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Course by: Laurence Riddle. E-mail the author

Properties of Active Sonar Matched Filtering

Module by: Laurence Riddle. E-mail the author

Summary: This module develops expressions for the response of a matched filter to ambient noise, reverberation and target echoes. A scattering function formulation is used to characterize propagation channels with multipath and Doppler spreading.

Properties of Active Sonar Matched Filtering

Introduction

Matched filters are used extensively in coherent active sonar. The output of a matched filter is used for detection, classification and localization. This document develops some properties of matched filters, including the SNR response in ambient noise and the response to reverberation.

In a matched filter for active sonar, we are integrating the echo plus interference times the echo’s replica. When an echo passes through the matched filter, we are cross-correlating the echo with a scaled version of the echo, so that the output is a scaled version of the auto-correlation of the echo corrupted by noise. The autocorrelation of the echo has a peak in time whose duration is approximately the inverse of the echo’s bandwidth.

For some waveforms (such as the Sinusoidal Frequency Modulation pulse) the autocorrelation function will have multiple peaks, termed ‘fingers’, due to the periodic structure of the pulse. Each autocorrelation finger has a time width approximately equal to the signal’s bandwidth.

Continuous Time Matched Filter

The echo is written as

e(t)=ERr(t)e(t)=ERr(t) size 12{e $$t$$ = sqrt {E rSub { size 8{R} } } r $$t$$ } {}, where 0Tr2(t)dt=10Tr2(t)dt=1 size 12{ Int cSub { size 8{0} } cSup { size 8{T} } {r rSup { size 8{2} } $$t$$ } ital "dt"=1} {}

This implies that the echo energy 0Te2(t)dt0Te2(t)dt size 12{ Int cSub { size 8{0} } cSup { size 8{T} } {e rSup { size 8{2} } $$t$$ ital "dt"} } {}is ERER size 12{E rSub { size 8{R} } } {}, measured in Pascal^2-seconds.

We can write the matched filter operation in continuous time as

m ( t ) = t t + T y ( σ ) r ( σ t ) m ( t ) = t t + T y ( σ ) r ( σ t ) size 12{m $$t$$ = Int cSub { size 8{t} } cSup { size 8{t+T} } {y $$σ$$ r $$σ - t$$ dσ} } {}

y(σ)y(σ) size 12{y $$σ$$ } {}is the receiver time series. In response to a target echo that arrives at TDTD size 12{T rSub { size 8{D} } } {}seconds and without noise or reverberation, the receiver output is y(t)=e(tTD)y(t)=e(tTD) size 12{y $$t$$ =e $$t - T rSub { size 8{D} }$$ } {}. The output of the matched filter becomes:

m ( t ) = t t + T e ( σ T D ) r ( σ t ) = E R t t + T r ( σ T D ) r ( σ t ) m ( t ) = t t + T e ( σ T D ) r ( σ t ) = E R t t + T r ( σ T D ) r ( σ t ) size 12{m $$t$$ = Int cSub { size 8{t} } cSup { size 8{t+T} } {e $$σ - T rSub { size 8{D} }$$ r $$σ - t$$ dσ} = sqrt {E rSub { size 8{R} } } Int cSub { size 8{t} } cSup { size 8{t+T} } {r $$σ - T rSub { size 8{D} }$$ r $$σ - t$$ dσ} } {}

Hence m(TD)=ERm(TD)=ER size 12{m $$T rSub { size 8{D} }$$ = sqrt {E rSub { size 8{R} } } } {}. The peak power output of the matched filter, m2(t)m2(t) size 12{m rSup { size 8{2} } $$t$$ } {}, in response to a echo is ERER size 12{E rSub { size 8{R} } } {}.

We determine the matched filter response to noise next. Assume the input noise is white with variance AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}:

E n ( t ) n ( s ) = AN 0 δ ( t s ) E n ( t ) n ( s ) = AN 0 δ ( t s ) size 12{E left lbrace n $$t$$ n $$s$$ right rbrace = ital "AN" rSub { size 8{0} } δ $$t - s$$ } {}

Note that the delta function has units of inverse seconds, and therefore AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}has units of Pascals^2/Hz, equivalent to a spectral density. From the definition of stationary random process autocorrelations and power spectral density, we know that the Fourier transform of the autocorrelation is the spectral density function for the random process. The Fourier transform of covariance becomes ej2πfτAN0δ(τ)=AN0ej2πfτAN0δ(τ)=AN0 size 12{ Int {e rSup { size 8{j2πfτ} } } ital "AN" rSub { size 8{0} } δ $$τ$$ dτ= ital "AN" rSub { size 8{0} } } {}, which is the spectral density of the noise.

E m ( t ) m ( s ) = E t t + T n ( σ ) r ( σ t ) t t + T n ( β ) r ( β t ) = AN 0 t t + T t t + T δ ( σ β ) r ( σ t ) r ( β t ) dσdβ = AN 0 t t + T r 2 ( σ t ) = AN 0 E m ( t ) m ( s ) = E t t + T n ( σ ) r ( σ t ) t t + T n ( β ) r ( β t ) = AN 0 t t + T t t + T δ ( σ β ) r ( σ t ) r ( β t ) dσdβ = AN 0 t t + T r 2 ( σ t ) = AN 0 alignl { stack { size 12{E left lbrace m $$t$$ m $$s$$ right rbrace =E left lbrace Int cSub { size 8{t} } cSup { size 8{t+T} } {n $$σ$$ r $$σ - t$$ dσ} Int cSub { size 8{t} } cSup { size 8{t+T} } {n $$β$$ r $$β - t$$ dβ} right rbrace ={}} {} # ital "AN" rSub { size 8{0} } Int cSub { size 8{t} } cSup { size 8{t+T} } {} Int cSub { size 8{t} } cSup { size 8{t+T} } {δ $$σ - β$$ r $$σ - t$$ r $$β - t$$ dσdβ} = ital "AN" rSub { size 8{0} } Int cSub { size 8{t} } cSup { size 8{t+T} } {r rSup { size 8{2} } $$σ - t$$ dσ={}} ital "AN" rSub { size 8{0} } {} } } {}

Thus, the noise power response of a matched filter is the input spectral density, AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}.

We conclude that the signal to noise ratio (SNR) at the output of a matched filter is the ratio of the echo energy to the noise spectral density, ER/AN0ER/AN0 size 12{E rSub { size 8{R} } / ital "AN" rSub { size 8{0} } } {}. This assumes that the noise is white, e.g. a flat spectral density at the input to the matched filter. This is a general result, independent of the signal waveform details, except for its energy ERER size 12{E rSub { size 8{R} } } {}.

