E&CE 418: Tutorial-6

Instructor: Prof. Xuemin (Sherman) ShenTA: Ning Zhang

Email: n35zhang@uwaterloo.caOffice hour for this week: 12:00 pm-2:00pm on Tue.

Extra office hours : 1:00 pm-5:00 pm on Wed.

Problem 1The signal is corrupted by additive white Gaussian noise with zero mean and power spectral density N0/2 watts/Hz. The optimum detector is a matched filter.

(a) Find and sketch the impulse response of the filter which is matched to the signal g(t)

(b) Derive an expression for the maximum output signal-to-noise ratio (SNR)0 and express it in terms of the parameters α, T and N0

(c) Consider the transmission of either s0(t)=g(t) or s1(t)=0 every T second interval, with equal probability, over a zero mean additive white Gaussian noise channel with power spectral density N0/2 watts/Hz. Sketch the pdf of the decision variable corresponding to s0(t) and s1(t) sent and derive the probability of error in making the decision as to whether s0(t) and s1(t) was sent.

(a) Find and sketch the impulse response of the filter which is matched to the signal g(t)

(b) Derive an expression for the maximum output signal-to-noise ratio (SNR)0 and express it in terms of the parameters α, T and N0

Problem 2Two equiprobable messages are transmitted on an AWGN channel with two-sided power spectral density N0/2. The signals are of the form

(a) Determine the structure of the optimal receiver.(b) Determine the probability of error of this binary system

Let u = y(T) be the decision variable, then the mean

Problem3The purpose of a radar system is basically to detect the presence of a target, and to extract useful information about the target. Suppose that in such a system, hypothesis H0 is that there is no target present, so that the received signal x(t)= w(t), where w(t) is white Gaussian noise with power spectral density N0/2. For hypothesis is H1 , a target is present, and x(t)= w(t)+ s(t),where s(t) is an echo produced by the target. Assumed that s(t) is completely known.

(a) Determine the structure of the optimal receiver.

(b) Evaluate the probability of false alarm defined as the probability that the receiver decides a target is present when it is not.

(c) Evaluate the probability of detection defined as the probability that the receiver decides a target is present when it is.

(d) Given that i) the cost of a false alarm is the same as that of failing to detect a target when it is present, and ii) the probability that there is a target is 0.1, determine the optimal decision threshold for the receiver.

(a) Optimal Receive Structure

Let u = y(T)be the decision variable

The mean

The variance

Decision threshold:

Let α be the decision threshold, α depends on Es and the probability that a target is present or not.

Decision rule:

(b) Evaluate the probability of false alarm defined as the probability that the receiver decides a target is present when it is not.

(c) Evaluate the probability of detection defined as the probability that the receiver decides a target is present when it is.

(d) Given that i) the cost of a false alarm is the same as that of failing to detect a target when it is present, and ii) the probability that there is a target is 0.1, determine the optimal decision threshold for the receiver.

Find the decision threshold such that the average cost of the radar system is minimized.

Average Cost

where

Let cost of false alarm=cost of failure to detect= c

Let the decision threshold, as mentioned in part

Find the value of β which minimizes the above cost function and use that value of β (let it be βmin) to calculate the optimal threshold (βmin*Es) for the receiver.