uh

i'm more talk about the use simple uh

multi antenna spectrum set

using the uh

data autocorrelation correlation function

and the uh

uh

so is the multiple antennas the noise variance at each antenna

uh that's the general model

i'm not gonna assume that the uh signal bargains uh gaussian possibly non gaussian

uh noise is assumed to be

first part of my talk at a time

extent

correlated gaussian noise

very complex

this proper

uh

the

nice very a different sensors can uh a lot to deaf

uh this problem has been what on by one bunch of people in different

forms

a

this paper here uh because of the need

title of paper or happening would buy

people including me

uh

so

this assume

the basic some so there is there on and as the title delay indicates is unclear good

so

the so each

sense can have different nice variance

and they use a generalized likelihood ratio approach

uh the basic assumption that is a a a a a a in addition to white gaussian noise the signal

itself is a gaussian as well as what

so we need i on the

like to a show all time samples are independent easy to write

like you you sure than a like to lies like to

sure

the next set of these two set of papers

they have a similar stuff but these two papers this you equal variances

and they also have to noise like to the sure approach

and again this assumed that the signal is by "'cause" you

and then the this paper has a a nice or are all the stuff available at that one extension

what i wanna do it i don't one assume that the signal is gaussian so i don't to be using

lies like to visual approach

i just go and straight and use the autocorrelation function

and the basic assumption be are using here is

that D

noise is spatially uncorrelated

so the cross-correlation correlation those signals

of the observation across different senses

is zero under the null hypothesis

could be non-zero under the uh

that's all

but they also out that up a go to all my stuff

my final result days

you put there result

a they have a to less like a short test which under low and my social conditions

the has an approximation which is exactly what i

in the case of like it's

okay so okay said think got thing about the the that like like a best is

that asymptotically you can use the will to room the related stuff

and you can you can

asymptotically the the most should will have a chi-square distribution

and the null hypothesis

a a central chi-square distribution

bear a i have be sensor so the the degrees of freedom is actually the number of unknown parameters under

H one

or of five parts

minus the number of one on put on this and of the not

do that

you get a

so asymptotically you have to do

oh the distribution and you can calculate the touch

knife i go back uh

do these two papers here

they exploit the fact

uh a that that the single signal with the one selected one

autocorrelation correlation function

under that the the uh in the absence of noise and of the alternative

the the with them is

that

they can they

compute the threshold

for

or

reasonable data a like using simple

okay there's no analytical way to compute that the threshold

they use this approximation but sort in their "'cause" this approximation is valid for very long

you have a we we have a like of

and the got or the probability of detection again that there's is a

bunch of results asymptotically it becomes a the chi-square just noncentral chi-square distribution

uh with the same because of it except the non-centrality parameter is a functional or

first them information matrix and

this

in my is they come what very simply there in in a very simple fashion

okay so what are gonna do is but simply take the uh uh estimate the uh

correlation function i i'm when you gonna use the correlation function of the data i zero like

but this is the new stuff

so the idea is the i and

the

idea a component of this so this is the

i you

sensor

and the G sensor cost correlated

so if

on that the null hypothesis X to a white complex gaussian i then turns out that the you like

the idea of component of that is complex gaussian asymptotically a zero game

a member the

noise is

uncorrelated spatial

okay so we now is not equal to J

it a zero mean

and the

uh

the variance of that the square of in here

this variance is this is the the noise variance under the

that for ten so this is the noise waiting for the G sense

okay

and they are assumed to be unknown so

uh

and plus if you

the this is a a B Y be me X P of the number of sensors

so the off diagonal terms either the log or triangle or or the low triangle

they are mutually in yeah asymptotically the be usually

okay so we don't that problem in two of the spectrum sensing whether there is a signal present or not

present

in two

oh this hypothesis testing problem so the

correlation function between in the uh the uh i in the j-th sensor

i

system i is spatially uncorrelated

they should be you don't is not a college check

no primary signal if the by me signal

this could be not

okay not it's not identically zero for i not jet

so we we use the large sample of correlation properties and B

consider this to the test statistic and how the statistic

these are the estimates

and if you yeah and and B do place the unknown is if we go back

i need this variance of be a list them by the estimate

and be it

okay and

compare is against trash

and as i mentioned it before if you go back to the national render rinse approach

are less like an sure approach and of the white a signal and white noise white gaussian signal in white

noise

under the low snr conditions it don't suck be pretty it's

see

but

so we want to you want to uh pick the special so for a given problem false alarm so we

look at lots of properties

and if you have a the true values here

then if we look at a single P i G I not equal G it's a

uh

chi-square distribution with two degrees of freedom

okay because asymptotically that's complex it

and it a some overall all to also so gonna a placed this by the estimated value

so kiss to i'm and all that stuff it's still is

asymptotically chi-squared distribution

we two do

but might that statistic is something and over all uh uh

possible page

non back not pairs

so if you do that

then it it becomes uh might that statistic i'm gonna be using

is to

chi-square distribution

but these many degrees of freedom

okay i'm using all

uh

payers money pace

and this is what you would got a you would have got a if you use the uh

like

uh

signal

some signal and use the

uh

we

i didn't have to use

but it's it's got

okay no we wanna to the detection probability

so a detection probability uh turns are

a we will use the the a low to do to my social calculations but uh under

alternative hypothesis and it doesn't have white signal but you have an expression for the