Discrete Time Matched Filters

Discrete time filters have nearly the same properties as continuous time filters. In discrete time, we assume an echo of e(k)=ERr(k)e(k)=ERr(k) size 12{e $$k$$ = sqrt {E rSub { size 8{R} } } r $$k$$ } {}, with k=1Tr2(k)=1k=1Tr2(k)=1 size 12{ Sum cSub { size 8{k=1} } cSup { size 8{T} } {r rSup { size 8{2} } $$k$$ } =1} {}. The discrete matched filter output to the input y(k) is given by:

m ( k ) = l = k k + T 1 y ( l ) r ( l k + 1 ) m ( k ) = l = k k + T 1 y ( l ) r ( l k + 1 ) size 12{m $$k$$ = Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } {y $$l$$ r $$l - k+1$$ } } {}

In response to the echo, y(k)=e(kTD)y(k)=e(kTD) size 12{y $$k$$ =e $$k - T rSub { size 8{D} }$$ } {}the output of the discrete time matched filter is

m ( k ) = l = k k + T 1 e ( l T D ) r ( l k + 1 ) = E R l = k k + T 1 r ( l T D ) r ( l k + 1 ) m ( k ) = l = k k + T 1 e ( l T D ) r ( l k + 1 ) = E R l = k k + T 1 r ( l T D ) r ( l k + 1 ) size 12{m $$k$$ = Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } {e $$l - T rSub { size 8{D} }$$ r $$l - k+1$$ } = sqrt {E rSub { size 8{R} } } Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } {r $$l - T rSub { size 8{D} }$$ r $$l - k+1$$ } } {}

Hence m(TD1)=ERm(TD1)=ER size 12{m $$T rSub { size 8{D} } - 1$$ = sqrt {E rSub { size 8{R} } } } {}. The peak power output of the matched filter, m2(t)m2(t) size 12{m rSup { size 8{2} } $$t$$ } {}, in response to a echo is ERER size 12{E rSub { size 8{R} } } {}.

We determine the discrete matched filter response to noise next. Assume the input noise is sampled white with variance AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}:

E n ( k ) n ( l ) = AN 0 δ kl E n ( k ) n ( l ) = AN 0 δ kl size 12{E left lbrace n $$k$$ n $$l$$ right rbrace = ital "AN" rSub { size 8{0} } δ rSub { size 8{ ital "kl"} } } {}

E m ( k ) m ( p ) = E l = k k + T 1 n ( l ) r ( l k + 1 ) i = p p + T 1 n ( i ) r ( i p + 1 ) = AN 0 l = k k + T 1 i = p p + T 1 δ li r ( l k + 1 ) r ( i p + 1 ) = AN 0 l = k k + T 1 r 2 ( l k + 1 ) = AN 0 E m ( k ) m ( p ) = E l = k k + T 1 n ( l ) r ( l k + 1 ) i = p p + T 1 n ( i ) r ( i p + 1 ) = AN 0 l = k k + T 1 i = p p + T 1 δ li r ( l k + 1 ) r ( i p + 1 ) = AN 0 l = k k + T 1 r 2 ( l k + 1 ) = AN 0 alignl { stack { size 12{E left lbrace m $$k$$ m $$p$$ right rbrace =E left lbrace Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } {n $$l$$ r $$l - k+1$$ Sum cSub { size 8{i=p} } cSup { size 8{p+T - 1} } {n $$i$$ r $$i - p+1$$ } } right rbrace ={}} {} # ital "AN" rSub { size 8{0} } Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } { Sum cSub { size 8{i=p} } cSup { size 8{p+T - 1} } {} } δ rSub { size 8{ ital "li"} } r $$l - k+1$$ r $$i - p+1$$ = ital "AN" rSub { size 8{0} } Sum cSub { size 8{l=k} } cSup { size 8{k+T - 1} } {r rSup { size 8{2} } $$l - k+1$$ } = ital "AN" rSub { size 8{0} } {} } } {}

Thus, the signal to noise ratio at the output of a discrete time matched filter is ER/AN0ER/AN0 size 12{E rSub { size 8{R} } / ital "AN" rSub { size 8{0} } } {}.

The matched filter compresses the echo signal to a pulse (or a series of pulses for waveforms such as SFM) with time width equal approximately to its inverse bandwidth, 1/BW.

Matched Filter Response to Reverberation

One model for reverberation assumes that the reverberation comes from distributed discrete scatterers, with density A(u)A(u) size 12{A $$u$$ } {}.

y(t)=ET0A(u)Γ(u)r(tτ(u))duy(t)=ET0A(u)Γ(u)r(tτ(u))du size 12{y $$t$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ r $$t - τ \( u$$ \) d} u} {},

A(u)A(u) size 12{A $$u$$ } {} is considered a random, spatial process that models the amplitude of the scattering that occurs at range u back to the receiver. We are assuming that the receiver has significant aperture, and that y(t) is the receiver response at the output of a beamformer. In this case, scattering is occurring from the patch of the ocean bottom or surface that lies at range u and within the receiver beamwidth in azimuth and elevation. Each patch of the bottom or surface will arrive at the receiver at a different time. Γ(u)Γ(u) size 12{Γ $$u$$ } {}is the transmission loss from the source to the scattering range (u) and back to the receiver. τ(u)τ(u) size 12{τ $$u$$ } {}is the total travel time from source to scatterer to receiver. As one can see, the reverberation is made up of many time delayed and amplitude scaled replicas of the transmitted waveform.

The matched filter response to the reverberation is

m R ( t ) = t t + T r ( σ t ) E T 0 A ( u ) Γ ( u ) r ( σ τ ( u ) ) d u = E T 0 A ( u ) Γ ( u ) t t + T r ( σ τ ( u ) ) r ( σ t ) dσd u m R ( t ) = t t + T r ( σ t ) E T 0 A ( u ) Γ ( u ) r ( σ τ ( u ) ) d u = E T 0 A ( u ) Γ ( u ) t t + T r ( σ τ ( u ) ) r ( σ t ) dσd u alignl { stack { size 12{m rSub { size 8{R} } $$t$$ = Int rSub { size 8{t} } rSup { size 8{t+T} } {r $$σ - t$$ sqrt {E rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ r $$σ - τ \( u$$ \) d} u} dσ={}} {} # sqrt {E rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ Int rSub { size 8{t} } rSup { size 8{t+T} } {r $$σ - τ \( u$$ \) r $$σ - t$$ } dσd} u {} } } {}

We define the transmitted waveform autocorrelation function as

χ ( τ ) = 0 T r ( t τ ) r ( t ) dt χ ( τ ) = 0 T r ( t τ ) r ( t ) dt size 12{χ $$τ$$ = Int rSub { size 8{0} } rSup { size 8{T} } {r $$t - τ$$ } r $$t$$ ital "dt"} {}

Recall, that by definition, χ(0)=0Tr2(t)dt=1χ(0)=0Tr2(t)dt=1 size 12{χ $$0$$ = Int rSub { size 8{0} } rSup { size 8{T} } {r rSup { size 8{2} } $$t$$ } ital "dt"=1} {}. In more general terms, we define the transmitted wideband signal ambiguity function as

χ WB ( τ , η ) = η r ( η ( t τ ) ) r ( t ) dt χ WB ( τ , η ) = η r ( η ( t τ ) ) r ( t ) dt size 12{χ rSub { size 8{ ital "WB"} } $$τ,η$$ = sqrt {η} Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {r rSup { size 8{*} } $$η \( t - τ$$ \) } r $$t$$ ital "dt"} {}

Note: Some authors define the ambiguity function as the magnitude squared value of this definition. Other authors choose different normalizations or the sign (-/+) on the delay term ττ size 12{τ} {}.

In the wideband signal ambiguity function, the Doppler effect is represented by the scaling factor ηη size 12{η} {}. In narrowband cases, the Doppler effect is represented by a frequency shift, φφ size 12{φ} {}. For a monostatic sonar the frequency shift is given by φ=2v/cφ=2v/c size 12{φ=2 ital "v/c"} {}, where vv size 12{v} {}is the radial velocity between the scattering object and the sonar system.

