ASSIGNMENT 2 ELEC620 Signal Analysis for Telecommunication Submission Guidelines: • Submit your assignment via iLearn in a single PDF file. • A 10% penalty will apply for each day late. • Assignments...

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ASSIGNMENT 2 ELEC620 Signal Analysis for Telecommunication Submission Guidelines: • Submit your assignment via iLearn in a single PDF file. • A 10% penalty will apply for each day late. • Assignments without cover sheet will not be accepted. • Maximum marks for this assignment are 50, and worth 10% of the assessment for this course. Due Date: Tuesday 2 October 2018 Question 1 [6 Marks] For the following signal x(t) x(t) = A |t| ≤ τ20 |t| > τ2 a) Find and plot Fourier transform X(ω). b) Energy spectrum density of the signal. c) Discuss the impact of changing A and τ on X(ω). Question 2 [4 Marks] Consider a periodic impulse train δTo(t) = ∞∑ k=−∞ δ(t − kTo) a) Determine the complex exponential Fourier series of δTo(t). b) Determine the trigonometric Fourier series representation of δTo(t). Question 3 [5 Marks] Show that the following expression (Modulation Theorem) represents a valid Fourier transform pair x(t)cos(ω0t)←→ 1 2 X(ω −ω0) + 1 2 X(ω −ω0) Hint: Use the time shifting property of Fourier transform. Question 4 [5 Marks] Prove Mathematically that time domain convolution is equivalent to frequency do- main multiplication and vice versa. Provide justification of each of the steps in- volved. 2 Figure 1: Question 6 Figure Question 5 [6 Marks] Consider an LTI system described by the following differential equation dy(t) dt +2y(t) = dx(t) dt a) Find the frequency response H(ω) of the system. b) Represent H(w) in time domain i.e., h(t). c) Find the output y(t) to each of the following input signals: • x(t) = e−tu(t) • x(t) = u(t) Hint: Use the table given in lecture slides to find the Fourier transforms. Question 6 [8 Marks] The extract shown in Fig. 1 shows how the energy spectral density is the Fourier transform of the auto correlation function. There are 7 equal (=) signs are involved where the previous result is transformed. For each of the equal sign give a written description of the transformation involved and what Fourier transform property or mathematical fact is involved. 3 Question 7 [6 Marks] For the following triangular signal x(t) −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 t x(t) a)Determine the energy of the signal x(t). b) What is the spectral energy density of x(t). c) State Parsvel’s theorem for this signal. Question 8 [10 Marks] The signal x(t) = cos(2π8t) is sampled at a rate of 3 samples/sec using ideal samples (impulses) to obtain a sampled signal xs(t). The sampled signal is then sent through an ideal low pass filter, with a transfer function H(2πf ) = 13rect ( f 3 ) . 1. Draw the Fourier transform, Xs(2πf ) of the sampled signal. Plot this as a function of f , measured in Hz. 2. What is the frequency of the output signal from the filter, in Hz?. Explain your answer. 3. Here in this scenario, after passing through a filter we are trying to reconstruct the signal. Has there been any distortion in the signal as compared to the original signal? If so, explain what the problem was? and how this distortion can be mitigated? 4