Lecture 6: DFT

XILIANG LUO 2014/10

Periodic Sequence  Discrete Fourier Series For a sequence with period N, we only need N DFS coefs

Discrete Fourier Series

DFS

Synthesis Analysis

Example  DFS of periodic impulse

DFS Properties Linearity: Shift:

DFS Properties Duality: Periodic Convolution:

DTFT of Periodic Signals

Sampling Fourier Transform Sample the DTFT of an aperiodic sequence: Let the samples be the DFS coefficients:

Sampling Fourier Transform DTFT definition: Synthesized sequence:

Sampling Fourier Transform Synthesized sequence:

Sampling Fourier Transform Sampling the DTFT of the above sequence with N=12, 7

Discrete Fourier Transform For a finite-length sequence, we can do the periodic extension: or DFT definition:

Discrete Fourier Transform DFT is just sampling the unit-circle of the DTFT of x[n]

DFT Properties  Linearity  Circular shift of a sequence  Duality

DFT Properties  Circular convolution

Compute Linear Convolution In DSP, we often need to compute the linear convolution of two sequences.

Considering the efficient algorithms available for DFT, i.e. FFT, we typically follow the following steps:

Compute Linear Convolution Linear convolution of two finite-length sequences of length L & P: How about circular convolution using length N=L+P-1?

Compute Linear Convolution Sampling DTFT of x[n] as DFS: one period

Compute Linear Convolution

Compute Linear Convolution DFT/IDFT linear conv w/ aliasing

Compute Linear Convolution Circular convolution becomes linear convolution!

LTI System Implementation

LTI System Implementation

Block convolution

LTI System Implementation

LTI System Implementation

Overlap-Add Method

Overlap-Save Method P-point impulse response: h[n] L-point sequence: x[n] L > P We can perform an L-point circular convolution as: 𝑃−1 𝑦 𝑛 = ℎ 𝑙 𝑥[ 𝑛 − 𝑙 𝐿 ] 𝑙=0 Observation: starting from sample: P-1, y[n] corresponds to linear convolution!

Overlap-Save Method

Overlap-Save Method

Overlap-Save Method