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