4.3 Properties of active sonar matched filtering  (Page 4/7)

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 $\chi (\tau ,\eta )$ 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 ( $\eta )$

$m(t,\eta )=\sqrt{\eta }\underset{t}{\overset{t+T/\eta }{\int }}y(\sigma )r(\eta (\sigma -t))\mathrm{d\sigma }$

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 $r(t)$ by $\sqrt{\eta }$ when using it for other Doppler hypotheses. This comes from the fact that

$\underset{0}{\overset{T/\eta }{\int }}{r}^2(\mathrm{\eta t})\text{dt}=1/\eta$

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

$m_R(t,\eta )=\sqrt{{\mathrm{\eta E}}_T}\underset{0}{\overset{\infty }{\int }}A(u)\Gamma (u){\int }_t^{t+T/\eta }r(\sigma -\tau (u))r(\eta (\sigma -t))\mathrm{d\sigma d}u$

The integral including the replicas, using the change of variables ${\sigma }^{\text{'}}=\sigma -t$ , can be written as

$I_{\delta ,\eta }={\int }_0^{T/\eta }r({\sigma }^{\text{'}}+\delta )r(\eta {\sigma }^{\text{'}})d{\sigma }^{\text{'}}$ , where $\delta =t-\tau (u)$

We can extend the limits of the integral, because $r(\eta {\sigma }^{\text{'}})$ is zero outside the integration limits. Extending limits and changing variables to ${\sigma }^{\text{'}\text{'}}={\sigma }^{\text{'}}+\delta$ yields

$I_{\delta ,\eta }={\int }_{-\infty }^{\infty }r({\sigma }^{\text{'}\text{'}})r(\eta ({\sigma }^{\text{'}\text{'}}-\delta ))d{\sigma }^{\text{'}\text{'}}=\chi (\delta ,\eta )/\eta$

Hence the range Doppler matched filter bank response to reverberation becomes

$m_R(t,\eta )=\sqrt{E_T}\underset{0}{\overset{\infty }{\int }}A(u)\Gamma (u)\chi (t-\tau (u),\eta )du$

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.