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An Extended Correlation Function of Two Random Variables Applied to Mobile Radio Transmission

01 December 1969

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If two random variables depend on only one common parameter, such as time or distance, conventional correlation formula can be applied to the two variables. However, if both of these variables involve not one but two common variable parameters, then the correlation formula found in the current literature is limited.* Since such cases occur in some of our mobile radio problems, as we discuss later, we need to define an extended correlation function and outline its properties and applications. [I. DERIVATION OF AN EXTENDED CORRELATION FUNCTION OF TWO RANDOM V A R I A B L E S INVOLVING TWO COMMON P A R A M E T E R S A conventional normalized correlation function of two random variables ri and r 2 , both of which are functions of one parameter * Prior to acceptance of this paper for publication, the author was advised that a similar concept was discovered independently by A. Papoulis in his recently published book. 1 3423 3424 T H E BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1969 d[that is, r,(d,) and r2(d2)] can be expressed as2 , /?,2((/, , d2) - m,mo PM , (i2) = a,a 2