# One-sided bandwidth of the Gaussian filter

Even though this question seems to be quite simple, I was surprised not to find it here.

As defined in the CCSDS (section 3.1.2), the impulse response of the Gaussian filter is given by

$$h(t) = \frac{1}{\sigma T \sqrt{2\pi}}e^{-\frac{t^2}{2\sigma^2T^2}}$$ where $$\sigma = \frac{\sqrt{\ln{2}}}{2 \pi B T}$$ $$B$$ is defined as

one-sided 3-dB bandwidth of the filter with impulse response $$h(t)$$.

What is exactly the meaning of "one-sided" in this context? What would be $$B$$ if it were the two-sided version instead?

A good reference can be found in Gaussian Pulse – FFT & PSD in Matlab & Python.

• Please edit your question to tell us where you found this. Either a link, or a book or article citation. If you can pull out a short excerpt, please quote it. This probably means the bandwidth from 0Hz to the 3dB point, but without more context, I cannot say. Nov 4 at 13:50
• @TimWescott done. Nov 4 at 13:54
• Typically that's just a convention: one-sided means "distance from the center frequency (0dB) to the -3 dB frequency". Two-sided would be the difference between the two -3dB points (below and above the center), so twice the one-sided bandwidth for a symmetrical filter (which the Gaussian is) Nov 4 at 16:10

Because that's a document that pertains to communications, where it makes sense to talk about signals that have negative frequencies, they're drawing a distinction between the one-sided bandwidth (from 0Hz to the positive-frequency $$-3\mathrm{dB}$$ point) vs. the two-sided bandwidth (from one $$-3\mathrm{dB}$$ point to the other).
If you haven't encountered doing signal processing with complex-valued signals (or, strictly, inphase/quadrature signal pairs), then the one-sided bandwidth, for a lowpass filter, is what you would call the "perfectly ordinary bandwidth". If you are building a radio system that translates some chunk of spectrum down to baseband with an I/Q demodulator, then -- because you have both inphase and quadrature parts -- it suddenly not only makes sense to have different signal components at positive vs. negative frequencies, but it also makes sense to have filters whose response is asymmetric around $$0 \mathrm{Hz}$$. For such filters, you care about the two-sided bandwidth.
In this particular case, the modulation signal is purely real, so the lowpass Gaussian filter is, of necessity, symmetrical around $$0 \mathrm{Hz}$$, so the one-sided bandwidth is the most sensible specification for it.
• This signal converted to bandpass and centered at the frequency $f_c$, would have a bandwidth equals to $2B$, which is $[f_c-B, f_c+B]$, right? Can I conclude that the bandpass signal have twice the bandwidth of its lowpass equivalent? Nov 5 at 16:49