4.2 Active sonar detection in ambient noise  (Page 4/4)

$p(y\mid \phi )=\frac{1}{{\pi }^N\text{det}(C)}\text{exp}\left(-y^H(C{)}^{-1}y\right)$

Active sonar likelihood function

We will find that the natural logarithm of the measurement likelihood ratio simplifies the detection expression considerably:

$\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}=\text{log}(\text{det}(C))-\text{log}(\text{det}(C+C_r))+y^H\left(C^{-1}-(C_r+C{)}^{-1}\right)y$

The determinant of $C$ is $\text{det}(C)=\text{det}(N_0{I}_{L-D_{\text{min}}})\text{det}(R)=(N_0{)}^{L-D_{\text{min}}}\text{det}(R)$

It is convenient to partition $C+C_r$ into sub-matrices compatible with r.

$C+C_r=\left[\begin{array}{cccc}R& 0& 0& 0\\ 0& N_0{I}_{D\left(h\right)-D_{\text{min}}}& 0& 0\\ 0& 0& N_0{I}_N+{\sigma }_{A(h)}^2{\text{ww}}^H& 0\\ 0& 0& 0& N_0{I}_{L-D(h)-N}\end{array}\right]$

The determinant of $C+C_r$ is

$\text{det}(C+C_r)=\text{det}(R)\text{det}(N_0{I}_{D(h)-D_{\text{min}}})\text{det}(N_0{I}_N+{\sigma }_{A(h)}^2{\text{rr}}^H)\text{det}(N_0{I}_{L-D(h)-N})$

Which becomes

$\text{det}(C+C_r)=(N_0{)}^{L-D_{\text{min}}-N}\text{det}(N_0{I}_N+{\sigma }_{A(h)}^2{\text{rr}}^H)\text{det}(R)$

Using Sylvester’s determinant theorem :

$\text{det}(I+\text{AB})=\text{det}(I+\text{BA})$

We obtain:

$\text{det}(C+C_r)=(N_0{)}^{L-D_{\text{min}}}(1+\frac{{\sigma }_{A(h)}^2{r}^{H}r}{N_0})\text{det}(R)$

Which equals

$\text{det}(C+C_r)=(N_0{)}^{L-D_{\text{min}}}(1+\frac{{\sigma }_{A(h)}^2}{N_0})\text{det}(R)$

Now we partition the observed ping history y into

$y=\left[\begin{array}{c}{y}_R\\ y_{\mathrm{N1}}\\ y_h\\ y_{\mathrm{N2}}\end{array}\right]$

so that:

$C^{-1}-(C_r+C{)}^{-1}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \frac{1}{N_0}{I}_N-(N_0{I}_N+{\sigma }_{A(h)}^2{\text{ww}}^H{)}^{-1}& \\ 0& 0& 0& 0\end{array}\right]$

The Woodbury matrix identity states:

$(A+\text{UCV}{)}^{-1}=A^{-1}-A^{-1}U(C^{-1}+{\text{VA}}^{-1}U{)}^{-1}{\text{VA}}^{-1}$

So that

$(N_0{I}_N+{\sigma }_{A(h)}^2{\text{ww}}^H{)}^{-1}=\frac{1}{N_0}{I}_N-\frac{1}{N_{{}_0}^2}\frac{{\text{ww}}^H}{\frac{1}{N_0}+\frac{1}{{\sigma }_{A(h)}^2}}$

And hence

$y^H\left(C^{-1}-(C_r+C{)}^{-1}\right)y=\frac{1}{N_0}\frac{{\sigma }_{A(h)}^2/N_0}{(1+\frac{{\sigma }_{A(h)}^2}{N_0})}{\mid w^H{y}_h\mid }^2$

So that the log likelihood function becomes

$\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}=-\text{log}(1+\frac{{\sigma }_{A(h)}^2}{N_0})+\frac{1}{N_0}\frac{{\sigma }_{A(h)}^2/N_0}{(1+\frac{{\sigma }_{A(h)}^2}{N_0})}{\mid w^H{y}_h\mid }^2$

