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

1 / 18

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

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'v (0) , v (1) , v (2) , v (3), v (4), v (5), v (6), v (7), ….., v ( N -2) , v ( N -1)' - aerona

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

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

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

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

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)

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)

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)

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

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

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

How to derive the optimal transform?

(O3)

Equation O1 can be rewritten as

(O4)

The covariance of f is given by

(O5)

b. Every term is important

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:

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 

How to derive the optimal transform?

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)

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.

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)

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

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.