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Problems

1. Consider a detection problem where under hypothesis H the PDF of the
0
signal x is p (x | H ) = α·exp(–x/a), 0 ≤ x ≤ ∞, while under hypothesis H 1
0
x
the PDF of the signal x is p (x | H ) = β·exp(–x/β), 0 ≤ x ≤ ∞, with β > α.
1
x
What are the likelihood ratio and log likelihood ratio for this problem?
2. Consider Neyman-Pearson detection of a constant A = 0.5 in uniform (not
Gaussian) white noise. Specifically, the PDF of the noise w[n] is

The measured signal is x[n]. Under hypothesis H (noise only), x[n] =
0
w[n]. Under hypothesis H , x[n] = A + w[n].
1
a. On one graph, sketch carefully the PDF of x[n] under each hypothesis,
p (x | H ) and p (x | H ). Label all significant values.
1
x
x
0
b. What is the likelihood ratio Λ(x) (not the log-likelihood ratio or the
likelihood ratio test) for this problem? It may be necessary to express
this in more than one region, i.e. “Λ(x) = expression 1 for α < x < b,”
etc.