4.2 Active sonar detection in ambient noise  (Page 3/4)

While receiving an echo, we will also receive ambient noise, $q(t)$ , which we will assume to be complex Gaussian noise, with constant power spectral density over the receiver’s bandwidth. The noise power spectral density over the receiver’s bandwidth BW is assumed to be $N_0$ Pascals^2/Hertz.

We receive L complex valued samples, $y$ as the ping history after heterodyning. For hypothesis h, the observation in discrete time is:

$y(\mathrm{k\Delta t})=q(\mathrm{k\Delta t})+r(\mathrm{k\Delta t}-\mathrm{2R}/c\mid h)$ for , $k=\mathrm{1,}\dots L$

where $\mathrm{\Delta t}$ is the digital sample rate after heterodyning, and q is a sample of the noise and reverberation. Note that $r(\mathrm{k\Delta t}-\mathrm{2R}(h)/c\mid h)=0$ when $\mathrm{2R}(h)/c>\mathrm{k\Delta t},$ because the echo is delayed. The delay for hypothesis $h$ in samples, is given by

$D(h)=\left[\frac{\mathrm{2R}}{\mathrm{c\Delta t}}\right]$

where [x] is the nearest integer to x. We choose the sample rate $\mathrm{\Delta t}$ to be small enough to satisfy the Nyquist sampling criteria for the received echo. We will assume that the non-zero part of the echo is N samples long.

Statistical model of the ping history

We will represent the sampled echo response as a partitioned vector:

$r(h)=\left[\begin{array}{c}{0}_{D(h)\mathrm{x1}}\\ \mathrm{Aw}\\ 0_{L-D(h)-N}\end{array}\right]$ ,

Where

$w=\left[\begin{array}{c}w(\mathrm{\Delta t})\\ ⋮\\ w(\mathrm{N\Delta t})\end{array}\right]$

and the sampled noise and interference as a vector

$q=\left[\begin{array}{c}{q}_1\\ ⋮\\ q_L\end{array}\right]$ ,

so that the sampled ping history becomes

$y=q+r(h)$

The echo is modeled as a known signal $w$ , with Gaussian random complex amplitude A, with zero mean and variance ${\sigma }_{A(h)}^2$ . We will assume that $w^{H}w=1$ , and that ${\mid A(h)\mid }^2$ is the energy of the echo, with units Pascals^2-seconds. Since ${\sigma }_{A(h)}^2$ is $E{\mid A(h)\mid }^2$ , it has units of Pascals^2-seconds as well. The amplitude of the echo is a function of the target location hypothesis $h$ . The location of $w$ in $r(h)$ depends on the location of the target through the time delay $D(h)$ .

Since each element of the random vector $\mathrm{Aw}$ is complex Gaussian, the random vector $\mathrm{Aw}$ has a complex Gaussian distribution. The probability density of $\mathrm{Aw}$ is Gaussian zero mean with covariance matrix ${\sigma }_{A(h)}^2{\text{ww}}^H$ . To see this, consider that

$E(\mathrm{Aw})=E(A)w=0_N$

The covariance of $\mathrm{Aw}$ is given by:

$E(\mathrm{Aw})(\mathrm{Aw}{)}^H=E({\text{AA}}^H){\text{ww}}^H={\sigma }_{A(h)}^2{\text{ww}}^H$

hence $r(h)$ is zero mean complex Gaussian with covariance matrix ${\sigma }_{A(h)}^2{\text{rr}}^H$ .

For the clutter only hypothesis $\phi ,$ $y=q$ .

We have sampled, heterodyned and possibly re-sampled the noise process $q(t)$ to form $q$ .

During the period where r is non-zero, $q$ is a sampled version of the ambient noise, represented as a N by 1 complex Gaussian noise random vector with zero mean and covariance matrix $(N_0)I_N$ . This is true because $\text{BW}\mathrm{\Delta t}\approx 1$ for complex Nyquist sampling of a band-limited signal.

Overall, the noise and reverberation $q$ is assumed to be complex Gaussian with zero mean and L by L covariance matrix $C$ .

Because we are assuming that the reverberation dies away before the echoes from the target search arrive, $C$ has the following partition:

$C=\left[\begin{array}{cc}R& 0\\ 0& N_{0}I\end{array}\right]$

Matrix R has dimensions of $D_{\text{min}}{\text{xD}}_{\text{min}}$ , the minimum delay where the echo interference is dominated by Ambient noise.

Under target hypothesis $h$ , $y$ is Gaussian with has zero mean and covariance matrix $C+{\sigma }_{A^2}{\text{rr}}^H$ .

The probability density of $y$ under $h$ becomes:

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

where $C_r={\sigma }_{A^2}{\text{rr}}^H$ .

Under the clutter hypothesis, $\phi ,$ y has zero mean and covariance matrix $C$ .The probability density of $y$ under $\phi$ becomes: