Sensitivity and bandwidth

 

There are two quantities to define how good the detector is:

·       The sensitivity of the detector is the amplitude of a signal that would be detected with unitary SNR.

·       The bandwidth of the detector is the measure of the frequency span where the SNR density plot (in the frequency domain) is above  times its maximum value.

 Where does the bandwidth come from?

It can be more conveniently understand if we refer all noise sources at the input of the detector, were a gravitational wave burst is a flat function of frequency.

Let  and  be the transfer function, respectively, of the signal and of the noise sources of the detector. Let also represent the power spectrum density of a single noise source,  beeing the corresponding cross power spewctrum density. The power spectrum density of the noise seen at the input of the detector would look like

In fact, the only non-vanishing cross correlation terms arises from the equivalent noise sources of the SQUID amplifier.

Now, suppose that the thermal noise of the bar, Sb, is the most important contribution. As , then  is just Sb, that is a flat white noise spectrum. In this case the SNR is constant over all frequencies, i.e. the bandwidth of the detector is not limited.

On the contrary, suppose that the leading term is the wide band noise of the amplifier, S0. Then : this function has two deep and narrow minima, corresponding to the two mechanical resonance frequencies of the system. The total bandwidth is of the order of the greater of the two the bandwidths of the modes, i.e. of the order of ~1mHz.

If all noise sources are taken in account altogether, the bandwidth will balance to an optimal value Dnș(2p·topt)-1, where topt is the optimum duration of a measurement, in order to smooth out the wide band noise without spreading too much the signal. This arguments are naïvely used when choosing the integration time of a lock-in. A full account of the noise properties of the detector leads to Wiener-Kolmogorov filtering theory.


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