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University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering
ECE 459: Communications I
Fall 2005
Problem Set 9 Signal Detection in Noise, Matched Filtering
Issued: Thursday, December 1st. Due: Thursday, Dec. 8th (beginning of lecture).
Reading from Lathi: Chapter 13, Sections 13.1–13.2.
Announcement: The Final Exam will be held on Saturday, December 17th, from 1:30pm to 4:30pm in 165 Everitt Laboratory. The exam will cover all material from the beginning of the term. For the exam, you can bring three 8. 5 × 11-inch double-sided sheets of handwritten notes. Calculators are allowed but will not be necessary.
Problem 9.
Consider the signal
p(t) =
2 A (^) Tt , 0 ≤ t ≤ T / 2 , − 2 A(1 − (^) Tt ) , T / 2 ≤ t ≤ T , 0 , otherwise.
(a) Sketch the impulse response of a filter that is matched to this signal. (b) Plot the output of the matched filter in part (a) when p(t) is the input. What is the maximum value of the output?
Problem 9.
In a binary on-off signaling scheme the transmitted signal s(t) is either set to zero (under hypothesis H 1 ), or is given by
p(t) =
{ At, 0 ≤ t ≤ T , 0 , otherwise
(under hypothesis H 0 ). The a priori probabilities for the two hypotheses are given by Pr(H 0 ) = p 0 and Pr(H 1 ) = p 1. The signal r(t) seen at the receiving end includes additive noise, i.e., r(t) = s(t) + n(t), where n(t) is a sample function of a white Gaussian random process N (t) with autocorrelation function RN N (τ ) = N 20 δ(τ ).
r(t) - h(t) g(t)
t = T
y = g(T )
@ @ (^) y -
‘H 1 ’
< ‘ H 0 ’
γ
Decision Device
In order to make a decision as to whether hypothesis H 0 or H 1 took place we use the approach shown below. The impulse response h(t) is given by h(t) = 1 for 0 ≤ t ≤ T (0 otherwise). At the decision device, the value y observed for random variable Y (i.e., the value of g(t) at time t = T ) is compared to a threshold γ. The decision is “H 0 ” if y > γ, and “H 1 ” otherwise.
(a) Find fY |H 0 (y|H 0 ), the conditional probability density of Y given hypothesis H 0.
(b) Choose γ so that the probability of error is minimized.
Problem 9.
r(t) - h(t) g(t)
t = 1
y = g(1)
@ @ (^) y -
‘H 1 ’
< ‘ H 0 ’
γ
Decision Device
Consider the binary hypothesis testing situation shown above. Signal r(t) = s(t) + n(t), where s(t) is a deterministic signal and n(t) is noise. Under hypothesis H 0 , signal s(t) is given by
p(t) =
{ sin( π 2 t) , 0 ≤ t ≤ 2 , 0 , otherwise.
Under hypothesis H 1 , signal s(t) is zero. Assume that the a priori probabilities for the two hypotheses are equal and that the noise n(t) is a sample function of a white Gaussian random process N (t) with average power N 0 /2. At the decision device, the value y observed for random variable Y is compared to a threshold γ. The decision is “H 0 ” if y > γ, and “H 1 ” otherwise.
(a) Let T = 1 and h(t) be given by
h(t) =
{ 1 , 0 ≤ t ≤ T , 0 , otherwise.
Find fY |H 0 (y|H 0 ) and fY |H 1 (y|H 1 ), the conditional probability densities of Y given each of the two hypotheses. Choose γ to minimize the probability of error and find the corre- sponding probability of error.
(b) Repeat part (a) for T = 2.
(c) Let T = 2. Choose h(t) and γ so that you minimize the probability of error. What is the corresponding probability of error?
Problem 9.
Consider the following communication scenario: the transmitter sends one bit (“0” or “1”) by setting the transmitted signal s(t) either to zero or to the signal p(t) = e−tu(t); the signal r(t)
The signal r(t) received at the receiver is corrupted by additive noise, i.e., r(t) = s(t) + n(t), where n(t) is a sample path of a (non-white) Gaussian random process N (t) with zero mean and autocorrelation function RN N (τ ) shown below.
RN N (τ ) =
2(2 − τ ), 0 ≤ τ ≤ 2 , 2(2 + τ ), − 2 ≤ τ ≤ 0 , 0 , otherwise.
The receiver is faced with the following binary hypothesis testing problem:
Hypothesis H 0 (“0” being transmitted) : r(t) = n(t) , Hypothesis H 1 (“1” being transmitted) : r(t) = p(t) + n(t).
The a priori probabilities for the two hypotheses are unequal with Pr(H 0 ) = 2 Pr(H 1 ). In order to make a decision as to whether hypothesis H 0 or H 1 took place, the receiver uses the system shown below.
r(t) - h(t) g(t)
t = 1
y = g(1)
@ @ (^) y -
‘H 1 ’
< ‘ H 0 ’
γ
Decision Device
We will analyze the performance of this detection scheme for the following receiver filter h(t):
h(t) =
{ 1 , 0 ≤ t ≤ 1 , 0 , otherwise.
(a) Find fY |H 0 (y|H 0 ), the conditional probability density of Y under hypothesis H 0.
(b) Find fY |H 1 (y|H 1 ), the conditional probability density of Y under hypothesis H 1.
(c) Choose the threshold γ so that the probability of error is minimized.
