Objectives

The aims of this assignment are:
1. To understand different signal models (course objectives 1 and 2)
2. To be able to model a discrete-time system (course objectives 3).
3. To design simple signal processing algorithms (course objective 4)

Students are expected to communicate their findings and ideas in a clear and logical manner.

Submission
• Assignments must:

- Be typed, not handwritten.
- Be submitted electronically via the Study Desk.
- Be submitted in PDF format, and less than 4M in size.
- Use the file naming format LastName-StudentNumber.pdf.
- State your name and student number at the top of the first page, leaving the remainder of the first page blank for marking.

• You are not permitted to use any "toolbox" functions from MATLAB. If in doubt, use which to see if a function you wish to use is part of an add-on toolbox or not.

• For each question, submit a written report, detailing your approach and discussing your findings. Your report should include diagrams, figures, source code, waveforms and/or images as appropriate for this assignment.

• Late assignments are not normally accepted. If you wish to apply for consideration for late submission, it must be done at least one week prior to the due date in writing or via email. Include documentary evidence of illness (a medical certificate) or additional work commitments (a written confirmation of changed work circumstances from your supervisor). For extension applications for other reasons, please contact the examiner at least 2 weeks in advance of the due date.

• Students are reminded of the penalties applying to plagiarism. Copying all or part of an assessment from another student, or from the web, is unacceptable. Plagiarism may result in loss of marks, or other penalties as determined by the Academic Misconduct Policy. Further helpful hints on how to correctly reference (and how to avoid plagiarism) may be found under the link Academic Honesty on the course Study Desk.

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Marking
Marks are awarded as per the marking guidelines at the end of each question. The breakdown of marks will be noted on the PDF file returned to you via the Study Desk. Where an explanation or description is specifically requested, your response will be assessed according

to the following:

You may use MATLAB to complete questions where you are required to plot waveforms or calculate results. You may use any MATLAB notes or tutorials provided in this course as a starting point. You must not use any MATLAB "toolbox" functions - that is, any which are not shown as "built-in" in response to the which command. Where MATLAB coding is required, show all your code for each question as part of your report.

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Part I
Question 1 - 40 Marks
Using the audiorecorder() function in MATLAB, make a recording of your voice. Use a sample rate of 8kHz, 16 bits per sample, and aim to record around 2 seconds worth.

Part (a) - 20 Marks
Quantize the audio samples down from the original 16 bits by removing the least-significant bit (LSB) from the waveform, to make a 15 bit recording. Repeat for 14, 13, 12 bits, down to 1 bit.

Explain how you would do this in MATLAB, and implement your approach.

Part (b) - 20 Marks
Listen to the recordings, and make a table with the number of bits and the corresponding quality assessment. This can simply be a subjective assessment; use terms such as ‘no perceptual difference', ‘minor noise present', ‘quite noisy', ‘poor quality' etc.

Calculate the signal-to-noise ratio in each case, and add that to your table. Explain how you calculated the SNR.

(a) Explanation & MATLAB coding 20
(b) Subjective assessment table with SNR 20

Total 40
Question 2 -

This question examines the addition of a (synthetic) echo to the voice recording, such as would be found in a reverberant room. To do this, use a difference equation of the form y(n) = x(n) + αy(n - D) (1) where x(n) is the input audio, y(n) is the output (echoed) audio, D is the echo delay (in samples), and α governs the amount of echo fed back.

Part (a) -
Explain how to convert the above equation into a form suitable for passing to MATLAB's filter() command. Use α = 0.8 and D = 2 samples to explain your reasoning.

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Part (b) -

A delay of 2 samples (as in the previous part) would not be audible. Select an echo delay of 0.2 seconds, and α = 0.4. Using your reasoning above, implement the reverberation equation, and listen to the result. Experiment with different values of the parameters α and D. In your report, include a plot of the waveform, and explain in your own words the physical significance of these parameters.

Part (c) -
Convert equation (1) to a z transfer function. Where are the poles located
(i) for D in general?
(ii) for the value of D found in part (b) above?

Part (d) -
Suppose the equation governing the reverberation is
y(n) = x(n) + αx(n - D) (2)
What is the physical significance of this form, as opposed to that used in equation (1) ? Implement an audio echo system based on equation (2), and listen to the results.
(a) Explain use of filter command 20
(b) Implement realistic filter, show waveform 20
(c) z transfer function and poles 20

(d) Explain nonrecursive transfer function 20

Part II
Question 3 - 20 Marks
Part (a) - 10 Marks
Generate and plot a sampled sine wave with fs=8kHz, of 4 seconds duration, with frequency ωo = π10 rad/samp and amplitude A = 1.2. The waveform equation is x(n) = A sin nω (3)

Explain the role of each of the variables in this equation. What is the true (Hertz) frequency generated in this case?

Part (b) -

Generate a Gaussian random signal vector, v(n), of the same length. Then generate a noisy signal of the form
y(n) = x(n) + αv(n) (4)

Listen to the resulting signal y(n) for various values of α. You will have to choose the value of α experimentally - try both small and large, and investigate the differences. Plot one of the waveforms, and comment briefly on your results.

(a) Plot clean sinusoidal waveform & comment 10

(b) Plot waveform with noise & comment 10

Question 4 -
A filter of the form
G(z) = z2(z - p)(z - p∗)(5) with p = re?ωn will be studied.

Part (a) - 20 Marks

With r = 0.95 and ωn =π10 , plot the time response to the input clean sinusoidal waveform generated in the previous question. Show both the transient and steady-state response.

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Part (b) -

Plot the frequency-domain response of the filter. Explain all your working, particularly how the z transfer function is converted to gain/phase plots.

Part (c) -

Find the gain and phase from the time-domain response of part (a), and compare to that expected from the frequency response in part (b). Are the results the same?

(a) Time response (transient+steady-state) 20
(b) Frequency response (gain+phase) 20
(c) Compare gains and phases, explain results 20