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

40

DFT Interpolation

normalised

41

Zero padding

42

Padded sequence

43

Zero-padding

N π

44

Zero-padding

just visualisation, not additional information!

45

Circular Convolution

circular convolution

xxxxxxxxx

m

46

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

50

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