<< Chapter < Page | Chapter >> Page > |
$h=(x,y)\in {H}^{\tau}$ if ${R}_{\text{min}}<\sqrt{(x-{x}_{r}{)}^{2}+(y-{y}_{r}{)}^{2}}<{R}_{\text{max}}$
Where we are assuming that the depth of the target is small when compared to its $(x,y)$ coordinates, the receiver is located at $({x}_{r},{y}_{r})$ . ${R}_{\text{min}}$ is the range at which the echo is noise, not reverberation limited, and ${R}_{\text{max}}$ is the farthest range of interest. For this problem, $h$ is an index into the target range from the sonar.
The sonar transmits the waveform $m(t)$ for each ping. In most sonar transmitters, the transmitted waveform is narrow-band, that is, the waveform bandwidth is much smaller than its center frequency, $f$ . This is true because efficient sonar transmitters use resonant mechanical and electrical components to provide maximum electrical to sound power transfer. An approximation therefore is to model the transmitted waveform as an amplitude modulated carrier:
$m(t)=\text{sin}(\mathrm{2\pi}\text{ft})w(t)$ , $t=(\mathrm{0,}T)$
We will assume that the target is motionless, so that Doppler effects can be ignored. We will assume that the sonar receiver is a single sensor, with no directionality characteristics. For each target location hypothesis $h=(x,y)$ we know approximately the received echo time series:
$g(t\mid h)=\text{Bm}(t-\mathrm{2R}/c)$
The amplitude $B$ is related to the propagation loss out to the target hypothesis location, and the reflection characteristics of the target. The time delay $\mathrm{2R}/c$ corresponds to the time it takes for the transmission waveform to reach the target and return to the sonar. $R$ is the range to the target and c is the effective speed of sound, when including refraction and boundary reflections.
The received echo is band-limited to approximately the same frequency band as the transmission. The receiver bandwidth may be greater than the transmitted bandwidth due to Doppler frequency shifts, but for the present, we are assuming that the target is not moving. Sonar receivers use heterodyne techniques to reduce the data storage of the ping history. The sonar receiver multiplies the ping history by a carrier signal ${e}^{-\mathrm{j2\pi}\text{ft}}$ to shift the positive frequency part of the received echo closer to DC. The resulting signal is then low pass filtered to eliminate the shifted negative frequency part of the ping history. Since the original ping history was real, the negative frequency part of the signal spectra carries no additional information. The result is a complex signal with a lower bandwidth, but retains all of the echo related information of the original ping history. This heterodyne process can be done in the analog or digital domain.
A target echo passing through the heterodyne part of the sonar receiver becomes:
$r(t\mid h)={\text{Ae}}^{\mathrm{j\theta}}w(t-\mathrm{2R}/c)$
The phase shift $\theta $ corresponds to the phase shift due to heterodyne operation; the uncertainty in propagation conditions; and the summation of multi-path arrivals with almost the same time delay, etc.
We will assume that the target echo amplitude, ${\text{Ae}}^{\mathrm{j\theta}}$ ,is a complex Gaussian random variable with zero mean and with standard deviation ${\sigma}^{2}(h)\text{.}$ We are modeling the echo as having the same waveform as the transmission, but with an uncertain phase and amplitude. This is assuming that the target echo amplitude obeys Swerling target type I statistics with unknown phase.
Notification Switch
Would you like to follow the 'Signal and information processing for sonar' conversation and receive update notifications?