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Part A Instructions Answer All 4 questions from Part A Each question in Part A carries 10 Marks. Total marks available in Part A are 40 Marks. Question A.1 Find the Discrete Fourier Transform of the sequence π‘₯(𝑛)= [1,2,4,5]. The Discrete Fourier Transform of π‘₯(𝑛) is: 𝑛=π‘βˆ’1 𝑋(π‘˜) = βˆ‘ π‘₯(𝑛)π‘’βˆ’π‘—2πœ‹π‘›π‘˜/𝑁 𝑛=0 Where: π‘˜ = 0,1,2, … . . 𝑁 βˆ’ 1 [10 Marks] Question A.2 a. Show whether the following system y(n)=0.4 x(n)+0.3 x(n-1)+0.2 x(n-2) is: i) Linear, ii) Time Invariant, iii) Causal and iv) Stable. [5 Marks] b. The discrete signal x(n)={…,0,1,2,4,6,…} is applied to a digital filter that has the discrete equation: y(n) = x(n)+4 x(n-1)+2 x(n-2)+ y(n-1) Find the first seven samples of the output y(n). [5 Marks] Question A.3 Consider two discrete time filters described by the following difference equations: (i) 𝑦(𝑛) = 1 π‘₯(𝑛) + 1 π‘₯(𝑛 βˆ’ 1) 5 5 (ii) 𝑦(𝑛) = π‘₯(𝑛) + 0.69 𝑦(𝑛 βˆ’ 1) For each filter: a. Find the transfer function 𝐻(𝑧) = π‘Œ(𝑧)/𝑋(𝑧) [4 Marks] b. Determine whether the filter is FIR or IIR [2 Marks] c. Find the frequency response H(ejw) [4 Marks] Question A.4 a. Calculate the auto-correlation sequence for the input sequence x(n) = [2,4,6,5]. The discrete autocorrelation is: 𝑛=π‘βˆ’1 π‘Ÿπ‘₯π‘₯ = βˆ‘ π‘₯(𝑛)π‘₯(𝑛 βˆ’ π‘˜) 𝑛=0 [8 Marks] b. Give an example of an application of auto-correlation. [2 Marks] Part B Instructions Answer All 3 questions from Part B. Each question in Part B carries 35 Marks. Total marks available in Part B are 105 Marks. Question B.1 a. A voice signal is sampled at a sampling frequency of 48 kHz. The bandwidth of interest for the audio signal extends from 0 kHz – 5 kHz. The sampled audio signal is interpolated by a factor of L as shown in Figure B.1. The resulting signal is then passed through a low pass anti-aliasing filter, before being decimated by a factor of M. The voice input signal has a bandwidth range from 0 kHz to 5 kHz, and the anti-imaging filter is an ideal low pass filter with a cut-off frequency of 10 kHz. Select suitable values for L and M to convert the sampling rate from 48kHz to 32kHz using minimum vales for L and M. Draw time and frequency domain representations of the signals at points 1, 2, 3 and 4. Voice Signal Sampled at 48kHz Output signal Sampled at 32kHz Figure B.1 [22 Marks] b. A digital LTI system has the impulse response h(n) = {…,0, 2, -3, 0,…} : find the output produced by the input x (n) = {…,0, 1, 2, 3,1, -1, 0,…}. [6 Marks] c. What effects can you expect from a finite word length of 64-bits in fixed and floating-point numbers? [7 Marks] Question B.2 a. Briefly describe any of the known methods for image compression, and illustrate how image compression can be achieved. [6 Marks] b. Produce a code that can efficiently represent each pixel in the image tile of FigureB.2.b. What compression ratio did you achieve compared to a simple 4-bit per pixel binary code? 5 6 6 6 5 6 7 7 7 8 7 9 9 9 10 10 Figure B.2.b [9 Marks] c. Quantise the image tile shown in Figure B.2.c.i, using the quantisation table given in Figure B.2.c.ii. Decode this new image tile using the same quantisation table Figure B.2.c.ii, and calculate the root mean square error of the decoded image. [9 Marks] 122 100 69 63 90 88 50 60 82 86 72 66 74 80 75 67 Figure B.2.c.i Image Tile 3 5 7 9 5 7 9 11 7 9 11 13 9 11 13 15 Figure B.2.c.ii Quantisation Table RMS Error is: 𝑅𝑀𝑆 πΈπ‘Ÿπ‘Ÿπ‘œπ‘Ÿ = 𝑖=𝑁 1 βˆšβˆ‘(π‘₯𝑖 βˆ’ π‘₯̂𝑖)2 𝑁 𝑖=1 Where: π‘₯𝑖 = Original values π‘₯̂𝑖 = Values after processing 𝑁 = Number of samples d. Design a new quantisation table to reduce the root mean square error of the encoding/decoding process. Calculate your new root mean square error. [11 Marks Question B.3 a. Design a FIR low pass filter with pass band edge frequency 1=/4, transition width =/2, and stop band attenuation at least 40 dB. The following formulas could be useful to you. Window Transition Width  Pass-band Ripple (dB) Stop-band Attenuation Window Function ο€­ N / 2 ο‚£ n ο‚£ N / 2 Rectangular 1.8Ο€/N 0.7416 21 dB 1 Bartlett 5.8Ο€/N 0.4752 25 dB 1 ο€­ 2 n N Hanning 6.2Ο€/N 0.0546 44 dB 0.5  0.5cos n 1 N / 2 Hamming 6.6Ο€/N 0.0194 53 dB 0.54  0.46 cos 2n N The impulse response of a low pass filter is: Ο‰c Ο€ Ο‰c sin(nΟ‰c) Ξ  nΟ‰c n = 0 n β‰  0 [18 Marks] b. Explain the operation of a 1-bit first order oversampling sigma-delta analogue to digital converter. [12 Marks] c. For a 0.6 V dc input demonstrate how a 1-bit sigma delta analogue to digital converter works by tabulating input, output and feedback terms over 7 clock cycles. [5 Marks]

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