ECE3084 Spring 2015Homework 1 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 s ignals 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). 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) 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. (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). (b) Draw a labeled plot of a portion of the periodic signal x(t) = =m{40 exp(j2000πt)}. Let the left limit of your horizontal axis correspond to t = 0, and choose the right limit of your horizontal axis such that you plot exactly three periods.1 (c) (Credit to where it is due: this part was inspired by an old MIT problem set.) Show that 1 − exp(ja) = 2 sin(a/2) exp(j[a − π]/2) 1 Throughout this course, when we say “period,” we mean “fundamental period.” Of course a signal with the property that x(t) = x(t + T0 ) will also have the property that x(t) = x(t + kT0 ), where k is an integer, so anything periodic with period T0 is also periodic with period 2T0 , 3T0 , 4T0 , etc. The fundamental period of a periodic signal is the smallest T0 such that x(t) = x(t + T0 ). (b) xb (t) = (t + 1)u(t − 1) − tu(t) − u(t − 2). 3rd Edition. draw a labeled sketch the following signals: (a) xa (t) = u(t + 1) − u(t − 1) + u(t − 3). 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. by Kamen and Heck. (c) xc (t) = 2(t − 1)u(t − 1) − 2(t − 2)u(t − 2) + t(t − 3)u(t − 3).) Problem (3) Using your brain instead of a calculator. Problem (2) Expressions of the form y(t) = x(At + B) can modify a signal in three ways: time shifting.We will use this property in the part of the course that covers Fourier transforms. Consider a function x(t) given by the graph: x(t) 1 t 0 12 Given this x(t). time scaling (stretching or compressing). and/or flipping. since it is easy to get confused. (Credit where it is due: this problem was adapted from Problem 1. You have to be careful when multiple such effects are happening at once.) Problem (3) Try your hand at simplifying the following expressions: Z t−3 (a) δ(τ + 9)dτ . t+2 .4 on page 39 of Fundamentals of Signals and Systems Using the Web and MATLAB. Z ∞ τ [u(τ − 2) − u(τ − 5)]dτ (Hint: you can get rid of the unit step (b) −∞ functions in the integrand if you incorporate their effects by changing the limits of the integral.e. since those parameters are constants. if the system has a particular property for a range of values.) (c) 1 X k=−1 Z t (d) −∞ 2 t sin πt 2 δ(t − k). i. and κ are real constants. β. If you need to constrain a parameter to satisfy the stated criterion. γ. give the least restrictive constraint. we will take off points if you constrain parameters that do not need to be constrained. write “NONE. only focus on the parameter or parameters that are relevant to the particular property you are being asked about in that particular question. γ or κ be function of t. i. which makes no sense at all. 1 [δ(τ − 3) + δ(τ − 5)]dτ . (c) Describe how the parameters must be constrained to make the system be causal. not t − 5!) dt Problem (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]. You do not need to say anything about the parameters that are irrelevant to that particular property.” Each part is independent of the others. (b) Describe how the parameters must be constrained to make the system be time invariant. β. (d) Describe how the parameters must be constrained to make the system be memoryless. in particular.e. In the questions below. Don’t make the same mistake!) (a) Describe how the parameters must be constrained to make the system be linear. the restrictions you state in (b) are not “on top of” the restrictions you list in your answer to (a). If no constraints are needed at all. . be sure to specify the whole range. τ d (e) {t4 u(t + 5)} (Careful — that is t + 5. (Warning: A common mistake on this problem is that students will try to make α. where the parameters α.