EE 350 Problem Set 4 Cover Sheet Spring 2023 Last Name Print First Name Print ID number Last 4 digits Section Submission deadlines 21 Please submit your solutions as a PDF le to the EE 350 CANVAS page by 11 59 pm on Tuesday February Problem Weight Score 16 17 18 19 20 Total 20 20 20 20 20 100 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 signi cant 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 16 20 points 1 Simplify the following expressions a 4 points t 1 t 1 u t 1 t 2 u t 1 cos t b 4 points u t 1 1 t e3t 3 ln e 2 t t 1 2 Evaluate the following integrals a 4 points R b 4 points R c 4 points R t h t d 1 t 1 d 0 e 1 d Problem 17 20 points 1 2 points Show that f t 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 f t g t f t g t c t 5 3 points Derive the identity 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 t have the same e ect on an arbitrary function f t that is du t dt t Z f t g t T dt f T at t 1 a Z at f t dt f 0 a Z t a f t dt where T is a real valued constant parameter 6 6 points Derive the identity That is the generalized functions at and t a have the same e ect on a given function f t In order to obtain this identity rst show that where the parameter a is a real number that can be either positive or negative Next nd an expression for in terms of f 0 and a Use the last two results to obtain the desired identity Note that because t t t is an even generalized function Problem 18 20 points Systems can be represented either by an ODE or an impulse response function Given either representation you can nd 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 5 points Derive the ODE representation of the system and show that it can be expressed as y y f 1 K Express the time constant and parameter K in terms of R1 R2 and C What is the physical signi cance of the parameter K 2 5 points Solve the ODE in part 1 to determine the zero state unit step response 3 5 points Determine the impulse response function h t of the circuit 4 5 points Determine the zero state unit step response using the convolution integral Check your answer against the result obtained in part 2 Using the graphical convolution method discussed in section 2 4 2 of the text and lecture nd and sketch y t f t h t for the following signals Problem 19 20 points 1 10 points 2 10 points f t u t 1 u t h t e t 2u t 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 fundamental period To may be expresses as a superposition of an in nite number of sinusoids f t ao an cos n ot bn sin n ot Xn 1 Xn 1 where a0 a1 a2 b1 b2 are real valued constant coe cients given by ao f t dt an f t cos n otdt bn f t sin n otdt 1 2 To ZTo To ZTo To ZTo 2 ao 0 5 an 0 bn 1 n and o 2 To As an example the coe cients for the periodic sawtooth waveform in Figure 2 are o 2 To As it not possible to numerically determine f t for an in nite number of terms consider an approximation that utilizes the rst N terms of the summation fN t ao an cos n ot bn sin n ot N Xn 1 N Xn 1 If N then where e t is the approximation error f t fN t e t Figure 2 Periodic sawtooth waveform with a fundamental period of 2 s Write an m le that 1 Plots f t over the interval 0 t 2 using the equation 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 nd fN that determines fN t given an integer value of n and the time vector from part 1 The syntax for the calling the function must be f t t 2 fn nd fn t N Realize the function using a For Loop 3 Using the MATLAB function nd fN determine vectors representing f1 t f10 t and f100 t and plot these functions in the gure containing f t using a dotted dash dotted and solid curve respectively Use a legend to distinguish the four curves in the gure and appropriately label the axes and title the plot To earn full credit for Problem 20 Include your name and section number at the top of m le and function le using the comment symbol Use the title command to appropriately label the gure 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 gure Use the MATLAB Publish tool to generate a PDF le that shows your m le and plot Append this PDF le to the PDF le containing your written solutions
View Full Document
Unlocking...