χ NB ( τ , φ ) = r ( t τ ) r ( t ) e j2 πφ t dt χ NB ( τ , φ ) = r ( t τ ) r ( t ) e j2 πφ t dt size 12{χ rSub { size 8{ ital "NB"} } $$τ,φ$$ = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {r rSup { size 8{*} } $$t - τ$$ } r $$t$$ e rSup { size 8{ - j2 ital "πφ"t} } ital "dt"} {}

One can show (Weiss) that the narrowband approximation to Doppler is valid if 2v/c<<1BT2v/c<<1BT size 12{2 ital "v/c""<<" { {1} over { ital "BT"} } } {}, where BB size 12{B} {}is the waveform bandwidth and TT size 12{T} {}is the duration. For one hundred (100) Hertz bandwidth waveforms that last for one (1) second, the speed of the target must be much less than 7.5 m/sec, or approximately 15 knots.

An important invariance property of the narrowband ambiguity function is that

χ NB ( τ , φ ) 2 dt = 1 χ NB ( τ , φ ) 2 dt = 1 size 12{ Int { lline χ rSub { size 8{ ital "NB"} } $$τ,φ$$ rline } rSup { size 8{2} } ital "dt"=1} {}

Using either definition of the signal ambiguity function we have

t t + T r ( σ τ ( u ) ) r ( σ t ) = 0 T r ( σ ' ( τ ( u ) t ) ) r ( σ ' ) d σ ' = χ ( τ ( u ) t , 0 ) t t + T r ( σ τ ( u ) ) r ( σ t ) = 0 T r ( σ ' ( τ ( u ) t ) ) r ( σ ' ) d σ ' = χ ( τ ( u ) t , 0 ) size 12{ Int rSub { size 8{t} } rSup { size 8{t+T} } {r $$σ - τ \( u$$ \) r $$σ - t$$ } dσ= Int rSub { size 8{0} } rSup { size 8{T} } {r $${ {σ}} sup { ' } - \( τ \( u$$ - t \) \) r $${ {σ}} sup { ' }$$ } d { {σ}} sup { ' }=χ $$τ \( u$$ - t,0 \) } {}

Therefore the matched filter response is

m R ( t ) = E T 0 A ( u ) Γ ( u ) χ ( τ ( u ) t , 0 ) d u m R ( t ) = E T 0 A ( u ) Γ ( u ) χ ( τ ( u ) t , 0 ) d u size 12{m rSub { size 8{R} } $$t$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ χ $$τ \( u$$ - t,0 \) d} u} {}

This expression for the matched filter response shows that the amount of reverberation at time t is directly related to the transmitted signal’s autocorrelation and energy source level (ESL). Wide band signals will have narrower autocorrelation peaks, and thus less reverberation amplitude.

Using the narrowband approximation for Doppler shifts allows efficient implementation of matched filter banks as generalized spectrogram analysis. One treats the replica as a “window function” in place of the more traditional Hanning or Hamming windows. The matched filter for narrowband Doppler shifts is given by:

m ( t , φ ) = t t + T y ( σ ) r ( σ t ) e j2 πφσ m ( t , φ ) = t t + T y ( σ ) r ( σ t ) e j2 πφσ size 12{m $$t,φ$$ = Int cSub { size 8{t} } cSup { size 8{t+T} } {y $$σ$$ r rSup { size 8{*} } $$σ - t$$ e rSup { size 8{ - j2 ital "πφσ"} } dσ} } {}

Which can be rewritten as:

m ( t , φ ) = e j2 πφ t y ( t + σ ' ) r ( σ ' ) e j2 πφ { σ ' d σ ' m ( t , φ ) = e j2 πφ t y ( t + σ ' ) r ( σ ' ) e j2 πφ { σ ' d σ ' size 12{m $$t,φ$$ =e rSup { size 8{j2 ital "πφ"t} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {y $$t+ { {σ}} sup { ' }$$ r rSup { size 8{*} } $${ {σ}} sup { ' }$$ e rSup { size 8{ - j2 ital "πφ {" ital {σ}} sup { ' }} } d { {σ}} sup { ' }} } {}

The spectrogram with window function w(σ)w(σ) size 12{w $$σ$$ } {} is given by:

y ( t + σ ' ) w ( σ ' ) e j2 πφ { σ ' d σ ' Φ ( t , φ ) = 2 y ( t + σ ' ) w ( σ ' ) e j2 πφ { σ ' d σ ' Φ ( t , φ ) = 2 size 12{Φ $$t,φ$$ = lline Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {y $$t+ { {σ}} sup { ' }$$ w $${ {σ}} sup { ' }$$ e rSup { size 8{ - j2 ital "πφ {" ital {σ}} sup { ' }} } d { {σ}} sup { ' }} rline rSup { size 8{2} } } {}

Hence, the squared envelope of the narrowband Doppler matched filter is a spectrogram with the window function being the conjugate of the transmitted waveform replica. This interpretation of narrowband matched filtering lends insight into the use of different window functions on transmitted waveforms. Often one adds a Hanning or Tukey window to the transmitted waveform. This windowing is necessary in some cases because the sonar transmitter cannot turn on and off instantly.

In spectral analysis, windows are used to control ‘spectral leakage’, which occurs because of the finite time window used for frequency analysis. Spectral leakage generates sidelobes from strong tones that mask low amplitude tones at different frequencies. In a matched filter using Doppler resolving waveforms, reverberation will be much stronger near zero Doppler than at other Doppler frequencies. The waveform windows help keep Doppler sidelobes of reverberation from masking the lower amplitude target echoes that may occur at high Doppler.

Using matched filters based on the narrowband approximation to process echoes with Doppler beyond the narrowband approximation limits will result in correlation loss. The correlation loss will result in a loss of Signal to Noise ratio for these echoes.

This presents a fundamental design decision for a sonar system that needs to process echoes with Doppler on the order of 15 knots or greater. If one uses “narrowband” processing for efficiency, then one has to limit the waveforms to those that satisfy the narrowband approximation. However, as shown in the earlier sections, having a larger bandwidth will reduce the autocorrelation time of the waveform and thus reduce the response to reverberation. This reverberation versus bandwidth property advocates the use of wideband waveforms and hence, broadband matched filtering. There are, however, waveforms that have low correlation loss across all Doppler shifts. These are known as hyperbolic frequency modulation (HFM) waveforms.

So, one can use narrowband processing, and restrict the waveforms to low bandwidth (1 Hertz say) waveforms such as a pulsed sine wave, and wideband waveforms with high Doppler tolerance, such as HFM. If one wants to use waveforms that have Doppler resolving power and high bandwidth, one needs to use broadband matched filtering.

Doppler Sensitive Waveform Matched Filtering

In some cases, one uses a signal that is Doppler sensitive, e.g. the signal ambiguity function χ(τ,η)χ(τ,η) size 12{χ $$τ,η$$ } {}is a strong function of the Doppler variable. Examples of these waveforms are CW, SFM and comb waveforms. Other waveforms are less sensitive to Doppler effects, beyond a time delay/Doppler coupling effect.

