Consider the problem of approximating a continuous analog signal X with a discre

Question

Consider the problem of approximating a continuous analog signal X with a discrete digital signal Y. To do this, we may divide the range of X into discrete intervals, [a_0, a_1), [a_1, a_2), etc. Then, if X takes on a value in the interval [a_0, a_1), we assign the value y_1to the discrete signal (predictor) Y. and so on for the other intervals. Suppose that X is distributed on [0,1] according to the PDF f(x) = 1.2x + 0.4. If we discretize by picking a_0 = 0. a_1 = 1/2 and a_2 = 1 how should we pick y_1 and y_2 to generate the best predictor of X (that is, to minimize E[(X - Y)^2]? In this case, what is the expectation? E[(X-Y)^2]

Explanation / Answer

The pdf of x on [0 1] is given by f(x) = 1.2x + 0.4

Let P(x) = E(X - Y ) = (X - Y)2 f(x) dx ( limit 0 to 1) = (x - 1.2 x - 0.4)2 f(x) dx ( limit 0 to 1)

= (x - 1.2 x - 0.4)2 ( 1.2x + 0.4) dx ( limit 0 to 1) = (0.036x3 + 0.208x2 +0.832 x + 0.34) dx ( limit 0 to 1)

After Integrating and putting the limit

We get P(x) = 0.009x4 + 0.069x3 +0.416x2 + 0.64x

Differentiating w.r.t x, three values of x and substituting x, we get different values of y, which is the required question.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.