0:00:13 | uh |
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0:00:15 | i'm more talk about the use simple uh |

0:00:18 | multi antenna spectrum set |

0:00:20 | using the uh |

0:00:22 | data autocorrelation correlation function |

0:00:25 | and the uh |

0:00:26 | uh |

0:00:27 | so is the multiple antennas the noise variance at each antenna |

0:00:32 | uh that's the general model |

0:00:38 | i'm not gonna assume that the uh signal bargains uh gaussian possibly non gaussian |

0:00:44 | uh noise is assumed to be |

0:00:49 | first part of my talk at a time |

0:00:51 | extent |

0:00:53 | correlated gaussian noise |

0:00:55 | very complex |

0:00:56 | this proper |

0:00:58 | uh |

0:01:00 | the |

0:01:01 | nice very a different sensors can uh a lot to deaf |

0:01:06 | uh this problem has been what on by one bunch of people in different |

0:01:12 | forms |

0:01:14 | a |

0:01:18 | this paper here uh because of the need |

0:01:22 | title of paper or happening would buy |

0:01:25 | people including me |

0:01:27 | uh |

0:01:28 | so |

0:01:28 | this assume |

0:01:30 | the basic some so there is there on and as the title delay indicates is unclear good |

0:01:35 | so |

0:01:35 | the so each |

0:01:37 | sense can have different nice variance |

0:01:40 | and they use a generalized likelihood ratio approach |

0:01:43 | uh the basic assumption that is a a a a a a in addition to white gaussian noise the signal |

0:01:47 | itself is a gaussian as well as what |

0:01:50 | so we need i on the |

0:01:52 | like to a show all time samples are independent easy to write |

0:01:56 | like you you sure than a like to lies like to |

0:01:58 | sure |

0:02:01 | the next set of these two set of papers |

0:02:05 | they have a similar stuff but these two papers this you equal variances |

0:02:09 | and they also have to noise like to the sure approach |

0:02:12 | and again this assumed that the signal is by "'cause" you |

0:02:15 | and then the this paper has a a nice or are all the stuff available at that one extension |

0:02:22 | what i wanna do it i don't one assume that the signal is gaussian so i don't to be using |

0:02:27 | lies like to visual approach |

0:02:29 | i just go and straight and use the autocorrelation function |

0:02:32 | and the basic assumption be are using here is |

0:02:35 | that D |

0:02:36 | noise is spatially uncorrelated |

0:02:39 | so the cross-correlation correlation those signals |

0:02:41 | of the observation across different senses |

0:02:44 | is zero under the null hypothesis |

0:02:46 | could be non-zero under the uh |

0:02:48 | that's all |

0:02:50 | but they also out that up a go to all my stuff |

0:02:54 | my final result days |

0:02:57 | you put there result |

0:02:59 | a they have a to less like a short test which under low and my social conditions |

0:03:04 | the has an approximation which is exactly what i |

0:03:07 | in the case of like it's |

0:03:12 | okay so okay said think got thing about the the that like like a best is |

0:03:16 | that asymptotically you can use the will to room the related stuff |

0:03:22 | and you can you can |

0:03:23 | asymptotically the the most should will have a chi-square distribution |

0:03:29 | and the null hypothesis |

0:03:31 | a a central chi-square distribution |

0:03:33 | bear a i have be sensor so the the degrees of freedom is actually the number of unknown parameters under |

0:03:38 | H one |

0:03:39 | or of five parts |

0:03:41 | minus the number of one on put on this and of the not |

0:03:45 | do that |

0:03:45 | you get a |

0:03:47 | so asymptotically you have to do |

0:03:49 | oh the distribution and you can calculate the touch |

0:03:52 | knife i go back uh |

0:03:55 | do these two papers here |

0:03:57 | they exploit the fact |

0:04:01 | uh a that that the single signal with the one selected one |

0:04:05 | autocorrelation correlation function |

0:04:07 | under that the the uh in the absence of noise and of the alternative |

0:04:12 | the the with them is |

0:04:13 | that |

0:04:14 | they can they |

0:04:15 | compute the threshold |

0:04:17 | for |

0:04:18 | or |

0:04:19 | reasonable data a like using simple |

0:04:22 | okay there's no analytical way to compute that the threshold |

0:04:25 | they use this approximation but sort in their "'cause" this approximation is valid for very long |

