4.3 Properties of active sonar matched filtering  (Page 3/7)

An important invariance property of the narrowband ambiguity function is that

$\int {\mid {\chi }_{\text{NB}}(\tau ,\phi )\mid }^2\text{dt}=1$

Using either definition of the signal ambiguity function we have

${\int }_t^{t+T}r(\sigma -\tau (u))r(\sigma -t)\mathrm{d\sigma }={\int }_0^{T}r({\sigma }^{\text{'}}-(\tau (u)-t))r({\sigma }^{\text{'}})d{\sigma }^{\text{'}}=\chi (\tau (u)-t,0)$

Therefore the matched filter response is

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

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:

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

Which can be rewritten as:

$m(t,\phi )=e^{\mathrm{j2}\text{πφ}t}\underset{-\infty }{\overset{\infty }{\int }}y(t+{\sigma }^{\text{'}})r^{}({\sigma }^{\text{'}})e^{{-\mathrm{j2}\text{πφ{}\sigma }^{\text{'}}}d{\sigma }^{\text{'}}$

The spectrogram with window function $w(\sigma )$ is given by:

$\begin{array}{}\underset{-\infty }{\overset{\infty }{\int }}y(t+{\sigma }^{\text{'}})w({\sigma }^{\text{'}})e^{{-\mathrm{j2}\text{πφ{}\sigma }^{\text{'}}}d{\sigma }^{\text{'}}\\ \Phi (t,\phi )={\mid \mid }^2\end{array}$

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.