0:00:15i'm more talk about the use simple uh
0:00:18multi antenna spectrum set
0:00:20using the uh
0:00:22data autocorrelation correlation function
0:00:25and the uh
0:00:27so is the multiple antennas the noise variance at each antenna
0:00:32uh that's the general model
0:00:38i'm not gonna assume that the uh signal bargains uh gaussian possibly non gaussian
0:00:44uh noise is assumed to be
0:00:49first part of my talk at a time
0:00:53correlated gaussian noise
0:00:55very complex
0:00:56this proper
0:01:01nice very a different sensors can uh a lot to deaf
0:01:06uh this problem has been what on by one bunch of people in different
0:01:18this paper here uh because of the need
0:01:22title of paper or happening would buy
0:01:25people including me
0:01:28this assume
0:01:30the basic some so there is there on and as the title delay indicates is unclear good
0:01:35the so each
0:01:37sense can have different nice variance
0:01:40and they use a generalized likelihood ratio approach
0:01:43uh the basic assumption that is a a a a a a in addition to white gaussian noise the signal
0:01:47itself is a gaussian as well as what
0:01:50so we need i on the
0:01:52like to a show all time samples are independent easy to write
0:01:56like you you sure than a like to lies like to
0:02:01the next set of these two set of papers
0:02:05they have a similar stuff but these two papers this you equal variances
0:02:09and they also have to noise like to the sure approach
0:02:12and again this assumed that the signal is by "'cause" you
0:02:15and then the this paper has a a nice or are all the stuff available at that one extension
0:02:22what i wanna do it i don't one assume that the signal is gaussian so i don't to be using
0:02:27lies like to visual approach
0:02:29i just go and straight and use the autocorrelation function
0:02:32and the basic assumption be are using here is
0:02:35that D
0:02:36noise is spatially uncorrelated
0:02:39so the cross-correlation correlation those signals
0:02:41of the observation across different senses
0:02:44is zero under the null hypothesis
0:02:46could be non-zero under the uh
0:02:48that's all
0:02:50but they also out that up a go to all my stuff
0:02:54my final result days
0:02:57you put there result
0:02:59a they have a to less like a short test which under low and my social conditions
0:03:04the has an approximation which is exactly what i
0:03:07in the case of like it's
0:03:12okay so okay said think got thing about the the that like like a best is
0:03:16that asymptotically you can use the will to room the related stuff
0:03:22and you can you can
0:03:23asymptotically the the most should will have a chi-square distribution
0:03:29and the null hypothesis
0:03:31a a central chi-square distribution
0:03:33bear a i have be sensor so the the degrees of freedom is actually the number of unknown parameters under
0:03:38H one
0:03:39or of five parts
0:03:41minus the number of one on put on this and of the not
0:03:45do that
0:03:45you get a
0:03:47so asymptotically you have to do
0:03:49oh the distribution and you can calculate the touch
0:03:52knife i go back uh
0:03:55do these two papers here
0:03:57they exploit the fact
0:04:01uh a that that the single signal with the one selected one
0:04:05autocorrelation correlation function
0:04:07under that the the uh in the absence of noise and of the alternative
0:04:12the the with them is
0:04:14they can they
0:04:15compute the threshold
0:04:19reasonable data a like using simple
0:04:22okay there's no analytical way to compute that the threshold
0:04:25they use this approximation but sort in their "'cause" this approximation is valid for very long
0:04:30you have a we we have a like of
0:04:33and the got or the probability of detection again that there's is a
0:04:36bunch of results asymptotically it becomes a the chi-square just noncentral chi-square distribution
0:04:42uh with the same because of it except the non-centrality parameter is a functional or
0:04:47first them information matrix and
0:04:52in my is they come what very simply there in in a very simple fashion
0:04:55okay so what are gonna do is but simply take the uh uh estimate the uh
