Pillai: Beam Forming

so [Applause] in beam forming you use more than one sensor to collect the data so you have a source somewhere we'll assume that the source is far enough so that by the time it comes to you the rays come to you they are sort of plane waves coming at you so the signal of course gets to the sensor which is closest to it so i'm going to call it st and then there will be a delay and let's say we have two sensors separated by d then you can clearly see depending on the direction of arrival so remember direction is perpendicular to the wavefront right so you get the direction with respect to the array remember array has a baseline so you can denote this angle or you can denote this angle whichever this is from the more side and so you can see if this distance is d uh what is this distance and you want that's the separation right the time it the waveform here i'm going to call it s of t so the waveform here is s of t minus some delay question is how much is the delay anybody what is the delay if this distance is d what is this distance if this angle is theta this is 90 minus theta if this is 90 this is theta then this distance is d sine theta right so tau is d sine theta over c right that's what did now if i assume this is a narrowband source with some amplitude a or some power p then you see st minus t naught is going to be so this is going to be s t e to the power minus j omega naught tau right but omega not tau is what omega naught is 2 pi over 2 pi f naught f naught is c over lambda and tau is d sine theta over c so c cancels so we can write this quantity as equal to generally it's written like this d over d is the separation but what matters is to avoid aliasing etc d over lambda by two usually this separation is uh relevant with respect to lambda by two so we'll normalize d over lambda by two uh so you see this the things are delayed the two sensors are separated by lambda by two or one unit which is lambda by two right so then this will be one this is a dimensionless quantity and in any case so usually if the sensors are lambda by two apart then this will be one unit two units three units etc and i'm going to call just to make things a symbol uh this is going to be uh let's say uh i and this i'm going to call it uh the whole thing i'm going to call it i omega so omega is points pi sine theta so of course the signal now looks like t e to the power minus j omega right remember omega is just a notation for uh theta is the unknown angle so if you have sensors at 2d 2d from here so another d another d etc so if this sensor is at 2d the last sensory it is at md and if i call this signal to be x1 the second sensor to be x2 and the last sensor to be xm of t so notice that the signal at the 2d will be here 2d divided by lambda by 2. so d over lambda by 2 is 1 unit so that will be 2 units etc so if i define x of t to be x 1 of t x 2 of t etcetera x m of t for a single source this will be first signal will be s of t then s of t e raised to minus j omega remember omega is this pi [Music] and this is s t e raised to minus j 2 omega etc and this we can write it in this this form ah so this is going to be s t 1 e raised to minus j omega so usually this is called a direction vector [Music] so it's a spatial direction vector because the actual direction is theta so you are collecting what happens is from a source you are collecting in tuples of data generally this is a clean data but generally the data also has noise right so you can add a at each sensor there is a uh depending on where the sensor is you can assume that there is also noise so let me call this to be a n vector so this is usually the form right so you have duplicates of the signal but if the signal is narrow band it is phase delayed and then you have all this noise wherever it is and this is just one source so if you have one source coming from direction theta k and another source coming from direction theta i s i of t uh another source from here another source from here let's say theta one and theta theta remember all the directions are with the way i have drawn with respect to the normal right so you could say that this is a situation where i have these are interferences this is your desired signal but everything is coming at you so you collect the data from so this is the mode four one so if you have multiple signals uh still you are collecting the data so it will be like this right so they have k sources uh still enough t so just to be clear ak omega will be or am a omega k would be 1 e raised to minus j omega k e raised to minus j to omega k all right so that's generally the form of the data some are interferences uh some may be one may be the desired signal so generally the goal is you want to suppress all the interferences and boost this signal whatever and sometimes these interferences may be actually multi-path the signal also goes here and comes up etc that means so the correlation between some of these s k's maybe one etcetera so just to show the advantage of an array let me take this single source case so why use an array so in this case x of t is of course st multiplied by a omega plus nt so you can see here you can already see if i want to coherently add the signal remember look here each each sensor there is a signal component and noise component right there is signal plus noise