# Amplifiers: Periodic and non-periodic signals

All the signal amplifiers (op amps) I've dealt with thus far use periodic signals. The idea of bandwidth is natural when dealing with these periodic signals because it is just a description of the range of signal frequencies.

What about when you have a non-periodic signal? Does amplification work in the same way? 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. Would I approach amplification for this signal in the same way? If so, what is the bandwidth in such a situation? Would it just be 1/(time width of signal)?

• Try speaking into a microphone. It surely is a nonperiodic signal. I might even go so far as to say that virtually all signals amplified around the world are nonperiodic. – PlasmaHH May 17 '16 at 15:34
• @PlasmaHH Then what would be the bandwidth of such a signal? I thought this information would be necessary to decide what center frequency to use if say you are using a band pass filter. – Denu May 17 '16 at 15:37
• potentially infinite. The question is: what bandwidth do you need to get the information across that you need. It is the same thing for everydays square waves. To accurately represent the step, you would need infinite bandwidth, but what you really want is just barely enough to recognize them as what they are on the other side. – PlasmaHH May 17 '16 at 15:51
• @PlasmaHH: Very nice. :) – EM Fields May 17 '16 at 19:50

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 $t=0$.

It's pretty straight forward,

$BW = \frac{0.35}{Tr}$

where Tr = 10% - 90% Rise time of the signal. For an "analog" type signal this would be the fastest rise time you can find.

All of the signal content will be within that Bandwidth. You may not need to design your amplifiers and filters to support that, and it depends upon sampling rates, aperture size etc. but if you need to support the the signal with good fidelity this is the calculation.

• I believe this depends on the type of response the system will have (i.e. Gaussian response for the formula you gave. It would be BW = .22/Tr for a One-stage low-pass RC network (wikipedia), etc. for other systems). – Denu May 18 '16 at 20:17