Sampling a narrowband signal with sampling frequency $F_s$ lower than twice the maximum
frequency of the signal does not always lead to aliasing.
In this example, you can adjust the sampling frequency to see how it affects the spectral
content of the sampled signal (in red). The original signal (in blue) is a narrowband
signal with central frequency $F_0=1000\ \mathrm{Hz}$ and bandwidth $B=200\ \mathrm{Hz}$.
Find which sampling frequencies are acceptable, and why.
Original signal
Sampled signal
In this example, aliasing occurs when $F_s$ lies in:
The generalized Shannon theorem states that to avoid aliasing when sampling a narrowband signal (with central frequency $F_0$ and bandwidth $B$), the sampling frequency must obviously be at least twice the bandwidth of the signal:
Yet, not all such frequencies are acceptable, given the possible overlap between positive and negative spectral images. This excludes the following sampling frequencies:
with $k$ integer but not 0. As a matter of fact, for $k$ odd, values of $F_s$ in these
intervals put the Nyquist frequency inside a spectral image; for $k$ even, they put the
zero frequency inside a spectral image. In both cases, this causes aliasing.
In our example, this clearly leads to excluding $F_s$ from:
So, in this example, the smallest possible sampling frequency is $440\ \mathrm{Hz}$.
Notice, by examining the spectral content of the sampled signal, and by listening to it,
that using sampling frequencies $F_s<2(F_0+B/2)$ leads to signals in the
$[0,F_s/2]$ band that are different from the original signal. However, since no aliasing
occurs, it is still possible to recover the original signal by further digital upsampling
and band-pass filtering.
Nota Bene: for convenience, the effect of the sampling frequency on the magnitude
of the sampled signal spectrum has been compensated for in the plot.