Signals and Systems - Electrical Engineering

170 C H A P T E R 3: The Laplace Transform

nExample 3.1

A problem in wireless communications is the so-calledmultipath effecton the transmitted message.

Consider the channel between the transmitter and the receiver as a system like the one depicted

in Figure 3.2. The sent messagex(t)does not necessarily go from the transmitter to the receiver

directly (line of sight) but it may take different paths, each with different length so that the signal

in each path is attenuated and delayed differently.^2 At the receiver, these delayed and attenuated

signals are added, causing a fading effect—given the different phases of the incoming signals their

addition at the receiver results in a weak or a strong signal, thus giving the sensation of the message

fading back and forth. Ifx(t)is the message sent from the transmitter, and the channel hasN

different paths with attenuation factors{αi}and corresponding delays{ti},i=0,...,N, use the

eigenfunction property to find the system function of the channel causing the multipath effect.

FIGURE 3.2

Block diagram of a wireless

communication channel causing a

multipath effect on the sent message

x(t). The messagex(t)is delayed

and attenuated when sent over

N+ 1 paths. The effect is similar to

that of an echo in acoustic signals.

+

α 0

α 1

αN

Delay t 0

Delay t 1

Delay tN

x(t) y(t)

Solution

The output of the channel or multipath system in Figure 3.2 can be written as

y(t)=α 0 x(t−t 0 )+α 1 x(t−t 1 )+···+αNx(t−tN) (3.4)

Considerings=σ+jas the variable, the response of the multipath system tox(t)=estisy(t)=

x(t)H(s), so that when replacing them in Equation (3.4), we get

x(t)H(s)=x(t)

[

α 0 e−st^0 +···+αNe−stN

]

(^2) Typically, there are three effects each path can have on the sent signal: The distance the signal needs to travel (in each path this is due
to reflection or refraction on buildings, structures, cars, etc.) determines how much it is attenuated and delayed (the longer the path,
the more attenuated and delayed with respect to the time it was sent) and the third effect is a frequency shift—or Doppler effect—that
is caused by the relative velocity between the transmitter and the receiver.