Putting the noise variance inside the observation vector yields for the log likelihood function

$\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}=-\text{log}(1+\frac{{\sigma }_{A(h)}^2}{N_0})+\frac{{\sigma }_{A(h)}^2/N_0}{(1+\frac{{\sigma }_{A(h)}^2}{N_0})}{\mid w^H\frac{y_h}{\sqrt{N_0}}\mid }^2$

The log-likelihood ratio shows that the assumed echo shape, $w^H$ , the assumed energy of the signal ${\sigma }_{A(h)}^2$ , the ambient noise density $N_0$ , and the expected location in time of the echo needed to select $y_h$ are the parameters required to evaluate the log-likelihood function. The assumed echo shape and energy can vary by hypothesis h, but the noise properties $N_0$ have been assumed to be the same for each decision hypothesis h.

Signal to noise ratio of the detector

The magnitude squared term, ${\mid w^H\frac{y_h}{\sqrt{N_0}}\mid }^2$ , is a matched filter, where the observations are cross-correlated with the signal template. The observations are normalized by the noise variance before cross correlation, which is a form of pre-whitening. The term $\frac{{\sigma }_{A(h)}^2}{N_0}$ is the Energy to Noise Density Ratio (ENR) of the detection problem. Note that the signal to noise ratio of the matched filter output can be written as:

$\frac{E\left\{{\mid w^H\frac{\mathrm{Aw}}{\sqrt{N_0}}\mid }^2\right\}}{E\left\{{\mid w^H\frac{q}{\sqrt{N_0}}\mid }^2\right\}}=\frac{\frac{{\sigma }_{A(h)}^2}{N_0}}{E\left\{\frac{w^H{\text{qq}}^{H}w}{N_0}\right\}}=\frac{\frac{{\sigma }_{A(h)}^2}{N_0}}{w^H\text{Iw}}=\frac{{\sigma }_{A(h)}^2}{N_0}=\text{ENR}$

This result is a general result for matched filters. A matched filter’s SNR is the Energy to Noise Density Ratio of the problem. The energy of a signal being detected is related to the average amplitude and the duration of the signal. The SNR output of the matched filter is independent of the details of the waveform being detected, only the signal energy and the noise spectral density determine the matched filter response.

The likelihood function is dependent only the ENR as well.

A-priori assumptions

Before the ping history is received, we assess the probability of each hypothesis, $p(h)$ and $p(\phi )$ . The a-priori information may have come from previous pings, or are probabilities assigned by the sonar system to begin a target search.

Using the logarithm of probability density function ratios simplifies the expressions:

$\text{log}\frac{p(h\mid y)}{p(\phi \mid y)}=\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}+\text{log}\frac{p^{-}(h)}{p^{-}(\phi )}$

Using the likelihood ratio notation,

$\text{log}\Lambda (h\mid y)=\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}+\text{log}{\Lambda }^{-}(h)$

Once we compute $\text{log}\Lambda (h\mid y)$ , we can declare a target is present, with confidence $p_T$ by computing:

Target Present if: $\underset{h\in H^{\tau }}{\int }\Lambda (h)\text{dh}>\frac{1-p_T}{p_T}$

Because the target hypothesis space contains many hypotheses, this detection problem is can be considered a composite hypothesis test.

An alternative approach to detection of a target with an unknown range is solved by finding the target range hypothesis with the greatest measurement likelihood as the detection statistic [Kay]. This is referred to as Generalized Maximum Likelihood Ratio Testing (GLRT).

Target Present if: $\underset{h}{\text{max}}\text{log}\frac{L(y\mid h)}{L(y\mid \phi )}>\gamma$

The GLRT approach is often easier to implement than the Bayes detection approach, because one avoids the integration/summation over a-priori probabilities.