## Criteria for matched filtering in AWGN

#### Consider the response of a signal S(t) plus noise n(t) passing through a detection filter with transfer function H(f). If the Fourier transform of the signal is S(f) then the time domain output of the filter so(t) due to the signal component alone is given by:
and the output signal power S is proportional to the square of the signal voltage, thus:
The power spectral density of the noise at the filter output N0(f) is given by the squared magnitude of the filter transfer function multiplied by the power spectral density of the input noise.
For AWGN, we know the noise spectral density is flat with a value N0 Watts/Hz, hence the output noise spectral density is:
N0(f) = N0|H(f)|2
The average noise power N is found by integrating the noise power density over all possible frequencies to give:
The goal of the matched filter is to make the signal to noise ratio at the sampling time t = T a maximum. A matched filter will thus need to optimise the S/N ratio given by:
In order to find the transfer function H(f) which maximises the S/N ratio we need to make use of a result known as Schwarz's inequality. Schwartz's inequality states that:
and also states that for the two sides of this expression to be equal, then:
X(f) = Y*(f)exp(-j2p fT)
Applying this relationship to the S/N expression for the filter output we get:
where we have made us of the fact that |exp(j2p fT)|=1
The value of the S/N ratio is maximized when this expression is equal. The Schwartz's inequality thus allows us to conclude that for optimum S/N ratio, that is a matched filter, then:
Hmatched(f)=S*(f)exp(-j2p fT)
The impulse response for this matched filter is thus given by:
Given that S*(f)=S(-f) for a real valued signal s(t) then:
This result tells us that the impulse response of a matched filter should be a time reversed and delayed version of the input symbol s(t).