The matched filter for burst detection

- Let us consider a signal in presence of noise .
The available information is the sum

is the measurement at the output of the low noise amplifier and is a random process with known properties. Let us start by applying to a linear filter which must be such to maximize the signal to noise ratio SNR at a given time (we emphasize the fact that we search the signal at a given time ).Indicating with the impulse response of the filter (to be determined) and with and respectively the convolutions of the signal and of the noise we have

The expectation of the noise after the filter, indicating with the power spectrum of the noise is

where is the Fourier transform of the unknown .At the output due to with Fourier transform is given by

We apply now the Schwartz' inequality to the integral and using the identity

we obtain

It can be shown that the equal sign holds if and only when

Applying this optimum filter to the data we obtain the maximum SNR

where and as already specified are, respectively, the Fourier transform of the signal and the power spectrum of the noise at the end of the electronic chain where the measurement is taken.Let us apply the above result to the case of measurements done at the end of a chain of two filters with transfer functions (representing the bar) and (representing the electronics).

Be the white spectrum of the brownian noise entering the bar and the white spectrum of the electronics noise. The total noise power spectrum is

The Fourier transform of the signal is

where is the Fourier transform of the GW signal at the bar entrance.The optimum filter will have the transfer function

where

We obtain

We apply now the above result to a delta GW with Fourier transform independent on . The integral can be easily solved if we consider that near the resonance we can approximate

We estimate now signal and noise just after the first transfer function, before the electronic wide-band noise. For the signal due to a delta excitation we have

where is the maximum signal. For the thermal noise we have

where is the mean square narrow-band noise .Introducing the signal energy we calculate (with the variable )

The effective noise temperature is