EE6617 Detection & Estimation

Homework #1

Due on Oct 22, 2024

Problem 1

Consider a binary hypothesis problem where the observed vector is given by

where nm ∼ N (0, σ2 ), π0 = 0.6, π1 = 0.4 with the following cost assignment c10 = 0.7, c00 = 0.3, c01 = 0.8, c11 = 0.2.

a) Perform. the likelihood ratio (LR) test and derive the optimum decision regions R1 and R0.

b) Express the probability of false alarm PF A and probability of miss detection PM as a function of N.

c) Calculate the minimum N that guarantees PF A to be less than 1%, given that A = σ. (Q−1 (0.01) = 2.3263)

Problem 2

Consider the conditional PDFs under H0 and H1 given by

a) Perform. the likelihood ratio (LR) test.

b) Derive the optimum decision regions R1 and R0 for both Λ0 > 0 and Λ0 < 0.

c) Find R1 and R0 explicitly, given that PF A = 8/1 · 10−3.

Problem 3

Let’s apply the SLRT method to a coin tossing problem, where the probability of a head is θ1 for H1, and θ0 for H0, respectively (θ1 > θ0). Let’s indicate the outcome of each coin tossing as Ii = 1 for a head, and Ii = 0 for a tail. We assume that all Ii are IID random variables. Denote Xm = Σm i=1 Ii as the total number of heads in a sequence of length m.

a) Find the LR function Λ(Xm) for a sequence of length m.

b) Express the two thresholds in terms of θ0 and θ1. Use the following expressions to find a, b, c in terms of A, B, θ0, and θ1

B < Λ(Xm) < A ⇒ ln B < Zm < ln A ⇒ b + cm < Xm < a + cm              (1)

c) For α = 0.05, β = 0.95, θ0 = 0.2 and θ1 = 0.6, find A and B.

d) Calculate the average number of required observations ¯nH1 and ¯nH0 .

Problem 4

Consider a binary hypothesis testing where the observation Y is modeled as follows

where X and V are independent exponential random variables with PDF as follows

a) Derive the likelihood probability of each hypothesis H0 and H1.

b) Use Bayes criterion to find the optimum decision rule and its decision regions R0 and R1.

c) Calculate the probability of false alarm PF A and correct detection PD in terms of λx, λv, then find a relationship between PD and PF A.

Problem 5

Consider a binary hypothesis testing where the observation Y has the following PDFs

Let’s assume b > a > 0.

a) Calculate the constants A0 and A1 in terms of parameters a and b.

b) Assume uniform. cost assignment cij = 1 − δij and find the optimum decision rule based on Bayes criterion.

c) Evaluate PD and PF A as a function of the detection threshold Λ0 = π1/π0. Assume b = 2a and sketch the ROC curve.