0:04:30 | you have a we we have a like of |

0:04:33 | and the got or the probability of detection again that there's is a |

0:04:36 | bunch of results asymptotically it becomes a the chi-square just noncentral chi-square distribution |

0:04:42 | uh with the same because of it except the non-centrality parameter is a functional or |

0:04:47 | first them information matrix and |

0:04:49 | this |

0:04:52 | in my is they come what very simply there in in a very simple fashion |

0:04:55 | okay so what are gonna do is but simply take the uh uh estimate the uh |

0:05:00 | correlation function i i'm when you gonna use the correlation function of the data i zero like |

0:05:06 | but this is the new stuff |

0:05:08 | so the idea is the i and |

0:05:10 | the |

0:05:11 | idea a component of this so this is the |

0:05:14 | i you |

0:05:14 | sensor |

0:05:15 | and the G sensor cost correlated |

0:05:18 | so if |

0:05:19 | on that the null hypothesis X to a white complex gaussian i then turns out that the you like |

0:05:26 | the idea of component of that is complex gaussian asymptotically a zero game |

0:05:31 | a member the |

0:05:32 | noise is |

0:05:33 | uncorrelated spatial |

0:05:35 | okay so we now is not equal to J |

0:05:37 | it a zero mean |

0:05:38 | and the |

0:05:39 | uh |

0:05:40 | the variance of that the square of in here |

0:05:43 | this variance is this is the the noise variance under the |

0:05:47 | that for ten so this is the noise waiting for the G sense |

0:05:51 | okay |

0:05:51 | and they are assumed to be unknown so |

0:05:53 | uh |

0:05:55 | and plus if you |

0:05:57 | the this is a a B Y be me X P of the number of sensors |

0:06:00 | so the off diagonal terms either the log or triangle or or the low triangle |

0:06:06 | they are mutually in yeah asymptotically the be usually |

0:06:13 | okay so we don't that problem in two of the spectrum sensing whether there is a signal present or not |

0:06:18 | present |

0:06:19 | in two |

0:06:20 | oh this hypothesis testing problem so the |

0:06:24 | correlation function between in the uh the uh i in the j-th sensor |

0:06:28 | i |

0:06:28 | system i is spatially uncorrelated |

0:06:31 | they should be you don't is not a college check |

0:06:34 | no primary signal if the by me signal |

0:06:36 | this could be not |

0:06:37 | okay not it's not identically zero for i not jet |

0:06:41 | so we we use the large sample of correlation properties and B |

0:06:45 | consider this to the test statistic and how the statistic |

0:06:50 | these are the estimates |

0:06:51 | and if you yeah and and B do place the unknown is if we go back |

0:06:55 | i need this variance of be a list them by the estimate |

0:06:59 | and be it |

0:07:02 | okay and |

0:07:03 | compare is against trash |

0:07:05 | and as i mentioned it before if you go back to the national render rinse approach |

0:07:10 | are less like an sure approach and of the white a signal and white noise white gaussian signal in white |

0:07:15 | noise |

0:07:16 | under the low snr conditions it don't suck be pretty it's |

0:07:19 | see |

0:07:23 | but |

0:07:23 | so we want to you want to uh pick the special so for a given problem false alarm so we |

0:07:28 | look at lots of properties |

0:07:30 | and if you have a the true values here |

0:07:33 | then if we look at a single P i G I not equal G it's a |

0:07:38 | uh |

0:07:38 | chi-square distribution with two degrees of freedom |

0:07:42 | okay because asymptotically that's complex it |

0:07:45 | and it a some overall all to also so gonna a placed this by the estimated value |

