4.3 Properties of active sonar matched filtering  (Page 2/7)

Discrete time matched filters

Discrete time filters have nearly the same properties as continuous time filters. In discrete time, we assume an echo of $e(k)=\sqrt{E_R}r(k)$ , with $\sum _{k=1}^T{r}^2(k)=1$ . The discrete matched filter output to the input y(k) is given by:

$m(k)=\sum _{l=k}^{k+T-1}y(l)r(l-k+1)$

In response to the echo, $y(k)=e(k-T_D)$ the output of the discrete time matched filter is

$m(k)=\sum _{l=k}^{k+T-1}e(l-T_D)r(l-k+1)=\sqrt{E_R}\sum _{l=k}^{k+T-1}r(l-T_D)r(l-k+1)$

Hence $m(T_D-1)=\sqrt{E_R}$ . The peak power output of the matched filter, $m^2(t)$ , in response to a echo is $E_R$ .

We determine the discrete matched filter response to noise next. Assume the input noise is sampled white with variance ${\text{AN}}_0$ :

$E\left\{n(k)n(l)\right\}={\text{AN}}_0{\delta }_{\text{kl}}$

$\begin{array}{}E\left\{m(k)m(p)\right\}=E\left\{\sum _{l=k}^{k+T-1}n(l)r(l-k+1)\sum _{i=p}^{p+T-1}n(i)r(i-p+1)\right\}=\\ {\text{AN}}_0\sum _{l=k}^{k+T-1}\sum _{i=p}^{p+T-1}{\delta }_{\text{li}}r(l-k+1)r(i-p+1)={\text{AN}}_0\sum _{l=k}^{k+T-1}{r}^2(l-k+1)={\text{AN}}_0\end{array}$

Thus, the signal to noise ratio at the output of a discrete time matched filter is $E_R/{\text{AN}}_0$ .

The matched filter compresses the echo signal to a pulse (or a series of pulses for waveforms such as SFM) with time width equal approximately to its inverse bandwidth, 1/BW.

Matched filter response to reverberation

One model for reverberation assumes that the reverberation comes from distributed discrete scatterers, with density $A(u)$ .

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

$A(u)$ is considered a random, spatial process that models the amplitude of the scattering that occurs at range u back to the receiver. We are assuming that the receiver has significant aperture, and that y(t) is the receiver response at the output of a beamformer. In this case, scattering is occurring from the patch of the ocean bottom or surface that lies at range u and within the receiver beamwidth in azimuth and elevation. Each patch of the bottom or surface will arrive at the receiver at a different time. $\Gamma (u)$ is the transmission loss from the source to the scattering range (u) and back to the receiver. $\tau (u)$ is the total travel time from source to scatterer to receiver. As one can see, the reverberation is made up of many time delayed and amplitude scaled replicas of the transmitted waveform.

The matched filter response to the reverberation is

$\begin{array}{}{m}_R(t)={\int }_t^{t+T}r(\sigma -t)\sqrt{E_T}\underset{0}{\overset{\infty }{\int }}A(u)\Gamma (u)r(\sigma -\tau (u))du\mathrm{d\sigma }=\\ \sqrt{E_T}\underset{0}{\overset{\infty }{\int }}A(u)\Gamma (u){\int }_t^{t+T}r(\sigma -\tau (u))r(\sigma -t)\mathrm{d\sigma d}u\end{array}$

We define the transmitted waveform autocorrelation function as

$\chi (\tau )={\int }_0^{T}r(t-\tau )r(t)\text{dt}$

Recall, that by definition, $\chi (0)={\int }_0^T{r}^2(t)\text{dt}=1$ . In more general terms, we define the transmitted wideband signal ambiguity function as

${\chi }_{\text{WB}}(\tau ,\eta )=\sqrt{\eta }{\int }_{-\infty }^{\infty }{r}^{}(\eta (t-\tau ))r(t)\text{dt}$

Note: Some authors define the ambiguity function as the magnitude squared value of this definition. Other authors choose different normalizations or the sign (-/+) on the delay term $\tau$ .

In the wideband signal ambiguity function, the Doppler effect is represented by the scaling factor $\eta$ . In narrowband cases, the Doppler effect is represented by a frequency shift, $\phi$ . For a monostatic sonar the frequency shift is given by $\phi =2\text{v/c}$ , where $v$ is the radial velocity between the scattering object and the sonar system.

${\chi }_{\text{NB}}(\tau ,\phi )={\int }_{-\infty }^{\infty }{r}^{}(t-\tau )r(t)e^{-\mathrm{j2}\text{πφ}t}\text{dt}$

One can show (Weiss) that the narrowband approximation to Doppler is valid if , where $B$ is the waveform bandwidth and $T$ is the duration. For one hundred (100) Hertz bandwidth waveforms that last for one (1) second, the speed of the target must be much less than 7.5 m/sec, or approximately 15 knots.