4.3 Properties of active sonar matched filtering  (Page 7/7)

$E\left\{S(\tau ,\phi )S({\tau }^{\text{'}},{\phi }^{\text{'}})\right\}=R_{\text{SS}}(\tau ,\phi )\delta (\tau -{\tau }^{\text{'}})\delta (\phi -{\phi }^{\text{'}})$

$R_{\text{SS}}(\tau ,\phi )$ 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:

$\begin{array}{}E\left\{m(t,\phi )m^{}(t,\phi )\right\}=\\ E_T\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\delta )e^{-\mathrm{j2\pi }(\phi -\delta )\tau }\chi (t-\tau ,\phi -\delta )\mathrm{d\tau d\delta }\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{S}^{}({\tau }^{\text{'}},{\delta }^{\text{'}})e^{\mathrm{j2\pi }(\phi -{\delta }^{\text{'}}){\tau }^{\text{'}}}{\chi }^{}(t-{\tau }^{\text{'}},\phi -{\delta }^{\text{'}})d{\tau }^{\text{'}}d{\delta }^{\text{'}}\end{array}$

Which becomes

$E\left\{{\mid m(t,\phi )\mid }^2\right\}=E_T\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{R}_{\text{SS}}(\tau ,\delta ){\mid \chi (t-\tau ,\phi -\delta )\mid }^2\mathrm{d\tau d\delta }$

Using two dimensional convolution notation **, this expression for the matched filter power $P(t,\phi )$ becomes

$P(t,\phi )=E\left\{{\mid m(t,\phi )\mid }^2\right\}=E_T{R}_{\text{SS}}(t,\phi )\text{**}{\mid \chi (t,\phi )\mid }^2$

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 ${\text{AN}}_0$ . The overall signal to interference ratio, for a target at range $t$ and Doppler $\phi$ is

$\text{SIR}=\frac{P_{\text{Target}}(t,\phi )}{P_{\text{Reverb/clutter}}(t,\phi )+{\text{AN}}_0}$

Which becomes,

$\text{SIR}(t,\phi )=\frac{R_{{}_{\text{SS}}}^{\text{Target}}(t,\phi )\text{**}{\mid \chi (t,\phi )\mid }^2}{R_{{}_{\text{SS}}}^{\text{Reverb/Clutter}}(t,\phi )\text{**}{\mid \chi (t,\phi )\mid }^2+\frac{{\text{AN}}_0}{E_T}}$

This expression can be simplified for different target and scattering conditions. For a point target at range $t_0$ and Doppler ${\phi }_0$ , the target scattering function becomes

$R_{\text{SS}}^{\text{Target}}(t,\phi )=\mathrm{S\delta }(t-t_0)\delta (\phi -{\phi }_0)$ .

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

$R_{\text{SS}}^{\text{Reverb/Clutter}}(t,\phi )=R_{t_0}{Q}_{\text{Revrb}}(\phi )$

$Q_{\text{Revrb}}(\phi )$ 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 $Q_{\text{Revrb}}(\phi )$ is normalized, so that:

$\int Q_{\text{Revrb}}(\phi )\mathrm{d\phi }=1$

With these assumptions we obtain

$\text{SIR}(t,\phi )=\frac{S^{\text{Target}}(t_0,{\phi }_0){\mid \chi (t-t_0,\phi -{\phi }_0)\mid }^2}{R_{{}_{t_0}}\iint Q_{\text{Reverb}}(\phi -\delta ){\mid \chi (\tau ,\phi )\mid }^2\mathrm{d\tau d\delta }+\frac{{\text{AN}}_0}{E_T}}$

Note that when the matched filter is matched in time and Doppler, then $t=t_0$ and $\phi ={\phi }_0$ , and the numerator is maximized:

