Digital Signal Processing Assignment - z-Transforms, Filters and Multirate Signal Processing Concepts

Introduction - The objective of this assessment is for each student to demonstrate understanding of the contents of lecture materials in the unit by applying the principles and concepts in lecture notes. Concepts covered by this assignment include z-transforms, system stability, impulse response of a filter and multirate signal processing concepts including down-sampling and up-sampling.

Questions - A digital system is given by the following transfer function:

H(z) = (z+1)/(z²+z+0.5)

1) Calculate the zero(s) and pole(s) of the transfer function.

2) Provide the zero(s) and pole(s) in polar form.

3) Plot the poles and zeros using the z-plane.

4) Is the system stable or unstable? Why?

5) Determine the system equation in the form of y[n] = _i=1∑ⁿa_iy[n-i] + _j=1∑^mb_jx[n-j]

This means clearly give the values of a_i and b_j, n and m for 1 ≤ i ≤ n and 1 ≤ j ≤ m.

6) Sketch the magnitude of the frequency response of the system between 0 and 2π.

7. Suppose the above transfer function is a filter, what type of filter is it?

8. Is the filter a low-pass, high-pass, band-pass, band-stop or all-pass?

9. Sketch a block diagram for implementing this system?

10. Determine the impulse response h[n] of the causal system (note: the input x[n] is a unit impulse).

11. Down-sample the system in the time domain by factor L = 2.

12. Up-sample the system in the time domain by factor M = 2.

Format of Report - Your report should be typed using a suitable equation editor. Hand written reports will not be given zero marks. Explain your results and give reasons for your answers.