In wireless communications, the effects of multi-path significantly 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 (called line of sight). Several copies of the signal, shifted in time and frequency as well as attenuated, are received—i.e., 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 signal x(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 − 2τ)

and let τ = 0.5 sec. Determine the number of samples corresponding to a delay of τseconds by using the sampling rate Fs (samples per second) given when the file handel is loaded.

To simplify matters just work with a signal of duration 1 sec; that is, generate a signal from handel with the appropriate number of samples. Plot the segment of the original handel signal x(t) and the signal y(t) to see the effect of multi-path. Use the MATLAB function sound to listen to the original and the received signals.