Proakis Matlab Comms

8 CHAPTER 1 SIGNALS AND LINEAR SYSTEMS

The MA1LAB script for plotting the discrete spectrum of the signal is given below

% MATIAB .\'cript j(Jr 1/fusmmve Problem f. ChtJpter I n"'[-20·1:20]; x=abs(sinc(n/2)); s!em(ll,x);

When the signal x(t) is described on one period between a and b. as shown in Figure 1.5, and the signal in the interval [a, b] is given in an m-fi!e, the Fourier series coefficients can be obtained using them-file fseries.m given below

_.Mii" function xx=fseries(funfcn,a,b,n,tol,p l,p2,p3) %FSER!ES Return.< rhe Fmmer uriu coefficient.<. % XX = FS£Rf£S(FUNFCN.A.B.N.TOL.Pl,P2.P31 % % %

% % % %

J=Sqrt{-1): args"'[ ];

funfcn = The SIVefl fUnctum. '" un m.-file It cun depend "'' up to three parameter.< pi, pl. und p3. The func:tion 1.< ~:n·o1

over llfl.t perwd ex/et~di!JK {rom 'u' 111 'b" xx = W'CI11r of" /en~th n + I of Frmuu Sene.< Coefficirmt.<. xxO. x.xl. x.xn pl. p2. pJ = pr.m1meu>r.< of fut~{cll

/It/ "' th<' ernw levtl

for nn=1 :nargin-5 args=[args,' , p' .int2str(nn)]:

"' :u-gs=[args, · J ·]:

t=b-a: xx(1)=evnl((' 1 I ( · ,num2str(t).' ) . • quad r fun fen, a, b, tol . [ I · .;u-gs]): for i"'1 :n

newfun=[' exp ( -j ~2 *pi *x" ( · .int2str(il. ·)I I '.num2str(t). · l l . • '.funfcn]. u(i+1)~val([ · 11 ( '.num2str(!),' ) . •quad(newfun, a, b, tol. ( I · .args]):

"'

lllustrative Problem 1.2 [The magnitude and the phase spectra] Determine and plot the discrete magnitude and phase spectra of the periodic signal x (t) with a period equal to 8 and defined as x(t) = A(t) for it! :::: 4.

-+!·"""·'§+ Since the signal is given by an m-file lambda.m, we can choose the interval (a, b]

[ -4, 4] and determine the coefficients. Note that them-file fseries.m determines the Fourier series coefficients for nonnegative values of n, but since here x(t) is real-valued, we have x_11 = x~. In Figure 1.6 the magnitude and the phase spectra of this signal are plotted for a choice of n = 24.

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I .2. Fourier Sencs 9

0/ "-.......// 'ZJ

Figure 5: ...\periodic ~igml

The MATLAB .\Cripl for determinmg 3nd plotting the magnitude and the phase spectra is given below.

_.Mii" % MATUB "·npt jol !1/u.<tmll>e Pn.blem 1. Ch<lpler I echo on functiOn=' lambda· a=-4; b"'4; n=24; tol=0.1; u=fseriesl funwon. a. h. n. toll. .\Xl=xx(n+1·-12); :.:.x l=[conj(xx 1 ).xx]; absxx l =abs(xx l l: pause % Prr.<S WIY key Ill ,,.,. d riot o/ ,,~ m<l).!1.!iUde <peel!'"'' nl=[-n:n]: stem(n l.absu I) tule( ·::he discrete rr.ag:~i t~.:de spec':.r'..:..-:-, · l pha.sexxl =angle(:u l l: pause % Prt.u uny ke_\" to <re 11 plot o/ the rlw.<r stem(n l.phasexx 1) tide(' the discre':.e phase spec':.rum ·)

--~~••••€i•=t·s•ea•¥i;'·'=''§ 1•~• Ulustrative Problem 1.3 [The magnitude and the phase spectra] Determine and plot the magnitude and the phase spectra of a periodic signal with a period equal to 12 that is given by

l ~~ x(t) = __ ,- '

Jiii in the interval I -6. 6]. A plot of this signal IS shown in Figure 1.7.