Signals and Systems - Electrical Engineering

356 CHAPTER 5: Frequency Analysis: The Fourier Transform

5.16. Periodic signal-equivalent representations

Applying the time and frequency shifts it is possible to get different but equivalent Fourier transforms of

periodic signals. Assume a period of a periodic signalx(t)of periodT 0 isx 1 (t), so that

x(t)=

∑

k

x 1 (t−kT 0 )

and as seen in Chapter 4 the Fourier series coefficients ofx(t)are found asXk=X 1 (jk 0 )/T 0 , so thatx(t)

can also be represented as

x(t)=

1

T 0

∑

k

X 1 (jk 0 )ejk^0 t

(a) Find the Fourier transform of the first expression given above forx(t)using the time-shift property.

(b)Find the Fourier transform of the second expression forx(t)using the frequency-shift property.

(c)Compare the two expressions and comment on your results.

5.17. Modulation property

Consider the raised-cosine pulse

x(t)=[1+cos(πt)](u(t+ 1 )−u(t− 1 ))

(a) Carefully plotx(t).

(b)Find the Fourier transform of the pulsep(t)=u(t+ 1 )−u(t− 1 ).

(c)Use the definition of the pulsep(t)and the modulation property to find the Fourier transform ofx(t)in

terms ofP()=F[p(t)].

5.18. Solution of differential equations

An analog averager is characterized by the relationship

dy(t)

dt

=0.5[x(t)−x(t− 2 )]

wherex(t)is the input andy(t)the output. Ifx(t)=u(t)− 2 u(t− 1 )+u(t− 2 ):

(a) Find the Fourier transform of the outputY().

(b)Findy(t)fromY().

5.19. Generalized AM

Consider the following generalization of amplitude modulation where instead of multiplying by a cosine we

multiply by a periodic signal with harmonic frequencies much higher than those of the message. Suppose

the carrierc(t)is a periodic signal with fundamental frequency 0 , let’s say

c(t)=

∑^6

k= 4

2 cos(k 0 t)

and that the message is a sinusoid of frequency 0 = 2 π, orx(t)=cos( 0 t).

(a) Find the AM signals(t)=x(t)c(t).

(b)Determine the Fourier transformS().

(c)What would be a possible advantage of this generalized AM system? Explain.