Signals and Systems - Electrical Engineering

Problems 115

1.20. Multipath effects, first part—MATLAB
In wireless communications, the effects ofmultipathsignificantly affect the quality of the received signal.
Due to the presence of buildings, cars, etc. between the transmitter and the receiver, the sent signal does
not typically go from the transmitter to the receiver in a straight path (calledline of sight). Several copies
of the signal, shifted in time and frequency as well as attenuated, are received—that is, the transmission
is done over multiple paths each attenuating and shifting the sent signal. The sum of these versions of
the signal appears quite different from the original signal given that constructive as well as destructive
effects may occur. In this problem we consider the time-shift of an actual signal to illustrate the effects of
attenuation and time shift. In the next problem we consider the effects of time and frequency shifting and
attenuation.
Assume that the MATLAB “handel.mat” signal is an analog signalx(t)that it is transmitted over three
paths, so that the received signal is

y(t)=x(t)+0.8x(t−τ)+0.5x(t− 2 τ)

and letτ=0.5seconds. Determine the number of samples corresponding to a delay ofτseconds by using

the sampling rateFs(samples per second) given when the file “handel.mat” is loaded.

To simplify matters, just work with a signal of duration 1 second—that is, generate a signal from “han-

del.mat” with the appropriate number of samples. Plot the segment of the original “handel.mat” signalx(t)

and the signaly(t)to see the effect of multipath. Use the MATLAB functionsoundto listen to the original

and the received signals.

1.21. Multipath effects, second part—MATLAB
Consider now the Doppler effect in wireless communications. The difference in velocity between the trans-
mitter and the receiver causes a shift in frequency in the signal, which is called theDoppler effect(e.g., this
is just like the acoustic effect of a train whistle as a train goes by).
To illustrate the frequency-shift effect, consider a complex exponentialx(t)=ej^0 t. Assume two paths:
One that does not change the signal, while the other causes the frequency shift and attenuation, resulting
in the signal

y(t)=ej^0 t+αej^0 tejφt

=ej^0 t

[

1 +αejφt

]

whereαis the attenuation andφis the Doppler frequency shift, which is typically much smaller than the

signal frequency. Let 0 =π,φ=π/ 100 , andα=0.7. This is analogous to the case where the received

signal is the sum of the line-of-sight signal and an attenuated signal affected by Doppler.

(a) Consider the termαejφt, a phasor with frequencyφ=π/ 100 to which we add 1. Use the MATLAB

plotting functioncompassto plot the addition 1 +0.7ejφtfor times from 0 to 256 sec, changing in

increments ofT=0.5sec.

(b)If we writey(t)=A(t)ej(^0 t+θ(t)), give analytical expressions forA(t)andθ(t), and compute and plot

them using MATLAB for the times indicated above.

(c)Compute the real part of the signal

y 1 (t)=x(t)+0.7x(t− 100 )ejφ(t−^100 )

That is, the effects of time and frequency delays, put together with attenuation, for the times indicated

in part (a). Use the functionsound(letFs= 2000 in this function) to listen to the different signals.

1.22. Beating or pulsation—MATLAB
An interesting phenomenon in the generation of musical sounds is beating or pulsation. SupposeNPdif-
ferent players try to play a pure tone, a sinusoid of frequency 160 Hz, and that the signal recorded is the