4.3 Properties of active sonar matched filtering

 Page 1 / 7
This module develops expressions for the response of a matched filter to ambient noise, reverberation and target echoes. A scattering function formulation is used to characterize propagation channels with multipath and Doppler spreading.

Properties of Active Sonar Matched Filtering

Introduction

Matched filters are used extensively in coherent active sonar. The output of a matched filter is used for detection, classification and localization. This document develops some properties of matched filters, including the SNR response in ambient noise and the response to reverberation.

In a matched filter for active sonar, we are integrating the echo plus interference times the echo’s replica. When an echo passes through the matched filter, we are cross-correlating the echo with a scaled version of the echo, so that the output is a scaled version of the auto-correlation of the echo corrupted by noise. The autocorrelation of the echo has a peak in time whose duration is approximately the inverse of the echo’s bandwidth.

For some waveforms (such as the Sinusoidal Frequency Modulation pulse) the autocorrelation function will have multiple peaks, termed‘fingers’, due to the periodic structure of the pulse. Each autocorrelation finger has a time width approximately equal to the signal’s bandwidth.

Continuous time matched filter

The echo is written as

$e\left(t\right)=\sqrt{{E}_{R}}r\left(t\right)$ , where $\underset{0}{\overset{T}{\int }}{r}^{2}\left(t\right)\text{dt}=1$

This implies that the echo energy $\underset{0}{\overset{T}{\int }}{e}^{2}\left(t\right)\text{dt}$ is ${E}_{R}$ , measured in Pascal^2-seconds.

We can write the matched filter operation in continuous time as

$m\left(t\right)=\underset{t}{\overset{t+T}{\int }}y\left(\sigma \right)r\left(\sigma -t\right)\mathrm{d\sigma }$

$y\left(\sigma \right)$ is the receiver time series. In response to a target echo that arrives at ${T}_{D}$ seconds and without noise or reverberation, the receiver output is $y\left(t\right)=e\left(t-{T}_{D}\right)$ . The output of the matched filter becomes:

$m\left(t\right)=\underset{t}{\overset{t+T}{\int }}e\left(\sigma -{T}_{D}\right)r\left(\sigma -t\right)\mathrm{d\sigma }=\sqrt{{E}_{R}}\underset{t}{\overset{t+T}{\int }}r\left(\sigma -{T}_{D}\right)r\left(\sigma -t\right)\mathrm{d\sigma }$

Hence $m\left({T}_{D}\right)=\sqrt{{E}_{R}}$ . The peak power output of the matched filter, ${m}^{2}\left(t\right)$ , in response to a echo is ${E}_{R}$ .

We determine the matched filter response to noise next. Assume the input noise is white with variance ${\text{AN}}_{0}$ :

$E\left\{n\left(t\right)n\left(s\right)\right\}={\text{AN}}_{0}\delta \left(t-s\right)$

Note that the delta function has units of inverse seconds, and therefore ${\text{AN}}_{0}$ has units of Pascals^2/Hz, equivalent to a spectral density. From the definition of stationary random process autocorrelations and power spectral density, we know that the Fourier transform of the autocorrelation is the spectral density function for the random process. The Fourier transform of covariance becomes $\int {e}^{\mathrm{j2\pi f\tau }}{\text{AN}}_{0}\delta \left(\tau \right)\mathrm{d\tau }={\text{AN}}_{0}$ , which is the spectral density of the noise.

$\begin{array}{}E\left\{m\left(t\right)m\left(s\right)\right\}=E\left\{\underset{t}{\overset{t+T}{\int }}n\left(\sigma \right)r\left(\sigma -t\right)\mathrm{d\sigma }\underset{t}{\overset{t+T}{\int }}n\left(\beta \right)r\left(\beta -t\right)\mathrm{d\beta }\right\}=\\ {\text{AN}}_{0}\underset{t}{\overset{t+T}{\int }}\underset{t}{\overset{t+T}{\int }}\delta \left(\sigma -\beta \right)r\left(\sigma -t\right)r\left(\beta -t\right)\mathrm{d\sigma d\beta }={\text{AN}}_{0}\underset{t}{\overset{t+T}{\int }}{r}^{2}\left(\sigma -t\right)\mathrm{d\sigma }={\text{AN}}_{0}\end{array}$

Thus, the noise power response of a matched filter is the input spectral density, ${\text{AN}}_{0}$ .

We conclude that the signal to noise ratio (SNR) at the output of a matched filter is the ratio of the echo energy to the noise spectral density, ${E}_{R}/{\text{AN}}_{0}$ . This assumes that the noise is white, e.g. a flat spectral density at the input to the matched filter. This is a general result, independent of the signal waveform details, except for its energy ${E}_{R}$ .

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!