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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.
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Lecture 6: DFT Xiliang Luo 2014/10
Periodic Sequence • Discrete Fourier Series For a sequence with period N, we only need N DFS coefs
DFS Synthesis Analysis
Example • DFS of periodic impulse
DFS Properties Linearity: Shift:
DFS Properties Duality: Periodic Convolution:
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 DFT/IDFT linear conv w/ aliasing
Compute Linear Convolution Circular convolution becomes linear convolution!
LTI System Implementation Block convolution
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: Observation: starting from sample: P-1, y[n] corresponds to linear convolution!
