The Sampling Theorem tells us that we can accurately sample signals which have all of their frequency content beneath the Nyquist Frequency (Half the Sampling Rate). Thanks to the Nyquist-Shannon theorem, we know that if we’re attempting to sample a signal containing frequencies higher than the Nyquist Frequency, our sampling process is flawed. What the theorem doesn’t tell us, is how exactly the sampling process breaks down when we attempt to sample signals which spill over the Nyquist limit. What actually happens when our input signals cross the Nyquist Limit?
Figure 1 depicts the sampling of a sine wave as it rises in frequency. In this case, our sampling rate is 24 Hz, so the Nyquist Limit is at 12 Hz. you’ll notice that once the input signal crosses the Nyquist Frequency something very strange begins to happen. After crossing the Nyquist Frequency, the sampled signal becomes a valid representation for not only the actual input wave (blue), but also a new sine wave (grey) which begins to decrease in frequency.
|Figure 1. Crossing the Nyquist Frequency|
If we don’t limit the frequency of the input signal, and instead allow it to travel over the Nyquist limit, our sampled signal will end up representing a sine wave which is flipped or reflected about the Nyquist Frequency. In practice, we will end up registering measurements for the grey signal which doesn’t actually exist. In the next two sections, we’ll look at two more examples of aliasing, and their relationship to the Sampling Theorem.