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ECE 102, Fall 2023 Homework #4 Department of Electrical and Computer Engineering Prof. Jonathan C. Kao University of California, Los Angeles TAs: Yang, Bruce, Shreyas Due Monday, 6 November 2023, by 11:59 pm to Gradescope Covers material up to lecture 9. 100 points total 1. (15 points) LTI systems Consider the following system: The system takes as input x ( t ), it first multiplies the input with e t , then sends it through an LTI system. The output of the LTI system gets multiplied by e − t to form the output y ( t ). (a) (5 points) Show that we can write y ( t ) as follows: y ( t ) = ( e t x ( t ) ) ∗ h ( t ) e − t (1) (b) (5 points) Use the definition of convolution to show that (1) can be equivalently written as: y ( t ) = Z ∞ −∞ h ′ ( τ ) x ( t − τ ) dτ (2) where h ′ ( t ) is a function to define in terms of h ( t ). (c) (5 points) Equation (2) represents a description of the equivalent system that maps x ( t ) to y ( t ). Show using (2) that the equivalent system is LTI and determine its impulse response h eq ( t ) in terms of h ( t ). 2. (10 points) Eigenfunctions and LTI systems Determine if the following signals are eigenfunctions of LTI systems (a) (5 points) f ( t ) = t 2 (b) (5 points) f ( t ) = e jwt u ( t ) 3. (28 points) Fourier Series (a) (18 points) Find the Fourier series coefficients for each of the following periodic signals: 1

i. (9 points) f ( t ) = sin(7 πt ) + 1 3 cos(4 πt ) ii. (9 points) f ( t ) is a periodic signal with period T = 1 s, where one period of the signal is defined as e − t for 0 < t < 1 s, as shown below. (b) (10 points) Suppose you have two periodic signals x ( t ) and y ( t ), of periods T 1 and T 2 respectively. Let x k and y k be the Fourier series coefficients of x ( t ) and y ( t ). i. (5 points) If T 1 = T 2 , express the Fourier series coefficients of z ( t ) = 3 x ( t ) + 2 y ( t ) in terms of x k and y k . ii. (5 points) If T 1 = 2 T 2 , express the Fourier series coefficients of w ( t ) = x ( t ) + y ( t ) in terms of x k and y k . 4. (34 points) Fourier series of transformation of signals (a) (15 points) Suppose that f ( t ) is a periodic signal with period T 0 , with the following Fourier series: f ( t ) = ∞ X k = −∞ c k e jkω 0 t Determine the period of each of the following signals, then express its Fourier series in terms of c k : i. (5 points) g ( t ) = 3 f ( t ) ii. (5 points) g ( t ) = f ( − at ) where a is a positive real number. iii. (5 points) g ( t ) = f ( t − t 0 ) (b) (5 points) Given two periodic signals and their corresponding Fourier series representa- tion as follows: x 1 ( t ) = 100 X k = − 100 cos ( kπ ) e jk 2 π 50 t x 2 ( t ) = 100 X k = − 100 jsin ( kπ 2 ) e jk 2 π 50 t Identify whether the signals is/are even. (c) (14 points) Suppose x ( t ) is periodic with period T and is specified in the interval 0 < t < T/ 4 as shown in figure 1. 2

Figure 1: x ( t ) in the interval 0 < t < T/ 4 Sketch x ( t ) in the interval 0 < t < T if i. (7 points) the Fourier series has only odd harmonics and x ( t ) is an even function ii. (7 points) the Fourier series has only odd harmonics and x ( t ) is an odd function 5. (13 points) Python (a) (6 points) Task 1 Write an python function that takes a set of Fourier series coefficients, a fundamental frequency, and a vector of output times, and computes the truncated Fourier series evaluated at these times. The declaration and help for the python file might be: def myfs (Dn,omega0,t): ''' Evaluates the truncated Fourier Series at times t Dn -- vector of Fourier series coefficients omega0 -- fundamental frequency t -- vector of times for evaluation ''' fn = myfs(Dn, omega0, t) ''' fn -- truncated Fourier series evaluated at t} ''' The output of the python function should be f N ( t ) = Σ N n = − N D n e jω 0 nt The length of the vector Dn should be 2 N + 1. You will need to calculate N from the length of Dn. (b) (7 points) Task 2 Verify the output of your routine by checking the Fourier series coefficients for the sawtooth waveform. The sawtooth signal is given by f(t) = t mod 1 described in the 3

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