Maximizing SNR in improper complex noise and applications to CDMA

The optimum linear time-invariant (LTI) filter that maximizes the signal-to-noise ratio (SNR) is, in general, the solution to an integral equation. This is the well-known matched filter. With regards to the detection of bandpass signals, we present the full form of the integral equation using the complex baseband representation of signals. The correct form of the complex integral equation includes, in addition to the autocorrelation function, a pseudoautocorrelation function which vanishes for most applications. However, in the presence of improper complex noise, overlooking the pseudoautocorrelation leads to a "nonoptimal" filter. To illustrate the concept, we apply the theory to a direct-sequence code-division multi-access (DS-CDMA) system in which improper complex noise arises. In the application, we derive the SNR maximizing filter and the nonoptimal filter and compare their near-far resistances.