In the cases where one is using Doppler sensitive waveforms, the matched filter is generalized to a matched filter bank, indexed by both time (range) and Doppler ( η)η) size 12{η \) } {}

m ( t , η ) = η t t + T / η y ( σ ) r ( η ( σ t ) ) m ( t , η ) = η t t + T / η y ( σ ) r ( η ( σ t ) ) size 12{m $$t,η$$ = sqrt {η} Int cSub { size 8{t} } cSup { size 8{t+T/η} } {y $$σ$$ r $$η \( σ - t$$ \) dσ} } {}

This allows one to search for targets that have relative motion to the source and receivers of the active sonar. When the received signal is a Doppler scaled echo, then the filter that matches the echo Doppler will be a matched filter for that echo, and obey the same SNR properties for echoes embedded in noise as the earlier discussion for stationary targets echoes. The replica is matched to the echo compression and time delay. To ensure energy consistency, we scale the zero Doppler replica r(t)r(t) size 12{r $$t$$ } {} by ηη size 12{ sqrt {η} } {} when using it for other Doppler hypotheses. This comes from the fact that

0 T / η r 2 ( ηt ) dt = 1 / η 0 T / η r 2 ( ηt ) dt = 1 / η size 12{ Int cSub { size 8{0} } cSup { size 8{T/η} } {r rSup { size 8{2} } $$ηt$$ ital "dt"} =1/η} {}

Now, using the reverberation model as before, we have the following expression for the matched filter bank response to reverberation:

m R ( t , η ) = ηE T 0 A ( u ) Γ ( u ) t t + T / η r ( σ τ ( u ) ) r ( η ( σ t ) ) dσd u m R ( t , η ) = ηE T 0 A ( u ) Γ ( u ) t t + T / η r ( σ τ ( u ) ) r ( η ( σ t ) ) dσd u size 12{m rSub { size 8{R} } $$t,η$$ = sqrt {ηE rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ Int rSub { size 8{t} } rSup { size 8{t+T/η} } {r $$σ - τ \( u$$ \) r $$η \( σ - t$$ \) } dσd} u} {}

The integral including the replicas, using the change of variables σ'=σtσ'=σt size 12{ { {σ}} sup { ' }=σ - t} {}, can be written as

Iδ,η=0T/ηr(σ'+δ)r(ησ')dσ'Iδ,η=0T/ηr(σ'+δ)r(ησ')dσ' size 12{I rSub { size 8{δ,η} } = Int rSub { size 8{0} } rSup { size 8{T/η} } {r $${ {σ}} sup { ' }+δ$$ r $$η { {σ}} sup { ' }$$ d} { {σ}} sup { ' }} {}, where δ=tτ(u)δ=tτ(u) size 12{δ=t - τ $$u$$ } {}

We can extend the limits of the integral, because r(ησ')r(ησ') size 12{r $$η { {σ}} sup { ' }$$ } {}is zero outside the integration limits. Extending limits and changing variables to σ''=σ'+δσ''=σ'+δ size 12{ { {σ}} sup { '' }= { {σ}} sup { ' }+δ} {}yields

I δ , η = r ( σ ' ' ) r ( η ( σ ' ' δ ) ) d σ ' ' = χ ( δ , η ) / η I δ , η = r ( σ ' ' ) r ( η ( σ ' ' δ ) ) d σ ' ' = χ ( δ , η ) / η size 12{I rSub { size 8{δ,η} } = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {r $${ {σ}} sup { '' }$$ r $$η \( { {σ}} sup { '' } - δ$$ \) d} { {σ}} sup { '' }=χ $$δ,η$$ /η} {}

Hence the range Doppler matched filter bank response to reverberation becomes

m R ( t , η ) = E T 0 A ( u ) Γ ( u ) χ ( t τ ( u ) , η ) d u m R ( t , η ) = E T 0 A ( u ) Γ ( u ) χ ( t τ ( u ) , η ) d u size 12{m rSub { size 8{R} } $$t,η$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ χ $$t - τ \( u$$ ,η \) d} u} {}

This expression shows that if the target has Doppler, and one uses a replica matched to the target Doppler, further suppression of reverberation is possible. This suppression of reverberation is without loss to matching the target echo or loss in noise limited performance.

Choosing waveforms that are broadband, and with roll-off with respect to Doppler in its signal ambiguity function, will optimize the active sonar’s processing in reverberation. For some waveforms (such as the Sinusoidal Frequency Modulation pulse) the signal ambiguity function will have multiple time delay peaks, termed ‘fingers’, due to the periodic structure of the pulse. Each autocorrelation finger has a time width approximately equal to the signal’s bandwidth. This will increase the reverberation response, relative to a waveform with a single autocorrelation response. However, the SFM waveform has a roll-off in Doppler as well. So for targets that have Doppler, there can be a Doppler shift where the roll-off in Doppler more than compensates for the additional autocorrelation peaks.

We will make the assumption that A(u) is wide sense stationary, that is its statistics are invariant over the range of u:

E A ( u ) A ( v ) = R A ( u v ) E A ( u ) A ( v ) = R A ( u v ) size 12{E left lbrace A rSup { size 8{*} } $$u$$ A $$v$$ right rbrace =R rSub { size 8{A} } $$u - v$$ } {}

Furthermore, we will assume that A(u) is spatially white, e.g. the scattering elements are uncorrelated with each other:

E A ( u ) A ( v ) = R A ( u v ) = R A δ ( u v ) E A ( u ) A ( v ) = R A ( u v ) = R A δ ( u v ) size 12{E left lbrace A $$u$$ rSup { size 8{*} } A $$v$$ right rbrace =R rSub { size 8{A} } $$u - v$$ =R rSub { size 8{A} } δ $$u - v$$ } {}

Now, these two assumptions, that the reflection coefficient statistics are independent of range and each differential patch is statistically independent of each other is only an approximation to the real situation. However, these approximations allow one to see the interaction of reverberation and waveform selection.

E m R 2 ( t , η ) = E E T 0 A ( u ) Γ ( u ) χ ( t τ ( u ) , η ) d u 0 A ( φ ) Γ ( φ ) χ ( t τ ( φ ) , η ) d φ E m R 2 ( t , η ) = E E T 0 A ( u ) Γ ( u ) χ ( t τ ( u ) , η ) d u 0 A ( φ ) Γ ( φ ) χ ( t τ ( φ ) , η ) d φ size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E left lbrace E rSub { size 8{T} } Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$u$$ Γ $$u$$ χ $$t - τ \( u$$ ,η \) d} u Int cSub { size 8{0} } cSup { size 8{ infinity } } {A $$φ$$ Γ $$φ$$ χ $$t - τ \( φ$$ ,η \) d} φ right rbrace } {}

Rearranging,

E m R 2 ( t , η ) = E T 0 0 E A ( u ) A ( φ ) Γ ( u ) Γ ( φ ) χ ( t τ ( u ) , η ) χ ( t τ ( φ ) , η ) d ud φ E m R 2 ( t , η ) = E T 0 0 E A ( u ) A ( φ ) Γ ( u ) Γ ( φ ) χ ( t τ ( u ) , η ) χ ( t τ ( φ ) , η ) d ud φ size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } Int cSub { size 8{0} } cSup { size 8{ infinity } } { Int cSub { size 8{0} } cSup { size 8{ infinity } } {E left lbrace A $$u$$ A $$φ$$ right rbrace Γ $$u$$ Γ $$φ$$ χ $$t - τ \( u$$ ,η \) χ $$t - τ \( φ$$ ,η \) d} } ital "ud"φ} {}