such each come each sensor except we can assume if the sensors are far enough you can you can as you may be able to assume that the noise is uncorrelated or partially correlated etc the best case is uncorrelated so put it far away so that they are uncorrelated you don't want to put the sensor on top of each other right so this is a vector right so let me i am just assuming one case so you can clearly see look here the signal is phase delayed so if i if i each sensor i cleverly multiply by the conjugate of this delay then i can add the signal in phase whereas the noise will be so i am going to create an output which is whatever is the conjugate of the thing i am going to multiply that by that right x i t e raised to if there is minus i'm going to put it by plus see if i do this you can see here or this will be oh each of this is multiplied by its conjugate so the signal here becomes m multiplied by s t whereas the noise of course is so this is the uh so if i look at the signal to noise ratio at the output this is the signal part so that's going to be m multiplied by s t squared average value divided by if i call this to be w of t the output noise that's going to be expected value of wt squared right output signal to noise ratio this is this whole thing is squared right this is squared this is squared so this is m squared p i'm going to call this to be p and the denominator is look at here w is this so this is going to be a double summation right double summation expected value is on nit let's say in kt then you have e to the power j i t i omega e to the power minus j k omega right this is star summation is on i k but i'm going to assume uh when i is not equal to k they are uncorrelated so when i equal to k they have equal value sigma squared when i equal to k this term goes away look at here so this simply becomes sigma squared multiplied by m because sigma squared is the value of this when i equal to k so m cancels you get m multiplied by p over sigma squared but that's the input to signal to noise ratio so that's snr input so you can see the advantage of using if you use the m sensors you can boost the signal to noise ratio by a factor of m the number of sensors provided the noise is uncorrelated so that's the simple thing why even use an array so generally what i have done here is called beam forming let me write this in a matrix form so notice that this expression if i want i can write this as [Music] so remember this is a so this i can write it as x1t [Applause] e raised to j so this should have been i minus 1 right because when i equal to 1 you want it to be 0 so this will be right i use this term i put it here so this is the look at here i already have a vector a is a column vector with the minus so what will be this one anybody how will you express this in terms of a right so that is a conjugate transpose omega multiplied by this vector this is x of t which we defined before so you can define it this way output so a is there x is there also so i am going to remove this space so this is the picture if i put if this is x1 etc of course i i hope you see the advantage but this is so you are taking previously in communication you have one sensor you are only taking advantage of time you sit there and collect data in time here you are sitting in collecting data both in time and space so space time adaptive processing is what your and then you want to adapt to the data if there are interferences you want to cancel it if there's a signal you want to boost it so what we are trying to do is space time adaptive processing adapt to what so signal or whatever right so the simplest processing is that whatever is the output you put a set of weights w1 wi or wm so your output is going to be sigma w k star x kt so this of course i can write it as but this is my x of t so the question is what weights to use so if i define a weight vector w to be w1 w2 etc wm so you can see this is the transpose of that so you have w star x of t of course in what i have done here is a special weight vector look here here the weight vector is w is a so this is called the beam former because you are you are collapsing all the phase differences or the phase changes of the single signal is all face aligned to be coherent the signal part whereas the noise is of course added incoherently right so if you if you pick a w to be a that's called the beam former so this is so the general expression is here and if you want to look at the output power we can look at it so p naught which is expected value of w t squared is expected value of w star x of t squared so that's going to be expected value of w star x of t and its conjugate transpose that's going to be x star w star so the expected value is only inside and this is the covariance matrix because remember this is a vector so you can write it like this so the output power turns out to be w transpose or w and the special case of if i choose w to be a particular direction that you are interested is interested in then that is beam forming along that direction because look at here i'll show you why it is called the beamformer so this is the output i hope you see what is r y r w is like this so r is what r x x is expected value of x of t x of t transpose remember x of t is what x 1 of t x 2 of t etc x m of t that's this its transpose is x 1 star etc x