EE 350 Problem Set 4 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Sectio n:Submission deadl ines:• Turn in the written solutions by 4:00 pm on Tuesday October 7 in the homework sl ot outside 121 EE East.Problem Weight Score16 2017 2018 2019 2020 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you colla bo rated with on this assignment.Reading assignment:• Lathi Chapter 2, Sections 2.3 and 2.4• Priemer Chapter 4Carefully read section 2.4-2 in the text which presents a graphical interpretation of convolution. This approachyields significant insight to the convolution operation and is an important tool that will be used in technical electivessuch as Introduction to Communications (EE 360), Discrete-Time Systems Analysis (EE 351 ), and Fundamentals ofDigital Signa l Processing (EE 453).Problem 16: (20 points)1. Simplify the following expressions:(a) (4 points) [δ(t − 1) δ(t + 1) + u(t − 1) δ(t − 2) u(t + 1)] cos(πt)(b) (4 points) u(t + 1)δ(1 − t)e3t−3+ln(π)+ e−(2−π)tδ(t − 1)2. Evaluate the following integrals:(a) (4 points)R∞−∞h(τ)δ(t − τ)dτ(b) (4 points)R∞−∞δ(τ − 1)δ(t + 1 − τ )dτ(c) (4 points)Rt0e−τδ(τ − 1)dτProblem 17: (20 points)1. (2 points) Show thatf(t) ∗ δ(t − T ) = f(t − T ).2. (3 points) If y(t) = f(t) ∗ h(t), show thatf(t) ∗ h(t − T ) = f(t − T ) ∗ h(t) = y(t − T ).3. (3 points) Show thatf(t) ∗ [g(t) + h(t)] = f(t) ∗ g(t) + f(t) ∗ h(t).4. (3 points) If f(t) ∗ g(t) = c(t), show the derivative property of convolution˙f(t) ∗ g(t) = f(t) ∗ ˙g(t) = ˙c(t).5. (3 points) Derive the identitydu(t)dt= δ(t),where u(t) is the unit-step function. In order to obtain this identity, you need to show that the functionalsg(t) = du/dt and δ(t) have the same effect on an arbitrary function f(t), that isZ∞−∞f(t)g(t − T )dt = f(T )where T is a real-valued constant parameter.6. (5 points) Derive the identityδ(at) =1|a|δ(t).That is, the generalized functions δ(at) and δ(t)/|a| have the same effect on a given function f(t). In order toobtain this identity, first show thatZ∞−∞δ(at)f(t)dt =f(0)|a|,where the parameter a is a real numb er that can be either positive or negative. Next, find an expression forZ∞−∞δ(t)|a|f(t)dtin terms of f(0) and |a|. Use the last two results to obtai n the desired identity. Note that because δ(−t) = δ(t),δ(t) is an even generalized function.Problem 18: (20points)Systems can be represented either by an ODE or an impulse response function. G iven either representation, you canfind the zero-state response for a given input. For example, consider the RL circuit in Figure 1.Figure 1: RL circuit with input voltage f(t) and output current y(t).1. (4 points) Derive the ODE representation of the system and show that it can be expressed as˙y +1τy =Kτf.Express the time constant τ and parameter K in terms of R1, R2, a nd L. What is the physical significance ofthe parameter K?2. (4 points) Solve the ODE in part 1 to determine the zero-state unit-step response.3. (6 points) Determine the impulse response function h(t) of the circuit.4. (6 p oints) Determine the zero-state unit-step response using the convolution integral. Check your answeragainst the result obtained in pa rt 2 .Problem 19: (20 points)Using the graphical convolution method discussed in section 2 .4-2 of the text and lecture, find and sketch y(t) =f(t) ∗ h(t) for the following signals.1. (10 points)f(t) = u(t + 1) − u(t)h(t) = e−t/2u(t)2. (10 points)f(t) = e−|t|h(t) = e−tu(t)Problem 20: (20 points) A future lecture demonstrates that any real-valued periodic signal f(t) with fundamentalperiod Tomay be expresses as a superposition of an infinite number of sinusoids,f(t) = ao+∞Xn=1ancos(n ωot) +∞Xn=1bnsin(n ωot),where a0, a1, a2, . . ., b1, b2. . . are real-valued constant coefficients given byao=1ToZTof(t)dtan=2ToZTof(t) cos nωotdtbn=2ToZTof(t) sin nωotdt,and ωo= 2π/To. As an example, the coefficients for the periodic sawtooth waveform in Figure 2 areao= 0.5an= 0bn= −1nπ,ωo= 2π/To= π. As it not possible to numerically determine f(t) for an infinite number of terms, consider anapproximation that utilizes the first N terms of the summati on,fN(t) = ao+NXn=1ancos(n ωot) +NXn=1bnsin(n ωot),If N < ∞ , thenf(t) = fN(t) + e(t),where e(t) is the approximation error.Figure 2: Periodic sawtooth waveform with a fundam ental period of 2 s.Write an m-file that1. Plots f(t) over the i nterval 0 ≤ t ≤ 2 using the equationf(t) = t/2.The time vector must consist of 10,000 points equally spaced between 0 and 2. Plot f(t) using a dashed blackcurve2. Write a MATLAB function find fN that determines fN(t) given an integer value of n and the time vector frompart (1). The syntax for the calling the function must befn = find fn(t,N);Realize the function using a For-Loop.3. Using the MATLAB function find fN, determi ne vectors representing f1(t), f10(t), and f100(t), and plot thesefunctions in the figure containing f(t) using a dotted, dash-dotted, and solid curve, respectively. Use a legendto distinguish the four curves in the figure, and a ppropriately label the axes and title the plot.To earn full credit for Prob lem 20:• Turn in the figure along with a copy of your m-file and function file.• Include your name and section number at the top of m-file and function file using the comment symbol %.• Use the title command to appropriately label the figure, for example, Problem 20.• Appropriately label the x and y axes; no credit is given for MATLAB plots whose axes are unlabeled!• Use the MATLAB command gtext to place your name and section name within the
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