I am working on a problem which raised what should be a simple question, but I'm stuck on it. The objective is to detect a shifted and scaled version of a known signal in the presence of Additive White Gaussian Noise (AWGN).
Think of a radar system. In a radar system, you know the transmitted signal. The returned system is delayed in time and scaled. I understand that a time-delay, only shifts the location of the matched filter's peak output. No problems there. But if the scaling is unknown, then how does one construct the reference signal?
Using a reference signal with the an amplitude that doesn't match the amplitude of the signal in the noisy data will mean that a threshold tests based on that reference signal will be incorrect.
I'm guessing that in a radar system, the noise power is approximately known, and so the power of the received signal is estimated by taking the power of the received signal and subtracting the noise power. That value could then be used to scale the reference signal. Still, that would only be and approximation, and it would still effect the probability of detection and false alarm in some way. Any help is appreciated. Thanks!