.
285
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