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
𝑦𝑛 =
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