DISCRETE-TIME SIGNAL PROCESSINGLECTURE 8 (DFT)

Husheng Li, UTK-EECS, Fall 2012

WHY DFT

The specification of filter is usually given by the tolerance scheme.

Discrete Fourier Transform (DFT) has both discrete time and discrete frequency.

DTFT has a continuous frequency, which is difficult to process using digital processors.

DFT has a fast computation algorithm: FFT.

DISCRETE FOURIER SERIES (DFS)

Consider a periodic sequence x(n) with period N. We have

Usually we define . Then, we have

PROPERTIES OF DFS

MORE PROPERTIES

FOURIER TRANSFORM OF PERIODIC SIGNALS

For periodic signals, the continuous-frequency Fourier transform is given by

SAMPLING THE FOURIER TRANSFORM

Consider an aperiodic sequence x(n) with Fourier transform X(w), we can do sampling:

NEW SEQUENCE

The sequence having DFS equaling the frequency domain sampling results from the aperiodic sampling.

DFT FOR FINITE-DURATION SEQUENCES Consider a finite-duration sequence

x(n) with length N. We can define its DFT as the DFS of the periodic sequence

where . The DFT is given by

EXAMPLE: DFT OF A RECTANGULAR PULSE

PROPERTIES OF DFT

COMPUTING CONVOLUTION USING DFT Since there is a fast computation

algorithm in DFT, we can compute convolution via DFT:

Compute the DFTs of both sequences Compute the product of both DFTs. Compute the output using IDFT. The length of DFT should be properly

chosen; otherwise we will see aliasing.