# Fractional Bandwidth of a Gaussian Amplitude Modulated Signal

a gaussian modulated sinusoidal signal may be expressed as $$x(t)=A\cdot e^{(j2\pi ft)}\cdot e^{\left[-\frac{1}{2\sigma^{2}}\cdot(t-t_{0})^{2}\right]}$$

Let's consider the case in which the gaussian peak is at 0 and the total amplitude is 1:

$$x(t)=e^{(j2\pi ft)}\cdot e^{\left[-\frac{1}{2\sigma^{2}}\cdot t^{2}\right]}$$

What is the fractional bandwidth of this function?

I'm asking this question since I think it's a quite used parameter. In fact, there are a proper function of MATLAB and a proper function of Python) which can generate this function, and both ask me to specify:

• the sine wave frequency (which is $$f$$);

• the signal fractional bandwidth

I think the last one is necessarily linked to the standard deviation $$\sigma$$. But by which mathematical relationship? It's a bandwidth in time domain (so the FWHM of the gaussian pulse: $$2.35\cdot \sigma$$) or the bandwidth in frequency domain (so ???)?

I'll give this a shot.

The Fourier Transform of a Gaussian is also a Gaussian. The standard deviations in each domain are related as $$\sigma_t \cdot \sigma_F = \frac{1}{2\pi}$$ The time standard deviation, $$\sigma_t$$ has units of time and the frequency domain standard deviation $$\sigma_F$$ has units of Hz.

We can define the "bandwidth" of a gaussion as the -3dB point, i.e the point where the energy has fallen to 50%. We solve for $$f(x) = \sqrt2$$ and we get

$$x_{-3dB} = \sigma \sqrt{ln(2)}$$

This holds in both domains. We can then find the -3dB frequency $$f_{-3dB}$$ as follows

$$f_{-3dB} = \sigma_F \sqrt{ln(2)} = \frac{\sqrt{ln(2)}}{2\pi\sigma_T}$$

The bandwidth is twice the -3dB frequency since, the energy extends above and below the modulation frequency. Fractional, I assume, means relative to modulation frequency, $$f_{mod}$$ and so we can define the fractional bandwidth as

$$B_{frac} = \frac{2f_{-3dB}}{f_{mod}} = \frac{\sqrt{ln(2)}}{\pi \cdot \sigma_T \cdot f_{mod}}$$

The scale factor depends on your exact definition of "bandwidth". For -3dB it comes out to be $$\sqrt{ln(2)}$$

The fact that it's modulated with a sinusoid doesn't change the FWHM bandwidth of your pulse – the $$e^{jx}$$ function has $$\left\lvert e^{jx}\right\rvert\equiv 1$$ at every point. That doesn't change the amplitude, so the FWHM of a sinusoid-modulated gaussian is just the same as of the unmodulated gaussian.

• What does "Fractional" mean in this context then?. I would have assumed that it means "ratio of bandwidth to carrier". – Hilmar Jan 9 at 13:29
• @Hilmar that is right, but I'm assuming this is all baseband, as we're talking complex sinusoids, which is why I'm referring this to the fractional bandwidth of the same system, if that baseband wasn't modulated. – Marcus Müller Jan 9 at 13:32
• With FWHM do you mean the pulse width in time domain, or the frequency bandwidth? – Kinka-Byo Jan 9 at 14:02
• @Kinka-Byo never heard it being used in time domain, so frequency domain. For a gaussian pulse, one is the inverse of the other, anyways. Point of my answer is anyway that it's the same, no matter how you define it. – Marcus Müller Jan 9 at 14:20