"Naive" or "raw" sampling is associated to the very intuitive idea of measuring the value of a phenomenon of interest over time in order to record its evolution. We have all done it in our lives, whether to monitor the temperature of a cake we are cooking or to keep track of weather patterns. In the previous lesson we have formalized this intuition and shown the conditions under which the sampling operation entails no loss of information and allows for perfect reconstruction, namely:

• the function is $$F_s$$-bandlimited (that is, the highest positive frequency is $$F_s/2$$)

• the sampling interval $$T_s$$ is smaller or equal to $$1/F_s$$

The mathematical derivations showed that sampling should in fact be interpreted as the decomposition of a continuous-time signal into a linear combination of sinc basis functions and that the samples are in fact the coefficients of this linear combination. The equivalence between sinc sampling and raw sampling of course only holds if the previous two conditions are met.

In signal processing practice, however, there are many situation in which this idealized setup does not hold:

• we may think the signal is bandlimited but in reality it is not

• we may not be able to sample fast enough

In these and in many other cases, the hypothesis of perfect bandlimitedness required by the sampling theorem is not fulfilled. We will therefore now study in detail what happens when we raw-sample a signal at a rate that is not adapted to its spectral support.

If we take samples of a continuous-time signal without fulfilling the conditions imposed by the sampling theorem, we will incur a phenomenon called aliasing . We have already encountered a simple example in the first part of this course, when we looked at the so-called wagonwheel effect: when looking at the video of a spoked wheel going forward, sometimes the wheel appears to rotate backwards.

In this lecture we are going to explore this in details, starting with sinusoids instead of wheels and then moving on to arbitrary signals. We will see that, if we sample a signal carelessly, the resulting aliasing error can be extremely disrupting.