1. 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

2. The Fourier Series is an example of an efficient representation of a signal. In fact take any continuous time signal with finite length :
Recall Parceval’s Theorem:

3. 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

4. No Discontinuities at the Boundaries
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

5. DCT II DCT II 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

6. Example:

7. all values with similar p(i): High Entropy
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

8. Example:
signal
Histogram
DCT
Histogram