### 4.6 Source amplitudes vs sensitivity

How does one compare the gravitational wave amplitude of astronomical sources with the instrumental
sensitivity and assess what sort of sources will be observable against noise? Comparisons are almost always
made in the frequency domain, since stationary noise is most conveniently characterized by its
PSD.
The simplest signal to characterize is a long-lasting periodic signal with a given fixed frequency . In
an observation time T, all the signal power is concentrated in a single frequency bin of width
. The noise against which it competes is just the noise power in the same bin, . The power
SNR is then , and the amplitude SNR is . This improves with
observation time as the square root of the time. The reason for this is that the noise is stationary, but
longer and longer observation times permit the signal to compete only with noise in smaller and smaller
frequency windows.

Of course, no expected gravitational-wave signal would have a single fixed frequency in the detector
frame, because the detector is attached to the Earth, whose motion produces frequency modulations.
But the principle of this SNR increase with time can still be maintained if one has a signal
model that allows one to exclude more and more noise from competing with the signal over
longer and longer periods of time. This happens with matched filtering, which we return to in
Section 5.

Short-lived signals have wider bandwidths, and long observation times are not relevant. To characterize
their SNR, it is useful to define the dimensionless noise power per logarithmic bandwidth, , which
we earlier called . The signal Fourier amplitude has dimensions of Hz^{–1}
and so the Fourier amplitude per logarithmic frequency, which is called the characteristic signal amplitude
, is dimensionless. This quantity should be compared with to obtain a rough
estimate of the SNR of the signal: SNR .