Aliasing: sampling at too low a rate

From the last section, we saw that two signals -- all zero and a sinusoid whose frequency is half the sampling interval, both resulted in the same set of samples: all zero. Thus, sampling is not always a reversible operation. We cannot always recover the continuous-time input from its samples.

As a more interesting and informative example, note from the relation

In[20]:=

  w = f T

that when we increase f by 1/T, the new discrete-time frequency is

In[21]:=

  (f+ 1/T) T = fT + 1 = w + 1

It follows that the samples will not change. To illustrate this:

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  A=1; f=1; p=0; T = 1/5; 
  p1 = PlotSampledSinusoid[10];

In[23]:=

  f = 1 + 1/T; 
  p2 = PlotSampledSinusoid[10];

These two input continuous-time sinusoids result in the same set of samples. This is called aliasing, meaning that two signals result in the same set of samples. We say that the second sinusoid's frequency is too high for the sampling rate. In fact, we require that

In[24]:=

  w = f T < 0.5, or 2f < 1/T

The way this is commonly stated: The sampling rate 1/T must be more than twice the frequency of the sinusoid.

More generally, if an input continuous-time signal consists of a superposition of sinusoids, no aliasing occurs in the sampling if the highest-frequency sinusoid has frequency less than half the sampling rate.

Up to Sampling and aliasing