### 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.

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