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Signals and Systems (Lab) Resource Person : Hafiz Muhammad Ijaz

Signals and Systems (Lab) Resource Person : Hafiz Muhammad Ijaz

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Signals and Systems (Lab) Resource Person : Hafiz Muhammad Ijaz

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  1. Signals and Systems (Lab) Resource Person : Hafiz Muhammad Ijaz COMSATS Institute of Information Technology Lahore Campus

  2. EXPERIMENT # 8 Fourier Transform and its Properties.

  3. In Previous Lab… • Introduction to Fourier Series • Complex Exponential Fourier Series Representation • Trigonometric Fourier series Representation • Properties of Fourier series • Linearity • Time Shifting • Time Reversal • Signal Multiplication • Parseval’s Identity

  4. In this Lab… • Introduction to Fourier Transform • How to compute Fourier Transform and Inverse Fourier Transform using MATLAB? • Implementation of Fourier Transform Pairs • Properties of Fourier Transform • Linearity • Time Shifting • Frequency Shifting • Scaling in Time and Frequency • Time Reversal • Fourier Transform of the Even and Odd Part of a Signal • Convolution in Time and Frequency • Parseval’s Theorem

  5. Introduction to Fourier Transform • The mathematical expression of Fourier transform is • The mathematical expression of Inverse Fourier transform is • The Fourier transform of a signal is called (frequency) spectrum.

  6. Example: • Plot the Fourier transform of the continuous time signal • x(t) = cos(t)

  7. Solution: syms t w x=cos(t) X=fourier (x,w) w1=[-4:0.05:4] X=subs(X,w,w1) for i=1:length(X) if X(i) == inf X(i) = 1 end end plot(w1,X) legend ('F[cos(t)]')

  8. Example: • Plot the Fourier transform of the signal • x(t) = sin(πt) / (πt)

  9. Example: Compute the Fourier transform of the function Solution: syms t w x=exp(-t^2); fourier(x) Xf=int(x*exp(-j*w*t),t,-inf,inf) xt=ifourier(Xf,t) • ans =pi^(1/2)/exp(w^2/4) • Xf =pi^(1/2)/exp(w^2/4) • xt= 1/exp(t^2)

  10. Fourier Transform Pairs • Verify the Fourier transform pair • where is a rectangular pulse of duration, given by

  11. Solution: syms t w T x=heaviside(t+T/2)-heaviside(t-T/2); xx=subs(x,T,4); subplot(2,1,1) ezplot(xx,[ -4 4]) legend('x(t)') x1=fourier(x,w) ww=[-10:.1:-.1 .1:.1:10]; X=subs(x1,w,ww) X=subs(X,T,4); subplot(2,1,2) plot(ww,X) xlabel('\Omega rad/s') legend('X(\Omega)')

  12. Properties of Fourier Transform

  13. Time and Frequency Shifting • Time shifting Property can be written as • Frequency shifting Property is given as

  14. Solution syms t w x=cos(t); t0=2; xt0=cos(t-t0); Left=fourier(xt0,w) X=fourier(x,w); Right=exp(-j*w*t0)*X • syms t w • x=cos(t); • w0=2; • Le=exp(j*w0*t)*x; • Left=fourier(Le,w) • X=fourier(x,w); • Right=subs(X,w,w-w0)

  15. Scaling in Time and Frequency • Scaling in Time domain is given as • Scaling in frequency can be written as

  16. Solution syms t w b=3; x=heaviside(t+1)-heaviside(t-1); ezplot(x,[-2 2]); legend('x(t)')

  17. FT of x(t)… X=fourier(x,w); ezplot(X,[-40 40]); legend('X(\Omega)') xlabel('\Omega')

  18. Signal x(bt) , b=3 xb=subs(x,t,b*t); ezplot(xb, [-2 2]); legend('x(bt), b=3');

  19. FT of x(bt) Xb=fourier(xb,w); ezplot(Xb, [-40 40]) legend('F(x(bt))') xlabel('\Omega')

  20. X(bw)… Ri=subs(X,w,w/b); Right=(1/abs(b))*Ri; ezplot(Right,[-40 40]); legend('(1/|b|)*X(\Omega/b)') xlabel('\Omega')

  21. Time Reversal Verify the time reversal property for the signal x(t)=t u(t)

  22. Solution syms t w x=t*heaviside(t); X=fourier(x,w) ; Right=subs(X,w,-w) x_t=subs(x,t,-t); Left=fourier(x_t,w) Right = - 1/w^2 - pi*i*dirac(w, 1) Left = - 1/w^2 + pi*i*dirac(-w, 1)

  23. Duality • Satisfy the Duality property for the signal mentioned below.

  24. Solution syms t w x=exp(-t)*heaviside(t); X=fourier(x) Xt=subs(X,w,t) Left=fourier(Xt) x_w=subs(x,t,-w); Right=2*pi*x_w Left = 2*pi*heaviside(-w)*exp(w) Right = 2*pi*heaviside(-w)*exp(w)

  25. Differentiation in Time and Frequency • For the signal x(t) • Differentiation in time domain is given as • Differentiation in Frequency domain can be written as

  26. Solution syms t w x=exp(-3*t)*heaviside(t); der=diff(x,t); Left=fourier(der,w) X=fourier(x,w); Right=j*w*X Left=fourier(t*x,w) der=diff(X,w); Right=j*der Left =1 - 3/(i*w + 3) Right =(i*w)/(i*w + 3) Left =1/(i*w + 3)^2 Right =1/(i*w + 3)^2

  27. Integration • For a given signal x(t) • Integration property can be written as • Satisfy the integration property for

  28. Solution syms t r w x=exp(r)*heaviside(-r)+exp(-r)*heaviside(r); integ=int(x,r,-inf,t); Left=fourier(integ,w) X=fourier(x,w); X0=subs(X,w,0); Right=(1/(j*w))*X+pi*X0*dirac(w) • Left = 2*pi*dirac(w) + ((1/(- 1 + w*i) - 1/(1 + w*i))*i)/w • Right = 2*pi*dirac(w) + ((1/(- 1 + w*i) - 1/(1 + w*i))*i)/w

  29. Convolution in Time and Frequency

  30. Solution syms t w x=exp(-t^2); Et=int((abs(x))^2,t,-inf,inf) ; eval(Et) X=fourier(x,w); Ew=(1/(2*pi))*int((abs(X))^2,w,-inf,inf); eval(Ew) ans =1.2533 ans =1.2533

  31. Parseval’s Theorem • Parsvel’s Identity can be written as • Satisfy the Parsvel’s Identity using input signal

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