4.3 Properties of active sonar matched filtering  (Page 5/7)

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

$E\left\{A^{}(u)A(v)\right\}=R_A(u-v)$

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

$E\left\{A(u{)}^{}A(v)\right\}=R_A(u-v)=R_A\delta (u-v)$

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.

$E\left\{m_R^2(t,\eta )\right\}=E\left\{E_T\underset{0}{\overset{\infty }{\int }}A(u)\Gamma (u)\chi (t-\tau (u),\eta )du\underset{0}{\overset{\infty }{\int }}A(\phi )\Gamma (\phi )\chi (t-\tau (\phi ),\eta )d\phi \right\}$

Rearranging,

$E\left\{m_R^2(t,\eta )\right\}=E_T\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}E\left\{A(u)A(\phi )\right\}\Gamma (u)\Gamma (\phi )\chi (t-\tau (u),\eta )\chi (t-\tau (\phi ),\eta )d\text{ud}\phi$

Using the covariance of the scattering elements we get,

$E\left\{m_R^2(t,\eta )\right\}=E_T\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{R}_A\delta (u-\phi )\Gamma (u)\Gamma (\phi )\chi (t-\tau (u),\eta )\chi (t-\tau (\phi ),\eta )d\text{ud}\phi$

Or,

$E\left\{m_R^2(t,\eta )\right\}=E_T{R}_A\underset{0}{\overset{\infty }{\int }}{\mid \Gamma (u)\mid }^2{\chi }^2(t-\tau (u),\eta )du$

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

$E\left\{m_R^2(t,\eta )\right\}=E_T{R}_A{\mid \Gamma (u_0)\mid }^2\underset{0}{\overset{\infty }{\int }}{\chi }^2(t-\tau (u),\eta )du$

Where $u_0$ is defined by $\tau (u_0)=t$ .

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 $\tau$ , where we assume that the chance of variable from u to $\tau$ is approximately given by $\tau =\mathrm{2u}/c$ , 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

$E\left\{m_R^2(t,\eta )\right\}=E_T{R}_A{\mid \Gamma (u_0)\mid }^{2}c/2\underset{0}{\overset{\infty }{\int }}{\chi }^2(t-\tau ,\eta )d\tau$

If we assume that the matched filter time t is greater than the signal duration T, then letting ${\tau }^{\text{'}}=t-\tau$ , we obtain

$E\left\{m_R^2(t,\eta )\right\}=E_T{R}_A{\mid \Gamma (u_0)\mid }^{2}c/2\underset{-\infty }{\overset{\infty }{\int }}{\chi }^2(t-\tau ,\eta )d\tau$

We define the Q-function of the waveform as

$Q(\eta )=\underset{-\infty }{\overset{\infty }{\int }}{\mid \chi ({\tau }^{\text{'}},\eta )\mid }^{2}d{\tau }^{\text{'}}$

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

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

$\underset{-\infty }{\overset{\infty }{\int }}{Q}_{\text{NB}}(\phi )\mathrm{d\phi }=\underset{-\infty }{\overset{\infty }{\int }}{\mid {\chi }_{\text{NB}}({\tau }^{\text{'}},\phi )\mid }^{2}d{\tau }^{\text{'}}\mathrm{d\phi }=1$

The wideband waveform Q function is approximately normalized to unity.

The reverberation response can be written as

$E\left\{m_R^2(t,\eta )\right\}=E_T{R}_A{\mid \Gamma (u_0)\mid }^{2}Q(\eta )c/2$

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.