EECS 2602 Winter Laboratory 3 Fourier series, Fourier transform and Bode Plots in MATLAB

Transcription

1 EECS 6 Winer 7 Laboraory 3 Fourier series, Fourier ransform and Bode Plos in MATLAB Inroducion: The objecives of his lab are o use MATLAB:. To plo periodic signals wih Fourier series represenaion. To obain and plo he oupu response signal wih periodic inpu signal 3. To learn and creae plos for he frequency response of a sysem (i.e. Bode plo) in MATALB There are 5 exercises in his lab. Noe: You are encouraged o complee he pre-lab exercises before you begin his lab. Periodic signals wih Fourier series represenaion Recall ha an arbirary periodic signal wih a fundamenal period T can be expressed as an exponenial Fourier series as follows: ( ) = n= x D n e jnω where he exponenial Fourier series coefficiens D n are calculaed as: D = x( ) n T T e jnω In an LTIC sysem wih impulse response h() as shown in Fig., he oupu response signal is hen given by: jnω y ( ) Dne H ( ω) ω = n ω n= = where H(ω) is he Fourier ransform of h() Hence, he oupu signal y() is anoher periodic signal wih he same fundamenal period T. d h() Fig. y() Laboraory exercise. a) Consider he following periodic signal, which is a full-wave recified sine-wave. Using he Fourier coefficiens ha you deermined from he pre-lab exercise, plo in MATALB ha considers he firs 4 erms in he Fourier series if T = seconds. Plo for [Noe: you should no simply ener erm by erm in MATLAB o complee he plo ]

2 EECS 6 Winer 7 -T T T b) Based on he Fourier coefficiens ha you deermined from he pre-lab exercise, if T = seconds, plo boh he magniude specrum and phase specrum of as a funcion of n for - n in MATLAB. Remember he magniude specrum is he magniude of he exponenial Fourier series coefficiens (i.e. D n ) and he phase specrum is he phase of he exponenial Fourier series coefficiens. Use MATLAB buil-in funcion sem()o plo all he poins in boh plos. Use subplo() o plo he magniude specrum in he upper plo and he phase specrum in he lower plo. Label boh plos. c) is now an inpu signal for an LTIC sysem wih impulse response h ( ), and he Fourier ransform of h() in his sysem is: H( ω) =. Deermine he oupu response signal 3.5 jω y(). Plo boh and he oupu signal y() on he same graph. Plo a few cycles. [Noe: you need o complee all he mahemaical analysis in MATLAB]. Laboraory exercise. a) Consider a symmerical square wave signal x () wih an ampliude of and a fundamenal period of T, he Fourier series is given as: 4 x sin πn ( ) = ( nω ) n=,3,5,... Deermine he exponenial Fourier series of his signal and plo i in MATLAB for -5 n 5. x () - b) If an LTIC sysem wih impulse response h () and he Fourier ransform of h () is given as: H ( ω) =, deermine he oupu response signal y (). Plo boh x () and he oupu jω signal y() on he same graph for -5 n 5.

