Signals and Systems - Electrical Engineering

5.7 Convolution and Filtering 331

so thatAX 0 =1, orA= 1 /X 0 =π/2, to get the output to have a unit amplitude. Although the

proposed filter is not realizable, the above provides what needs to be done to obtain a dc source

from a full-wave rectified signal. n

nExample 5.14

Windowingis a time-domain process by which we select part of a signal. This is done by multiplying

the signal by a “window” signal w(t). Consider the rectangular window

w(t)=u(t+1)−u(t−1) 1 > 0

For a given signalx(t), the windowed signal is given by

y(t)=x(t)w(t)

Discuss how windowing relates to the convolution property.

Solution

Windowing is the dual of filtering. In this case, the signaly(t)has the support determined by the

window, or− 1 ≤t≤ 1 , and as such it is zero outside this interval. The window gets rid of parts

of the signal outside its support. The signaly(t)can be written as

y(t)=w(t)x(t)=w(t)

1

2 π

∫∞

−∞

X(ρ)ejρtdρ

=

1

2 π

∫∞

−∞

X(ρ)w(t)ejρtdρ

Considering the integral an infinite summation, the Fourier transform ofy(t)is

Y()=

1

2 π

∫∞

−∞

X(ρ)F[w(t)ejρt]dρ

=

1

2 π

∫∞

−∞

X(ρ)W(−ρ)dρ

using the frequency-shifting property. Thus, we have that the windowing, or multiplication in the

time domain,y(t)=w(t)x(t)givesY()as the convolution ofX()=F[x(t)] and

W()=F[w(t)]=

1

s

[

e^1 s−e−^1 s

]

s=j=

2 sin(1)