One-sided bandwidth of the Gaussian filter

Question

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.

Answer 1

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.