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h = ( x , y ) H τ size 12{h= \( x,y \) in H rSup { size 8{τ} } } {} if R min < ( x x r ) 2 + ( y y r ) 2 < R max size 12{R rSub { size 8{"min"} }<sqrt { \( x - x rSub { size 8{r} } \) rSup { size 8{2} } + \( y - y rSub { size 8{r} } \) rSup { size 8{2} } }<R rSub { size 8{"max"} } } {}

Where we are assuming that the depth of the target is small when compared to its ( x , y ) size 12{ \( x,y \) } {} coordinates, the receiver is located at ( x r , y r ) size 12{ \( x rSub { size 8{r} } ,y rSub { size 8{r} } \) } {} . R min size 12{R rSub { size 8{"min"} } } {} is the range at which the echo is noise, not reverberation limited, and R max size 12{R rSub { size 8{"max"} } } {} is the farthest range of interest. For this problem, h size 12{h} {} is an index into the target range from the sonar.

Sonar receiver model

The sonar transmits the waveform m ( t ) size 12{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 size 12{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 ) = sin ( ft ) w ( t ) size 12{m \( t \) ="sin" \( 2πital "ft" \) w \( t \) } {} , t = ( 0, T ) size 12{t= \( 0,T \) } {}

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 ) size 12{h= \( x,y \) } {} we know approximately the received echo time series:

g ( t h ) = Bm ( t 2R / c ) size 12{g \( t \lline h \) = ital "Bm" \( t - 2R/c \) } {}

The amplitude B size 12{B} {} is related to the propagation loss out to the target hypothesis location, and the reflection characteristics of the target. The time delay 2R / c size 12{2R/c} {} corresponds to the time it takes for the transmission waveform to reach the target and return to the sonar. R size 12{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 j2π ft size 12{e rSup { size 8{ - j2πital "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 h ) = Ae w ( t 2R / c ) size 12{r \( t \lline h \) = ital "Ae" rSup { size 8{jθ} } w \( t - 2R/c \) } {}

The phase shift θ size 12{θ} {} 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, Ae size 12{ ital "Ae" rSup { size 8{jθ} } } {} ,is a complex Gaussian random variable with zero mean and with standard deviation σ 2 ( h ) . size 12{σrSup { size 8{2} } \( h \) "." } {} 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.

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Bob Reply
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?
Damian Reply
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Renato
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Stoney Reply
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Kyle
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Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Abigail
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Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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Devang Reply
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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