We know from previous lectures that any signal can be decomposed into a sum of scaled and shifted sinusoids of varying frequencies. We can represent this decomposition on two plots:
We can be a bit more parsimonious by representing both magnitude and phase using complex numbers. This will then give us a single plot, but the magnitude and phase are the properties that we are usually concerned with in practice.
Consider an point segment of a signal . If is generated by sampling a continous-time (CT) signal with sampling period , the corresponding sampling frequency is:
The Nyquist sampling theorem says that the maximum frequency that you can see in a signal is going to be half the sampling frequency:
x is an point segment of , one way to determine its
frequency content is to take its discrete Fourier transform (DFT) by using the
fast Fourier transform (FFT) to compute it:
>> X=fft(x);This gives us an component vector , which in general will be complex valued. The magnitudes of these complex numbers are a sampled and quantized version of the magnitude response mentioned above, and the phases of these complex numbers are a sampled and quantized version of the phase response.
If we index
X by the variable , the first  component
represents the average (DC) value of the original signal. The second 
component gives the frequency content at Hz. The
component corresponds to or .
From previous lectures, we know that changing the discrete-time frequency by 1 does nothing to the time domain representation of a signal. That is, and are the same discrete-time (DT) signal. So the DFT of a signal should be periodic with period 1; we cannot tell the difference in the time domain between a sinusoid at frequency and a sinusoid at frequency , so the frequency content at should be the same as that at .
But where is the periodicity? It is reflected in the fact that the DFT is periodic with period ; we have sampled the frequency response at points.
It turns out that sampling a signal in the time domain forces the frequency domain representation of that signal to be periodic. However, we are also sampling in the frequency domain. This forces the time domain signal to be periodic. This topic will be addressed in more depth when you take EE123.
One final note: if the signal that we DFTed was real valued, the magnitude response would be even about the th component, and the phase response would be odd.