Amplifiers: Periodic and non-periodic signals
You can approximate the spectrum of an aperiodic signal by taking the fourier transform of a finite sample of the signal. Or, more practically, taking the discrete fourier transform (DFT) of a finite sequence of samples of the signal. Generally, some kind of windowing should be applied to get a spectrum that represents the signal more than the artifacts of the finite sampling duration.

I like to recommend R. W. Hamming, Digital Filters, as an excellent and accessible reference on windowing for spectral estimation, although any digital signal processing text will cover this topic.


  I'm currently dealing with signals that are very sharp peaks, with small peak widths (i.e. less than a microsecond - nanosecond). This signal is non-periodic and describes single photon detection.


You're in luck, because this is one of the easiest types of signal to estimate the spectrum of, because windowing won't significantly change its shape, provided the window function is sufficiently wider than the pulse width.

If you have a model for the shape of the impulse response, say \$v(t)=v_0 e^{-at}u(t)\$, you can estimate the bandwidth of your receiver by just taking the fourier transform of this response function for a single pulse at \$t=0\$.