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An in-depth analysis of hypothesis testing in communications engineering, focusing on minimizing error probability. It covers the concept of hypothesis testing, the use of gaussian distributions, and the derivation of the optimal threshold (γ) for decision making. The document also includes examples of digital and optical communication systems and the application of the likelihood ratio test.
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Prepared by: Suneil Hosmane
Based on Lectures by Prof. Hadjicostis
University of Illinois at Urbana-Champaign
Department of Electrical and Computer Engineering
© Copyright 2005 Christoforos Hadjicostis. All rights reserved.
Hypothesis testing is a recurrent theme in many disciplines, especially that of
Communications Engineering. In Digital Communications, the fundamental idea behind
hypothesis testing is the following: We have two hypotheses H 0
and H 1
which could
represent respectively a “0” or “1” being transmitted. If we are given the probability of
receiving a “0” given that H 0
occurred (meaning that we actually sent a 0) and the
probability of receiving a “1” given that H 1
occurred, how do we decide which signal was
actually sent?
We are given a Digital Communication System that transmits a single bit (0 or 1).
During transmission, noise (represented by a R.V. N ) is introduced into the system. On
the receiving end, the receiver is presented with two hypotheses. Hypothesis H 0
states
that a “0” was sent, and Hypothesis H 1
states that a “1” was sent. If we are also given the
information below, how do we determine which bit was actually received so that we
minimize the probability of error?
0
: A A + n
1
: - A - A + n
n
Assume that a priori, the two hypotheses have the following probabilities:
Pr(H 0
0,
Pr(H 1
1,
0
1
And that the noise N is a Gaussian R.V.
f N
(n) = Ν (0, σ
2
n
exp(-
2 π
2
2
σ σ
Under H 0
: Under H 1
z|H
(z|H 0
) is Ν (A, σ
2
z|H
(z|H 1
) is Ν (-A, σ
2
Note: These pictures are a little deceiving. The Gaussian distribution does not have a
finite bound like depicted in the pictures above. It actually extends from – infinity to +
z = Z
We can then solve for
γ
and obtain our optimal solution for the threshold test:
0
γ
1
Notice that even if we choose
γ
in an optimal fashion, it is still unclear at this point that
the above rule will be the best we can do in terms of minimizing the probability of error.
This is so because we arbitrarily set the rule to be of the form
γ
It turns out that to minimize the probability of error we need to use a MAP rule.
Choose H 0
if:
Pr(H 0
| Z = z) > Pr(H 1
| Z = z)
Choose H 1
if:
Pr(H 0
| Z = z) < Pr(H 1
| Z = z)
If we arrange terms, we can arrive at the following rule:
1
0
∙f z|H
(z | H 0
1
∙f z|H
(z | H 1
0
0
0
1
z|H 1
z|H 0
1
0
This is know as the Likelihood Ratio Test
1
By looking at the weighted probabilities of P 0
∙f z|H
(z | H 0
) and P 1
∙f z|H
(z | H 1
we can choose which hypothesis occurred.
Let us use the Likelihood Ratio Test to find the optimal decision rule in the previous
example.
0
0
1
2
2
2
2
1
0
0
1
2
2
2
2
1
0
0
1
2
2
2
2
1
0
0
1
2
2 2
2
2 2
1
0
0
1
2
1
0
1
2
so we have now solved for our
γ
1
γ
= ln( )
1
2
σ
0
1
k
1 1
k
0 0
1
0
0
1
k
1
0
1
0
1
0
0
1
0
1
k
1
0
1
0
0
1
0
1
k
1
0
1
0
0
1
0
1
1
0
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
( ) ( ) ( ) ( )
( 0 ) ( 1 )
1 1 0 0
1
0
where
( ) ( ) ( ) ( )
( 0 ) ( 1 )
1 1 0 0
lnu ln u
ln ln 1 - u ln ln 1 - u
c
1
Since c can be any real number but k is an integer:
0
(if
then “H 0
” is selected, otherwise if
then “H 1
1
” is selected.)
So what is the error probability?
Error Probability
Pr(Error) = Pr(Error ∩ H 0
) + Pr(Error ∩ H 1
= Pr(H 0
) ∙ Pr(Error | H 0
) + Pr(H 1
) ∙ Pr(Error | H 1
0
∙ Pr(Error | H 0
1
∙ Pr(Error | H 1
∑
−
=
1
0
c
k
k
P u u
∑
∞
=
k c
k
P 1 ( 1 u 1 ) u 1
( )
( )
1
1
c
1
0 1
0
c
0
0
1 - u
u
1 - u
1 - u
1 - u
= 1 - u ⋅ ⋅ P + ⋅ ⋅ P
