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While receiving an echo, we will also receive ambient noise, q ( t ) size 12{q \( t \) } {} , which we will assume to be complex Gaussian noise, with constant power spectral density over the receiver’s bandwidth. The noise power spectral density over the receiver’s bandwidth BW is assumed to be N 0 size 12{N rSub { size 8{0} } } {} Pascals^2/Hertz.

We receive L complex valued samples, y size 12{y} {} as the ping history after heterodyning. For hypothesis h, the observation in discrete time is:

y ( kΔt ) = q ( kΔt ) + r ( kΔt 2R / c h ) size 12{y \( kΔt \) =q \( kΔt \) +r \( kΔt - 2R/c \lline h \) } {} for , k = 1, L size 12{k=1, dotslow L} {}

where Δt size 12{Δt} {} is the digital sample rate after heterodyning, and q is a sample of the noise and reverberation. Note that r ( kΔt 2R ( h ) / c h ) = 0 size 12{r \( kΔt - 2R \( h \) /c \lline h \) =0} {} when 2R ( h ) / c > kΔt , size 12{2R \( h \) /c>kΔt,} {} because the echo is delayed. The delay for hypothesis h size 12{h} {} in samples, is given by

D ( h ) = 2R cΔt size 12{D \( h \) = left [ { {2R} over {cΔt} } right ]} {}

where [x] is the nearest integer to x. We choose the sample rate Δt size 12{Δt} {} to be small enough to satisfy the Nyquist sampling criteria for the received echo. We will assume that the non-zero part of the echo is N samples long.

Statistical model of the ping history

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

r ( h ) = 0 D ( h ) x1 Aw 0 L 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 ) size 12{w= left [ matrix { w \(Δt \) {} ## dotsvert {} ##w \( NΔt \) } right ]} {}

and the sampled noise and interference as a vector

q = q 1 q L 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 ) size 12{y=q+r \( h \) } {}

The echo is modeled as a known signal w size 12{w} {} , with Gaussian random complex amplitude A, with zero mean and variance σ A ( h ) 2 size 12{σrSub { size 8{A \( h \) } } rSup { size 8{2} } } {} . We will assume that w H w = 1 size 12{w rSup { size 8{H} } w=1} {} , and that 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 σ A ( h ) 2 size 12{σrSub { size 8{A \( h \) } } rSup { size 8{2} } } {} is E 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 h size 12{h} {} . The location of w size 12{w} {} in r ( h ) size 12{r \( h \) } {} depends on the location of the target through the time delay D ( h ) size 12{D \( h \) } {} .

Since each element of the random vector Aw size 12{Aw} {} is complex Gaussian, the random vector Aw size 12{Aw} {} has a complex Gaussian distribution. The probability density of Aw size 12{Aw} {} is Gaussian zero mean with covariance matrix σ A ( h ) 2 ww H size 12{σrSub { 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 size 12{E \( Aw \) =E \( A \) w=0 rSub { size 8{N} } } {}

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

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} } =σrSub { size 8{A \( h \) } } rSup { size 8{2} } bold "ww" rSup { size 8{H} } } {}

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

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

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

During the period where r is non-zero, q 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 ( N 0 ) I N size 12{ \( N rSub { size 8{0} } \) I rSub { size 8{N} } } {} . This is true because BW Δt 1 size 12{ ital "BW"Δt approx 1} {} for complex Nyquist sampling of a band-limited signal.

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

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

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 D min xD min 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 h size 12{h} {} , y size 12{y} {} is Gaussian with has zero mean and covariance matrix C + σ A 2 rr H size 12{C+σrSub { size 8{A} rSup { size 8{2} } } bold "rr" rSup { size 8{H} } } {} .

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

p ( y h ) = 1 π N det ( C r + C ) exp y H ( C r + C ) 1 y size 12{p \( y \lline h \) = { {1} over {πrSup { 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 C r = σ A 2 rr H size 12{C rSub { size 8{r} } =σrSub { size 8{A} rSup { size 8{2} } } bold "rr" rSup { size 8{H} } } {} .

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

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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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