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

EXPERIMENT # 8 Fourier Transform and its Properties.

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

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

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.

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

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 plot(w1,X) legend ('F[cos(t)]')

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

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)

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

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)')

Properties of Fourier Transform

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

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)

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

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

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

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

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

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')

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

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)

Duality Satisfy the Duality property for the signal mentioned below.

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)

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

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

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

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*dir ac(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

Convolution in Time and Frequency

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

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