10.3.1 Correlation Receiver

The problem may be characterized as follows

where S _k ( t) is a known deterministic signal with energy E _k, such that

and W( t) is an additive white Gaussian noise process with mean zero and power spectral density N ₀ / 2, or of covariance (autocorrelation) function

The M signals may be dependent and correlated with autocorrelation coefficients

As before, we need to find a set of orthonormal basis functions in order to expand the received process Y( t); that is W( t) into the Karhunen-Lo ve expansion, since C _yy( t,u) = C _ww( t,u).

Using the Gram-Schmidt orthogonalization procedure, we can find a set of Kbasis functions, K ? M, if only K signals { s _k( t)} are linearly independent out of the original M signals. Once the complete set of K orthonormal functions { ? _t( j)}, j = 1,2, , K, are obtained, we generalize the corresponding coefficients by

From (10.29), the signals S _k( t), k = 1,2, , M, may be written as

where S _kj is as defined in (8.36). Substituting (10.94) into...