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

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

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  1. 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)

  2. 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

  3. 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

  4. 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

  5. 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)

  6. 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)

  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)

  8. 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

  9. 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

  10. 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

  11. How to derive the optimal transform? (O3) Equation O1 can be rewritten as (O4) The covariance of f is given by (O5)

  12. 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:

  13. 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 

  14. 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)

  15. 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.

  16. 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)

  17. 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

  18. 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.

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