Using the covariance of the scattering elements we get,

E m R 2 ( t , η ) = E T 0 0 R A δ ( u φ ) Γ ( u ) Γ ( φ ) χ ( t τ ( u ) , η ) χ ( t τ ( φ ) , η ) d ud φ E m R 2 ( t , η ) = E T 0 0 R A δ ( u φ ) Γ ( u ) Γ ( φ ) χ ( t τ ( u ) , η ) χ ( t τ ( φ ) , η ) d ud φ size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } Int cSub { size 8{0} } cSup { size 8{ infinity } } { Int cSub { size 8{0} } cSup { size 8{ infinity } } {R rSub { size 8{A} } δ $$u - φ$$ Γ $$u$$ Γ $$φ$$ χ $$t - τ \( u$$ ,η \) χ $$t - τ \( φ$$ ,η \) d} } ital "ud"φ} {}

Or,

E m R 2 ( t , η ) = E T R A 0 Γ ( u ) 2 χ 2 ( t τ ( u ) , η ) d u E m R 2 ( t , η ) = E T R A 0 Γ ( u ) 2 χ 2 ( t τ ( u ) , η ) d u size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } R rSub { size 8{A} } Int cSub { size 8{0} } cSup { size 8{ infinity } } { lline Γ $$u$$ rline rSup { size 8{2} } χ rSup { size 8{2} } $$t - τ \( u$$ ,η \) d} u} {}

To see this more clearly, assume that the transmission loss term Γ(u)Γ(u) size 12{Γ $$u$$ } {}is approximately constant over the transmitted signal’s correlation time and receiver’s beam pattern. Then we obtain

E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 0 χ 2 ( t τ ( u ) , η ) d u E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 0 χ 2 ( t τ ( u ) , η ) d u size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } R rSub { size 8{A} } lline Γ $$u rSub { size 8{0} }$$ rline rSup { size 8{2} } Int cSub { size 8{0} } cSup { size 8{ infinity } } {χ rSup { size 8{2} } $$t - τ \( u$$ ,η \) d} u} {}

Where u0u0 size 12{u rSub { size 8{0} } } {}is defined by τ(u0)=tτ(u0)=t size 12{τ $$u rSub { size 8{0} }$$ =t} {}.

If we assume that the time delay varies smoothly with respect to range, we can replace the integration over u with an integration over time delay ττ size 12{τ} {}, where we assume that the chance of variable from u to ττ size 12{τ} {}is approximately given by τ=2u/cτ=2u/c size 12{τ=2u/c} {}, where c is the speed of sound. This is assuming an approximate monostatic geometry, or that the patch of reverberation is far away relative to the source receiver separation.

We then get

E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 c / 2 0 χ 2 ( t τ , η ) d τ E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 c / 2 0 χ 2 ( t τ , η ) d τ size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } R rSub { size 8{A} } lline Γ $$u rSub { size 8{0} }$$ rline rSup { size 8{2} } c/2 Int cSub { size 8{0} } cSup { size 8{ infinity } } {χ rSup { size 8{2} } $$t - τ,η$$ d} τ} {}

If we assume that the matched filter time t is greater than the signal duration T, then letting τ'=tττ'=tτ size 12{ { {τ}} sup { ' }=t - τ} {}, we obtain

E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 c / 2 χ 2 ( t τ , η ) d τ E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 c / 2 χ 2 ( t τ , η ) d τ size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } R rSub { size 8{A} } lline Γ $$u rSub { size 8{0} }$$ rline rSup { size 8{2} } c/2 Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {χ rSup { size 8{2} } $$t - τ,η$$ d} τ} {}

We define the Q-function of the waveform as

Q ( η ) = χ ( τ ' , η ) 2 d τ ' Q ( η ) = χ ( τ ' , η ) 2 d τ ' size 12{Q $$η$$ = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { lline χ $${ {τ}} sup { ' },η$$ rline rSup { size 8{2} } d} { {τ}} sup { ' }} {}

Note that Q(η)Q(η) size 12{Q $$η$$ } {}has units of seconds^2. We call a waveform with a sharp peak in Q(η)Q(η) size 12{Q $$η$$ } {}as a Doppler Sensitive Waveform (DSW). A sine wave pulse will have a sharp peak in Q(η)Q(η) size 12{Q $$η$$ } {} for instance.

When the narrowband ambiguity function is used the Q function is normalized:

Q NB ( φ ) = χ NB ( τ ' , φ ) 2 d τ ' = 1 Q NB ( φ ) = χ NB ( τ ' , φ ) 2 d τ ' = 1 size 12{ Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {Q rSub { size 8{ ital "NB"} } $$φ$$ } dφ= Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { lline χ rSub { size 8{ ital "NB"} } $${ {τ}} sup { ' },φ$$ rline rSup { size 8{2} } d} { {τ}} sup { ' }dφ=1} {}

The wideband waveform Q function is approximately normalized to unity.

The reverberation response can be written as

E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 Q ( η ) c / 2 E m R 2 ( t , η ) = E T R A Γ ( u 0 ) 2 Q ( η ) c / 2 size 12{E left lbrace m rSub { size 8{R} } rSup { size 8{2} } $$t,η$$ right rbrace =E rSub { size 8{T} } R rSub { size 8{A} } lline Γ $$u rSub { size 8{0} }$$ rline rSup { size 8{2} } Q $$η$$ c/2} {}

Clearly, the best waveform to use for detection depends on the assumed target velocity. Waveforms such as HFM and LFM have low Q-functions that are relatively constant across Doppler. Doppler sensitive waveforms often have lower Q-functions at higher Doppler shifts than LFM and HFM, much higher Q functions near zero Doppler. To best search for targets, one needs waveforms optimized for both low and high Doppler targets.

So far, this has been a deterministic description of the matched filter response to reverberation.

Channel Doppler Effects on Reverberation

In reality, the reflection coefficient or the transmission loss term will be time varying (as well as spatially varying) because of the surface of the ocean having waves, and the internal thermal structure of the ocean channel will be time varying.

For bottom reverberation, we will assume that the reflection coefficient is time invariant. In shallow water at low frequencies (< 2000 Hz, say) the bottom reverberation dominates over surface reverberation. However, the acoustic propagation through the sound channel and specular reflection from the ocean surface introduces a time varying component to the reverberation formation process.

To derive the results needed for channel Doppler effects, we will restrict ourselves to the narrowband model.

The matched filter is given by:

m ( t , φ ) = t t + T y ( σ ) r ( σ t ) e j2 πφσ m ( t , φ ) = t t + T y ( σ ) r ( σ t ) e j2 πφσ size 12{m $$t,φ$$ = Int cSub { size 8{t} } cSup { size 8{t+T} } {y $$σ$$ r rSup { size 8{*} } $$σ - t$$ e rSup { size 8{ - j2 ital "πφσ"} } dσ} } {}

Since r(t)=0r(t)=0 size 12{r $$t$$ =0} {}for t<0t<0 size 12{t<0} {}and t>Tt>T size 12{t>T} {}, we extend the limits of integration for the matched filter response to:

m ( t , φ ) = y ( σ ) r ( σ t ) e j2 πφσ m ( t , φ ) = y ( σ ) r ( σ t ) e j2 πφσ size 12{m $$t,φ$$ = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {y $$σ$$ r rSup { size 8{*} } $$σ - t$$ e rSup { size 8{ - j2 ital "πφσ"} } dσ} } {}

We define the effects of reverberation, targets, clutter and the acoustic channel, via a spreading function S(τ,φ)S(τ,φ) size 12{S $$τ,φ$$ } {} acting on the transmitted waveform:

y ( t ) = E T S ( τ , φ ) e j2 πφ t r ( t τ ) dτdφ y ( t ) = E T S ( τ , φ ) e j2 πφ t r ( t τ ) dτdφ size 12{y $$t$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,φ$$ e rSup { size 8{j2 ital "πφ"t} } r $$t - τ$$ dτdφ} } } {}

This expression does not include contributions of ambient noise, only scattering phenomena. The spreading function S(τ,φ)S(τ,φ) size 12{S $$τ,φ$$ } {}defines the acoustic scattering, as a function of delay ττ size 12{τ} {}and Doppler shift φφ size 12{φ} {} for the sonar reception. The spreading function is a random variable, changing due to surface waves and time varying refraction effects (internal waves) in the sound channel.

