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This module describes the random process statistics of a typical sonar receiver, where the sonar data is frequency shifted and then sampled to a digital time series.

Active sonar receiver model

Consider the sonar receiver processing chain shown below:

The sonar array input is heterodyned, low-pass filtered, sampled and scaled to generate a discrete time set of samples of the noise plus signal.

The input noise n ( t ) size 12{n \( t \) } {} is a real-valued, wide sense stationary random process with power spectral density P nn ( f ) size 12{P rSub { size 8{ ital "nn"} } \( f \) } {} . Because n ( t ) size 12{n \( t \) } {} is wide sense stationary, the power spectral density and the autocorrelation function are related by

R nn ( τ ) = P nn ( f ) e j2πfτ df size 12{R rSub { size 8{ ital "nn"} } \( τ \) = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {P rSub { size 8{ ital "nn"} } \( f \) e rSup { size 8{ - j2πfτ} } ital "df"} } {}

We assume that n ( t ) size 12{n \( t \) } {} is zero mean. A complex carrier e j2π ft size 12{e rSup { size 8{j2π ital "ft"} } } {} is applied to the random process n ( t ) size 12{n \( t \) } {} resulting in a complex random process z ( t ) . size 12{z \( t \) "." } {} The mean value of z ( t ) size 12{z \( t \) } {} is given by:

E { z ( t ) } = E { n ( t ) e j2π ft } = E { n ( t ) } e j2π ft = 0 size 12{E lbrace z \( t \) rbrace =E lbrace n \( t \) e rSup { size 8{j2π ital "ft"} } rbrace =E lbrace n \( t \) rbrace e rSup { size 8{j2π ital "ft"} } =0} {}

The covariance of z ( t ) size 12{z \( t \) } {} is given by

E z ( t ) z ( s ) = e j2πf ( t s ) E n ( t ) n ( s ) = e j2πf ( t s ) R nn ( t s ) size 12{E left lbrace z \( t \) z rSup { size 8{*} } \( s \) right rbrace =e rSup { size 8{j2πf \( t - s \) } } E left lbrace n \( t \) n \( s \) right rbrace =e rSup { size 8{j2πf \( t - s \) } } R rSub { size 8{ ital "nn"} } \( t - s \) } {} ,

which shows that z ( t ) size 12{z \( t \) } {} is wide sense stationary as well.

z ( t ) size 12{z \( t \) } {} is passed through a band-pass filter to produce x ( t ) size 12{x \( t \) } {} . The frequency response of the band-pass filter is assumed to be low-pass with a bandwidth of B/2 size 12{"B/2 "} {} Hertz. That is we will assume that the filter transfer function H BPF ( f ) size 12{H rSub { size 8{ ital "BPF"} } \( f \) } {} is given by:

H ( f ) = { 1, f < B / 2 0, f > B / 2 size 12{H \( f \) = left lbrace matrix { 1, lline f rline<B/2 {} ## 0, lline f rline>B/2 } right none } {}

The resulting x ( t ) size 12{x \( t \) } {} is a wide sense stationary random process with zero-mean and power spectral density:

P xx ( f ) { N 0 / 2, f < B / 2 0, f > B / 2 size 12{P rSub { size 8{ ital "xx"} } \( f \) approx left lbrace matrix { N rSub { size 8{0} } /2, lline f rline<B/2 {} ## 0, lline f rline>B/2 } right none } {} , (1)

Where we have assumed that the bandwidth of the receiver is small relative to the center frequency of the signal we are trying to detect, B / f 0 << 1 size 12{B/f rSub { size 8{0} } "<<"1} {} . The power spectral density of x ( t ) size 12{x \( t \) } {} can then be approximated by the power spectral density of the noise near f 0 size 12{f rSub { size 8{0} } } {} :

P nn ( f ) P nn ( f 0 ) = N 0 / 2, f f 0 < B / 2 size 12{P rSub { size 8{ ital "nn"} } \( f \) approx P rSub { size 8{ ital "nn"} } \( f rSub { size 8{0} } \) =N rSub { size 8{0} } /2, lline f - f rSub { size 8{0} } rline<B/2} {}

