Theory of the DFT

Discrete Fourier Transform

If we have a continuous signal over the range as shown here,

 and we take the continuous fourier transform to find it's the frequency content. What we get is two impulses in the frequency domain at the frequency of the sine wave.

This is ONLY true if we have an infinite signal. In reality we never have an infinite sine wave, but rather a signal that exists only for time T. We can represent this by multiplying the infinite sine wave by a window function, w(t), that exists only for time T.

 This gives us the following signal of multiplying the two signals x(t)w(t):

This multiplication of x(t) and w(t) as time signals will have produced a new frequency response which can be worked out by convolving the frequency responses of x(t) and w(t). ( Remember that multiplication in the time domain is convolution in the frequency domain, and vice versa.)

 The frequency response of the window function is :

 And so the frequency domain convolution gives us:

This picture shows the frequency response of the finite sine wave.

This is only an approximation to the ideal frequency response. The only way to improve the resolution is to increase the data length, the length of the window function. So far we have only dealt with continuous time, we want to know what to do with discrete signals. First thing that needs done is to sample them. This is equivalent to multiplying the time signal by a series of impulses c(t) e.g.:

 which has the frequency response of:

which is the frequency response of the series of impulses

Sampling our finite sine wave gives us the signal x(t)*w(t)*c(t) which has the frequency response X(w)*W(w)*C(w).

Sampled sine wave

Frequency response of the sine wave

If you look at the frequency response of the sampled signal, you will see that it is periodic at the sampling frequency. There is still one problem left to deal with. The frequency responce is still continuous even though the original time signal was discrete. We need a discrete frequency responce to be of any use in a computer, so we simply multiply it in the frequency domain by a sequence of impulses,and there we go, the frequency responce of our sample time signal.......or is it ?

That last step where we multiplied in the frequency domain, requires us to convolve the discrete time sequence with the sequence of impulses in the time domain. What does this do to our sampled discrete sequence ?

What happens is that the sampled sequence becomes periodic the length of the sampled sequence eg If the input signal is

 Then the discrete sampled sequence for the DFT is

This picture shows the input signal as the DFT sees it.

This is a VERY important point. When interpreting the frequency domain results you have to remember that they are ONLY truly mathematically correct for the periodic input signal.

If we take our samples in frequency as representing our continuous wave then the ASSUMPTION is that the input signal is zero from MINUS INFINITY to -T/2, then the signal to T/2, then from T/2 to INFINITY is zero, and doesn't exist for all time.

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