Discrete-Time Fourier Transform
For infinite length sequences - in practice, we don't have an infinite amount of data so we'll also study the Discrete Fourier Transform for finite data sequences.
Recall that we wrote the sampled signal . We calculated its Fourier Transform using the Fourier Transform for a periodic function.
We do the following:
Find the Continuous Time Fourier Transform of .
Using superposition, find the CT Fourier Transform of .
Now, you just calculated that
Let and make a change of variables (we'll talk more about this later -- it relates the discrete-frequency variable to the continuous frequency variable via the sampling period ) and we get:
Formula to calculate inverse DTFT (this is similar to the Fourier Series):
Ex. 1 Find where , . What if ?
Ex. 2 , . Find .
What if ?
Ex. 3 Rectangular pulse, . Find .
Show that this filter has a linear phase term.
We already saw the DTFT as the Z-transform of evaluated on the unit
circle when we discussed the frequency response:
If the ROC for the Z-transform contains the unit circle, we can get DTFT from the Z-transform by substitution (compare the DTFT of with its Z-transform).
We'll see that the DTFT exists in cases where the ROC of the Z-transform does not include the unit circle (e.g. for periodic discrete-time signals) -analogous to the CT Fourier Transform and Laplace Transform.