Lecture 6: DFTXILIANG 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 TransformSynthesized sequence:

Sampling Fourier Transform

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

Discrete Fourier TransformFor 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 PropertiesCircular 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 typicallyfollow 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 ImplementationBlock convolution

LTI System Implementation

LTI System Implementation

Overlap-Add Method

Overlap-Save MethodP-point impulse response: h[n]L-point sequence: x[n]

L > P

We can perform an L-point circular convolution as:

𝑦 [𝑛 ]=∑𝑙=0

𝑃−1

h [ 𝑙 ] 𝑥 [ (𝑛− 𝑙 )𝐿]

Observation: starting from sample: P-1, y[n] corresponds to linear convolution!

Overlap-Save Method

Overlap-Save Method

Overlap-Save Method