What is the correlation receiver and how does it achieve optimal signal detection in noise?

The correlation receiver achieves optimal signal detection in noise by computing the correlation between the received signal and a locally generated copy of the expected signal (a template or reference), producing a maximum output SNR when the received signal matches the template. The correlation receiver implements the matched filter in the time domain. The operation: multiply the received signal r(t) by the reference signal s(t) and integrate over the signal duration T: output = integral from 0 to T of r(t) x s(t) dt. If r(t) = s(t) + n(t) (signal plus noise): the output = integral of s^2(t) dt + integral of s(t) x n(t) dt. The first term is the signal energy (deterministic, constant). The second term is the noise contribution (zero mean, random). The output SNR is: SNR = 2E/N0, where E is the signal energy (integral of s^2(t) dt) and N0 is the noise power spectral density. This is the maximum achievable SNR for any linear receiver. The correlation receiver is optimal because no other linear receiver architecture can produce a higher output SNR. Applications include: radar pulse detection (the received echo is correlated with the transmitted pulse shape), spread spectrum receivers (DSSS receivers correlate with the PN code to despread the signal), GPS receivers (the GPS signal is below the noise floor; correlation with the known C/A code extracts the signal with up to 43 dB of processing gain), and digital communication receivers (correlate with each possible symbol waveform to determine which symbol was transmitted).