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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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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