Wide-sense stationary processes - Statistical Signal Processing of Complex-Valued Data - page 220

f 2

x ∗ x ∗ x

xx ∗ x

x ∗ xx

f 1

x ∗ xx ∗

xx ∗ x ∗

xxx ∗

Figure 8.4 Support of bispectra with different conjugation patterns for a bandlimited analytic
signal x ( t ) = u ( t ) + j u ( t ). The principal domain of the real signal u ( t )isgray.

The observation in this example can be generalized: moment spectra of order n
for analytic signals can be forced to be zero depending on the number of conjugates
q and the nonzero bandwidth of d

ξ ( f ). Let f min and f max denote the minimum and

maximum frequencies at which d ( f ) is nonzero. In order to obtain a nonzero moment

spectrum M x , ♦ ( f ), there must be overlap between the support of the random hypercube
d

ξ

ξ 1 ( ξ n − 1 ( ξ n ( − n f T 1 ). The lowest nonzero

1 f 1 ) ··· d n − 1 f n − 1 ) and the support of d

ξ i (

frequency of d i f i ), i = 1 ,..., n − 1, is f min if i = 1 and − f max if i =∗ , and

similarly, the highest nonzero frequency is f max if i = 1 and − f min if i =∗ .Takefirst

the case in which q ≥ 1, so we can assume without loss of generality that n =∗ . Then

we obtain the required overlap if both

( n − q ) f min − ( q − 1) f max < f max (8.100)

and

−

q ) f max −

−

1) f min > f min .

( n ( q (8.101)

Now, if q = 0, then n = 1 and we require

( n − 1) f min < − f min (8.102)

and

( n − 1) f max > − f max , (8.103)

which shows that ( 8.100 ) and ( 8.101 ) also hold for q 0. Since one of the inequalities

( 8.100 ) and ( 8.101 ) will always be trivially satisfied, a simple necessary (not sufficient)
condition for a nonzero M x , ♦ ( f )is

=

( n − q ) f min < qf max , if 2 q ≤ n ,

(8.104)

( n − q ) f max > qf min , if 2 q > n .