EE321 Sixth Homework Assignment
@ Kefu Xue, Ph.D., Sep. 2001 to Aug. 2007
1FourierTransforms
1. Find Fourier transforms of the following functions
(a) f1(t)=3e−3tu(t−2)
(b) f2(t)=e−2t[u(t)−u(t−2)]
(c) f3(t)=2rect(t−5
2)
(d) f4(t)=rect(t+1
2)+rect(t−1
2)
(e) f5(t)=f4(2t)
(f) f6(t)=2sinc(4πt)
(g) f7(t)=f2(t−2)
(h) Given Fourier transforms of δ(t)is 1, use Fourier transforms duality and frequency
shift properties to prove that Fourier transform of ejω0tis 2πδ(ω−ω0).
(i) f8(t)=rect(t
T)cos(ω0t)
(j) f9(t)=t[u(t−1) −u(t−5)]
2. Find inverse Fourier transform of the following functions:
(a) F1(ω)=rect(2ω−6π
4)
(b) F2(ω)=3e−2a|ω|
(c) F3(ω)=4cos(τω)
(d) F4(ω)=3sinc(5ω)
3. Sketch the following functions:
(a) f(t)=2·sinc(2πt−2
6π)
(b) F(ω)=rect(2ω−6π
4)
4. A system transfer function is H(s)=s2+5s+4 and input signal is x(t)=2cos(20πt)
using the convolution property of the Fourier transform to
(a) find the Fourier transform of the output signal y(t),and
(b) Sketch the magnitude and phase spectrum. (hint: F(ω)·δ(ω−ω0)=F(ω0)·
δ(ω−ω0))
(c) Can you tell the steady state response yss(t)from the above answers and what is
it?