m star t expected value so this is going to be a matrix will be r 1 1 r 1 2 etc r 1 m r i j here so r i j is you are actually correlating everything across space and time this is x i of t x j of t star see the averaging on the time is here i and j represent space so it's a correlation across space both space and time so this is a co this is the space time covariance matrix this expect this expected value is on time but expected values on spatial different spatial different array components so you take advantage or you take the correlation in of both the aspects in a single receiver you don't have the advantage of space here you bring in both so you should be able to do better than just operating with one sensor that's the whole point so this is the space time covariance matrix and then of course if if you know the structure of s you put it into this but let's proceed generally so p b naught is if you p b remember so this is remember this expression so the output power uh using a particular sense uh weight vector is going to look like this if you use a particular weight vector the output is always w transpose rw so the question is what is the weight vector easiest one is remember i don't know what this is of course the weight vector has to depend on if you want to suppress the signals coming from these directions and boost something in this direction i need to know where the things are coming from and if you have no knowledge then you say i i know this is the direction i want to look so then i can remember earlier we said if you if you use this weight vector it is the same as aligning the face for things coming from that direction if you do not know which direction thickness coming from you just scan electronically this is what is electronic beam scanning at the airports etc one direction another direction except you do it fast enough so the weights will keep changing depending on which angle you are looking at so you can scan the sky so let's use this so p b naught is going to be what this is going to be a star omega naught or x x a omega not right so let me take a single source and see why this is called the beam former you will see so suppose x of t is s t so this is your x so here the covariance matrix is look at here this is going to be this is a scalar so this is going to be expected value of s t squared multiplied by a a transpose because how do you find the covariance matrix look here it's expected value of x x transpose that's going to be sa multiplied by star a star s star i picked up here a a star is here plus expected value of n n star so that's the noise covariance matrix i am going to assume that the noise is all of this form this is assuming that the noise is uncorrelated from element to element and its equal variance at each sensor if the variances are different you put sigma 1 squared sigma 2 squared sigma n squared etcetera but i am taking the simplistic case so this is the case if for a single source i can write this as so this is for a single source so let me put this expression here so then you can see this becomes p multiplied by its complex conjugate squared plus sigma squared a star a a star a is m right what is a star a anybody a star any omega a star a is what remember a is where a is a is just a face vector right right right so 1 e raised to minus j omega so what is a star a m right so sometimes to normalize i could have done it we also could have defined sometimes a is defined as so let us leave it like this so a star a is m so you get this one so let us look at this pattern this is what a vector like this multiplied by a vector like this here an element is going to be e raised to minus j omega here an element is e raised to plus j omega naught right i so i hope you see that when you multiply this with this what do you get sigma e to the power minus j yes do you see this or you don't i i wrote for a star omega not that's this so this entry is 1 1 uh so this multiplied by this is plus sign this is minus sign the only difference is omega n omega naught so this will be here plus a star plus sigma squared m right so this is not going this is just like a base now this you should be there's a p here right this you should be able to sum this what is the sum of this this is a what is the sum of this anybody so here this is exactly the same form where r is if i define r to be e raised to j minus j omega minus omega naught right so what is the sum of this what did you say this is only a minus 1 right so this is r to the power m over 1 minus r so let's not worry about this sigma squared let's just change the base to or we can add it later so this is going to be p 1 minus e to the power minus j m omega minus omega naught divided by 1 minus e raised to minus j omega minus omega naught absolute value squared absolute value squared plus sigma squared i so that comes out to be let me pull out half of this outside you you know that trick right if you pull out half of this so this reads now sine m omega minus omega naught divided by by sine omega minus omega naught by 2 here you have e to the power j m minus 1 omega minus omega not right but that e to the power doesn't matter because it's absolute value squared so this expression is known as the beam former whatever you remember this is a function of omega omega is the either way omega naught is the