Let's design a Neyman-Pearson decision rule
of size αα for the
problem
ℋ
0
:
x∼𝒩0σ2I
ℋ
0
:
x
0
σ
2
I
ℋ
1
:
x∼𝒩μ1σ2I
ℋ
1
:
x
μ
1
σ
2
I
where
μ>0
μ
0
,
σ2>0
σ
2
0
are known,
0=0…0T
0
0
…
0
,
1=1…1T
1
1
…
1
are NN-dimensional
vectors, and II
is the N
N×NN identity
matrix. The likelihood ratio is

Λx=∏
n
=1N12πσ2e−
x
n
−μ22σ2∏
n
=1N12πσ2e−
x
n
22σ2=e−∑
n
=1N
x
n
−μ22σ2e−∑
n
=1N
x
n
22σ2=e12σ2∑
n
=1N2
x
n
μ−μ2=e1σ2(−Nμ22+μ∑
n
=1N
x
n
)
Λ
x
n
1
N
1
2
σ
2
x
n
μ
2
2
σ
2
n
1
N
1
2
σ
2
x
n
2
2
σ
2
n
1
N
x
n
μ
2
2
σ
2
n
1
N
x
n
2
2
σ
2
1
2
σ
2
n
1
N
2
x
n
μ
μ
2
1
σ
2
N
μ
2
2
μ
n
1
N
x
n

(4)
To simplify the test further we may apply the natural
logarithm and rearrange terms to obtain

t≡∑
n
=1N
x
n
≷
ℋ
0
ℋ
1
σ2μlnη+Nμ2≡γ
t
n
1
N
x
n
≷
ℋ
0
ℋ
1
σ
2
μ
η
N
μ
2
γ
We have used the assumption
μ>0
μ
0
. If
μ<0
μ
0
, then division by
μμ is not a
monotonically increasing operation, and the inequalities
would be reversed.

The test statistic

tt is

sufficient for the unknown
mean. To set the threshold

γγ, we write the
false-alarm probability (size) as

P
F
=Prt>γ=∫
f
0
td
t
P
F
t
γ
t
t
t
γ
f
0
t
To evaluate

P
F
P
F
, we need to know the density of

tt under

ℋ
0
ℋ
0
. Fortunately,

tt
is the sum of normal variates, so it is again normally
distributed. In particular, we have

t=Ax
t
A
x
, where

A=1T
A
1
, so

t∼𝒩A0A(σ2I)AT=𝒩0Nσ2
t
A
0
A
σ
2
I
A
0
N
σ
2
under

ℋ
0
ℋ
0
. Therefore, we may write

P
F
P
F
in terms of the

Q-function as

P
F
=QγNσ
P
F
Q
γ
N
σ
The threshold is thus determined by

γ=NσQ-1α
γ
N
σ
Q
α
Under

ℋ
1
ℋ
1
, we have

t∼𝒩A1A(σ2I)AT=𝒩NμNσ2
t
A
1
A
σ
2
I
A
N
μ
N
σ
2
and so the detection probability (power) is

P
D
=Prt>γ=Qγ−NμNσ
P
D
t
γ
Q
γ
N
μ
N
σ
Writing

P
D
P
D
as a function of

P
F
P
F
, the ROC curve is given by

P
D
=QQ-1
P
F
−Nμσ
P
D
Q
Q
P
F
N
μ
σ
The quantity

Nμσ
N
μ
σ
is called the

signal-to-noise
ratio. As its name suggests, a larger SNR
corresponds to improved performance of the Neyman-Pearson
decision rule.

In the context of signal processing, the
foregoing problem may be viewed as the problem of detecting a
constant (DC) signal in

additive white
Gaussian noise:

ℋ
0
:
x
n
=
w
n
,
n
=
1
,
…
,
N
ℋ
0
:
x
n
w
n
,
n
=
1
,
…
,
N
ℋ
1
:
x
n
=A+
w
n
,
n
=
1
,
…
,
N
ℋ
1
:
x
n
A
w
n
,
n
=
1
,
…
,
N
where

AA is a known, fixed
amplitude, and

w
n
∼𝒩0σ2
w
n
0
σ
2
. Here

AA corresponds
to the mean

μμ in the
example.