Detection and Estimation Theory

University of Tehran
Instructor: Dr. Ali Olfat Spring 2020
Homework 4
Due : 99/2/1

Problem 1
Consider the model
1
Yk = θ 2 sk Rk + Nk , k = 1, 2, ..., n
where s1 , s2 , . . . , sn is a known signal sequence, θ ≥ 0 is a constant, and R1 , R2 , . . . , Rn , N1 , N2 , . . . , Nn
are i.i.d. N (0, 1) random variables.

(a) Consider the hypothesis pair

H0 :θ = 0
H1 :θ = A

where A is a known positive constant. Describe the structure of the Neyman-

(b) Consider now the hypothesis pair

H0 :θ = 0
H1 :θ > 0

Under what conditions on s1 , s2 , . . . , sn does a UMP test exist?

(c) For the hypothesis pair of part 2 with s1 , s2 , . . . , sn general, find the locally most
powerful detector.
 2

Problem 2
Suppose we have observations Yk = Nk + θSk , k = 1, 2, ..., n, where N ∼ N (0, I) and
where S1 , S2 , ..., Sn are i.i.d. random variables, independent of N and each taking on
the values of +1 and −1 with equal probabilities of 21 .

(a) Find the likelihood ratio for testing H0 : θ = 0 versus H1 : θ = A , where A is

(b) For the case n = 1, find the Neyman-Pearson rule and corresponding detection
probability for false alarm probability α ∈ (0, 1), for hypotheses of part a.

(c) Is there a UMP test of H0 : θ = 0 versus H0 : θ 6= 0 in this model? If so, why

Problem 3
The distribution of ri on the two hypotheses is (ri are independent under both hy-
potheses)
ri |Hk ∼ N (mk , σk2 ), i = 1, 2, ..., N andk = 0, 1

(a) Find the LRT. Express the test in terms of the following quantities :
N
X
Iα = ri
i=1
XN
Iβ = ri2
i=1

(b) Draw the decision regions in the Iα , Iβ -plane for the case in which

2m0 =m1 > 0

(c) For the special m0 = 0 and σ1 = σ0 , compute the ROC.

Detection and Estimation Theory HW #4

Problem 4
Consider the following ternary hypothesis testing problem with two- dimensional
observation Y = (Y1 , Y2 )T :

H0 : Y = N, H1 : Y = s + N, H2 : Y = −s + N

where s = √1 (1, 1)T and the noise vector N is Gaussian N (0, Σ) with covariance
2
matrix
1 14
 
1
4
1

(a) Assuming that all hypotheses are equally probable, show that the minimum
error probability rule can be written as

 0 sT Σ−1 y ≥ η
δ ∗ (y) = 1 −η ≤ sT Σ−1 y ≤ η
2 sT Σ−1 y ≤ −η


(b) Specify the value of η that minimizes the error probability and find the minimum
error probability.

(c) Assuming now that we are free to choose the signal s subject to the constraint
||s||2 ≤ 1, comment on whether the preceding error probability can be improved
upon.

Problem 5
Consider the M -ary decision problem: (Γ = Rn )

H0 : Y = N + s0
H1 : Y = N + s1
.
.
.
HM −1 : Y = N + sM −1

where s0 , s1 , ..., sM −1 are known signals with equal energies, ||s0 ||2 = ||s1 ||2 = ... =
||sM −1 ||2 .

(a) Assuming N ∼ N (0, σ 2 I), find the decidion rule achieving minimum error prob-
ability when all hypothesis are equally likely.

Detection and Estimation Theory HW #4

(b) Assuming further that the signals are orthogonal, show that the minimum error
probability is given by:
Z ∞
1 2
pe = 1 − √ [Φ(x)]M −1 e−(x−d) /2 dx
2π −∞

where d2 = ||s0 ||/σ 2 .

Detection and Estimation Theory HW #4