Problem 1 – DWT Decomposition and Reconstruction
Generate a signal x(t), which is a sum of two sinusoids at 5 Hz and 50 Hz, with sampling interval Ts= 1/1000 or Sampling Frequency fs = 1000 Hz.
a. Perform Discrete Wavelet Transform based 4 level Decomposition using Haar Wavelet
b. Plot the Coefficients (approximation & details) obtained from the decomposition and observe the number of coefficients at each level. (Hint: Use “stem” to plot the coefficients)
c. Prove that the Energy is preserved in the decomposition. Provide the approximate frequency bands corresponding to the coefficients (Hint: Energy of signal = sum of energy of approximation coefficients and sum of energy of the detail coefficients at all levels)
d. Reconstruct the signal from the approximation and detail coefficients by performing a scale based thresholding and plot the respective reconstructed signals. (Hint: To reconstruct signal only using approximation coefficients, set all detail coefficients at all the other scales to zero and vice versa)
Repeat a to d using db2, db3, coil 2 and coil 3 wavelets. Comment on the best wavelet choices for isolating the sinusoids.
Problem 2 – Interval Based Thresholding
Generate a frequency modulated sinusoid signal x(t) given below with sampling interval Ts= 1/1000 or Sampling Frequency fs = 1000 Hz.
x(t) = sin (5*pi*t) , t ≤ 0.5
sin (50*pi*t) , 0.5 < t ≤ 1
a. Perform Discrete Wavelet Transform based 4 level Decomposition using db1, db2, db4 Wavelets and plot respective approximation and detail coefficients.
b. Perform interval based thresholding to isolate the frequency components in the given signal.
c. Reconstruct and plot the thresholded signal. Comment on which wavelet would yield a better reconstruction
Problem 3 – Discontinuity Detection
Load scddvbrk signal from matlab wavelet toolbox. Use “load scddvbrk”.
a. Plot the signal, its first and second derivatives. b. Perform a 1 level DWT decomposition using db1, db2, db3,
b. Perform a 1 level DWT decomposition using db1, db2, db3, db 4 wavelets and plot the respective approximation and detail coefficients. (Hint: Observe the detail coefficients)
c. Comment on which wavelet is best for detecting the discontinuity in the signal derivatives.