ELG3 1 2 5 Signal and System Analysis Lab5: Fourier series: - - PowerPoint PPT Presentation

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elg3 1 2 5 signal and system analysis lab5 fourier series
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ELG3 1 2 5 Signal and System Analysis Lab5: Fourier series: - - PowerPoint PPT Presentation

Feb 28, 2023 106 likes •234 views

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.

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School of Information Technology and Engineering (SITE) Lab5: Fourier series: Synthesis of signals TA: Jungang Liu Fall 2010 ELG3 1 2 5 Signal and System Analysis

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Outline

  • 1. Continuous-time Fourier series and its truncated

version.

  • 2. Discrete-time Fourier series.
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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Continuous-time Fourier Series

  • Fourier series representation

Synthesis equation: Analysis equation:

∫ =

− T t jk k

e t x T a ) ( 1

ω

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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

  • f complex exponentials (or from cosines) weighted by

the Fourier series coefficients can also be written by:

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

  • 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. Continuous-time Fourier Series

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Continuous-time Fourier Series Truncated version Note:

  • Na odd;
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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Discrete-time Fourier Series Fourier series representation Synthesis equation Analysis equation

∑ =

> =< − N n n jk k

e n x N a ] [ 1

ω

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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.

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Example (Fourier series for a square wave)

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Example (cont) Note: In Matlab, use iteration to do the synthesis.

  • 4
  • 3
  • 2
  • 1

1 2 3 4

.5 .5 1 1 .5

  • 4
  • 3
  • 2
  • 1

1 2 3 4

.5 .5 1 1 .5

  • 4
  • 3
  • 2
  • 1

1 2 3 4

.5 .5 1 1 .5

T=4;T1=0.5;omega0=2*pi/T; t=-4:0.001:4;a_0=2*T1/T; x_approx=ones(1,length(t))*a_0; N_a=9; x_approx1=x_approx; for k=1:(N_a-1)/2 a_k1=sin(k*omega0*T1)/(k*pi); x_approx1=x_approx1+2*abs(a_k1)*... cos(k*omega0.*t+angle(a_k1)); end %Please write codes for %N_a=27 and N_a=271;then subplot(3,1,1); plot(t,x_approx1),grid subplot(3,1,2); plot(t,x_approx2),grid subplot(3,1,3); plot(t,x_approx3),grid 9 27 271 N_a=

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ELG3125 Signal and System Analysis Fall 2010

School of Information Technology and Engineering

Enjoy Fourier Series……