EE 350 Problem Set 4 Cover Sheet Fall 2016 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: Submission deadlines: Turn in the written solutions by 4:00 pm on Tuesday October 4 in the homework slot outside 121 EE East. Problem Weight 15 18 16 15 17 20 18 16 19 16 20 15 Total 100 Score The 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 collaborated with on this assignment.
Reading assignment: Lathi Chapter 2, Sections 2.3 and 2.4 Carefully read section 2.4-2 in the text which presents a graphical interpretation of convolution. This approach yields significant insight to the convolution operation and is an important tool that will be used in technical electives such as Introduction to Communications (EE 360), Discrete-Time Systems Analysis (EE 351), and Fundamentals of Digital Signal Processing (EE 453). Problem 15: (18 points) Dynamic second-order linear time-invariant physical systems are often encountered in system analysis and control engineering. Consider a system whose output y(t) is related to the input f(t) by the second-order differential equation d 2 y dt + 2ζω dy 2 n dt + ω2 n y(t) = Kω2 nf(t), (1) where ζ is the dimensionless damping ratio and ω n 0 is the natural frequency. When 0 ζ < 1 the characteristic roots are complex conjugates where the damped frequency ω d is and In this case the homogeneous solution has the form where the coefficients C 1 and C 2 are complex valued. λ 1 = σ + jω d λ 2 = σ jω d, ω d = ω n 1 ζ 2 σ = ζω n. y h (t) = C 1 e ( σ+jωd)t + C 2 e ( σ jωd)t 1. (6 points) Assuming that the numbers θ, A, B, α, ω, and t are real, while c is complex, use Euler s formula, e jθ = cos(θ) + j sin(θ), to derive the following identities: (a) (3 points) (c/2)e (α+jω)t + (c /2)e (α jω)t = c e αt cos(ωt + c) Hint: Use the fact that c + c = 2 Re {c}. (b) (3 points) Acos(ωt) + B sin(ωt) = A 2 + B 2 cos(ωt tan 1 (B/A)) Hint: Use the result from part 1(c). 2. (3 points) Because the coefficients and initial conditions of the ODE in equation (1) are real numbers, the homogeneous solution y h (t) must be real-valued. This forces C 1 = C 2, that is,the undetermined coefficients C 1 and C 2 must be complex conjugates. Using this fact, along with the results in part 1, show that y h (t) = Ae σt cos(ω d t) + Be σt sin(ω d t), where A and B are real-valued coefficients. Find expressions for A and B in terms of C 1. 3. (3 points) Using the result in part 2, solve for the unit-step response when 0 ζ < 1 and show that ( y(t) = K [1 e σt cos ω d t + σ )] sin ω d t u(t). ω d Observe that the real (σ) and imaginary (ω d ) part of the complex roots determine the rate of exponential decay and the frequency of oscillation, respectively. Due to the oscillatory behavior of y(t), the system response is said to be underdamped. 4. (6 points) A LTI system with input f(t) and output y(t) is represented by the ordinary differential equation ÿ + 4ẏ + 5y = 10f(t) Determine the zero-state unit-step response of the system for t 0.
Problem 16: (15 points) 1. Simplify the following expressions: (a) (3 points) [δ(t 1) δ(t + 1) + u(t 1) δ(t 2) u(t + 1)]cos(πt) (b) (3 points) u(t + 1)δ(1 t)e 3t 3+ln(π) + e (2 jπ)t δ(t 1) 2. Evaluate the following integrals: (a) (3 points) h(τ)δ(t τ)dτ (b) (3 points) δ(τ 1)δ(t + 1 τ)dτ (c) (3 points) t 0 e τ δ(τ 1)dτ Problem 17: (20 points) 1. (2 points) Show that f(t) δ(t T) = f(t T). 2. (3 points) If y(t) = f(t) h(t), show that f(t) h(t T) = f(t T) h(t) = y(t T). 3. (3 points) Show that f(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 5. (3 points) Derive the identity f(t) g(t) = f(t) ġ(t) = ċ(t). du(t) dt = δ(t), where u(t) is the unit-step function. In order to obtain this identity, you need to show that the functionals g(t) = du/dt and δ(t) have the same effect on an arbitrary function f(t), that is where T is a real-valued constant parameter. 6. (6 points) Derive the identity f(t)g(t T)dt = f(t) δ(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 to obtain this identity, first show that δ(at)f(t)dt = f(0) a, where the parameter a is a real number that can be either positive or negative. Next, find an expression for δ(t) a f(t)dt in terms of f(0) and a. Use the last two results to obtain the desired identity. Note that because δ( t) = δ(t), δ(t) is an even generalized function.
Problem 18: (16 points) Systems can be represented either by an ODE or an impulse response function. Given either representation, you can find the zero-state response for a given input. For example, consider the RC circuit in Figure 2. Figure 1: RC 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 ẏ + 1 τ y = K τ f. Express the time constant τ and parameter K in terms of R 1, R 2, and C. What is the physical significance of the parameter K? 2. (4 points) Solve the ODE in part 1 to determine the zero-state unit-step response. 3. (4 points) Determine the impulse response function h(t) of the circuit. 4. (4 points) Determine the zero-state unit-step response using the convolution integral. Check your answer against the result obtained in part 2. Problem 19: (16 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. (8 points) f(t) h(t) = u(t + 1) u(t) = e t/2 u(t) 2. (8 points) f(t) = e t h(t) = e t u(t)
Problem 20: (15 points) A future lecture demonstrates that any real-valued periodic signal f(t) with fundamental period T o may be expresses as a superposition of an infinite number of sinusoids, f(t) = a o + a n cos(n ω o t) + b n sin(n ω o t), where a 0, a 1, a 2,..., b 1, b 2... are real-valued constant coefficients given by a o = 1 T o T o f(t)dt a n = 2 T o T o f(t) cos nω o tdt b n = 2 T o T o f(t) sin nω o tdt, and ω o = 2π/T o. As an example, the coefficients for the periodic sawtooth waveform in Figure 2 are a o = 0.5 a n = 0 b n = 1 nπ, ω o = 2π/T o = π. As it not possible to numerically determine f(t) for an infinite number of terms, consider an approximation that utilizes the first N terms of the summation, N N f N (t) = a o + a n cos(n ω o t) + b n sin(n ω o t), If N <, then where e(t) is the approximation error. f(t) = f N (t) + e(t), Figure 2: Periodic sawtooth waveform with a fundamental period of 2 s. Write an m-file that 1. Plots f(t) over the interval 0 t 2 using the equation f(t) = t/2. The time vector must consist of 10,000 points equally spaced between 0 and 2. Plot f(t) using a dashed black curve 2. Write a MATLAB function find fn that determines f N (t) given an integer value of n and the time vector from part (1). The syntax for the calling the function must be Realize the function using a For-Loop. fn = find fn(t,n); 3. Using the MATLAB function find fn, determine vectors representing f 1 (t), f 10 (t), and f 100 (t), and plot these functions in the figure containing f(t) using a dotted, dash-dotted, and solid curve, respectively. Use a legend to distinguish the four curves in the figure, and appropriately label the axes and title the plot.
To earn full credit for Problem 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 figure.