# Bandwidth of product of two signals

POINT 1: According to convolution theorem, if g1(t) has a bandwidth B1 and g2(t) has a bandwidth B2, then product g1(t)g2(t) has a bandwidth = B1 + B2.

POINT 2: Now, consider an AM waveform s(t)=g(t)cos(2pifct). If g(t) has bandwidth = B, how come s(t) has bandwidth = 2B.

So point 2 does not follow point 1. Why is that. Are there any limitations on convolution theorem in frequency domain

• POINT 1 is true only for low-pass signals, not in general. The product of two bandpass signals of bandwidths $B_1$ and $B_2$, e.g. $G_1(f)$ is nonzero only for $|f| \in [f_1, f_1+B1$ and $G_2(f)$ is nonzero only for $|f| \in [f_2, f_2+B2]$, is not a signal of bandwidth $B_1+B_2$. Exercise: Work out what $G_1(f)\star G_2(f)$ is in this case. So, Point 2 does not in any way contradict the false assertion of Point 1 since at least one signal in Point 2 is a bandpass signal. – Dilip Sarwate Apr 7 '18 at 17:56

There is a relationship that is sometimes called the Classical Fourier Uncertainty Principle but has other names as well.

https://en.wikipedia.org/wiki/Uncertainty_principle#Signal_processing

The point to make is that if a signal exists for only a finite duration, it's Fourier Transform requires the full $-\infty$ to $\infty$ frequency range. Does that mean it has an infinite bandwidth? Conversely a signal where the the Fourier transform is strictly zero outside of a range, must have infinite duration.

Actual observations of signals strongly imply that physical signals have a finite duration so the notion of bandwidth is largely empirical and associated with context and utility. Rules of Thumb are useful and common.

Filters are often designed to be passband, so the 3dB down criteria is a useful concept. Real signals have complex hermitian symmetric negative spectrum, but mostly by convention and when the term "bandwidth" has utility, only the positive spectrum is considered.

The mathematical distinctions between single and double sideband, with and suppressed carrier are well understood.

https://en.wikipedia.org/wiki/Amplitude_modulation

Convolution in the Frequency Domain holds. Using an empirical concept outside of its applicable context can lead to perceived mathematical inconsistencies.

Also your point 1 is another empiricism. A criteria such as a 3dB bandwidths for $g1(t)$ and $g2(t)$ will not be true in general for the resulting product $\mathcal{F}\{g1(t)g2(t)\}$ 3dB criteria, but is often useful a Rule of Thumb.

https://en.wikipedia.org/wiki/Rule_of_thumb

"Bandwidth" in this context is interpreted as the largest frequency component of a signal. So, the message signal $g(t)$ has bandwidth $B$, and the carrier has bandwdith $f_c$. The AM modulated signal has bandwidth $B+f_c$.