Total marks: 100                                          Due Date: April 28, 2023 at 16:00 AEST

Note: This assignment is worth 20% of the final course mark. Please submit your answers as a single pdf file via Gradescope, including your name and student number. Solutions, including equations, should be typed. Explain your solutions as if you are trying to teach a  peer.   Demonstrate your insight and understanding.   Answering an entire question with bare equations, lone numbers or without any explanation is not acceptable.  Marks may be reduced if an answer is of poor quality, demonstrates little effort or significant misunderstanding. If you have used code in your solution, please include it in your report.

Questions

Question 1.  Z-Transforms           (20 marks)

(a) Find the Z-transform X(z), the ROC, the poles and the zeroes for the following

sequences:

i) x[n] = ( )n u[n] +()n u[n]   (5 marks)

ii) x[n] = ( )n u[n] +()n u[−n − 1]    (7 marks)

(b) Find the inverse Z-transform of the following X(z):

i) X (z) = 1 + z−1 + z−3 , |z| > 0   (2 marks)

ii) X [z] =  , |z| > 2   (2 marks)

(c) Using the Z-transform, verify the following equations:

i) x[n] ∗ 6 [n] = x[n]   (2 marks)

ii) x[n] ∗ 6 [n − m] = x[n − m]   (2 marks)

You are allowed to use the Z-transform pairs presented in Table 11.1 from Lathi’s book.

Question 2.  Analogue Systems           (10 marks)

(a) Find the frequency response (magnitude and phase) of a system whose transfer function is

2s + 0.3

s + 7  .    (2 marks)

(b) Find the system response, y(t), if the input f(t) is:

i) cos(−t)   (2 marks)

ii) cos(10t − 45◦ )   (2 marks)

iii) sin(t)   (2 marks)

iv) sin(0.5t + 30◦ )   (2 marks)

Question 3.  Bilinear Transform                                                                 (10 marks)

Use the bilinear z-transform to design a digital second order Chebyshev high-pass filter with the following specifications:

• 0.5 dB ripple in the pass band

• cut-off frequency of 400 Hz

• sampling frequency of 1000 Hz

(a) What is the analogue frequency ωa  that maps to the specified discrete cut-off fre- quency ωd ?       (2 marks)

(b) What is the transfer function of the de-normalised prototype analogue high-passfilter?                      (3 marks)

(c) What is the transfer function of the resulting digital high-pass filter?        (5 marks)

Question 4.  Digital Filter Design                                                             (30 marks)

The human voice generates frequencies within the typical range of 100 Hz to 3500 Hz. To transmit voice calls over a telecommunications network, companies will filter and modulate the input sound signals to fit many voice calls into their available bandwidth.

(a) As a new employee at Australia’s newest telecommunications company Fonstra, you

have been tasked with designing a digital low-pass filter to isolate each incoming voice call. Your system must have a stopband which starts at 4000 Hz, no more than 3 dB of ripple in the passband and a minimum attentuation of 40 dB in the stopband. Your boss would like you to compare the specifications between basing the design on a Butterworth filter vs. an Elliptic filter.

Select an appropriate data sampling frequency and design the lowest-order Butter- worth and Elliptic filters that satisfy these criteria.

i) State your chosen sampling frequency   (1 mark)

ii) State the order of each filter    (2 marks)

iii) Plot the magnitude and phase response of the filters   (4 marks)

iv) Plot the poles and zeros of the filters in the z-plane    (2 marks)

(b) Your excellent work at Fonstra caught the attention of rival company Codafone, who

immediately offer to hire you for a significant salary increase (you accept of course).

At Codafone, they want to cut costs and squeeze even more voice calls into their available bandwidth. Your new boss thinks you can do this by reducing the passband to between 500 Hz and 3000 Hz, setting the stopbands at ±400 Hz either side of the passband and requiring only a minimum attenuation of 10 dB in the stopbands. There should still be no more than 3 dB ripple in the passband. Again, compare the specifications between basing the design on a Butterworth filter vs. an Elliptic filter.

Select an appropriate data sampling frequency and design the lowest-order Butter- worth and Elliptic filters that satisfy these criteria.

i) State your chosen sampling frequency   (1 mark)

ii) State the order of each filter   (2 marks)

iii) Plot the magnitude and phase response of the filters    (4 marks)

iv) Plot the poles and zeros of the filters in the z-plane     (2 marks)

(c) Discuss the differences between the resulting filters based on the specifications pro- vided by Fonstra and Codafone.

i) Include in your discussion a comparison of filter order, transition band, ripple and phase response.       (8 marks)

ii) Explain the practical implications of Codafone’s filter specifications (passband and minimum attentuation) on the resulting voice call quality.          (4 marks)

(a)

(b)

Figure 1: The original image (a), and the image corrupted with high frequency noise (b). The coloured horizontal lines indicate the line of pixels corresponding to the intensity plots in Figure 2.

Question 5. Image Denoising                                                                     (30 marks)

A live TV broadcast of the latest wildlife documentary from famous natural historian Navid Eatonborough has been corrupted with high frequency noise.  The original signal should show footage of a newly discovered animal species (Lama sonabilius, see Figure 1a). Unfortunately, the Corporation of Busy Broadcasters (CBB) only receives the noisy image shown in Figure 1b.  Use your knowledge of digital filtering and convolution to help the CBB denoise the signal and recover the original image.

In Matlab, load the .mat file attached to Problem Set 2. The file contains two variables: imdata, which is the original greyscale image, and noisy image vec which is the corrupted image transmitted to the CBB. Note that noisy image vec is a vectorised image. i.e. if the image is of size [m,n], then the vectorised image is of size [1,mn].  The first row of pixels in the image corresponds to the first 1: n elements of the vector, the second row corresponds to the next n + 1: 2n elements, and so on. Pixel intensities are all normalised to lie in [0, 1].

Attach your code in your submission.

(a) You have been told that the noise corrupting the broadcast is exhibiting frequencies at and above 0.2 px−1 . Design a brick wall filter with an appropriate cut-off frequency. What is its transfer function in the pixel domain?                                    (3 marks)

(b) Implement your filter in Matlab as a finite impulse response filter.  Hint:  a window

size of −10: 10 is a good place to start.

Convolve your filter with the noisy image using the Matlab function conv with the SHAPE parameter set to ‘same’. Remember to rescale the result back to [0, 1].

Plot the

resulting

filtered image.

(5 marks)

(a)

(b)