3 EECS 6 Winer 7 c) Suppose he impulse response h () is modified and is given as: H ( ω) = jω. Plo boh x () and he oupu signal y () on he same graph for -5 n 5. Commen on he differences ha you observe in he new oupu response signal. Creaing Bode plos (i.e. gain and phase responses) in MATLAB The Fourier complex expression (we usually call his he ransfer funcion ) H(ω) provides a complee descripion of he LTIC sysems. In many applicaions, he graphical plos of and versus frequency (ω) are used o analyze he characerisics of he LTIC sysem. The magniude specrum response funcion is also referred o as he gain response of he sysem, while he phase specrum is referred o as he phase response of he sysem. Here, we inroduce Bode plos, where a logarihmic scale is used for he frequency ω axis for boh gain and phase responses of he sysem. In Bode plos, he magniude H( ω) in decibels (dbs) and phase H ( ω) are ploed as funcions of frequency ω using a logarihmic scale. Use of a logarihmic scale, wih base, on he frequency ω axis has he advanage of allowing a wider range of frequencies o be ploed, wih he lower frequencies represened a a higher resoluion. Hence, in a Bode plo, he magniude H( ω) is expressed in dbs as H( ω) log and is ploed as a funcion of log ω. The phase Bode plo hen plos he phase H( ω) as a funcion of log ω. We will spend more ime in he lecures o discuss how o skech and creae a Bode plo. In MATLAB, here is a bode funcion ha is used o skech he Bode plo. he Bode plo of he following Fourier ransfer funcion H(ω): Suppose we wan o creae H ( ω). 3 jω = () The above Fourier ransfer funcion is expressed in MATLAB in erms of he Laplace variable s = jω. In he nex chaper, we will show ha he independen variable s represens he enire complex plane and leads o he generalizaion of he Fourier ransfer funcion ino an alernaive ransfer funcion, known as Laplace ransfer funcion. Hence, subsiuing ino () resuls in: H ( s). 3s = () Given H(s), he Bode plos are obained in MATLAB using he following insrucions: numcoefficien = ; % coefficiens of he numeraor dencoefficien = [-3.]; % coefficiens of he denominaor sysem = f(numcoefficien, dencoefficien)% specify he ransfer funcion bode(sysem, {., }); % skech he Bode plo of he ransfer funcion grid; 3

4 EECS 6 Winer 7 In he above MATLAB insrucions, we have used new buil-in funcions: f()and bode(). The funcion f() specifies he LTIC sysem H(s) in erms of he coefficiens of he polynomials of s in he numeraor and denominaor. The buil-in funcion bode() skeches he Bode plos. I acceps inpu argumens. The firs inpu argumen sys is used o represen he LTIC sysem, while he second inpu argumen {., } specifies he frequency range,.radians/sec o radians/sec, used o skech he Bode plos. When he above insrucions are execued in MATLAB, you will see he following from he MATLAB Command window, as well as he Bode plos of sysem. sysem = s -. Coninuous-ime funcion. ransfer Magniude (db) Phase (deg) Bode Diagram Frequency (rad/s) Noe ha he second inpu argumen in bode()can be omied and he bode insrucion will reurn a Bode plo wih auomaically adjused frequency range deermined by MATLAB. You can also plo muliple Bode plos on he same graph using he following srucure: bode(sysem, sysem,, sysemn); Laboraory exercise 3. a) Consider he wo impulse response funcions again in exercise b) and c), give he Bode plos of boh funcions and plo hem on he same graph. Label your plos. b) Now, wih he help of he Bode plos, explain he differences ha you observed in y() in exercises b) and c). c) Plo he oupu response signal y() if he inpu signal is given by he square wave signal in exercise, and he impulse response is given by he cascaded connecion of h b () and h c (), ha is: h b () h c () y() where ( ω) and ( ω) H b = jω H b = jω 4

5 EECS 6 Winer 7 Laboraory exercise 4. Consider he circui in Fig., which is an LTIC sysem ha can be represened by a s order differenial equaion relaing and y(). If R =, R L =, C = µf, based on he Fourier ransfer funcion ha you deermined from he pre-lab exercise, obain he Bode plos of his LTIC sysem (i.e. y(s)/x(s)) from MATLAB. R C R L y() Fig. Laboraory exercise 5. a) Consider he circui in Fig. 3, which is an LTIC sysem ha can be represened by a nd order differenial equaion relaing and y(). If R L = Ω, L = mh, C = µf, based on he Fourier ransfer funcion ha you deermined from he pre-lab exercise, obain he Bode plos of his LTIC sysem (i.e. y(s)/x(s)) from MATLAB. b) Now R L is increased o Ω, wih L and C remain he same, obain he Bode plos of his LTIC sysem again from MATLAB. Wha did you observe? Explain. L C R y() Fig. 3 Submiing your M-files and published repor: You can submi your files and repor o eecs6@gmail.com or hardcopy repor. Files o be submied: 5

6 EECS 6 Winer 7. Main M-file: he main file should conain your soluions in exercise 5.. Published repor of he main M-file in pdf (in your M-file, selec he PUBLISH Tab and click ) 6