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.
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=1∫0Tr2(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)dt∫0Te2(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
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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(t−TD)y(t)=e(t−TD) size 12{y \( t \) =e \( t - T rSub { size 8{D} } \) } {}. The output of the matched filter becomes:
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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} } } {}:
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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)dτ=AN0∫ej2πfτAN0δ(τ)dτ=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.
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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 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)=1∑k=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:
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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(k−TD)y(k)=e(k−TD) size 12{y \( k \) =e \( k - T rSub { size 8{D} } \) } {}the output of the discrete time matched filter is
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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(TD−1)=ERm(TD−1)=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} } } {}:
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size 12{E left lbrace n \( k \) n \( l \) right rbrace = ital "AN" rSub { size 8{0} } δ rSub { size 8{ ital "kl"} } } {}
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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.
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)=ET∫0∞A(u)Γ(u)r(t−τ(u))duy(t)=ET∫0∞A(u)Γ(u)r(t−τ(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
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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
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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
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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.
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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
∫
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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
∫
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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
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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
dσ
m
(
t
,
φ
)
=
∫
t
t
+
T
y
(
σ
)
r
(
σ
−
t
)
e
−
j2
πφσ
dσ
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.
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
)
)
dσ
m
(
t
,
η
)
=
η
∫
t
t
+
T
/
η
y
(
σ
)
r
(
η
(
σ
−
t
)
)
dσ
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'=σ−tσ'=σ−t 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=t−τ(u)δ=t−τ(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'=t−ττ'=t−τ 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
)
dφ
=
∫
−
∞
∞
∣
χ
NB
(
τ
'
,
φ
)
∣
2
d
τ
'
dφ
=
1
∫
−
∞
∞
Q
NB
(
φ
)
dφ
=
∫
−
∞
∞
∣
χ
NB
(
τ
'
,
φ
)
∣
2
d
τ
'
dφ
=
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.
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
dσ
m
(
t
,
φ
)
=
∫
t
t
+
T
y
(
σ
)
r
(
σ
−
t
)
e
−
j2
πφσ
dσ
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
dσ
m
(
t
,
φ
)
=
∫
−
∞
∞
y
(
σ
)
r
(
σ
−
t
)
e
−
j2
πφσ
dσ
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
dσ
dτdδ
m
(
t
,
φ
)
=
E
T
∫
−
∞
∞
∫
−
∞
∞
∫
−
∞
∞
S
(
τ
,
δ
)
e
j2
πδσ
r
(
σ
−
τ
)
r
(
σ
−
t
)
e
−
j2
πφσ
dσ
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
)
dσ
dτdδ
m
(
t
,
φ
)
=
E
T
∫
−
∞
∞
∫
−
∞
∞
∫
−
∞
∞
S
(
τ
,
δ
)
e
−
j2π
(
φ
−
δ
)
σ
r
(
σ
−
τ
)
r
(
σ
−
t
)
dσ
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'=σ−tσ'=σ−t 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
)
)
dσ
dτdδ
m
(
t
,
φ
)
=
E
T
∫
−
∞
∞
∫
−
∞
∞
∫
−
∞
∞
S
(
τ
,
δ
)
e
−
j2π
(
φ
−
δ
)
(
σ
'
+
τ
)
r
(
σ
'
)
r
(
σ
'
−
(
t
−
τ
)
)
dσ
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
e−j2π(φ−δ)e−j2π(φ−δ) 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
