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Correlation - Similarity between adjacent samples. v (0) , v (1) , v (2) , v (3), v (4), v (5), v (6), v (7), ….., v ( N -2) , v ( N -1). A sample can be predicted from its neighbor(s). Transform with good decorrelation property.

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slide1

Correlation - Similarity between adjacent samples

v(0), v(1), v(2), v(3), v(4), v(5), v(6), v(7), ….., v(N-2), v(N-1)

A sample can be predicted from its neighbor(s)

slide2

Transform with good decorrelation property

After transform, a coefficient is less predictable from its neighbor(s)

V(0)

V(1)

V(2)

V(3)

V(4)

V(5)

V(6)

V(7)

Magnitude of frequency components

slide3

Energy Compactness - Importance of each sample

v(0), v(1), v(2), v(3), v(4), v(5), v(6), v(7), ….., v(N-2), v(N-1)

All samples are important

slide4

Energy Compactness - Importance of each sample

v(0), v(1), v(2), v(3), v(4), v(5), v(6), v(7), ….., v(N-2), v(N-1)

Any missing sample causes large distortion

All samples are important

slide5

Energy Compactness - Importance of each sample

e.g. DFT samples

V(0)

V(1)

V(2)

V(3)

V(4)

V(5)

V(6)

V(7)

v(0)

v(1)

v(2)

v(3)

v(4)

v(5)

v(6)

v(7)

slide6

Energy Compactness - Importance of each sample

V(0)

V(1)

V(2)

V(3)

V(4)

V(5)

V(6)

V(7)

v(0)

v(1)

v(2)

v(3)

v(4)

v(5)

v(6)

v(7)

slide7

Energy Compactness

The signal can be constructed with the first 3 samples with good approximation

V(0)

V(1)

V(2)

V(3)

V(4)

V(5)

V(6)

V(7)

v(0)

v(1)

v(2)

v(3)

v(4)

v(5)

v(6)

v(7)

slide8

Good Energy Compactness

All information is concentrated in a small number of elements in the transformed domain

KLT has the best Energy Compactness and Decorrelation Properties

slide9

Two scenarios

1. Given a time function, find the transform that gives maximum energy compactness.

2. Given a transform, find the time sequence that gives maximum energy compactness.

If a time function has all the energy concentrated in the low frequency region, the sideloops are suppressed

slide10

How to derive the optimal transform?

Given a signal

f(n), define the mean and autocorrelation as

and

(O1)

Assume f(n) is wide-sense stationary, i.e. its statistical properties are constant with changes in time

(O2)

Define

and

slide11

How to derive the optimal transform?

(O3)

Equation O1 can be rewritten as

(O4)

The covariance of f is given by

(O5)

slide12

a. Adjacent terms are related

b. Every term is important

a. Adjacent terms are unrelated

b. Only the first few terms are important

How to derive the optimal transform?

The signal is transform to its spectral coefficients

Comparing the two sequences:

slide13

How to derive the optimal transform?

The signal is transform to its spectral coefficients

similar to f, we can define the mean, autocorrelation and covariance matrix for 

slide14

How to derive the optimal transform?

a. Adjacent terms are related

a. Adjacent terms are unrelated

Adjacent terms are uncorrelated if every term is only correlated to itself, i.e., all off-diagonal terms in the autocorrelation function is zero.

Define a measurement on correlation between samples:

(O6)

slide15

How to derive the optimal transform?

We assume that the mean of the signal is zero. This can be achieved simply by subtracting the mean from f if it is non-zero.

The covariance and autocorrelation matrices are the same after the mean is removed.

slide16

How to derive the optimal transform?

b. Only the first few terms are important

b. Every term is important

Note:

If only the first L-1 terms are used to reconstruct the signal, we have

(O7)

slide17

How to derive the optimal transform?

If only the first L-1 terms are used to reconstruct the signal, the error is

(O8)

The energy lost is given by

(O9)

but,

(O10)

hence

slide18

How to derive the optimal transform?

Eqn. O10 is valid for describing the approximation error of a single sequence of signal data f. A more generic description for covering a collection of signal sequences is given by:

(O11)

An optimal transform mininize the error term in eqn. O11. However, the solution space is enormous and constraint is required. Noted that the basis functions are orthonormal, hence the following objective function is adopted.