# Lecture 8: Discrete-Time Fourier Transform (DTFT)

Recall that the Z-transform is

The DTFT is defined as

Clearly, if is causal, the DTFT is the Z-transform evaluated on the unit circle. (Just like the FT is the Laplace transform evaluated on the axis.) Observe that is -periodic:

Again we see this idea of the periodic repetition of the spectrum in the frequency domain and the source of aliasing.

Note that is a function of continuous frequency -- we are not looking at samples of the spectrum (as for the DFT), and hence the function does not have to be periodic.

But,
is periodic, and hence,
has a Fourier
series representation: its F.S. coefficient are just the samples
. But this representation allows us to write down an
**
inverse DTFT
**
: to go from
to
, simply stick things
in the formula for the F.S. coefficients, where the period of the
function is

(Need we point out that, yet again, in going from time domain to frequency domain the exponent has negative sign, and in going from frequency domain to time domain the exponent has a positive sign?) Notationally we will write

And, as for other transforms, we can talk about the amplitude and phase spectra.

(Comment on spectral leakage).