4.3 Properties of active sonar matched filtering  (Page 6/7)

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:

$m(t,\phi )=\underset{t}{\overset{t+T}{\int }}y(\sigma )r^{}(\sigma -t)e^{-\mathrm{j2}\text{πφσ}}\mathrm{d\sigma }$

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

$m(t,\phi )=\underset{-\infty }{\overset{\infty }{\int }}y(\sigma )r^{}(\sigma -t)e^{-\mathrm{j2}\text{πφσ}}\mathrm{d\sigma }$

We define the effects of reverberation, targets, clutter and the acoustic channel, via a spreading function $S(\tau ,\phi )$ acting on the transmitted waveform:

$y(t)=\sqrt{E_T}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\phi )e^{\mathrm{j2}\text{πφ}t}r(t-\tau )\mathrm{d\tau d\phi }$

This expression does not include contributions of ambient noise, only scattering phenomena. The spreading function $S(\tau ,\phi )$ defines the acoustic scattering, as a function of delay $\tau$ and Doppler shift $\phi$ 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 $\tau$ region of non-zero spreading function, $S_{\text{Target}}(\tau ,\phi )$ . Reverberation will have an extended $\tau$ region with significant $S_{\text{Reverb}}(\tau ,\phi )$ . 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

$m(t,\phi )=\sqrt{E_T}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\delta )e^{\mathrm{j2}\text{πδσ}}r(\sigma -\tau )r^{}(\sigma -t)e^{-\mathrm{j2}\text{πφσ}}\mathrm{d\sigma }\mathrm{d\tau d\delta }$

Which equals

$m(t,\phi )=\sqrt{E_T}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\delta )e^{-\mathrm{j2\pi }(\phi -\delta )\sigma }r(\sigma -\tau )r^{}(\sigma -t)\mathrm{d\sigma }\mathrm{d\tau d\delta }$

Letting ${\sigma }^{\text{'}}=\sigma -t$ , we obtain

$m(t,\phi )=\sqrt{E_T}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\delta )e^{-\mathrm{j2\pi }(\phi -\delta )({\sigma }^{\text{'}}+\tau )}r({\sigma }^{\text{'}})r^{}({\sigma }^{\text{'}}-(t-\tau ))\mathrm{d\sigma }\mathrm{d\tau d\delta }$

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

$m(t,\phi )=\sqrt{E_T}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}S(\tau ,\delta )e^{-\mathrm{j2\pi }(\phi -\delta )\tau }\chi (t-\tau ,\phi -\delta )\mathrm{d\tau d\delta }$

The response of the matched filter is a“twisted convolution”of the spreading function and the waveform ambiguity function. The exponential $e^{-\mathrm{j2\pi }(\phi -\delta )}$ performs the twisting. Note that if the waveform ambiguity function was“perfect”, that is a single peak,

$\chi (\tau ,\phi )=\delta (\tau )\delta (\phi )$

Then the matched filter response would become:

$m(t,\phi )=\sqrt{E_T}S(t,\phi )+n(t,\phi )$

Where $n(t,\phi )$ 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 $\chi (\mathrm{0,0})=1$ , 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):