$\text{SIR}(t_0,{\phi }_0)=\frac{S^{\text{Target}}(t_0,{\phi }_0)}{R_{{}_{t_0}}\iint Q_{\text{Reverb}}({\phi }_0-\delta ){\mid \chi (\tau ,{\phi }_0)\mid }^2\mathrm{d\tau d\delta }+\frac{{\text{AN}}_0}{E_T}}$

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

$Q(\phi )=\int {\mid \chi (\tau ,\phi )\mid }^2\mathrm{d\tau }$

Using this definition, we obtain:

$\text{SIR}(t,\phi )=\frac{S^{\text{Target}}(t_0,{\phi }_0){\mid \chi (t-t_0,\phi -{\phi }_0)\mid }^2}{R_{{}_{t_0}}{Q}_{\text{Reverb}}(\phi )\ast Q(\phi )+\frac{{\text{AN}}_0}{E_T}}$

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 $Q_{\text{HFM}}$ . The convolution with the reverberation Q function becomes the constant $Q_{\text{HFM}}$ (because of the normalization of $Q_{\text{Reverb}}$ ) :

$Q_{\text{Reverb}}(\phi )\ast Q(\phi )=Q_{\text{HFM}}\int Q_{\text{Reverb}}(\phi -{\phi }^{\text{'}}){\mathrm{d\phi }}^{\text{'}}=Q_{\text{HFM}}$

With this approximation the signal to interference ratio becomes:

${\text{SIR}}_{\text{HFM}}(t,\phi )=\frac{S^{\text{Target}}(t_0,{\phi }_0){\mid \chi (t-t_0,\phi -{\phi }_0)\mid }^2}{R_{{}_{t_0}}{Q}_{\text{HFM}}+\frac{{\text{AN}}_0}{E_T}}$

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:

$Q_{\text{Reverb}}(\phi )\ast Q(\phi )=\int Q_{\text{Reverb}}(\phi -{\phi }^{\text{'}})Q_{\text{DSW}}({\phi }^{\text{'}}){\mathrm{d\phi }}^{\text{'}}\approx \underset{-1/\mathrm{2T}}{\overset{1/\mathrm{2T}}{\int }}{Q}_{\text{Reverb}}(\phi -{\phi }^{\text{'}})\text{Td}{\phi }^{\text{'}}\approx Q_{\text{Re}\text{verb}}(\phi )$

Therefore, the signal to interference ratio becomes

${\text{SIR}}_{\text{DSW}}(t,\phi )=\frac{S^{\text{Target}}(t_0,{\phi }_0){\mid \chi (t-t_0,\phi -{\phi }_0)\mid }^2}{R_{{}_{t_0}}{Q}_{\text{Reverb}}(\phi )+\frac{{\text{AN}}_0}{E_T}}$

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:

$\text{SE}=\text{ESL}+\text{TS}-{\text{TL}}_{\text{ST}}-{\text{TL}}_{\text{TR}}-({\text{RL}}_0\oplus {\text{AN}}_0)-\text{DT}$

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 $\text{10}\text{*log}\text{10}(E_R)=\text{ESL}+\text{TS}-{\text{TL}}_{\text{ST}}-{\text{TL}}_{\text{TR}}$ . 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 $\text{ESL}+\text{TS}-{\text{TL}}_{\text{ST}}-{\text{TL}}_{\text{TR}}$ .

${\text{AN}}_0$ 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. ${\text{AN}}_0$ and spectral density have units of ${\text{Pa}}^2/\text{Hz}$ , which is an energy quantity.

The matched filter noise output is ${\text{RL}}_0\oplus {\text{AN}}_0$ . The operation $\oplus$ 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.

${\text{RL}}_0\oplus {\text{AN}}_0=\text{10}{\text{log}}_{\text{10}}{\text{10}}^{{\text{RL}}_0/\text{10}}+{\text{10}}^{{\text{AN}}_0/\text{10}}$

${\text{RL}}_0$ is the reverberation level in a 1 Hz band, or equivalently, the reverberation level when measured at the output of the matched filter.