0:05:00correlation function i i'm when you gonna use the correlation function of the data i zero like
0:05:06but this is the new stuff
0:05:08so the idea is the i and
0:05:11idea a component of this so this is the
0:05:14i you
0:05:15and the G sensor cost correlated
0:05:18so if
0:05:19on that the null hypothesis X to a white complex gaussian i then turns out that the you like
0:05:26the idea of component of that is complex gaussian asymptotically a zero game
0:05:31a member the
0:05:32noise is
0:05:33uncorrelated spatial
0:05:35okay so we now is not equal to J
0:05:37it a zero mean
0:05:38and the
0:05:40the variance of that the square of in here
0:05:43this variance is this is the the noise variance under the
0:05:47that for ten so this is the noise waiting for the G sense
0:05:51and they are assumed to be unknown so
0:05:55and plus if you
0:05:57the this is a a B Y be me X P of the number of sensors
0:06:00so the off diagonal terms either the log or triangle or or the low triangle
0:06:06they are mutually in yeah asymptotically the be usually
0:06:13okay so we don't that problem in two of the spectrum sensing whether there is a signal present or not
0:06:19in two
0:06:20oh this hypothesis testing problem so the
0:06:24correlation function between in the uh the uh i in the j-th sensor
0:06:28system i is spatially uncorrelated
0:06:31they should be you don't is not a college check
0:06:34no primary signal if the by me signal
0:06:36this could be not
0:06:37okay not it's not identically zero for i not jet
0:06:41so we we use the large sample of correlation properties and B
0:06:45consider this to the test statistic and how the statistic
0:06:50these are the estimates
0:06:51and if you yeah and and B do place the unknown is if we go back
0:06:55i need this variance of be a list them by the estimate
0:06:59and be it
0:07:02okay and
0:07:03compare is against trash
0:07:05and as i mentioned it before if you go back to the national render rinse approach
0:07:10are less like an sure approach and of the white a signal and white noise white gaussian signal in white
0:07:16under the low snr conditions it don't suck be pretty it's
0:07:23so we want to you want to uh pick the special so for a given problem false alarm so we
0:07:28look at lots of properties
0:07:30and if you have a the true values here
0:07:33then if we look at a single P i G I not equal G it's a
0:07:38chi-square distribution with two degrees of freedom
0:07:42okay because asymptotically that's complex it
0:07:45and it a some overall all to also so gonna a placed this by the estimated value
0:07:50so kiss to i'm and all that stuff it's still is
0:07:53asymptotically chi-squared distribution
0:07:55we two do
0:07:56but might that statistic is something and over all uh uh
0:08:00possible page
0:08:03non back not pairs
0:08:05so if you do that
0:08:08then it it becomes uh might that statistic i'm gonna be using
0:08:12is to
0:08:14chi-square distribution
0:08:15but these many degrees of freedom
0:08:17okay i'm using all
0:08:21payers money pace
0:08:23and this is what you would got a you would have got a if you use the uh
0:08:30some signal and use the
0:08:34i didn't have to use
0:08:35but it's it's got
0:08:38okay no we wanna to the detection probability
0:08:41so a detection probability uh turns are
0:08:44a we will use the the a low to do to my social calculations but uh under
0:08:49alternative hypothesis and it doesn't have white signal but you have an expression for the
0:08:55a correlation function something like this
0:08:57and we make it big that a lot of it are basically take all the rooms
0:09:02uh a square minus speech rooms
0:09:04a a out of
0:09:05or were to don't sort of this make a big but out of it
0:09:08and asymptotically also this is uh gaussian
0:09:13a a a bit uh this meeting
0:09:16and this mean is
0:09:18the contribution of the mean is coming from the uh by means
0:09:22but asymptotically it's not calm
0:09:25okay it
0:09:26it's not a it's a
0:09:29it's not a problem or it's not a circular lisa
0:09:32okay so the the real part is not going to put a
