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 nonvanishing cross correlation terms arises from the equivalent
noise sources of the SQUID amplifier. Now,
suppose that the thermal noise of the bar, S_{b}, is the most
important contribution. As
, then
is just S_{b}, 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, S_{0}. 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 Dnº(2p·t_{opt})^{1},
where t_{opt} 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 lockin. A full
account of the noise properties of the detector leads to WienerKolmogorov
filtering theory. 