The result of raw sampling a complex exponential depends on its frequency $$f_0$$ and on the value of the sampling frequency $$F_s$$; there are three cases:

• if $$F_s > 2f_0$$, we are within the condition of the sampling theorem and everything is fine

• if $$f_0 < F_s < 2f_0$$, we perceive and reconstruct a sinusoid with negative frequency, i.e., the signal goes in the opposite direction

• if $$F_s < f_0$$, we perceive and reconstruct a sinusoid of low frequency, in other words, frequencies above a threshold are mapped back to low frequencies.

In the last two cases we are in the presence of aliasing.

For generic signals, it turns out that the the spectrum of a raw sampled signal is (up to a scaling factor) the sum of an infinite number of translated copies of the original continuous-time spectrum. The translation "step" for each copy is equal to $$F_s$$ so, if the maximum frequency in the signal is greater than $$F_s/2$$, there will be overlap between the different repetitions and the resulting spectrum will be corrupted by aliasing.

The error introduced by aliasing cannot be undone in the discrete-time domain, so aliasing must be avoided before sampling. This is done by first lowpass filtering the signal with a filter with cutoff frequency $$F_s/2$$. This operation can be interpreted within the setup of the Hilbert space we have studied in the previous module. This initial filtering operation is an orthogonal projection of the signal onto the subspace of bandlimited signals, from which it can be perfectly sampled and reconstructed at sampling frequency $$1/T_s$$