# How Matched filter is transformed due to frequency modulation?

Intro:

Consider a signal source S which emits a PAM modulated signal, bits encoded as +g(t) and -g(t) pulses.

The matched filter for S is f - f is a function of g.

Then S is fed to an FM modulator and yields a modulated signal, S', let's call the corresponding matched filter of the modulated signal f'.

Question:

If f is known how can f' be computed ?

• @jhegedus The PAM signal can be expressed as $$\sum_{k=0}^{N}a_k g(t-kT),$$ where $a_k$ is either 1 or -1, and $T$ is the symbol rate. The data $a_k$ can only be recovered with a matched filter if the pulses $g(t-kT)$ form an orthonormal set for all integers $k$. When the PAM signal is frequency-modulated, the orthogonality of the pulses is lost. This is why the best approach is to get the signal back to baseband, and then apply the matched filter. Most books on digital communications (e.g. Sklar) cover this subject. – MBaz Jan 15 '16 at 16:18