# 2.4 White gaussian noise  (Page 2/3)

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Expanding the dot product, $r^T{s}_{i}=\sum_{l=0}^{L-1} r(l){s}_{i}(l)$ another signal processing interpretation emerges. The dot product now describes a finite impulse response (FIR) filteringoperation evaluated at a specific index. To demonstrate this interpretation, let $h(l)$ be the unit-sample response of a linear, shift-invariant filterwhere $h(l)=0$ for $l< 0$ and $l\ge L$ . Letting $r(l)$ be the filter's input sequence, the convolution sum expresses the output. $(r(k), h(k))=\sum_{l=k-L-1}^{k} r(l)h(k-l)$ Letting $k=L-1$ , the index at which the unit-sample response's last value overlaps the input's value at the origin, we have $(k, , (r(k), h(k)))=\sum_{l=0}^{L-1} r(l)h(L-1-l)$ If we set the unit-sample response equal to the index-reversed, then delayed signal $h(l)={s}_{i}(L-1-l)$ , we have $(k, , (r(k), {s}_{i}(L-1-k)))=\sum_{l=0}^{L-1} r(l){s}_{i}(l)$ which equals the observation-dependent component of the optimal detector's sufficient statistic. depicts these computations graphically.

The sufficient statistic for the ${i}^{\mathrm{th}}$ signal is thus expressed in signal processing notation as $(k, , (r(k), {s}_{i}(L-1-k)))-\frac{{E}_{i}}{2}$ . The filtering term is called a matched filter because the observations are passed through a filter whose unit-sample response "matches" that of the signalbeing sought. We sample the matched filter's output at the precise moment when all of the observations fall within thefilter's memory and then adjust this value by half the signal energy. The adjusted values for the two assumed signals aresubtracted and compared to a threshold.

To compute the performance probabilities, the expressions should be simplified in the ways discussed in the hypothesis testingsections. As the energy terms are known a priori they can be incorporated into the threshold with the result $\sum_{l=0}^{L-1} r(l)({s}_{1}(l)-{s}_{0}(l))\underset{{}_{0}}{\stackrel{{}_{1}}{}}^{2}\ln +\frac{{E}_{1}-{E}_{0}}{2}$ The left term constitutes the sufficient statistic for the binary detection problem. Because the additive noise is presumed Gaussian,the sufficient statistic is a Gaussian random variable no matter which model is assumed. Under ${}_{i}$ , the specifics of this probability distribution are $(\sum_{l=0}^{L-1} r(l)({s}_{1}(l)-{s}_{0}(l)), (\sum {s}_{i}(l)({s}_{1}(l)-{s}_{0}(l)), ^{2}\sum ({s}_{1}(l)-{s}_{0}(l))^{2}))$ The false-alarm probability is given by ${P}_{F}=Q(\frac{^{2}\ln +\frac{{E}_{1}-{E}_{0}}{2}-\sum {s}_{0}(l)({s}_{1}(l)-{s}_{0}(l))}{\sum ({s}_{1}(l)-{s}_{0}(l))^{2}^{\left(\frac{1}{2}\right)}})$ The signal-related terms in the numerator of this expression can be manipulated with the false-alarm probability (and thedetection probability) for the optimal white Gaussian noise detector succinctly expressed by ${P}_{F}=Q(\frac{\ln +\frac{1}{2^{2}}\sum ({s}_{1}(l)-{s}_{0}(l))^{2}}{\frac{1}{}\sum ({s}_{1}(l)-{s}_{0}(l))^{2}^{\left(\frac{1}{2}\right)}})$ ${P}_{F}=Q(\frac{\ln -\frac{1}{2^{2}}\sum ({s}_{1}(l)-{s}_{0}(l))^{2}}{\frac{1}{}\sum ({s}_{1}(l)-{s}_{0}(l))^{2}^{\left(\frac{1}{2}\right)}})$

Note that the only signal-related quantity affecting this performance probability (and all of the others)is the ratio of energy in the difference signal to the noise variance. The larger this ratio, the better (smaller) the performance probabilities become. Note that thedetails of the signal waveforms do not greatly affect the energy of the difference signal. For example, consider the case wherethe two signal energies are equal ( ${E}_{0}={E}_{1}=E$ ); the energy of the difference signal is given by $2E-2\sum {s}_{0}(l){s}_{1}(l)$ . The largest value of this energy occurs when the signals are negatives of each other, with the difference-signalenergy equaling $4E$ . Thus, equal-energy but opposite-signed signals such as sine waves, square-waves, Bessel functions,etc. all yield exactly the same performance levels. The essential signal properties that do yield goodperformance values are elucidated by an alternate interpretation. The term $\sum ({s}_{1}(l)-{s}_{0}(l))^{2}$ equals $({s}_{1}-{s}_{0})^{2}$ , the $L^{2}$ norm of the difference signal. Geometrically, the difference-signal energy is the same quantity as the square ofthe Euclidean distance between the two signals. In these terms, a larger distance between the two signals will mean betterperformance.

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
can nanotechnology change the direction of the face of the world
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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