Proof of the cross-spectrum inequality

Journal of Sound and Vibration (1977) 51(4), 561-562

LETTERS

TO THE EDITOR

PROOF OF THE CROSS-SPECTRUM INEQUALITY For any pair of stationary random processes {x(t)} and {y(t)), the Cross-Spectrum Inequality states that, for any value off,

[G~,(f)[ 2 _-
(1)

where G~(f) and Gr(f ) are appropriately defined autospectral density functions and G~y(f) is their associated cross-spectral density function. A physical input/output system proof for this result is shown in Chapter 5 in the book by Bendat and Piersol, reference [1], which applies to both ordinary and multiple coherence functions. Advanced mathematical proofs appear in references [2, 3]. A simple indirect mathematical proof is claimed in equation (3.83) of reference [1]. Unfortunately, the steps outlined to prove equation (3.83) are not correct. The purpose of this note is to supply details for a new derivation which might provide engineering insight. As shown in equations (3.100) and (3. I01) of reference [I], cross-spectral density functions G~,(f) and autospectral density functions G~(f), Gy(f) can be defined for any f > 0 by 2

G~,(f) = r-.~lim-~E[X*(f) Yk(f)], 2

=

7E[ix

(2)

(:)l,j, (3)

Gr(f) = lira T~eo

2

E/I YKf)lz],

where Xk(f) and Yk(f) are certain finite Fourier transforms Of Xk(t) and yk(t), respectively, obtained from truncated records of length T. The quantity X*(f) is the complex conjugate of Xk(f) and E / ] denotes the expected value over the index k of the terms in the brackets. The function Gxy(f) can also be expressed as Gx,(f) = IGx,(f)l e -JO~,($),

(4)

which defines both the magnitude factor IG~,(f)I and the phase factor O~r(f) for any f > 0. It follows also from equations (2) and (4) that

G,x(f) = G*~y(f)= [Gx,(f)l eJ~ ~:~.

(5)

Consider now the quantities Xk(f) and Yk(f)e j~ For any real constants a and b, the expected value over the index k is greater than or equal to zero."

E[laXRf) + b YRf)eJ~

>--0.

(6)

The term inside the brackets is a~lXk(.f)[ z +

ab[X*(f) Yk(f)e Jax~:~+ X~(f) Y*(f) e -J~,($~] q" b2[ YRf)I 2 > 0.

(7)

562

LETTERSTO THE EDITOR

By taking the expectation of equation (6) over the index k, multiplying by (2[T) and letting T increase without bound, there results a 2 G~,(f) + ab[G,,,(f)e j~

+ Gyx(f)e -J~

+ b 2 G , ( f ) >=O.

(8)

But, from equations (4) and (5), Gxy(f) e j~

+ %x ( f ) e -J~

= 2[G~,(f)l.

(9)

Hence a 2 G,,(f) + 2ablG,,y(f)] + b 2 Gy(f) > O.

(I0)

It now follows exactly as in the earlier proof given for the Cross-Correlation Inequality, equation (3.50) of reference [1], that [G~,(f)l 2 =< G , , ( f ) G r ( f ) .

(11)

This gives the desired result in a direct way. ACKNOWLEDGMENT I would like to thank R. H. Burros for his correspondence on this matter. J.S. BENDAT

J. S. Bendat Company, 833 Moraga Drive, Los Angeles, California 90049, U.S.A. (Received 3 November 1976)

REFERENCES 1. J. S. BENIgAr and A. G. PIERSOL1971 Random Data: Analysis attd Measurement Procedures. New York: John Wiley and Sons, Inc. 2. J. M. JENKXNSand D. G. WATTS 1968 Spectral Analysis and its Applications. San Francisco : Holden-Day. 3. L. H. KOOPMANS 1974 The Spectral Analysis of Time Series. New York: Academic Press.