PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

344 Practical MATLAB® Applications for Engineers

F8 =

exp(i*w)*(pi*Dirac(w)-i/w)-exp(-i*w)*(pi*Dirac(w)-i/w)

>> F9 = fourier(sym(‘Heaviside(t)’),t,w) % part (i)

F9 =

pi*Dirac(w)-i/w

R.4.72 The MATLAB function f = ifourier(F), where F is a symbolic object with an inde-
pendent variable w, returns the inverse FT of F, which is a function of x, denoted
by F(w) → f(x). The MATLAB command f = ifourier(F, w, t) returns the function f(t),
given the symbolic function F(w).

R.4.73 For example, use MATLAB to obtain the inverse FTs of the following functions:

a. F1 = pi * Dirac(w − 3) + pi * Dirac(w + 3), default version

b. F2 = 1/2 * 2^(1/2) * pi * (Dirac(w − 3) + i * Dirac(w − 3) + Dirac(w + 3) − i * Dirac(w + 3)

(w → t), default and factor

c. F3 = −i * pi * Dirac(w − 3) + i * pi * Dirac(w + 3), (w → t)

d. F4 = i * pi * (Heaviside(−w) − Heaviside(w)); (w → t)

e. F5 = 1/(1 + i * w), (w → t)

f. F6 = 1/(1 + i * w)^2, (w → t)

g. F7 = 2/(1 + w^2), (w → t)

h. F8 = exp(i * w) * (pi * Dirac(w) −i/w) − exp(−i * w) * (pi * Dirac(w) −i/w), (w → t) and

simplify

i. F9 = pi * Dirac(w) −i/w, (w → t) and simplify.

MATLAB Solution

>> sym t w

>> F1 = pi*Dirac(w-3)+pi*Dirac(w+3);

>> f1 = ifourier(F1)

f1 =

1/2*exp(3*i*x)+1/2*exp(-3*i*x)

>> F2 =1/2*2^(1/2)*pi*(Dirac(w-3)+i*Dirac(w-3)+Dirac(w+3)- i*Dirac(w+3));

>> f2 = ifourier(F2,t,w)

f2 =

1/2*2^(1/2)*pi*(Dirac(w-3)+i*Dirac(w-3)+Dirac(w+3)-i*Dirac(w+3))

*Dirac(w)

>> factor(f2) % observe that f2 is a function of x

ans =

-1/4*2^(1/2)*(-exp(3*i*x)-i*exp(3*i*x)-exp(- 3*i*x)+i*exp(-3*i*x))

>> F3 = -i*pi*Dirac(w-3)+i*pi*Dirac(w+3);

>> f3 = ifourier(F3,w,t)

f3 =

1/2*i*(-exp(3*i*t)+exp(-3*i*t))