If x ( t ) size 12{x \( t \) } {} has a power spectral density given by Eq-1, then the autocorrelation function of x ( t ) size 12{x \( t \) } {} becomes:

R xx ( τ ) = E x ( t ) x ( t + τ ) = B / 2 B / 2 N 0 2 e j2πfτ df = N 0 2 sin πBτ πτ size 12{R rSub { size 8{ ital "xx"} } \( τ \) =E left lbrace x \( t \) x rSup { size 8{*} } \( t+τ \) right rbrace = Int cSub { size 8{ - B/2} } cSup { size 8{B/2} } { { {N rSub { size 8{0} } } over {2} } } e rSup { size 8{j2πfτ} } ital "df"= { {N rSub { size 8{0} } } over {2} } { {"sin"πBτ} over { ital "πτ"} } } {}

Note that R xx ( 0 ) = N 0 B 2 size 12{R rSub { size 8{ ital "xx"} } \( 0 \) = { {N rSub { size 8{0} } B} over {2} } } {} .

Now if we choose a sampling interval Δt = 1 / B size 12{Δt=1/B} {} ; then the samples at kΔt size 12{kΔt} {} have an autocorrelation given by

E { x ( kΔt ) x ( lΔt ) } = N 0 2 sin ( πB ( k l ) Δt ) π ( k l ) Δt = BN 0 2 δ kl size 12{E lbrace x \( kΔt \) x rSup { size 8{*} } \( lΔt \) rbrace = { {N rSub { size 8{0} } } over {2} } { {"sin" \( πB \( k - l \) Δt \) } over {π \( k - l \) Δt} } = { { ital "BN" rSub { size 8{0} } } over {2} } δ rSub { size 8{ ital "kl"} } } {} ,

Hence x ( kΔt ) , k = 0,1, . . . size 12{x \( kΔt \) ,k=0,1, "." "." "." } {} is a discrete time, wide sense stationary, white noise with intensity BN 0 2 size 12{ { { ital "BN" rSub { size 8{0} } } over {2} } } {} .

For matched filtering applications, we scale the output of the Analog to Digital conversion process by Δt = 1 / B size 12{Δt=1/B} {} to conserve the signal energy over a time interval T . size 12{T "." } {} This creates the discrete time process y k = x ( kΔt ) Δt size 12{y rSub { size 8{k} } =x \( kΔt \) sqrt {Δt} } {} .

To see this, consider that

E 0 T x ( t ) 2 dt = 0 T E { x ( t ) 2 } dt = 0 T R xx ( 0 ) dt = BTN 0 2 size 12{E left lbrace Int cSub { size 8{0} } cSup { size 8{T} } { lline x \( t \) rline rSup { size 8{2} } ital "dt"} right rbrace = Int cSub { size 8{0} } cSup { size 8{T} } {E lbrace lline x \( t \) rline rSup { size 8{2} } rbrace ital "dt"} = Int cSub { size 8{0} } cSup { size 8{T} } {R rSub { size 8{ ital "xx"} } \( 0 \) ital "dt"} = { { ital "BTN" rSub { size 8{0} } } over {2} } } {}

And

E k = 1 k = T Δt y ( k ) 2 = k = 1 k = T Δt E y ( k ) 2 = k = 1 k = T Δt BN 0 2 Δt = BTN 0 2 size 12{E left lbrace Sum cSub { size 8{k=1} } cSup { size 8{k= { {T} over {Δt} } } } { lline y \( k \) rline rSup { size 8{2} } } right rbrace = Sum cSub { size 8{k=1} } cSup { size 8{k= { {T} over {Δt} } } } {E left lbrace lline y \( k \) rline rSup { size 8{2} } right rbrace } = Sum cSub { size 8{k=1} } cSup { size 8{k= { {T} over {Δt} } } } { { { ital "BN" rSub { size 8{0} } } over {2} } Δt={}} { { ital "BTN" rSub { size 8{0} } } over {2} } } {} .

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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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