ELG3 1 2 5 Signal and System Analysis Lab5: Fourier series: Synthesis of signals TA: Jungang Liu Fall 2010 School of Information Technology and Engineering (SITE)
Outline 1. Continuous-time Fourier series and its truncated version. 2. Discrete-time Fourier series. ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Continuous-time Fourier Series • Fourier series representation Synthesis equation: Analysis equation: 1 − ω jk t = ∫ 0 ( ) a x t e k T T ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Continuous-time Fourier Series � Synthesis equation The synthesis or reconstruction of signal x ( t ) from a sum of complex exponentials (or from cosines) weighted by the Fourier series coefficients can also be written by: ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Continuous-time Fourier Series • Truncated version If instead of using an infinite amount of terms, the summation is truncated to Na terms (with Na odd here), we then obtain the following approximation. ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Continuous-time Fourier Series � Truncated version Note: � Na odd; � ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Discrete-time Fourier Series � Fourier series representation Synthesis equation Analysis equation 1 − ω jk n = ∑ 0 a x [ n ] e k N =< > n N ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Discrete-time Fourier Series � No need for truncated version because N is finite already. ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Example (Fourier series for a square wave) ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Example (cont) T=4;T1=0.5;omega0=2*pi/T; 1 .5 N_a= t=-4:0.001:4;a_0=2*T1/T; 1 9 x_approx=ones(1,length(t))*a_0; 0 .5 N_a=9; 0 x_approx1=x_approx; -0 .5 -4 -3 -2 -1 0 1 2 3 4 for k=1:(N_a-1)/2 1 .5 a_k1=sin(k*omega0*T1)/(k*pi); 1 x_approx1=x_approx1+2*abs(a_k1)*... 27 0 .5 cos(k*omega0.*t+angle(a_k1)); 0 end -0 .5 -4 -3 -2 -1 0 1 2 3 4 %Please write codes for %N_a=27 and N_a=271;then 1 .5 subplot(3,1,1); plot(t,x_approx1),grid 1 subplot(3,1,2); plot(t,x_approx2),grid 0 .5 271 subplot(3,1,3); plot(t,x_approx3),grid 0 -0 .5 -4 -3 -2 -1 0 1 2 3 4 Note: In Matlab, use iteration to do the synthesis. ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
Enjoy Fourier Series…… ELG3125 Signal and System Analysis Fall 2010 School of Information Technology and Engineering
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