Target echoes will have a small ττ size 12{τ} {}region of non-zero spreading function, STarget(τ,φ)STarget(τ,φ) size 12{S rSub { size 8{"Target"} } $$τ,φ$$ } {}. Reverberation will have an extended ττ size 12{τ} {} region with significant SReverb(τ,φ)SReverb(τ,φ) size 12{S rSub { size 8{"Reverb"} } $$τ,φ$$ } {}. The Doppler shift for both reverberation and targets will be related to receiver and source motion, as well as Doppler spreading due to surface and internal waves. The target will have additional Doppler contributions from its own motion.

Substituting the spreading function description to the sonar response into the matched filter we obtain

m ( t , φ ) = E T S ( τ , δ ) e j2 πδσ r ( σ τ ) r ( σ t ) e j2 πφσ dτdδ m ( t , φ ) = E T S ( τ , δ ) e j2 πδσ r ( σ τ ) r ( σ t ) e j2 πφσ dτdδ size 12{m $$t,φ$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,δ$$ e rSup { size 8{j2 ital "πδσ"} } r $$σ - τ$$ } } r rSup { size 8{*} } $$σ - t$$ e rSup { size 8{ - j2 ital "πφσ"} } dσ} dτdδ} {}

Which equals

m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) σ r ( σ τ ) r ( σ t ) dτdδ m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) σ r ( σ τ ) r ( σ t ) dτdδ size 12{m $$t,φ$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,δ$$ e rSup { size 8{ - j2π $$φ - δ$$ σ} } r $$σ - τ$$ } } r rSup { size 8{*} } $$σ - t$$ dσ} dτdδ} {}

Letting σ'=σtσ'=σt size 12{ { {σ}} sup { ' }=σ - t} {}, we obtain

m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) ( σ ' + τ ) r ( σ ' ) r ( σ ' ( t τ ) ) dτdδ m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) ( σ ' + τ ) r ( σ ' ) r ( σ ' ( t τ ) ) dτdδ size 12{m $$t,φ$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,δ$$ e rSup { size 8{ - j2π $$φ - δ$$ $${ {σ}} sup { ' }+τ$$ } } r $${ {σ}} sup { ' }$$ } } r rSup { size 8{*} } $${ {σ}} sup { ' } - \( t - τ$$ \) dσ} dτdδ} {}

Using the definition of the narrowband ambiguity function, the matched filter response becomes

m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) τ χ ( t τ , φ δ ) dτdδ m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) τ χ ( t τ , φ δ ) dτdδ size 12{m $$t,φ$$ = sqrt {E rSub { size 8{T} } } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,δ$$ e rSup { size 8{ - j2π $$φ - δ$$ τ} } χ $$t - τ,φ - δ$$ dτdδ} } } {}

The response of the matched filter is a “twisted convolution” of the spreading function and the waveform ambiguity function. The exponential ej2π(φδ)ej2π(φδ) size 12{e rSup { size 8{ - j2π $$φ - δ$$ } } } {}performs the twisting. Note that if the waveform ambiguity function was “perfect”, that is a single peak,

χ ( τ , φ ) = δ ( τ ) δ ( φ ) χ ( τ , φ ) = δ ( τ ) δ ( φ ) size 12{χ $$τ,φ$$ =δ $$τ$$ δ $$φ$$ } {}

Then the matched filter response would become:

m ( t , φ ) = E T S ( t , φ ) + n ( t , φ ) m ( t , φ ) = E T S ( t , φ ) + n ( t , φ ) size 12{m $$t,φ$$ = sqrt {E rSub { size 8{T} } } S $$t,φ$$ +n $$t,φ$$ } {}

Where n(t,φ)n(t,φ) size 12{n $$t,φ$$ } {} is the response of the matched filter to ambient noise. In this sense, the matched filter is estimating the spreading function of the channel, with targets, clutter and reverberation all part of the spreading function. Note, however that χ(0,0)=1χ(0,0)=1 size 12{χ $$0,0$$ =1} {} , so the ambiguity function cannot become a delta function.

Now, the power output of the matched filter is desired, so that Signal to Interference Ratios and similar quantities can be predicted. We will make statistical assumptions about the spreading function. The assumptions are that the spreading function is wide sense stationary and uncorrelated. This implies that the signals being processed are statistically stationary and that the scatterers are uncorrelated; so that (Van Trees, III, Ch 13):

E S ( τ , φ ) S ( τ ' , φ ' ) = R SS ( τ , φ ) δ ( τ τ ' ) δ ( φ φ ' ) E S ( τ , φ ) S ( τ ' , φ ' ) = R SS ( τ , φ ) δ ( τ τ ' ) δ ( φ φ ' ) size 12{E left lbrace S $$τ,φ$$ S $${ {τ}} sup { ' }, { {φ}} sup { ' }$$ right rbrace =R rSub { size 8{ ital "SS"} } $$τ,φ$$ δ $$τ - { {τ}} sup { ' }$$ δ $$φ - { {φ}} sup { ' }$$ } {}

RSS(τ,φ)RSS(τ,φ) size 12{R rSub { size 8{ ital "SS"} } $$τ,φ$$ } {}is known as the scattering function of the active sonar scenario. The description of the target, reverberation and clutter statistics are captured in this expression.

Using this definition, we obtain for the power of the matched filter:

E m ( t , φ ) m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) τ χ ( t τ , φ δ ) dτdδ S ( τ ' , δ ' ) e j2π ( φ δ ' ) τ ' χ ( t τ ' , φ δ ' ) d τ ' d δ ' E m ( t , φ ) m ( t , φ ) = E T S ( τ , δ ) e j2π ( φ δ ) τ χ ( t τ , φ δ ) dτdδ S ( τ ' , δ ' ) e j2π ( φ δ ' ) τ ' χ ( t τ ' , φ δ ' ) d τ ' d δ ' alignl { stack { size 12{E left lbrace m $$t,φ$$ m rSup { size 8{*} } $$t,φ$$ right rbrace ={}} {} # E rSub { size 8{T} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S $$τ,δ$$ e rSup { size 8{ - j2π $$φ - δ$$ τ} } χ $$t - τ,φ - δ$$ dτdδ} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {S rSup { size 8{*} } $${ {τ}} sup { ' }, { {δ}} sup { ' }$$ e rSup { size 8{j2π $$φ - { {δ}} sup { ' }$$ { {τ}} sup { ' }} } χ rSup { size 8{*} } $$t - { {τ}} sup { ' },φ - { {δ}} sup { ' }$$ d { {τ}} sup { ' }d { {δ}} sup { ' }} } {} } } {}

