301, ECE Homework #2, due date: 9/07/2011 http://cobweb.ecn.purdue.edu/chihw/11ECE301F/11ECE301F.html
Question 1: [Basic] Consider two functions f (t) and g(t) described as follows. 2 if -2 t < 0 f (t) = 1 if 0 t < 4 0 otherwise 3 + t if -3 t < 0 g(t) = 3 if 0 t < 2 . 0 otherwise Find the value of
-
(1)
(2)
g(1 - t)f (t)dt.
Question 2: [Basic] Review of Trigonometry: Suppose -/2 0 and 3/2, and cos() = 0.2 sin() = -0.4. Find the values of cos( + ) and sin( - ).
Question 3: [Basic] Review of complex numbers: Let j be the imaginary number, i.e., j 2 = -1. Suppose 2 + 2j = ea+bj 5 e3+ 3 j = c + dj. Find the values of a, b, c, and d.
Question 4: [Basic] Suppose A is a 3 by 3 matrix. Consider a linear system that outputs y = Ax where x R3 is the input signal. To be more precise, x is a column vector of dimensional 3, and y R3 is the output column vector of dimension 3. Further assume that we know that When x1 = (1, 0, 0)T , the output is y1 = (1, 2, 3)T .
When x2 = (0, 1, 0)T , the output is y2 = (3, -2, 1)T . When x3 = (0, 0, 1)T , the output is y3 = (1, -1, -3)T . What is the output y = Ax when the input is x = (1, 2, 4)T ? (Hint: Use the linearity of the system.)
Question 5: [Basic] p. 59, Problem 1.21. (a)(d).
Question 6: [Basic] p. 59, Problem 1.22. (a), (d), (g), (h).
Question 7: [Advanced] In class, we have shown how to construct a new signal y(t) from an existing signal x(t) by time shift, time reversal and time scaling. Namely, y(t) = x(t - t0 ), or y(t) = x(-t), or y(t) = x(t). These time transformations can be considered as "systems" as well since it takes x(t) as input and outputs a signal y(t). Show that these three time different transformation systems are linear.
Question 8: [Basic] p. 61, Problem 1.25 (a)(c).
Question 9: [Advanced] An important theorem says that every signal x(t) can be expressed as the sum of an even signal xe (t) and an odd signal xo (t). This theorem can be proved by the following steps. We can always write x(t) + x(-t) x(t) - x(-t) + . 2 2 Answer the question: Why does the above equality hold? x(t) = Show that in the above equality, the first term of the right-hand side even signal, and the second term x(t)-x(-t) is an odd function. 2
x(t)+x(-t) 2
is an
[Optional] You should ask yourself Q1: why the statement "every signal x(t) can be expressed as the sum of an even and an odd signals" is important. Hint: Think about the following question: If we want to design a set of "good test signals," we definitely would like our test signals to be able to cover all possible unknown signals. Q2: What exactly do we mean by covering all possible unknown signals? Q3: Is there any connection between the answers to Q1 and Q2?
Question 10: [Basic] p. 60, Problem 1.23(a). p. 61, Problem 1.24(b)
Question 11: [Basic] Consider a function f (t) such that f (t) = 1 if t 4 and f (t) = 0 otherwise. Find the expression of
h() =
-
eat+jbt f (t)e-jt dt,
(3)
where a is a constant that is strictly larger than zero.
Question 12: [Advanced] p. 63, Problem 1.32 and p. 64 Problem 1.33. You only need to answer the true/false questions. For the statement that is not true, produce a counterexample to it. There is no need to discuss the fundamental periods. Remark: p. 71 in the textbook contains some very important discussions about the polar form of a complex number.