0:07:50 | so kiss to i'm and all that stuff it's still is |

0:07:53 | asymptotically chi-squared distribution |

0:07:55 | we two do |

0:07:56 | but might that statistic is something and over all uh uh |

0:08:00 | possible page |

0:08:03 | non back not pairs |

0:08:05 | so if you do that |

0:08:08 | then it it becomes uh might that statistic i'm gonna be using |

0:08:12 | is to |

0:08:14 | chi-square distribution |

0:08:15 | but these many degrees of freedom |

0:08:17 | okay i'm using all |

0:08:19 | uh |

0:08:21 | payers money pace |

0:08:23 | and this is what you would got a you would have got a if you use the uh |

0:08:27 | like |

0:08:28 | uh |

0:08:29 | signal |

0:08:30 | some signal and use the |

0:08:32 | uh |

0:08:32 | we |

0:08:34 | i didn't have to use |

0:08:35 | but it's it's got |

0:08:38 | okay no we wanna to the detection probability |

0:08:41 | so a detection probability uh turns are |

0:08:44 | a we will use the the a low to do to my social calculations but uh under |

0:08:49 | alternative hypothesis and it doesn't have white signal but you have an expression for the |

0:08:55 | a correlation function something like this |

0:08:57 | and we make it big that a lot of it are basically take all the rooms |

0:09:02 | uh a square minus speech rooms |

0:09:04 | a a out of |

0:09:05 | or were to don't sort of this make a big but out of it |

0:09:08 | and asymptotically also this is uh gaussian |

0:09:11 | but |

0:09:13 | a a a bit uh this meeting |

0:09:15 | okay |

0:09:16 | and this mean is |

0:09:18 | the contribution of the mean is coming from the uh by means |

0:09:22 | but asymptotically it's not calm |

0:09:25 | okay it |

0:09:26 | it's not a it's a |

0:09:28 | but |

0:09:29 | it's not a problem or it's not a circular lisa |

0:09:32 | okay so the the real part is not going to put a |

0:09:35 | team at all |

0:09:36 | compliment a which does not sit however |

0:09:39 | if i and low snr condition |

0:09:42 | then it's approximately complex not seem so that's what i |

0:09:46 | basically the of the low snr conditions the or the only is |

0:09:51 | in |

0:09:51 | the mean becomes not |

0:09:53 | okay the variance doesn't T |

0:09:56 | so |

0:09:56 | if you use that that it's uh |

0:09:58 | complex gaussian and based on that you go back to the same test to just |

0:10:04 | and for large sample length |

0:10:06 | the test at this will become noncentral chi-square uh distribution |

0:10:11 | with non-centrality parameter a that which is given by this |

0:10:14 | but if use |

0:10:16 | this is a general expression but if for use uh a low snr assumption |

0:10:20 | then it becomes something like this it you if simplified |

0:10:24 | uh |

0:10:25 | what all the stuff and there is |

0:10:27 | so simple |

0:10:28 | so |

0:10:28 | i'm using the same expression except that |

0:10:31 | these |

0:10:32 | a a correlation functions are now with the primary signal press |

0:10:36 | okay so |

0:10:37 | do this that and use the nice variances as but he by have the effect of the pie |

0:10:41 | okay okay and eyes |

0:10:43 | not it to J this is not |

0:10:46 | and so now it's test the comes and of the |

0:10:50 | uh alternative hypothesis in the presence of the primary user signal |

0:10:55 | asymptotically is a noncentral chi-square distribution the same degrees of it |

0:10:59 | so |

0:11:00 | okay you you would have gotten a similar result |

0:11:03 | if you are assume that it was white gaussian signal white gaussian noise |

0:11:07 | and use the fixed |

0:11:11 | "'kay" is also you simulation sample |

0:11:15 | you of and and it's |

0:11:16 | and i'm many this although the results of that valid for the general channel |

0:11:19 | so mate |

0:11:20 | flat fading channel with independent components complex gaussian to basically each |

0:11:25 | uh uh a component is really fit |

0:11:28 | and i using a qpsk a to use a signal |

0:11:32 | and the nice variances |

0:11:34 | are multi pulse of some fixed variance to this good |

0:11:38 | is not null |

0:11:39 | significance to this |

0:11:42 | and i need an snr some sort of calculating this in my my they should each |

