# 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.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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.
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