EE3210 Lab 3: Periodic Signal Representation by Fourier Series

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1 City University of Hong Kong Department of Electronic Engineering EE321 Lab 3: Periodic Signal Representation by Fourier Series Prelab: Read the Background section. Complete Section 2.2(b), which asks you to derive the exponential Fourier series coefficients for x(t) defined in (6). Verification: The Warm-Up section must be completed during your assigned lab time. The steps marked Instructor Verification must also be signed off during the lab time. When you have completed a step that requires verification, simply demonstrate the step to the instructor. Lab Report: It is only necessary to turn in a report on the Experiment section with graphs and explanations. 1. Background The Fourier Series representation applies to periodic signals. The Fourier synthesis equation for a periodic signal x(t) = x(t + T ) is where 2π / T x t = D e jnωt ( ) n n=, (1) ω = is the fundamental frequency. To determine the Fourier series coefficients from a time-domain formula for the signal over one period, we must evaluate the analysis integral for every integer value of n: D n = T 1 T jnω t x( t) e dt, (2) where T = 2π / ω is the fundamental period. The power of a real periodic signal is defined as 1 T 2 P x = x ( t) dt. (3) T Parseval s theorem says that it can be expressed in terms of its exponential Fourier series coefficients as n= 2 P x = D n. (4)

2 2. Warm-up 2.1 Getting Familiar with fplot and stem MATLAB provides a function called fplot, which can be used to plot a function. Try the following: fplot( x.^2,[,1]) Type help fplot to learn more on it. Consider the following function: t t < 1 x ( t) = 2 t 1 t < 2 (5) otherwise This function can be represented as an inline function as follows: x = inline( t.*((t>=)&(t<1))+(2-t).*((t>=1)&(t<2)), t ); (a) Plot x(t) over the interval -1 t 3 using fplot. MATLAB provides a function called stem, which can be used to plot discrete sequence. Type help stem in MATLAB to learn how to use it. Perform the following exercises to gain more experience on it. (b) Type the following in MATLAB: x = 1:1:1; stem(x) (c) Then type the following in MATLAB: y = x.^2; stem(y) (d) Then type the following in MATLAB: t = [ ]; stem(t, y(t)) 2.2 Plotting Exponential Fourier Spectra Consider the following periodic signal x(t) defined over one period to be: = t / 2 x( t) e for t < π. (6) (a) Plot x(t) over the time interval t < π using the function fplot().

3 (b) Write an M-file for plotting x(t) over any specified time interval. The first few statements of the M-file should look like function exp_plot(lim) % usage: % lim = [xmin xmax] controls the x-axis limits. % lim = [xmin xmax ymin ymax] controls also the y-axis limits. Hint: Use the mod() function which is defined as a remainder function that gives a positive number. Type help mod in MATLAB to get more information. Note that it is possible to plot x(t) in a single line. Use the function exp_plot() to plot x(t) over the time interval 5 t 5. Set the y-axis limits to and 2. (c) Derive analytically the exponential Fourier series coefficients for x(t). (d) Plot its amplitude spectrum and phase spectrum as a function of ω in a two-panel subplot. Notice that ω = nω. Use the MATLAB function stem to plot the spectra over the range 1 ω 1. (e) Which spectrum is an odd function of ω? Which one is an even function? (f) Type the following in MATLAB: quad( exp(-x/2).^2,, pi) / pi What is the answer? What is its physical meaning? Can the answer be obtained by examining the coefficients of the exponential Fourier series? How? (g) Use two different methods to find the power of x(t) numerically. Make sure that you get the same result.

4 3. Experiment 3.1 Fourier Synthesis: Sum of Complex Exponentials In this project, we are going to determine Fourier series representations for periodic waveforms, synthesize the signals, and then plot them. In general, the limits on the sum in (1) are infinite, but for our computational purposes, we must restrict the limits to be a finite number N, which then gives the 2N+1 term approximation: N jn t x( t) = D n e ω. (7) n= N Write a MATLAB function called syn_fourier that implements the above computation. When we use it for Fourier synthesis, the vector of frequencies will consist of frequencies that are all integer multiples of the fundamental frequency. In addition, we must include both the positive and negative frequency components. Therefore, the input vector of complex amplitudes dk will be a vector of length 2N+1 containing the Fourier coefficients in the order { a a, K, a, a, a,, a } N, ( N 1) 1 1 K N and the vector fk should contain the harmonic frequencies { Nf, ( N 1) f, K, f,, f, K Nf }., The first few statements of syn_fourier.m are shown below: function xx = syn_fourier(tt, dk, fk) % SYN_FOURIER Function to synthesize a sum of complex exponetials % over the time range given by tt % usage: % xx = syn_fourier(tt, dk, fk) % tt = vector of time instants % dk = vector of complex Fourier coefficients % fk = vector of frequencies (having the same length as ak) % xx = vector of synthesized signal values. You may use the waveform in the previous example to debug your function. For example, you can make a plot by calling the function via: fplot( syn_fourier, [T1, T2], [], [] [], dk, fk).

5 3.2 Plotting a Periodic Function Consider the 2π-periodic function given by t / A t < A x ( t) = 1 A t < π (8) π t < 2π As A, x(t) approaches a square wave; as A π, x(t) approaches a type of sawtooth wave. This function can be represented in MATLAB as follows: (a) Represent x(t) as an inline function. Then plot it over the interval 1 x 7 using fplot. Let A = π / 2. Set the y-axis limits to.2 and 1.2. (b) Derive analytically the exponential Fourier series coefficients for x(t). (c) (d) Generate a two-panel subplot. Let A = π / 2. Plot x(t) and x 2 (t) together on one of the panels. Then plot x(t) and x 1 (t) together on the second panel. In both subplots, set the y-axis limits to.2 and 1.2. Does the Gibbs phenomenon occur? Estimate the percentage of overshoot for these two cases. Can it be reduced by increasing N? Repeat part (c) using A = π /64. Do you see the same phenomenon as that in part (c)? Explain your observation.

6 Lab 3 Instructor Verification Sheet Staple this page to the end of your lab report. Name: _ Date: Part 2.1 (a) Make a plot using fplot. Part 2.2 (b) Make a plot of x(t). Part 2.2 (c) Write down the exponential Fourier series coefficients for x(t) in the space below: Part 2.2 (d) Plot the amplitude and phase spectrum for x(t). Part 2.2 (g) Write down the power of x(t) here: