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Introduction

Detection theory applies optimal model evaluation to signals ( Helstrom , Poor , van Trees ). Usually, we measure a signal in the presence of additive noiseover some finite number of samples. Each observed datum is of the form s l n l , where s l denotes the l th signal value and n l the l th noise value. In this and in succeeding sections of thischapter, we focus the general methods of evaluating models.

Detection of signals in gaussian noise

For the moment, we assume we know the joint distribution of the noise values. In most cases, the various models for theform of the observations - the hypothesis - do not differ because of noise characteristics. Rather, the signal componentdetermines model variations and the noise is statistically independent of the signal; such is the specificity ofdetection problems in contrast to the generality of model evaluation. For example, we may want to determine whether asignal characteristic of a particular ship is present in a sonar array's output (the signal is known) or whether no shipis present (zero-valued signal).

To apply optimal hypothesis testing procedures previously derived, we first obtain a finite number L of observations r l , l 0 L 1 . These observations are usually obtained from continuous-timeobservations in one of two ways. Two commonly used methods for passing from continuous-time to discrete-time are known: integrate-and-dump and sampling . These techniques are illustrated in .

Integrate-and-Dump
Sampling
The two most common methods of converting continuous-time observations into discrete-time ones are shown. In the upperpanel, the integrate-and-dump method is shown: the input is integrated over an interval of duration and the result sampled. In the lower panel, the sampling method merelysamples the input every seconds.

Integrate-and-dump

In this procedure, no attention is paid to the bandwidth of the noise in selecting the sampling rate. Instead, thesampling interval is selected according to the characteristics of the signalset. Because of the finite duration of the integrator, successive samples are statistically independent when thenoise bandwidth exceeds 1 Consequently, the sampling rate can be varied to some extent while retaining this desirable analytic property.

Sampling

Traditional engineering considerations governed the selection of the sampling filter and the sampling rate. Asin the integrate-and-dump procedure, the sampling rate is chosen according to signal properties. Presumably, changesin sampling rate would force changes in the filter. As we shall see, this linkage has dramatic implications onperformance.

With either method, the continuous-time detection problem of selecting between models (a binary selection is used here forsimplicity) 0 : r t s 0 t n t 0 t T 1 : r t s 1 t n t 0 t T where s i t denotes the known signal set and n t denotes additive noise modeled as a stationary stochasticprocess

We are not assuming the amplitude distribution of the noise to be Gaussian.
is converted into the discrete-time detection problem 0 : r l s l 0 n l 0 l L 1 : r l s l 1 n l 0 l L where the sampling interval is always taken to divide the observation interval T : L T . We form the discrete-time observations into a vector: r r 0 r L 1 . The binary detection problem is to distinguish between two possible signals present in the noisy outputwaveform. 0 : r s 0 n 0 : r s 1 n To apply our model evaluation results, we need the probabilitydensity of r under each model. As the only probabilistic component of theobservations is the noise, the required density for the detection problem is given by p r i r p n r s i and the corresponding likelihood ratio by r p n r s 1 p n r s 0 Much of detection theory revolves about interpreting thislikelihood ratio and deriving the detection threshold (either threshold or ).

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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salma
Commplementary angles
Idrissa Reply
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Uday
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salma
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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