I am used to Time-Bandwidth product describing the processing gain or improvement in SNR for a spread spectrum signal using a matched filter.
One could view the bandwidth $B$ as the sample rate (complex) and the time $T$ is the integration time and thus there are $TB$ samples. Thus, the matched filter produces a gain of $10\log_{10}N$, with $N$ being the number of samples.
So with all of this in mind, suppose a bandpass filter removes all noise power except for the signal bandwidth, thus the noise power is $kTB$. The signal power is the amplitude $A$ squared, $A^2$. If we over sample the signal by some factor $M$, then I want to say that the processing gain is now $MTB$ because there are $M$ more samples.
I also have worked some theory of this as well as a MATLAB script and all is indicating that it is true. Am I missing something? Why not over sample signals to provide more processing gain using a longer matched filter?