Lesson 17

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:

- Discrete in time but continuous in frequency and periodic
- Spectrum of discrete signal
- Will use the DTFT to motivate the DFT by taking the DTFT of a windowed data segment
- Will compare the DTFT of a discrete signal with the Continuous Time Fourier Transform of a sampled continuous time signal

Formula to calculate inverse DTFT (this is similar to the Fourier Series):

where DTFT is periodic in frequency with period . Why? Because is periodic with period .

Not all DTFTs converge due to the infinite sum.

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.

Find for

and show that the filter has a linear phase term.

**Z-Transform**

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.