0:09:35team at all
0:09:36compliment a which does not sit however
0:09:39if i and low snr condition
0:09:42then it's approximately complex not seem so that's what i
0:09:46basically the of the low snr conditions the or the only is
0:09:51the mean becomes not
0:09:53okay the variance doesn't T
0:09:56if you use that that it's uh
0:09:58complex gaussian and based on that you go back to the same test to just
0:10:04and for large sample length
0:10:06the test at this will become noncentral chi-square uh distribution
0:10:11with non-centrality parameter a that which is given by this
0:10:14but if use
0:10:16this is a general expression but if for use uh a low snr assumption
0:10:20then it becomes something like this it you if simplified
0:10:25what all the stuff and there is
0:10:27so simple
0:10:28i'm using the same expression except that
0:10:32a a correlation functions are now with the primary signal press
0:10:36okay so
0:10:37do this that and use the nice variances as but he by have the effect of the pie
0:10:41okay okay and eyes
0:10:43not it to J this is not
0:10:46and so now it's test the comes and of the
0:10:50uh alternative hypothesis in the presence of the primary user signal
0:10:55asymptotically is a noncentral chi-square distribution the same degrees of it
0:11:00okay you you would have gotten a similar result
0:11:03if you are assume that it was white gaussian signal white gaussian noise
0:11:07and use the fixed
0:11:11"'kay" is also you simulation sample
0:11:15you of and and it's
0:11:16and i'm many this although the results of that valid for the general channel
0:11:19so mate
0:11:20flat fading channel with independent components complex gaussian to basically each
0:11:25uh uh a component is really fit
0:11:28and i using a qpsk a to use a signal
0:11:32and the nice variances
0:11:34are multi pulse of some fixed variance to this good
0:11:38is not null
0:11:39significance to this
0:11:42and i need an snr some sort of calculating this in my my they should each
0:11:46and uh i'm just gonna take an average snr
0:11:49so the at a signal power some the power signal power or or or or right hand as divide by
0:11:54some of the noise power
0:11:57i all the score an average S
0:12:02and i compared with than and it with the standard energy detector
0:12:06and this is the uh what what i call the uh time domain like to should test
0:12:11oh these are the the uh what original paper so mentioned it people back to
0:12:20these two papers are pretty much the same
0:12:22okay so i'm comparing these to not
0:12:26this paper that is that the same
0:12:29okay i i under the condition that i
0:12:35these guys don't allow the noise variance to be different that
0:12:39they are have this to say
0:12:40it's a
0:12:41and they compute the eigenvalues of a correlation
0:12:46and you this is the largest eigenvalue of that that this is the so
0:12:50some of
0:12:51all the i i use a compare is a special
0:12:54and again what this one
0:12:56and it to detect there as well as for this uh glrt
0:12:59the special has to be computed
0:13:03a simple results
0:13:08and and that's what i showing here the the
0:13:12this you were operating characteristic for one hundred and twenty eight
0:13:16uh samples
0:13:17minus seven every say
0:13:20some of all set
0:13:21signal power across all sense
0:13:24some of my is of all sensor
0:13:26this is
0:13:27uh this is the
0:13:29the glrt assuming
0:13:31equal will so
0:13:33at all sensors
0:13:34this is the energy detector and this is the
0:13:37are correlation function
0:13:40and this is the uh uh a same set except i'm changing my S and not
0:13:44one a probability of false alarm one zero one
0:13:47and this is the
0:13:48proposed stuff
0:13:50and this is the energy detector this is the glrt
0:13:54this the energy detector for
0:13:56if if you're threshold is based upon the noise variances are simulations but if it all white to you also
0:14:03it's sensitive to the selection of the touch it sensitive to
0:14:07the pressure selection is based upon your knowledge of the mice variance if it noise one
0:14:12it can the form is good
0:14:16and this these are the is that for the uh okay
