A quantizer takes in a real valued sample that cannot be represented digitally, and converts it...

  

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A quantizer takes in a real valued sample that cannot be represented digitally, and converts it to one of M possible levels. A quantizer with M = 2B levels is able to map each real valued sample at its input into a digital value at its output that can be represented with log2 (M) = B bits. Since many real valued inputs may be mapped to the same quantization level, quantization is said to be lossy: that is, some information about the input signal is destroyed during quantization and can never be recovered. A common figure of merit for a quantizer is the so-called quantization error, which is defined as the expected Mean Squared Error (MSE) between the quantizer's input and its output. 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 according to the quantization function Q(X) illustrated in the figure below and defined mathematically as follows: Q(x) 4 3- 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))² = (1.75 – 1.5)² = 1/16. 2 1 Q(X) = 0 0.5, 1.5, 2.5, 3.5, 1 if X = [0, 1), if X = [1,2), if X = [2,3), if X = [3, 4], 2 3 Figure 1: Uniform Quantizer 4 X (a) Determine the PDF of X. (b) Determine the PMF of Y = Q(X). Hint: From the problem statement, py (1.5) = P(Y = 1.5) = P(Q(X) = 1.5) = P(1 ≤ X < 2) = f²_₁ ƒx (x) dx. Use similar logic to also find py (0.5), py (2.5), and py (3.5). x=1 (c) Find fxy(x|i), i.e., the conditional PDF of X given that Y = i, for all i = {0.5, 1.5, 2.5, 3.5}. Hint: The event {Y = 1.5} is equivalent to the event {1 ≤ X <2}. Use similar logic for the events {Y = 0.5}, {Y = 2.5}, and {Y = 3.5}. (d) Find E [XY = i], i.e., the expected value of X given that Y = i, for all i = {0.5, 1.5, 2.5, 3.5}. (e) Find the quantization error €į = E [(X – Y)²|Y = i], i.e., the variance of X given that Yi, for all i = {0.5, 1.5, 2.5, 3.5}. A quantizer takes in a real valued sample that cannot be represented digitally, and converts it to one of M possible levels. A quantizer with M = 2B levels is able to map each real valued sample at its input into a digital value at its output that can be represented with log2 (M) = B bits. Since many real valued inputs may be mapped to the same quantization level, quantization is said to be lossy: that is, some information about the input signal is destroyed during quantization and can never be recovered. A common figure of merit for a quantizer is the so-called quantization error, which is defined as the expected Mean Squared Error (MSE) between the quantizer's input and its output. 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 according to the quantization function Q(X) illustrated in the figure below and defined mathematically as follows: Q(x) 4 3- 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))² = (1.75 – 1.5)² = 1/16. 2 1 Q(X) = 0 0.5, 1.5, 2.5, 3.5, 1 if X = [0, 1), if X = [1,2), if X = [2,3), if X = [3, 4], 2 3 Figure 1: Uniform Quantizer 4 X (a) Determine the PDF of X. (b) Determine the PMF of Y = Q(X). Hint: From the problem statement, py (1.5) = P(Y = 1.5) = P(Q(X) = 1.5) = P(1 ≤ X < 2) = f²_₁ ƒx (x) dx. Use similar logic to also find py (0.5), py (2.5), and py (3.5). x=1 (c) Find fxy(x|i), i.e., the conditional PDF of X given that Y = i, for all i = {0.5, 1.5, 2.5, 3.5}. Hint: The event {Y = 1.5} is equivalent to the event {1 ≤ X <2}. Use similar logic for the events {Y = 0.5}, {Y = 2.5}, and {Y = 3.5}. (d) Find E [XY = i], i.e., the expected value of X given that Y = i, for all i = {0.5, 1.5, 2.5, 3.5}. (e) Find the quantization error €į = E [(X – Y)²|Y = i], i.e., the variance of X given that Yi, for all i = {0.5, 1.5, 2.5, 3.5}.