a correlation function something like this

and we make it big that a lot of it are basically take all the rooms

uh a square minus speech rooms

a a out of

or were to don't sort of this make a big but out of it

and asymptotically also this is uh gaussian

but

a a a bit uh this meeting

okay

and this mean is

the contribution of the mean is coming from the uh by means

but asymptotically it's not calm

okay it

it's not a it's a

but

it's not a problem or it's not a circular lisa

okay so the the real part is not going to put a

team at all

compliment a which does not sit however

if i and low snr condition

then it's approximately complex not seem so that's what i

basically the of the low snr conditions the or the only is

in

the mean becomes not

okay the variance doesn't T

so

if you use that that it's uh

complex gaussian and based on that you go back to the same test to just

and for large sample length

the test at this will become noncentral chi-square uh distribution

with non-centrality parameter a that which is given by this

but if use

this is a general expression but if for use uh a low snr assumption

then it becomes something like this it you if simplified

uh

what all the stuff and there is

so simple

so

i'm using the same expression except that

these

a a correlation functions are now with the primary signal press

okay so

do this that and use the nice variances as but he by have the effect of the pie

okay okay and eyes

not it to J this is not

and so now it's test the comes and of the

uh alternative hypothesis in the presence of the primary user signal

asymptotically is a noncentral chi-square distribution the same degrees of it

so

okay you you would have gotten a similar result

if you are assume that it was white gaussian signal white gaussian noise

and use the fixed

"'kay" is also you simulation sample

you of and and it's

and i'm many this although the results of that valid for the general channel

so mate

flat fading channel with independent components complex gaussian to basically each

uh uh a component is really fit

and i using a qpsk a to use a signal

and the nice variances

are multi pulse of some fixed variance to this good

is not null

significance to this

and i need an snr some sort of calculating this in my my they should each

and uh i'm just gonna take an average snr

so the at a signal power some the power signal power or or or or right hand as divide by

some of the noise power

i all the score an average S

and i compared with than and it with the standard energy detector

and this is the uh what what i call the uh time domain like to should test

oh these are the the uh what original paper so mentioned it people back to

i'm

these two papers are pretty much the same

okay so i'm comparing these to not

this paper that is that the same

okay i i under the condition that i

use

because

okay

these guys don't allow the noise variance to be different that

they are have this to say

it's a

and they compute the eigenvalues of a correlation

and you this is the largest eigenvalue of that that this is the so

some of

all the i i use a compare is a special

and again what this one

and it to detect there as well as for this uh glrt

the special has to be computed

so

a simple results

this

and and that's what i showing here the the

oh

this you were operating characteristic for one hundred and twenty eight

uh samples

minus seven every say

some of all set

signal power across all sense

some of my is of all sensor

this is

more

uh this is the

the glrt assuming

equal will so

at all sensors

this is the energy detector and this is the

are correlation function

and this is the uh uh a same set except i'm changing my S and not

one a probability of false alarm one zero one

and this is the

proposed stuff

and this is the energy detector this is the glrt

this the energy detector for

if if you're threshold is based upon the noise variances are simulations but if it all white to you also

it's sensitive to the selection of the touch it sensitive to

the pressure selection is based upon your knowledge of the mice variance if it noise one

all

it can the form is good

and this these are the is that for the uh okay

the for these two

well i it's

the uh i i really fading

but for my probability of detection calculation uh that's based on

fix

channel

okay so these results are for the fixed channel it's say

noncentral chi-square distribution so that's what i

so in here

so here i fixed the channel the magnitude at a channel did you want the face can change

and the solid curve is the simulation result and the task is

vertical stuff

and this is for her twenty eight at

oh probably due false so what you're one

and same noise mismatch the the four sensors

and this is for twenty five set

okay so the everything is based on

and samples is and this is the solid

simulations

the dashed

the is the uh

uh you

so that's pretty much what i have in my paper

so you have some more time

yeah okay so

a a i would show you what happens you can extend this is not a

conference paper

so no i'm but a lot of the nice to be close so

basically what you using is a nice is spatially uncorrelated cross

but

i'm wise

it can go like so that's what it is

and i is the same

okay so the a than than other losses

the correlation function of zero lag

a a i not equal to G zero five not equal to J

it's not at C

so what what happens is that a nice way is not G

a

the under tree

no hypothesis

if you estimate this bad

it's nice to is in my previous stuff

what i is entered to for it is white gaussian eyes

this is nonzero zero and meet weird is it but if it is

colour nice

then a a nice it is it's high and it also depends upon the

the the uh

correlation structure of the nice it it's

so i i i this is i it sense structure this is a G S the structure

this is a lack

okay so i have something it'll or

oh different lacks had my capital and this is is is the upper model on the

correlation

okay i'm assuming that

you on "'em" it goes to zero and the rest of the stuff is exactly what i do for we

we modification we is this

okay

and again X it is the chi-square distribution with the same number of degrees of freedom

and if you do that it looks very fine

okay this is

was stuff

the energy detector

and this is the energy detector a the uh

nice estimate mismatch

and if we

this is not this is this is this design for a probability of false one shoe

however

if we if you assume much white nice to apply the generalized likelihood ratio test

but white

uh a noise and white signal then the probably the false let me gonna get is

much high point one the

so that

to basically goes best are not in me in two

the correlation structure

it is but modifying it

this place

that's all

we have a couple of minutes for

question

from the speech

is there in question

yeah

yeah a i'm not i'm not i'm gonna i'm not using the approach

it get it may have it be correlated

it it as a matter now have trying to simulations zone

because

at your like

it has to be non sit at different uh the lexus

yeah

like i it is so

i i no i'm not exploit

a a have just not short one uh a said that

your paper and your

technique is improving the spec to existing literature

uh taken into account

the possibility of different

uh but as for the noise on each sensor

so uh

is it

um

can you give some example were in practice we could have

different sensors with different noise uh

at the receiver side i mean i you

assuming that

each each each one of your sense of could have be different from the other one

yeah are are be the calibration could

goal for with time

that's it that

but

okay

so uh uh

and and a question we have still one

one mean

if we want to

supplied

yeah

no

there on

yeah

okay

as i T

the speaker