# Connexions

You are here: Home » Content » Signal and Information Processing for Sonar » Sonar Receiver Model

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Laurence Riddle. E-mail the author

# Sonar Receiver Model

Module by: Laurence Riddle. E-mail the author

Summary: 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)n(t) size 12{n $$t$$ } {}is a real-valued, wide sense stationary random process with power spectral density Pnn(f)Pnn(f) size 12{P rSub { size 8{ ital "nn"} } $$f$$ } {}. Because n(t)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 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)n(t) size 12{n $$t$$ } {}is zero mean. A complex carrier ej2πftej2πft size 12{e rSup { size 8{j2π ital "ft"} } } {}is applied to the random process n(t)n(t) size 12{n $$t$$ } {}resulting in a complex random process z(t).z(t). size 12{z $$t$$ "." } {} The mean value of z(t)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 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)z(t) size 12{z $$t$$ } {}is given by

Ez(t)z(s)=ej2πf(ts)En(t)n(s)=ej2πf(ts)Rnn(ts)Ez(t)z(s)=ej2πf(ts)En(t)n(s)=ej2πf(ts)Rnn(ts) 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)z(t) size 12{z $$t$$ } {}is wide sense stationary as well.

z(t)z(t) size 12{z $$t$$ } {}is passed through a band-pass filter to produce x(t)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 B/2 size 12{"B/2 "} {}Hertz. That is we will assume that the filter transfer function HBPF(f)HBPF(f) size 12{H rSub { size 8{ ital "BPF"} } $$f$$ } {}is given by:

H ( f ) = { 1, f < B / 2 0, f > B / 2 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)x(t) size 12{x $$t$$ } {}is a wide sense stationary random process with zero-mean and power spectral density:

Pxx(f){N0/2,f<B/20,f>B/2Pxx(f){N0/2,f<B/20,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/f0<<1B/f0<<1 size 12{B/f rSub { size 8{0} } "<<"1} {}. The power spectral density of x(t)x(t) size 12{x $$t$$ } {}can then be approximated by the power spectral density of the noise near f0f0 size 12{f rSub { size 8{0} } } {}:

P nn ( f ) P nn ( f 0 ) = N 0 / 2, f f 0 < B / 2 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)x(t) size 12{x $$t$$ } {} has a power spectral density given by Eq-1, then the autocorrelation function of x(t)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 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 Rxx(0)=N0B2Rxx(0)=N0B2 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/BΔt=1/B size 12{Δt=1/B} {}; then the samples at kΔtkΔt size 12{kΔt} {} have an autocorrelation given by

E{x(kΔt)x(lΔt)}=N02sin(πB(kl)Δt)π(kl)Δt=BN02δklE{x(kΔt)x(lΔt)}=N02sin(πB(kl)Δt)π(kl)Δt=BN02δkl 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,...x(kΔt),k=0,1,... size 12{x $$kΔt$$ ,k=0,1, "." "." "." } {} is a discrete time, wide sense stationary, white noise with intensity BN02BN02 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/BΔt=1/B size 12{Δt=1/B} {}to conserve the signal energy over a time interval T.T. size 12{T "." } {} This creates the discrete time process yk=x(kΔt)Δtyk=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 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

Ek=1k=TΔty(k)2=k=1k=TΔtEy(k)2=k=1k=TΔtBN02Δt=BTN02Ek=1k=TΔty(k)2=k=1k=TΔtEy(k)2=k=1k=TΔtBN02Δt=BTN02 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} } } {}.

## Content actions

EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks