BE2M31DSP seminar
Principal Component Analysis and Karhunen-Loeve Transform
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Computation of eigenvalues and eigenvectors of an elementary matrix
- Determine eigenvalues and eigenvectors of elementary matrix with values [ 1 0.5 ; 0.5 1 ],
- Analyze orthogonality of eigenvectors.
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Computation of covariance matrix, its eigenvalues and eigenvectors, and bases of KLT
- Estimate the covariance matrix of 20 realizations of given signal (see below, for particular realiztions use the segmentation with the frame length of 200 samples with a step of 50 samples)
- Use the following signals:
- s1 - one sinusoid with very small dithering white noise (parameters of sinuoid f=821 Hz, fs=8000 Hz, and scaling constant of 0.001 for normalized Gaussian white noise),
- s2 - signal with two sinusoids and very small dithering white noise (parameters of sinusoids f1=532 Hz, f2=640 Hz, fs=8000 Hz, and scaling constant of 0.001 for normalized Gaussian white noise),
- s1+n1, s2+n2 - above mentioned signals with Gaussain white noise of higher level (use the following values of scaling constant 0.1, 0.5, and 1),
- r1 - speech signal vm0.bin - raw data
without header, fs=16000 Hz,
use function loadbin.m to load it into MATLAB,
- r1+n1 - noisy speech signal, same noise levels as for sinusoids above.
- Compute eigenvalues and eigenvectors of covariance matrices for given signals, check the orthogonality of eigenvectors.
- Results:
- eigenvalues of covariance matrix for signal s1
- 4 the most significant eigenvectors of covariance matrix for signal s1
- scalar multiplication between 4 the most significant eigenvectors (orthogonality check),
- eigenvalues and 4 the most significant eigenvectors of covariance matrix for signal s2
- eigenvalues and 4 the most significant eigenvectors of covariance matrix for signal s1+n1 and s2+n1 respectively - observe differences in eigenvalues and shapes of eigenvectors for varying level of white noise.
- Determine the basis of Karhunen-Loeve Transform (KLT) and determine KLT for a particular realization.
- Result:
- KLT of selected short=time frame of signal s1
- KLT of selected short=time frame of signal s2
- KLT of selected short=time frame of signal r1
- Inverse transform using increasing number of KLT components.
- Result:
- Sythesis (inverse transform) of the frame s1 using 1,2,3,4,5,6 components of KLT spectrum,
- Sythesis (inverse transform) of the frame s2 using 1,2,3,4,5,6 of KLT spectrum
- Sythesis (inverse transform) of the frame s2 using 1,2,3,4,5,6 of KLT spectrum
obtained with the basis for signal s1.
- Comparison of compression properties of KLT, DFT and DCT
- Choose one realization of selected signal from above mentioned list, computed the basis of KLT for given signal and compute for given realization:
- KLT transform
- DFT spectrum
- DCT spectrum
- Compare compression properties of these three transforms,
i.e. reconstruct the signal from finite-number of components to achieve 95% of original-signal power.
- NOTE. Realize the selection of components by zeroing in the domain of KLT,
DFT or DCT spectrum.
- ATTENTION. Do not forget to take into account the symmetry of DFT.
- Result:
- Required number of components for a compression of signals s2 and r1
using KLT, DFT or DCT.