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Course by: Laurence Riddle. E-mail the author

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.

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} } } {}.

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