1. The properties of convolution is listed in the table 1; Verify the properties of convolution for the given signal by:

x[n]=n{u[n-10]-u[n+15]}

h[n]=(0.5)ⁿ {u[n]-u[n-10] }

h₁ [n] = cos(0.05πn) {u[n]-u[n-21]}

h₂ [n] = 2δ[n+3]+δ[n-1]-3δ[n-5].

a) Numerical computation of convolution sum.

b) Verify the result obtained in part(a) by using MATLAB function convand plot the output of each property.

2. a) The pole - zero diagram in figure 2 corresponds to the Z-transform [X(z)] of a causal sequence (x[n]). Sketch the pole-zero diagram of Y(z), where y[n] = x[-n+5]. Also, determine the region of convergence for Y(z).

b) Discuss the applications of Multirate Digital Signal processing or Explain the need of Multirate Signal Processing with suitable Example.

Note: Support your answers with appropriate explanation and figures. Provide references also.

3. Given the realization of IIr filter in Figure 3.

a) Obtain the difference equation relating g[n]to x[n] and g[n] to y[n].

b) Determine the transfer function H(z)for the filter.

4. In this problem use Lena image to compute the output.

a) Load the Lena image in MATLAB and display using imshow function.                                                  

b) Consider the 1D (single dimensional) impulse responseUsing it performs 1D convolution along the each row of the Lena image and display the resulting blurred image. Comment on the result.

c) Using the image obtained in part (b) perform 1D convolution along the each column of the Lena image and display the resulting blurred image. Compare this image with the image in part (b) and comment on the results.