More on DFT
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39 More on DFT Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner
description
3F3 – Digital Signal Processing (DSP), January 2009-2010, lecture slides 2a, Dr Elena Punskaya, Cambridge University Engineering Department
Transcript of More on DFT
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More on DFT
Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
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DFT Interpolation
normalised
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Zero padding
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Padded sequence
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Zero-padding
N π
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Zero-padding
just visualisation, not additional information!
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Circular Convolution
circular convolution
xxxxxxxxx
m
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Example of Circular Convolution
1
2 0
3
5 4
Circular convolution of x1={1,2,0} and x2={3,5,4} clock-wise anticlock-wise
1
2 0
3
5 4
1
2 0
5
4 3
1
2 0
4
3 5 0 spins 1 spin 2 spins
y(0)=1×3+2×4+0×5 y(1)=1×5+2×3+0×4 y(2)=1×4+2×5+0×3
folded sequence
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3
…
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Example of Circular Convolution
clock-wise anticlock-wise
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IDFT
m +
+
+ ++
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Standard Convolution using Circular Convolution
It can be shown that circular convolution of the padded sequence corresponds to the standard convolution
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Example of Circular Convolution
1
2 0 3
5 0
clock-wise anticlock-wise 1
2 3
5 0
0 spins
y(0)=1×3+2×0+0×4+0×5
folded sequence
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3
…
0 4
4
0
1
2 0 5
4 3 1 spin
0
0
1
2 0 4
0 5 2 spins
3
0
0 y(0)=1×5+2×3+0×0+0×4 y(0)=1×4+2×5+0×3+0×0
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Standard Convolution using Circular Convolution
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Proof of Validity
Circular convolution of the padded sequence corresponds to the standard convolution
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Linear Filtering using the DFT
FIR filter:
DFT and then IDFT can be used to compute standard convolution product and thus to perform linear filtering.
Frequency domain equivalent:
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Summary So Far
• Fourier analysis for periodic functions focuses on the study of Fourier series
• The Fourier Transform (FT) is a way of transforming a continuous signal into the frequency domain
• The Discrete Time Fourier Transform (DTFT) is a Fourier Transform of a sampled signal
• The Discrete Fourier Transform (DFT) is a discrete numerical equivalent using sums instead of integrals that can be computed on a digital computer
• As one of the applications DFT and then Inverse DFT (IDFT) can be used to compute standard convolution product and thus to perform linear filtering