Which becomes

E m ( t , φ ) 2 = E T R SS ( τ , δ ) χ ( t τ , φ δ ) 2 dτdδ E m ( t , φ ) 2 = E T R SS ( τ , δ ) χ ( t τ , φ δ ) 2 dτdδ size 12{E left lbrace lline m $$t,φ$$ rline rSup { size 8{2} } right rbrace =E rSub { size 8{T} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {R rSub { size 8{ ital "SS"} } $$τ,δ$$ lline χ $$t - τ,φ - δ$$ rline rSup { size 8{2} } dτdδ} } } {}

Using two dimensional convolution notation **, this expression for the matched filter power P(t,φ)P(t,φ) size 12{P $$t,φ$$ } {} becomes

P ( t , φ ) = E m ( t , φ ) 2 = E T R SS ( t , φ ) ** χ ( t , φ ) 2 P ( t , φ ) = E m ( t , φ ) 2 = E T R SS ( t , φ ) ** χ ( t , φ ) 2 size 12{P $$t,φ$$ =E left lbrace lline m $$t,φ$$ rline rSup { size 8{2} } right rbrace =E rSub { size 8{T} } R rSub { size 8{ ital "SS"} } $$t,φ$$ "**" lline χ $$t,φ$$ rline rSup { size 8{2} } } {}

Now, let us model the acoustic sonar problem as target, clutter/reverberation and noise. The matched filter power response to ambient noise was shown earlier to be AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}. The overall signal to interference ratio, for a target at range tt size 12{t} {}and Doppler φφ size 12{φ} {} is

SIR = P Target ( t , φ ) P Reverb/clutter ( t , φ ) + AN 0 SIR = P Target ( t , φ ) P Reverb/clutter ( t , φ ) + AN 0 size 12{ ital "SIR"= { {P rSub { size 8{"Target"} } $$t,φ$$ } over {P rSub { size 8{"Reverb/clutter"} } $$t,φ$$ + ital "AN" rSub { size 8{0} } } } } {}

Which becomes,

SIR ( t , φ ) = R SS Target ( t , φ ) ** χ ( t , φ ) 2 R SS Reverb/Clutter ( t , φ ) ** χ ( t , φ ) 2 + AN 0 E T SIR ( t , φ ) = R SS Target ( t , φ ) ** χ ( t , φ ) 2 R SS Reverb/Clutter ( t , φ ) ** χ ( t , φ ) 2 + AN 0 E T size 12{ ital "SIR" $$t,φ$$ = { {R rSub { size 8{ {} rSub { size 6{ ital "SS"} } } } rSup {"Target"} size 12{ $$t,φ$$ "**" lline χ $$t,φ$$ rline rSup {2} }} over { size 12{R rSub { {} rSub { size 6{ ital "SS"} } } rSup {"Reverb/Clutter"} size 12{ $$t,φ$$ "**" lline χ $$t,φ$$ rline rSup {2} } size 12{+ { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } } {}

This expression can be simplified for different target and scattering conditions. For a point target at range t0t0 size 12{t rSub { size 8{0} } } {}and Doppler φ0φ0 size 12{φ rSub { size 8{0} } } {}, the target scattering function becomes

RSSTarget(t,φ)=(tt0)δ(φφ0)RSSTarget(t,φ)=(tt0)δ(φφ0) size 12{R rSub { size 8{ ital "SS"} } rSup { size 8{"Target"} } $$t,φ$$ =Sδ $$t - t rSub { size 8{0} }$$ δ $$φ - φ rSub { size 8{0} }$$ } {}.

Often, we can assume that the reverberation scattering function is constant in the vicinity of the target, so that

R SS Reverb/Clutter ( t , φ ) = R t 0 Q Revrb ( φ ) R SS Reverb/Clutter ( t , φ ) = R t 0 Q Revrb ( φ ) size 12{R rSub { size 8{ ital "SS"} } rSup { size 8{"Reverb/Clutter"} } $$t,φ$$ =R rSub { size 8{t rSub { size 6{0} } } } Q rSub {"Revrb"} size 12{ $$φ$$ }} {}

QRevrb(φ)QRevrb(φ) size 12{Q rSub { size 8{"Revrb"} } $$φ$$ } {} describes the Doppler roll-off of the reverberation. It will be affected by the source, receiver and ocean motion. In this characterization, we are ignoring “discrete clutter”, e.g. target like responses from bottom features.

We will assume that QRevrb(φ)QRevrb(φ) size 12{Q rSub { size 8{"Revrb"} } $$φ$$ } {}is normalized, so that:

Q Revrb ( φ ) = 1 Q Revrb ( φ ) = 1 size 12{ Int {Q rSub { size 8{"Revrb"} } $$φ$$ dφ=1} } {}

With these assumptions we obtain

SIR ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ δ ) χ ( τ , φ ) 2 dτdδ + AN 0 E T SIR ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ δ ) χ ( τ , φ ) 2 dτdδ + AN 0 E T size 12{ ital "SIR" $$t,φ$$ = { {S rSup {"Target"} size 12{ $$t rSub {0} } size 12{,φ rSub {0} } size 12{$$ lline χ $$t - t rSub {0} size 12{,φ - φ rSub {0} } size 12{$$ } rline rSup {2} }} over { size 12{R rSub { {} rSub { size 6{t rSub {0} } } } iInt {} size 12{Q rSub {"Reverb"} } size 12{ $$φ - δ$$ lline χ $$τ,φ$$ rline rSup {2} } size 12{dτdδ+ { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } } {}

Note that when the matched filter is matched in time and Doppler, then t=t0t=t0 size 12{t=t rSub { size 8{0} } } {}and φ=φ0φ=φ0 size 12{φ=φ rSub { size 8{0} } } {}, and the numerator is maximized:

SIR ( t 0 , φ 0 ) = S Target ( t 0 , φ 0 ) R t 0 Q Reverb ( φ 0 δ ) χ ( τ , φ 0 ) 2 dτdδ + AN 0 E T SIR ( t 0 , φ 0 ) = S Target ( t 0 , φ 0 ) R t 0 Q Reverb ( φ 0 δ ) χ ( τ , φ 0 ) 2 dτdδ + AN 0 E T size 12{ ital "SIR" $$t rSub { size 8{0} } ,φ rSub { size 8{0} }$$ = { {S rSup {"Target"} size 12{ $$t rSub {0} } size 12{,φ rSub {0} } size 12{$$ }} over {R rSub { {} rSub { size 6{t rSub {0} } } } iInt {} size 12{Q rSub {"Reverb"} } size 12{ $$φ rSub {0} } size 12{ - δ$$ lline χ $$τ,φ rSub {0} size 12{$$ } rline rSup {2} } size 12{dτdδ+ { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } {}

The denominator can be simplified further by using the definition of the waveform Q function:

Q ( φ ) = χ ( τ , φ ) 2 Q ( φ ) = χ ( τ , φ ) 2 size 12{Q $$φ$$ = Int { lline χ $$τ,φ$$ rline } rSup { size 8{2} } dτ} {}

Using this definition, we obtain:

SIR ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ ) Q ( φ ) + AN 0 E T SIR ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ ) Q ( φ ) + AN 0 E T size 12{ ital "SIR" $$t,φ$$ = { {S rSup {"Target"} size 12{ $$t rSub {0} } size 12{,φ rSub {0} } size 12{$$ lline χ $$t - t rSub {0} size 12{,φ - φ rSub {0} } size 12{$$ } rline rSup {2} }} over { size 12{R rSub { {} rSub { size 6{t rSub {0} } } } size 12{Q rSub {"Reverb"} } size 12{ $$φ$$ *Q $$φ$$ + { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } } {}

