Assignment signals and systems - what is the physical

Question

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.

Question 1

Part I

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)

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)

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 using your data from the previous question, and add that to your table. Explain how you calculated the SNR, and comment on whether it agrees with theoretical predictions.
(a) Explanation & MATLAB coding
(b) Subjective assessment table with SNR

Question 2

This question examines the addition of a (synthetic) echo to the voice recording, such as would be found in a reverberant room.

Part (a)

Use a difference equation of the form

y(n) = x(n) + αy(n - D)

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. 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 one sample plot of the waveform, and explain what you found as you tried varying the parameters.

Part (b)

Suppose the equation governing the reverberation is

y(n) = x(n) + αx(n - D)

What is the physical significance of this form, as opposed to that used in equation? Implement an audio echo system based on equation (2), and listen to the results. Comment on what you have found.
(a) Implement echo filter, plot and discuss
(b) Explain equation form, comment & discuss

Part II

Question 3

This question examines discrete-time signal generation, for deterministic and random wave- forms.

Part (a)

Generate and plot a sampled sine wave with fs=8kHz, of 4 seconds duration, with frequency

ω_o = π/10 rad/samp and amplitude A = 1.2. The waveform equation is

x(n) = A sin nω

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)

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

(b) Plot waveform with noise & comment

Question 4

This question examines discrete-time filters. A filter of the form

G(z) = z² (z - p)(z - p∗)

with poles defined by p = re^ωn will be studied.

Part (a)

With r = 0.95 and ω_n = π/10, plot the time response to the input clean sinusoidal waveform generated in the previous question. Show both the transient (initial) and steady-state (after the initial transient) responses.

Part (b)

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

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). Compare the results obtained using these two methods.
(a) Time response (transient+steady-state)
(b) Frequency response (gain+phase) and compare