Experiment No. 7 Aim: To determine frequency domain representation of CT an d DT periodic signals. Sources/Software’s required: MATLAB Theory: Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform DFT! is the family member used "ith digiti#ed signals discrete$time signal!. A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these t"o features generates the four categories, described belo".
%eriodic$Continuous& 'ere the e(amples include& sine "a)es, square "a)es, and any "a)eform that repeats itself in a regular pattern from negati)e to positi)e infinity. This )ersion of the Fourier transform is called the Fourier series. Aperiodic$C Aperiodic$Contin ontinuous& uous& This includes, includes, for e(ample, e(ample, decaying decaying e(ponentials e(ponentials and the *aussian *aussian cur)e. These signals e(tend to both positi)e and negati)e infinity "ithout repeating in a periodic pattern. The Fourier Transform Transform for this type of signal is simply simply called the Fourier Transform. Transform. Aperiodic$Discrete& These signals are only defined at discrete points bet"een positi)e and negati)e infinity, and do not repeat themsel)es in a periodic fashion. This type of Fourier transform is called the Discrete Time Fourier Transform. %eriodic$Discrete& These are discrete signals that repeat themsel)es in a periodic fashion from negati)e to positi)e infinity. This class of Fourier Transform is sometimes called the Discrete Time Fourier +eries. ----------------------------------------------------------------------------------------------------------------N !A" E#E$%SE
uestion syms k t wo=1; ak=(-1/2)^abs(k); yk=ak.*exp(j.*k.*wo.*t); z=symsum(yk,k,-10,10); ezplot(z);
uestion syms k n; wo=p/!; ak=(1/").*(nt((1)*exp(-j*k*wo*t),t,2,!)#nt((-1)*exp(-j*k*wo*t),t,$,")); %&o 'onet symbol' om nto aay/matx om z=subs(ak,-20+20); subplot(2,1,1); stem(-20+20,abs(z)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(z)); xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
uestion / syms k n; wo=p./23; ak=1 # 'os($.*p.*k/4) # sn(3.*p.*k/23); yk=ak.*exp(j.*k.*wo.*n); z=symsum(yk,k,0,20); %&o 'onet symbol' om nto aay/matx om z1=subs(z,-20+20); subplot(2,1,1); stem(-20+20,abs(z1)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(z1)); xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
&'ST !A" E#E$%SE
uestion syms k n; wo=p./!"; xn=1 # 'os(!.*p.*n/12) # sn(2.*p.*n/!"); ak=xn.*exp(-j.*k.*wo.*n); z=(1/32).*symsum(ak,n,0,20); %&o 'onet symbol' om nto aay/matx om ak=subs(z,-20+20); subplot(2,1,1); stem(-20+20,abs(ak)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(ak)); xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
uestion a) syms k n; wo=p./2; ak=(1/5)*nt((1)*exp(-j*k*wo*t),t,-1,1);
%&o 'onet symbol' om nto aay/matx om z=subs(ak,-20+20); subplot(2,1,1); stem(-20+20,abs(z)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(z)); xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
b) syms k n; wo=p; ak=(1/2)*nt((sn(p*t))*exp(-j*k*wo*t),t,0,2); %&o 'onet symbol' om nto aay/matx om z=subs(ak,-20+20); subplot(2,1,1); stem(-20+20,abs(z)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(z));
xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
') syms k n; wo=p./2; ak=(1/5)*nt(sn(p*t)*exp(-j*k*wo*t),t,-1,1); %&o 'onet symbol' om nto aay/matx om z=subs(ak,-20+20); subplot(2,1,1); stem(-20+20,abs(z)); xlabel(k); ylabel(abs(k)); ttle(mpltue pe'tum); subplot(2,1,2); stem(-20+20,anle(z)); xlabel(k); ylabel(anle(k)); ttle(ase pe'tum);
!EA$NN( ')T%'*ES:
This e(periment ga)e us programming techniques to sol)e problems based on determination of frequency domain representation in CT 0 DT periodic signals.. 1e also learnt ho" to estimate errors using basic calculus concepts and results, as "ell as "riting programs to implement the numerical methods "ith a soft"are pac2age, Matlab.
