Transforms: Problems 12.1, 12.2, 12.3, 12.4

12.1

Show the discrete fourier transform is unitary.

12.2

Calculate the action of the inverse wavelet transform on a vector of length 2^12 of zeros with ones in the 5th and 30th entries, using Daubechies fourth-order coefficients. Python script

The left shows the preimage of the desired vector -- a strange function. The right shows the preimage of a linear ramp down from 1 over the domain.

12.3

Consider three component vector with two gaussian, independent components and the third is a sum of the first two. Calculate covariance matrix and its eigenvalues, and verify numerically. Use PCA to find a set of variables with diagonal covariance and no zero eigenvalues. Python script

12.4

Generate a uniformly random two dimensional data set and mix with a matrix. Use PCA to whiten and find the independent components with a constrast function. Python script

MAS.864 2014