3. A random voltage X which is uniformly distributed between a and b (a < b) is to be quantized into a discrete random variable Y = q (X) which has K ≥ 2 levels. The quantization error is E = X − Y = X − q(X) . Your answers below should be for general K except where you are asked to sketch something for particular K.

(a) Find the quantizer function q (x) such that the quantization intervals are all the same length, and the quantization value assigned to a quantization interval is the midpoint of the interval. Sketch the quantizer function q (x) for K = 2, 3.

(b) Find the quantization error function g (x) = x − q (x). Sketch the quantization error function g (x) for K = 2, 3.

(c) Find the pdf of the quantization error E = g (X).

(d) Find the mean square quantization error E E2 . Do this computation using the pdf for X, and also using the pdf for E