0:11:46 | and uh i'm just gonna take an average snr |

0:11:49 | so the at a signal power some the power signal power or or or or right hand as divide by |

0:11:54 | some of the noise power |

0:11:57 | i all the score an average S |

0:12:02 | and i compared with than and it with the standard energy detector |

0:12:06 | and this is the uh what what i call the uh time domain like to should test |

0:12:11 | oh these are the the uh what original paper so mentioned it people back to |

0:12:19 | i'm |

0:12:20 | these two papers are pretty much the same |

0:12:22 | okay so i'm comparing these to not |

0:12:26 | this paper that is that the same |

0:12:29 | okay i i under the condition that i |

0:12:31 | use |

0:12:33 | because |

0:12:34 | okay |

0:12:35 | these guys don't allow the noise variance to be different that |

0:12:39 | they are have this to say |

0:12:40 | it's a |

0:12:41 | and they compute the eigenvalues of a correlation |

0:12:46 | and you this is the largest eigenvalue of that that this is the so |

0:12:50 | some of |

0:12:51 | all the i i use a compare is a special |

0:12:54 | and again what this one |

0:12:56 | and it to detect there as well as for this uh glrt |

0:12:59 | the special has to be computed |

0:13:02 | so |

0:13:03 | a simple results |

0:13:05 | this |

0:13:08 | and and that's what i showing here the the |

0:13:11 | oh |

0:13:12 | this you were operating characteristic for one hundred and twenty eight |

0:13:16 | uh samples |

0:13:17 | minus seven every say |

0:13:20 | some of all set |

0:13:21 | signal power across all sense |

0:13:24 | some of my is of all sensor |

0:13:26 | this is |

0:13:27 | more |

0:13:27 | uh this is the |

0:13:29 | the glrt assuming |

0:13:31 | equal will so |

0:13:33 | at all sensors |

0:13:34 | this is the energy detector and this is the |

0:13:37 | are correlation function |

0:13:40 | and this is the uh uh a same set except i'm changing my S and not |

0:13:44 | one a probability of false alarm one zero one |

0:13:47 | and this is the |

0:13:48 | proposed stuff |

0:13:50 | and this is the energy detector this is the glrt |

0:13:54 | this the energy detector for |

0:13:56 | if if you're threshold is based upon the noise variances are simulations but if it all white to you also |

0:14:03 | it's sensitive to the selection of the touch it sensitive to |

0:14:07 | the pressure selection is based upon your knowledge of the mice variance if it noise one |

0:14:11 | all |

0:14:12 | it can the form is good |

0:14:16 | and this these are the is that for the uh okay |

0:14:19 | the for these two |

0:14:20 | well i it's |

0:14:21 | the uh i i really fading |

0:14:24 | but for my probability of detection calculation uh that's based on |

0:14:29 | fix |

0:14:29 | channel |

0:14:30 | okay so these results are for the fixed channel it's say |

0:14:34 | noncentral chi-square distribution so that's what i |

0:14:37 | so in here |

0:14:38 | so here i fixed the channel the magnitude at a channel did you want the face can change |

