electrical assignmnet signal system

Full Name: Lab Section: ECE 3500 (Fall 2015) – HW #1 Due Date: Sept. 3, 2015

Homework learning objectives: By the end of this homework, you should be able to: • Intuitively understand the fundamental properties / operations of signals • Sketch continuous-time signals and discrete-time signals

Question #1: (2 pts) How many hours did you spend on this homework?

Question #2: (6 pts) Let x(t) = 2 sin �� 4 t � be a continuous-time signal.

(a) Sketch the signal x(t) for −3 < t < 3. (b) Compute the energy of x(t) [for all time].

(c) Compute the power of x(t) [for all time].

(d) Is x(t) an energy signal, a power signal, or neither?

(e) Is x(t) an even signal, an odd signal, or neither.

(f) Is x(t) causal?

Question #3: (6 pts) Let x(t) =

� ����

����

0 if t � −1 2t if − 1 < t < 0 2−t if 0 < t < 1

0 if t � 1

be a continuous-time signal.

(a) Sketch the signal x(t) for −3 < t < 3. (b) Compute the energy of x(t) [for all time].

(c) Compute the power of x(t) [for all time].

(d) Is x(t) an energy signal, a power signal, or neither?

(e) Is x(t) an even signal, an odd signal, or neither.

(f) Is x(t) causal?

Question #4: (6 pts) Let x[n] = (−1)(n−1) be a discrete-time signal.

(a) Sketch the signal x[n] for −3 < n < 3. (b) Compute the energy of x[n] [for all time].

(c) Compute the power of x[n] [for all time].

(d) Is x[n] an energy signal, a power signal, or neither?

(e) Is x[n] an even signal, an odd signal, or neither.

(f) Is x[n] causal?

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Question #5: (4 pts) Let x(t) be the following continuous-time signal.

(a) Sketch y(t) = x(t − 4) (b) Sketch z(t) = y(4t)

(c) Sketch r(t) = x(4t − 4) (d) Sketch q(t) = x(4(t − 4))

Question #6: (3 pts) Let x(t) and y(t) be the following continuous-time signals.

(a) Sketch x(1 − t) (b) Sketch y(3t − 6) (c) Sketch x(1 − t) − 2y(3t − 6)

Question #7: (4 pts) Consider discrete-time signals x1[n], x2[n], and x3[n] below.

(a) Write x2(t) in terms of x1(t) [e.g., x2(t) = 2x1(t)]

(b) Write x1(t) in terms of x2(t)

(c) Write x3(t) in terms of x1(t)

(d) Write x1(t) in terms of x3(t)

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Question #8: (3 pts) A very powerful property of any signal x(t) (or x[n]) is that it can be represented by the sum of an odd signal xo(t) (or xo[n]) and an even signal xe(t) (or xe[n]):

x(t) = xo(t) + xe(t) or x[n] = xo[n] + xe[n] .

Consider the signal x[n] below.

(a) Find the xo[n] and xe[n] components that sum to x[n]. Require that the amplitudes of xo[n] and xe[n] can only be 1 or −1. Ignore n = 0.

(b) Assume the 1 and −1 values represents binary 1’s and 0’s, respectively. Decode xo[n] and xe(t), assuming t = 3 is the lowest binary digit (i.e., 2

0 digit) and t = −3 is the largest binary digit (i.e., 26 digit). Ignore n = 0.

(c) The knowledge that x[n] = xo[n] + xe[n] (and related concepts) is important for communica- tions technology. Why do you think that is?

Question #9: (2 pts) Throughout this class, we refer to our signals as either continuous-time sig- nals or discrete-time signals. This “time-centric” description is largely used for intuitive simplicity. Many signals measured and analyzed in the world are not time signals.

(a) What is an example of a signal that is not a time signal? What is this signal used for?

(b) Is your example signal causal (in general)?

Question #10: (4 pts) Let Ex be the energy of x[n].

(a) Determine the energy of y[n] = 2x[n]

(b) Determine the energy of y[n] = x[n]/Ex

(c) Determine the energy of y[n] = 2x[n]/Ex

(d) How might the results of parts (b) and (c) be useful when designing a signal?

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