We will represent the sampled echo response as a partitioned vector:

r(h)=0D(h)x1Aw0L−D(h)−Nr(h)=0D(h)x1Aw0L−D(h)−N size 12{r \( h \) = left [ matrix {
0 rSub { size 8{D \( h \) x1} } {} ##
Aw {} ##
0 rSub { size 8{L - D \( h \) - N} }
} right ]} {},

Where

w
=
w
(
Δt
)
⋮
w
(
NΔt
)
w
=
w
(
Δt
)
⋮
w
(
NΔt
)
size 12{w= left [ matrix {
w \( Δt \) {} ##
dotsvert {} ##
w \( NΔt \)
} right ]} {}

and the sampled noise and interference as a vector

q=q1⋮qLq=q1⋮qL size 12{q= left [ matrix {
q rSub { size 8{1} } {} ##
dotsvert {} ##
q rSub { size 8{L} }
} right ]} {},

so that the sampled ping history becomes

y
=
q
+
r
(
h
)
y
=
q
+
r
(
h
)
size 12{y=q+r \( h \) } {}

The echo is modeled as a known signal
ww size 12{w} {}, with Gaussian random complex amplitude A, with zero mean and variance
σA(h)2σA(h)2 size 12{σ rSub { size 8{A \( h \) } } rSup { size 8{2} } } {}. We will assume that
wHw=1wHw=1 size 12{w rSup { size 8{H} } w=1} {}, and that
∣A(h)∣2∣A(h)∣2 size 12{ lline A \( h \) rline rSup { size 8{2} } } {} is the energy of the echo, with units Pascals^2-seconds. Since
σA(h)2σA(h)2 size 12{σ rSub { size 8{A \( h \) } } rSup { size 8{2} } } {}is
E∣A(h)∣2E∣A(h)∣2 size 12{E lline A \( h \) rline rSup { size 8{2} } } {}, it has units of Pascals^2-seconds as well. The amplitude of the echo is a function of the target location hypothesis
hh size 12{h} {}. The location of
ww size 12{w} {}in
r(h)r(h) size 12{r \( h \) } {}depends on the location of the target through the time delay
D(h)D(h) size 12{D \( h \) } {}.

Since each element of the random vector
AwAw size 12{Aw} {} is complex Gaussian, the random vector
AwAw size 12{Aw} {}has a complex Gaussian distribution. The probability density of
AwAw size 12{Aw} {} is Gaussian zero mean with covariance matrix
σA(h)2wwHσA(h)2wwH size 12{σ rSub { size 8{A \( h \) } } rSup { size 8{2} } bold "ww" rSup { size 8{H} } } {}. To see this, consider that

E
(
Aw
)
=
E
(
A
)
w
=
0
N
E
(
Aw
)
=
E
(
A
)
w
=
0
N
size 12{E \( Aw \) =E \( A \) w=0 rSub { size 8{N} } } {}

The covariance of
AwAw size 12{Aw} {} is given by:

E
(
Aw
)
(
Aw
)
H
=
E
(
AA
H
)
ww
H
=
σ
A
(
h
)
2
ww
H
E
(
Aw
)
(
Aw
)
H
=
E
(
AA
H
)
ww
H
=
σ
A
(
h
)
2
ww
H
size 12{E \( Aw \) \( Aw \) rSup { size 8{H} } =E \( ital "AA" rSup { size 8{H} } \) bold "ww" rSup { size 8{H} } =σ rSub { size 8{A \( h \) } } rSup { size 8{2} } bold "ww" rSup { size 8{H} } } {}

hence
r(h)r(h) size 12{r \( h \) } {}is zero mean complex Gaussian with covariance matrix
σA(h)2rrHσA(h)2rrH size 12{σ rSub { size 8{A \( h \) } } rSup { size 8{2} } bold "rr" rSup { size 8{H} } } {}.

For the clutter only hypothesis
φ,φ, size 12{φ,} {}y=qy=q size 12{y=q} {}.

We have sampled, heterodyned and possibly re-sampled the noise process
q(t)q(t) size 12{q \( t \) } {}to form
qq size 12{q} {}.

During the period where r is non-zero,
qq size 12{q} {}is a sampled version of the ambient noise, represented as a N by 1 complex Gaussian noise random vector with zero mean and covariance matrix
(N0)IN(N0)IN size 12{ \( N rSub { size 8{0} } \) I rSub { size 8{N} } } {}. This is true because
BWΔt≈1BWΔt≈1 size 12{ ital "BW"Δt approx 1} {} for complex Nyquist sampling of a band-limited signal.

Overall, the noise and reverberation
qq size 12{q} {} is assumed to be complex Gaussian with zero mean and L by L covariance matrix
CC size 12{C} {}.

Because we are assuming that the reverberation dies away before the echoes from the target search arrive,
CC size 12{C} {} has the following partition:

C
=
R
0
0
N
0
I
C
=
R
0
0
N
0
I
size 12{C= left [ matrix {
R {} # 0 {} ##
0 {} # N rSub { size 8{0} } I{}
} right ]} {}

Matrix R has dimensions of
DminxDminDminxDmin size 12{D rSub { size 8{"min"} } ital "xD" rSub { size 8{"min"} } } {}, the minimum delay where the echo interference is dominated by Ambient noise.

Under target hypothesis
hh size 12{h} {} ,
yy size 12{y} {} is Gaussian with has zero mean and covariance matrix
C+σA2rrHC+σA2rrH size 12{C+σ rSub { size 8{A} rSup { size 8{2} } } bold "rr" rSup { size 8{H} } } {}.

The probability density of
yy size 12{y} {}under
hh size 12{h} {} becomes:

p(y∣h)=1πNdet(Cr+C)exp−yH(Cr+C)−1yp(y∣h)=1πNdet(Cr+C)exp−yH(Cr+C)−1y size 12{p \( y \lline h \) = { {1} over {π rSup { size 8{N} } "det" \( C rSub { size 8{r} } +C \) } } "exp" left ( - y rSup { size 8{H} } \( C rSub { size 8{r} } +C \) rSup { size 8{ - 1} } y right )} {},

where
Cr=σA2rrHCr=σA2rrH size 12{C rSub { size 8{r} } =σ rSub { size 8{A} rSup { size 8{2} } } bold "rr" rSup { size 8{H} } } {}.

Under the clutter hypothesis,
φ,φ, size 12{φ,} {} y has zero mean and covariance matrix
CC size 12{C} {}.The probability density of
yy size 12{y} {}under
φφ size 12{φ} {} becomes:

p
(
y
∣
φ
)
=
1
π
N
det
(
C
)
exp
−
y
H
(
C
)
−
1
y
p
(
y
∣
φ
)
=
1
π
N
det
(
C
)
exp
−
y
H
(
C
)
−
1
y
size 12{p \( y \lline φ \) = { {1} over {π rSup { size 8{N} } "det" \( C \) } } "exp" left ( - y rSup { size 8{H} } \( C \) rSup { size 8{ - 1} } y right )} {}