0:14:19the for these two
0:14:20well i it's
0:14:21the uh i i really fading
0:14:24but for my probability of detection calculation uh that's based on
0:14:30okay so these results are for the fixed channel it's say
0:14:34noncentral chi-square distribution so that's what i
0:14:37so in here
0:14:38so here i fixed the channel the magnitude at a channel did you want the face can change
0:14:44and the solid curve is the simulation result and the task is
0:14:48vertical stuff
0:14:50and this is for her twenty eight at
0:14:52oh probably due false so what you're one
0:14:54and same noise mismatch the the four sensors
0:14:59and this is for twenty five set
0:15:02okay so the everything is based on
0:15:04and samples is and this is the solid
0:15:08the dashed
0:15:09the is the uh
0:15:11uh you
0:15:12so that's pretty much what i have in my paper
0:15:16so you have some more time
0:15:18yeah okay so
0:15:20a a i would show you what happens you can extend this is not a
0:15:23conference paper
0:15:25so no i'm but a lot of the nice to be close so
0:15:28basically what you using is a nice is spatially uncorrelated cross
0:15:34i'm wise
0:15:35it can go like so that's what it is
0:15:37and i is the same
0:15:39okay so the a than than other losses
0:15:42the correlation function of zero lag
0:15:44a a i not equal to G zero five not equal to J
0:15:47it's not at C
0:15:49so what what happens is that a nice way is not G
0:15:54the under tree
0:15:55no hypothesis
0:15:57if you estimate this bad
0:15:59it's nice to is in my previous stuff
0:16:02what i is entered to for it is white gaussian eyes
0:16:04this is nonzero zero and meet weird is it but if it is
0:16:08colour nice
0:16:09then a a nice it is it's high and it also depends upon the
0:16:14the the uh
0:16:15correlation structure of the nice it it's
0:16:18so i i i this is i it sense structure this is a G S the structure
0:16:22this is a lack
0:16:23okay so i have something it'll or
0:16:26oh different lacks had my capital and this is is is the upper model on the
0:16:32okay i'm assuming that
0:16:33you on "'em" it goes to zero and the rest of the stuff is exactly what i do for we
0:16:38we modification we is this
0:16:40and again X it is the chi-square distribution with the same number of degrees of freedom
0:16:45and if you do that it looks very fine
0:16:47okay this is
0:16:48was stuff
0:16:49the energy detector
0:16:51and this is the energy detector a the uh
0:16:53nice estimate mismatch
0:16:55and if we
0:16:56this is not this is this is this design for a probability of false one shoe
0:17:01if we if you assume much white nice to apply the generalized likelihood ratio test
0:17:06but white
0:17:07uh a noise and white signal then the probably the false let me gonna get is
0:17:11much high point one the
0:17:14so that
0:17:15to basically goes best are not in me in two
0:17:18the correlation structure
0:17:20it is but modifying it
0:17:22this place
0:17:24that's all
0:17:28we have a couple of minutes for
0:17:31from the speech
0:17:32is there in question
0:17:44yeah a i'm not i'm not i'm gonna i'm not using the approach
0:17:47it get it may have it be correlated
0:17:49it it as a matter now have trying to simulations zone
0:17:54at your like
0:17:55it has to be non sit at different uh the lexus
0:17:58like i it is so
0:18:00i i no i'm not exploit
0:18:07a a have just not short one uh a said that
0:18:10your paper and your
0:18:11technique is improving the spec to existing literature
0:18:15uh taken into account
0:18:16the possibility of different
0:18:18uh but as for the noise on each sensor
0:18:22so uh
0:18:24is it
0:18:25can you give some example were in practice we could have
0:18:29different sensors with different noise uh
0:18:32at the receiver side i mean i you
0:18:34assuming that
0:18:35each each each one of your sense of could have be different from the other one
0:18:39yeah are are be the calibration could
0:18:41goal for with time
0:18:45that's it that
0:18:48so uh uh
0:18:50and and a question we have still one
0:18:51one mean
0:18:53if we want to
0:19:04there on
0:19:11as i T
0:19:13the speaker