3F3 – Digital Signal Processing (DSP), January 2009-2010, lecture slides 2a, Dr Elena Punskaya, Cambridge University Engineering Department

1. 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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2. DFT Interpolation
normalised
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3. Zero padding
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4. Padded sequence
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5. Zero-padding
π
N
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6. Zero-padding
just visualisation, not additional information! 44

7. Circular Convolution
xxxxxxxxx
m
circular convolution
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8. Example of Circular Convolution
Circular convolution of x1={1,2,0} and x2={3,5,4}
clock-wise anticlock-wise
1
3
5 4
0 2 folded sequence
y(0)=1×3+2×4+0×5 y(1)=1×5+2×3+0×4 y(2)=1×4+2×5+0×3
1 1 1
3 5 4
0 spins 1 spin 2 spins
…
5 4 4 3 3 5
0 2 0 2 0 2
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3 46

9. Example of Circular Convolution
clock-wise anticlock-wise
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10. IDFT
m +
+
+
+
+
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11. 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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12. Example of Circular Convolution
clock-wise anticlock-wise
1
3
0 2 5 0
4
folded sequence
y(0)=1×3+2×0+0×4+0×5 y(0)=1×5+2×3+0×0+0×4 0 y(0)=1×4+2×5+0×3+0×0
1 1 1
3 5 4
0 spins 1 spin 2 spins
0 5 0 2 0 4 3 2 0 0 5 2
4 0 3
…
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3 50
0 0 0

13. Standard Convolution using Circular Convolution
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14. Proof of Validity
Circular convolution of the padded sequence corresponds to the standard
convolution
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15. Linear Filtering using the DFT
FIR filter:
Frequency domain equivalent:
DFT and then IDFT can be used to compute standard convolution
product and thus to perform linear filtering.
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16. 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 54