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Statistical Signal Processing Homework Assignment for ECE 567, Spring 2008, Assignments of Electrical and Electronics Engineering

A homework assignment for the statistical signal processing course (ece 567) offered in spring 2008. The assignment includes five problems related to maximum likelihood testing, neyman-pearson testing, and minimizing energy consumption for signal transmission in the presence of gaussian noise. Students are expected to find solutions for problems such as finding the ml test to distinguish between two hypotheses, determining the neyman-pearson test for a given pfa, and calculating the optimal transmission amplitude and number of pulses.

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Pre 2010

Uploaded on 08/18/2009

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ECE 567 STATISTICAL SIGNAL PROCESSING SPRING 2008
Homework Assignment #1
Due: 6 February 2008
1. Find the Maximum Likelihood (ML) test to distinguish between H0and H1if
p(x|H0) = 1
2exp(−|x|)
p(x|H1) = 1
2exp(−|xA|)
when A > 0. What is PDand PF A?
2. Let p(x|H0) and p(x|H1) be as in problem (1) (with A > 0 as in that problem). Find
the Neyman-Pearson test (to distinguish between H0and H1) that achieves PF A = 0.1.
What is PD? Does your test depend on the value of A? Explain.
3. Suppose that we observe a single sample xdrawn from either
H0:p(x) = 1
2exp(−|x|)
H1:p(x) = 1
2πexp(1
2x2).
(a) Plot ln(Λ(x)) versus x.
(b) Determine the form of the NP test.
4. A signal is to be transmitted in Npulses of amplitude Ain the presence of Gaussian
noise. The received signal x(n) is
H0:x(n) = w(n), n = 0, . . . , N 1
H1:x(n) = A+w(n), n = 0, . . . , N 1
where H0represents the situation with no signal and H1the situation with a trans-
mission. Assume that each w(n) N(0, σ2), and that the {w(n)}N1
n=0 are i.i.d. with
σ2= 1.
We want to minimize NA2(the energy to transmit Aover Nsamples) while achieving
PDof at least 0.9 at the receiver while PFA is no greater than 0.01.
What Nand Ado you choose?

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ECE 567 STATISTICAL SIGNAL PROCESSING SPRING 2008

Homework Assignment # Due: 6 February 2008

  1. Find the Maximum Likelihood (ML) test to distinguish between H 0 and H 1 if

p(x|H 0 ) =

exp(−|x|)

p(x|H 1 ) =

exp(−|x − A|)

when A > 0. What is PD and PF A?

  1. Let p(x|H 0 ) and p(x|H 1 ) be as in problem (1) (with A > 0 as in that problem). Find the Neyman-Pearson test (to distinguish between H 0 and H 1 ) that achieves PF A = 0.1. What is PD? Does your test depend on the value of A? Explain.
  2. Suppose that we observe a single sample x drawn from either

H 0 : p(x) =

exp(−|x|)

H 1 : p(x) =

2 π

exp(−

x^2 ).

(a) Plot ln(Λ(x)) versus x. (b) Determine the form of the NP test.

  1. A signal is to be transmitted in N pulses of amplitude A in the presence of Gaussian noise. The received signal x(n) is

H 0 : x(n) = w(n), n = 0,... , N − 1 H 1 : x(n) = A + w(n), n = 0,... , N − 1

where H 0 represents the situation with no signal and H 1 the situation with a trans- mission. Assume that each w(n) ∼ N (0, σ^2 ), and that the {w(n)}N n=0^ −^1 are i.i.d. with σ^2 = 1. We want to minimize N A^2 (the energy to transmit A over N samples) while achieving PD of at least 0.9 at the receiver while PF A is no greater than 0.01. What N and A do you choose?