When using a waveform with a Q function much wider than the reverberation Q function, (such as an HFM), the waveform Q function can be replaced by a constant, such as QHFMQHFM size 12{Q rSub { size 8{ ital "HFM"} } } {}. The convolution with the reverberation Q function becomes the constant QHFMQHFM size 12{Q rSub { size 8{ ital "HFM"} } } {}(because of the normalization of QReverbQReverb size 12{Q rSub { size 8{"Reverb"} } } {}) :

Q Reverb ( φ ) Q ( φ ) = Q HFM Q Reverb ( φ φ ' ) ' = Q HFM Q Reverb ( φ ) Q ( φ ) = Q HFM Q Reverb ( φ φ ' ) ' = Q HFM size 12{Q rSub { size 8{"Reverb"} } $$φ$$ *Q $$φ$$ =Q rSub { size 8{ ital "HFM"} } Int {Q rSub { size 8{"Reverb"} } $$φ - φ rSup { size 8{'} }$$ } dφ rSup { size 8{'} } =Q rSub { size 8{ ital "HFM"} } } {}

With this approximation the signal to interference ratio becomes:

SIR HFM ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q HFM + AN 0 E T SIR HFM ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q HFM + AN 0 E T size 12{ ital "SIR" rSub { size 8{ ital "HFM"} } $$t,φ$$ = { {S rSup {"Target"} size 12{ $$t rSub {0} } size 12{,φ rSub {0} } size 12{$$ lline χ $$t - t rSub {0} size 12{,φ - φ rSub {0} } size 12{$$ } rline rSup {2} }} over { size 12{R rSub { {} rSub { size 6{t rSub {0} } } } size 12{Q rSub { ital "HFM"} } size 12{+ { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } } {}

When the waveform is Doppler sensitive, and it’s Q function is narrower than the Reverberation Q function, we can approximate the waveform Q function by a rectangle of height T and width 1/T centered at zero Doppler. Than the convolution becomes:

Q Reverb ( φ ) Q ( φ ) = Q Reverb ( φ φ ' ) Q DSW ( φ ' ) ' 1 / 2T 1 / 2T Q Reverb ( φ φ ' ) Td φ ' Q Re verb ( φ ) Q Reverb ( φ ) Q ( φ ) = Q Reverb ( φ φ ' ) Q DSW ( φ ' ) ' 1 / 2T 1 / 2T Q Reverb ( φ φ ' ) Td φ ' Q Re verb ( φ ) size 12{Q rSub { size 8{"Reverb"} } $$φ$$ *Q $$φ$$ = Int {Q rSub { size 8{"Reverb"} } $$φ - φ rSup { size 8{'} }$$ } Q rSub { size 8{ ital "DSW"} } $$φ rSup { size 8{'} }$$ dφ rSup { size 8{'} } approx Int cSub { size 8{ - 1/2T} } cSup { size 8{1/2T} } {Q rSub { size 8{"Reverb"} } $$φ - φ rSup { size 8{'} }$$ } ital "Td"φ rSup { size 8{'} } approx Q rSub { size 8{"Re" ital "verb"} } $$φ$$ } {}

Therefore, the signal to interference ratio becomes

SIR DSW ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ ) + AN 0 E T SIR DSW ( t , φ ) = S Target ( t 0 , φ 0 ) χ ( t t 0 , φ φ 0 ) 2 R t 0 Q Reverb ( φ ) + AN 0 E T size 12{ ital "SIR" rSub { size 8{ ital "DSW"} } $$t,φ$$ = { {S rSup {"Target"} size 12{ $$t rSub {0} } size 12{,φ rSub {0} } size 12{$$ lline χ $$t - t rSub {0} size 12{,φ - φ rSub {0} } size 12{$$ } rline rSup {2} }} over { size 12{R rSub { {} rSub { size 6{t rSub {0} } } } size 12{Q rSub {"Reverb"} } size 12{ $$φ$$ + { { ital "AN" rSub {0} } over { size 12{E rSub {T} } } } }} } } } {}

We see that the best waveform for enhancing signal to interference ratio depends on the environmental Q function, and the assumed target Doppler.

The Active Sonar Equation

The active sonar equation expresses the signal excess (SE) which is the part of the target signal to noise ratio that exceeds the sonar’s detection threshold (DT). In decibel quantities, it is given by:

SE = ESL + TS TL ST TL TR ( RL 0 AN 0 ) DT SE = ESL + TS TL ST TL TR ( RL 0 AN 0 ) DT size 12{ ital "SE"= ital "ESL"+ ital "TS" - ital "TL" rSub { size 8{ ital "ST"} } - ital "TL" rSub { size 8{ ital "TR"} } - $$ital "RL" rSub { size 8{0} } ⊕ ital "AN" rSub { size 8{0} }$$ - ital "DT"} {}

We are assuming that the active sonar uses a matched filter for detection. In the sonar equation, the transmitted energy signal level (ESL) is the sound pressure squared and integrated over the transmitted pulse length. The energy of the received echo (known as the echo energy level) is 10*log10(ER)=ESL+TSTLSTTLTR10*log10(ER)=ESL+TSTLSTTLTR size 12{"10""*log""10" $$E rSub { size 8{R} }$$ = ital "ESL"+ ital "TS" - ital "TL" rSub { size 8{ ital "ST"} } - ital "TL" rSub { size 8{ ital "TR"} } } {}. Note that the echo energy level can be computed using a received echo pulse length, which due to sound channel dispersion can be longer than the transmitted pulse.

Using the properties of matched filters, the matched filter generates an output due to the target echo with peak power level of ESL+TSTLSTTLTRESL+TSTLSTTLTR size 12{ ital "ESL"+ ital "TS" - ital "TL" rSub { size 8{ ital "ST"} } - ital "TL" rSub { size 8{ ital "TR"} } } {}.

AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {}is the ambient noise level in a 1 Hz band, and is assumed to be constant across the matched filter’s bandwidth, e.g. it is twice the spectral density of the ambient noise. AN0AN0 size 12{ ital "AN" rSub { size 8{0} } } {} and spectral density have units of Pa2/HzPa2/Hz size 12{ ital "Pa" rSup { size 8{2} } / ital "Hz"} {}, which is an energy quantity.

The matched filter noise output is RL0AN0RL0AN0 size 12{ ital "RL" rSub { size 8{0} } ⊕ ital "AN" rSub { size 8{0} } } {}. The operation size 12{⊕} {} corresponds to power addition. Power addition converts the quantities back to units of power (Pa^2, volts^2, etc), adds the two power like quantities, and then reconverts back into decibel.

RL 0 AN 0 = 10 log 10 10 RL 0 / 10 + 10 AN 0 / 10 RL 0 AN 0 = 10 log 10 10 RL 0 / 10 + 10 AN 0 / 10 size 12{ ital "RL" rSub { size 8{0} } ⊕ ital "AN" rSub { size 8{0} } ="10""log" rSub { size 8{"10"} } left ("10" rSup { size 8{ ital "RL" rSub { size 6{0} } /"10"} } +"10" rSup { ital "AN" rSub { size 6{0} } /"10"} right )} {}

RL0RL0 size 12{ ital "RL" rSub { size 8{0} } } {}is the reverberation level in a 1 Hz band, or equivalently, the reverberation level when measured at the output of the matched filter.

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