GEORGIA INSTITUTE OF TECHNOLOGYSCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 3084 Summer 2014 Problem Set #1 Assigned: 15-May-13 Due Date: 22-May-13 Your homework is due at the start of class on Thursday, May 22. Note that the notation we will use in lecture, on homeworks, and on tests is slightly different than that used in the textbook by Chen. Here, to maintain continuity with ECE2026, we use u(t) to represent the unit-step function, and x(t) to represent signals in general and system inputs in particular. Chen, as well as many control system engineers, prefer to use u(t) to represent system inputs. To avoid conflict, many alternative notations for unit-step functions have been used; Chen uses q(t). You may turn in your homework up to one day late, by 3:00 PM the following day. A 30% penalty will be assessed on late homeworks (even homeworks turned in on the day it is due but not at the start of class, although the penalty might be slightly less at our discretion). We understand that sometimes multiple assignments hit at once, or other life events intervene, and hence you have to make some tough choices. We’d rather let you turn something in late, with some points off, than have a “no late assignments accepted at all” policy, since the former encourages you to still do the assignment and learn something from it, while the latter just grinds down your soul. The somewhat aggressive late penalty is not intended to be harsh – it’s intended to encourage you to get things in relatively on time (or just punt if you have to and not leave it hanging over you all semester) so that you can move on to assignments for your other classes. Also, there is the practical matter that we cannot accept homeworks after solutions are posted, and we would like to post solutions shortly after both sections have submitted their homework. Please refrain from looking at backfiles of homework and exam solutions – i.e., “word” in Georgia Tech parlance – from previous versions of ECE2025, ECE2026, or ECE3084, beyond your own materials assembled while taking those classes and any old material we explicitly provide to you. If you get stuck, please come get help from your professor or TA before consulting a backfile. PROBLEM 1.1: As a warm-up, this problem reviews material from ECE2026, particularly Euler’s formula and complex numbers. Recall that Euler’s formula said that exp(ja) = cos(a) + j sin(a), and that this could be used to derive the “inverse” Euler’s formulas cos(a) = [exp(a) + exp(−a)]/2 and sin(a) = [exp(a) − exp(−a)]/(2j). The notations <e{z} and =m{z} √ represent taking the real and imaginary part of a complex number z. If z = a + jb, where j = −1 and a and b are real, then <e{z} = a and =m{z} = b (notice that you do not include the j when taking the imaginary part). Do all parts of this problem without using a calculator or a computer.1 (a) Find (j j )j , i.e. j to the power of j to the power of j. (Hint: first convert j to its “polar form” j = A exp(jφ), for appropriate values of A and φ). 1 I guess calculators are really just tiny, specialized computers, but let’s move on. ” we mean “fundamental period.2 (c) (Credit to where it is due: this part was inspired by an old MIT problem set. where k is an integer. You have to be careful when multiple such effects are happening at once. since it is easy to get confused. . Let the left limit of your horizontal axis correspond to t = 0. 3rd Edition. draw a labeled sketch each of the following functions y(t): (a) ya (t) = x(−t) (b) yb (t) = x(t + 2) (c) yc (t) = x(3t + 2) (d) yd (t) = x(1 − 4t) (Credit to where it is due: this problem was adapted from an old MIT problem set.) Show that 1 − exp(ja) = 2 sin(a/2) exp(j[a − π]/2) We will use this property in the part of the course that covers Fourier transforms.(b) Draw a labeled plot of a portion of the periodic signal x(t) = =m{40 exp(j2000πt)}. so anything periodic with period T0 is also periodic with period 2T0 . 3T0 .2: Expressions of the form y(t) = x(At + B) can modify a signal in three ways: time shifting.” Of course a signal with the property that x(t) = x(t + T0 ) will also have the property that x(t) = x(t + kT0 ). when we say “period. draw a labeled sketch the following signals: (a) xa (t) = u(t + 1) − u(t − 1) + u(t − 3) (b) xb (t) = (t + 1)u(t − 1) − tu(t) − u(t − 2) (c) xc (t) = 2(t − 1)u(t − 1) − 2(t − 2)u(t − 2) + t(t − 3)u(t − 3) (Credit where it is due: this problem was adapted from Problem 1. The fundamental period of a periodic signal is the smallest T0 such that x(t) = x(t + T0 ). by Kamen and Heck.) PROBLEM 1. and/or flipping. Consider a function x(t) given by the graph: x(t) 1 t 0 12 Given this x(t). and choose the right limit of your horizontal axis such that you plot exactly three periods. 4T0 .) 2 Throughout this course.3: Using your brain instead of a calculator. time scaling (stretching or compressing).4 on page 39 of Fundamentals of Signals and Systems Using the Web and MATLAB. PROBLEM 1. etc. You do not need to say anything about the parameters that are irrelevant to that particular property. since those parameters are constants. i. Don’t make the same mistake!) (a) Describe how the parameters must be constrained to make the system be linear. if the system has a particular property for a range of values. give the least restrictive constraint.5: Consider a continuous-time system whose output y(t) relates to the input x(t) according to the equation y(t) = cos(α[t − β])[x(t − γ) + κ].PROBLEM 1. where the parameters α.) (c) 1 X 2 t sin k=−1 Z t (d) −∞ (e) πt 2 δ(t − k) 1 [δ(τ − 3) + δ(τ − 5)]dτ τ d 4 {t u(t + 5)} (Careful — that’s t + 5. which makes no sense at all. γ or κ be function of t. we will take off points if you constrain parameters that do not need to be constrained. In the questions below. . If you need to constrain a parameter to satisfy the stated criterion. i. the restrictions you state in (b) are not “on top of” the restrictions you list in your answer to (a). (Warning: A common mistake on this problem is that students will try to make α.” Each part is independent of the others. and κ are real constants.4: Try your hand at simplifying the following expressions: Z t−3 δ(τ + 9)dτ (a) t+2 Z ∞ τ [u(τ − 2) − u(τ − 5)]dτ (Hint: you can get rid of the unit step functions in the integrand (b) −∞ if you incorporate their effects by changing the limits of the integral. If no constraints are needed at all. (d) Describe how the parameters must be constrained to make the system be memoryless.e. β. γ. write “NONE. in particular.e. (c) Describe how the parameters must be constrained to make the system be causal. not t − 5!) dt PROBLEM 1. β. (b) Describe how the parameters must be constrained to make the system be time invariant. only focus on the parameter or parameters that are relevant to the particular property you are being asked about in that particular question. be sure to specify the whole range.