A camera picks up a analog signal with intensity X that is uniformly distributed on [0,4]. The analog signal is quantized into 4 levels according to the quantization function Q(x) illustrated in the figure below and defined mathematically as follows:

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A camera picks up an analog signal with intensity X that is uniformly distributed on [0, 4]. The analog signal is quantized into 4 levels ac- cording to the quantization function Q(X) illustrated in the figure below and defined mathematically as follows: if X € [0,1), if X € (1,2), 0.5 1.5 Q(X): 2.5 if X € [2,3), 3.5 if X € [3, 4], In words, if the analog signal has intensity X, then it is quantized to the value Q(X). For example, using the above quantization function, if X = 1.75, then Q(1.75) = 1.5 and the squared error between its input X and output Q(X) is (X- Q(X))? = (1.75 – 1.5)² = 1/16. Q(x) 4 3 2 2 Figure 1: Uniform Quantizer (a) Determine the PDF of X. (b) Determine the PMF of Y = Q(X). (c) Find fx{Y=i} (x), i.e., the conditional PDF of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. (d) Find E [X|Y = i], i.e., the expected value of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. (e) Find the quantization error e; = variance of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. E [(X - Y) |Y =i], ie., the %3D A camera picks up an analog signal with intensity X that is uniformly distributed on [0, 4]. The analog signal is quantized into 4 levels ac- cording to the quantization function Q(X) illustrated in the figure below and defined mathematically as follows: if X € [0,1), if X € (1,2), 0.5 1.5 Q(X): 2.5 if X € [2,3), 3.5 if X € [3, 4], In words, if the analog signal has intensity X, then it is quantized to the value Q(X). For example, using the above quantization function, if X = 1.75, then Q(1.75) = 1.5 and the squared error between its input X and output Q(X) is (X- Q(X))? = (1.75 – 1.5)² = 1/16. Q(x) 4 3 2 2 Figure 1: Uniform Quantizer (a) Determine the PDF of X. (b) Determine the PMF of Y = Q(X). (c) Find fx{Y=i} (x), i.e., the conditional PDF of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. (d) Find E [X|Y = i], i.e., the expected value of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. (e) Find the quantization error e; = variance of X given that Y = i, for all i e {0.5, 1.5, 2.5, 3.5}. E [(X - Y) |Y =i], ie., the %3D