0:14:44 | and the solid curve is the simulation result and the task is |

0:14:48 | vertical stuff |

0:14:50 | and this is for her twenty eight at |

0:14:52 | oh probably due false so what you're one |

0:14:54 | and same noise mismatch the the four sensors |

0:14:59 | and this is for twenty five set |

0:15:02 | okay so the everything is based on |

0:15:04 | and samples is and this is the solid |

0:15:07 | simulations |

0:15:08 | the dashed |

0:15:09 | the is the uh |

0:15:11 | uh you |

0:15:12 | so that's pretty much what i have in my paper |

0:15:16 | so you have some more time |

0:15:18 | yeah okay so |

0:15:20 | a a i would show you what happens you can extend this is not a |

0:15:23 | conference paper |

0:15:25 | so no i'm but a lot of the nice to be close so |

0:15:28 | basically what you using is a nice is spatially uncorrelated cross |

0:15:33 | but |

0:15:34 | i'm wise |

0:15:35 | it can go like so that's what it is |

0:15:37 | and i is the same |

0:15:39 | okay so the a than than other losses |

0:15:42 | the correlation function of zero lag |

0:15:44 | a a i not equal to G zero five not equal to J |

0:15:47 | it's not at C |

0:15:49 | so what what happens is that a nice way is not G |

0:15:53 | a |

0:15:54 | the under tree |

0:15:55 | no hypothesis |

0:15:57 | if you estimate this bad |

0:15:59 | it's nice to is in my previous stuff |

0:16:02 | what i is entered to for it is white gaussian eyes |

0:16:04 | this is nonzero zero and meet weird is it but if it is |

0:16:08 | colour nice |

0:16:09 | then a a nice it is it's high and it also depends upon the |

0:16:14 | the the uh |

0:16:15 | correlation structure of the nice it it's |

0:16:18 | so i i i this is i it sense structure this is a G S the structure |

0:16:22 | this is a lack |

0:16:23 | okay so i have something it'll or |

0:16:26 | oh different lacks had my capital and this is is is the upper model on the |

0:16:31 | correlation |

0:16:32 | okay i'm assuming that |

0:16:33 | you on "'em" it goes to zero and the rest of the stuff is exactly what i do for we |

0:16:38 | we modification we is this |

0:16:40 | okay |

0:16:40 | and again X it is the chi-square distribution with the same number of degrees of freedom |

0:16:45 | and if you do that it looks very fine |

0:16:47 | okay this is |

0:16:48 | was stuff |

0:16:49 | the energy detector |

0:16:51 | and this is the energy detector a the uh |

0:16:53 | nice estimate mismatch |

0:16:55 | and if we |

0:16:56 | this is not this is this is this design for a probability of false one shoe |

0:17:00 | however |

0:17:01 | if we if you assume much white nice to apply the generalized likelihood ratio test |

0:17:06 | but white |

0:17:07 | uh a noise and white signal then the probably the false let me gonna get is |

0:17:11 | much high point one the |

0:17:14 | so that |

0:17:15 | to basically goes best are not in me in two |

0:17:18 | the correlation structure |

0:17:20 | it is but modifying it |

0:17:22 | this place |

0:17:24 | that's all |

0:17:28 | we have a couple of minutes for |

0:17:30 | question |

0:17:31 | from the speech |

0:17:32 | is there in question |

0:17:36 | yeah |

0:17:44 | yeah a i'm not i'm not i'm gonna i'm not using the approach |

0:17:47 | it get it may have it be correlated |

0:17:49 | it it as a matter now have trying to simulations zone |

0:17:53 | because |

0:17:54 | at your like |

0:17:55 | it has to be non sit at different uh the lexus |

0:17:57 | yeah |

0:17:58 | like i it is so |

0:18:00 | i i no i'm not exploit |

0:18:07 | a a have just not short one uh a said that |

0:18:10 | your paper and your |

0:18:11 | technique is improving the spec to existing literature |

0:18:15 | uh taken into account |

0:18:16 | the possibility of different |

0:18:18 | uh but as for the noise on each sensor |

0:18:22 | so uh |

0:18:24 | is it |

0:18:25 | um |

0:18:25 | can you give some example were in practice we could have |

0:18:29 | different sensors with different noise uh |

0:18:32 | at the receiver side i mean i you |

0:18:34 | assuming that |

0:18:35 | each each each one of your sense of could have be different from the other one |

0:18:39 | yeah are are be the calibration could |

0:18:41 | goal for with time |

0:18:45 | that's it that |

0:18:46 | but |

0:18:47 | okay |

0:18:48 | so uh uh |

0:18:50 | and and a question we have still one |

0:18:51 | one mean |

0:18:53 | if we want to |

0:18:54 | supplied |

0:18:57 | yeah |

0:19:03 | no |

0:19:04 | there on |

0:19:06 | yeah |

0:19:10 | okay |

0:19:11 | as i T |

0:19:13 | the speaker |