Matched filtering in improper complex noise and applications to DS-CDMA

Based on the criterion of maximizing signal to noise ratio (SNR), the optimum matched filter is, in general, the solution to a real integral equation. With regards to the detection of narrowband, bandpass signals, we present the full form of the integral equation based upon complex envelopes. The correct form of the integral equation includes a complementary or pseudo-autocorrelation function which can be ignored for most matched filtering applications. However, in the presence of improper complex noise, overlooking the function leads to a "non-optimal" matched filter. To illustrate the concept, we apply the theory to a simple direct-sequence code-division multiple-access (DS-CDMA) system.