Fourier Transform and Data Compression

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Fourier Transform and Data Compression
Problem compress audio and video data for more
efficient storage/transmission.
General Approach
Transform
Quantize
Encode
LOSSLESS
LOSSY
Inverse Transform
Decode
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The Fourier Series is an example of an efficient
representation of a signal. In fact take any
continuous time signal with finite length
Recall Parcevals Theorem
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We can approximate a continuous time signal
arbitrarily closely by a finite set of
coefficients.
However large errors at discontinuities.
Example
Large errors at the boundaries, since we assume
the signal to be extended periodically.
discontinuities
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We can easily solve this problem by expanding the
signal
No Discontinuities at the Boundaries
From the symmetry, all Fourier Coefficients are
real. Therefore
with
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This is easily extended to discrete time
signals. There are several ways of defining the
DCT (Discrete Cosine Transform). For example
with
DCT II
DCT II
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Example
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Divide the signal into blocks of length
N and take the DCT within each block
all values with similar p(i) High Entropy
DFT or DCT
DFT
DFT
DFT
Small values have large p(i), large values have
small p(i) Low Entropy
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Example
signal
Histogram
DCT
Histogram