true angle omega is the angle you are you want omega naught is where you want to focus on the way i wrote and omega is the angle you are coming from and remember this expression is well known right this is a well-known expression plus the noise term is here just a base base gets lifted so let's plot that quantity remember omega note is where i am trying to focus i don't know where i should focus but there is a source coming at me at omega at the direction omega right what is omega anybody omega don't forget omega is pi sine so this is where the source is i don't know it but i am going to i am trying to focus here but my idea is i will keep focusing everywhere but look at this pattern so you know this basically this is a digital sink right so this exp only this matters so let me read this now onwards i'm going to read this difference to be omega because that's the variable but you can see this this will where will they speak anyone if i keep doing this where is this going to peak at omega equal to omega naught so if you if i keep forming the beam i the beam will have maximum so when i keep going through this this is going to have the peak value here right so so this is called the main beam and these are called the solid loops so you can see if even if i do this so long as there is only one source in noise a dominant source then if i keep looking at the different directions when i i know that this beam is going to peak when i hit the true direction so this shape all i need to so let me assume it is coming all i need to do is study this shape so look here i'm going to reclaim this to be omega this variable to be omega or you can use a different variable but so long as we know what we are doing so beam bottom line is in beamformer the gain pattern so that's the gain this is the advantage due to an array so it has got to spill over from side lobes and so on so of course the problem is if there is a so when you are looking here if there is a source sitting here this can leak through the side lobe this interference is also going to come in here through this side loop if there is an interface so we'll see how to suppress it etc but let's take the so the main thing is this beam formula is sine m omega by 2 over so this is the characteristics of the beam formula so if i plot this so what's the value at omega equal to 0 look here what's the value of omega equal to 0. yeah because you can use whatever right so to avoid to scale it i'm just going to put an m here just a scaling factor what's the maximum value now at omega equal to 0 i only did it so that this value is 1 now so the shape is because of this square it will be like this so what does this mean these are this is the side lobe so if you use beam for me you get a gain pattern which looks like this that's the beam width because that's the main beam width this is the so and this quantity is the peak side loop so what do you what would be ideally you want the peak side load to be anybody you want to suppress it and you want the beam width to be what main beam would to be as narrow as possible and this value is 1 so this is going to be lower so what do you expect if m increases what do you expect for the peak side lobe and the main beam would to happen beam width will become and the peaks are low huh what happens to the peak side row what we cannot get well let's see what uh what happened so oh let's see what happens to the so let's compute both the quantities and see what remember this is we haven't done any complicated processing because for any complicated processing you need more information here we are i'm saying i'm going to look everywhere it's like what we do with the eye i look everywhere to see where there is somebody right scanning right so electronic scanning all you have to do is change the weight vectors that you can do increment the weight vectors by something right so of course g at 0 is 1. so where is this is easy to compute right where is this value anybody is to compute the main beam with this will be when the numerator goes to 0 right yeah so this m omega naught by two should be equal to pi this is the mean width half beam width right so this means omega naught is 2 pi as you said divided by m so since omega naught is pi sine theta naught is 2 pi divided by m so if you want in terms of theta naught is going to be sine inverse of 2 over m but the whole point is this goes to 0 as m goes to m increases right so you know the beam width goes to 0 so in omega it is 2 pi over m that's a half side half if you include both the sides it's twice that okay right so beam it decreases as m goes to m becomes large okay so that's good let's compute the peak side lobe also so i'll do a rough calculation so if this 2 is the this point is 2 pi over m what is this point anybody yeah you can eyeball it that's 4 pi over m right because this is equal to that equation becomes instead of pi it becomes 2 pi so this becomes 4 pi over f so what will be this is not exactly this correctly but roughly what is this point anybody yeah this is 3 pi over m right not exactly because to write to find the peak you have which you can do you take the derivative of this quantity then you will get a non-linear equation tangent and so on but so a slight adjustment but this is enough so let's compute the g at 3 pi over m so that's going to be sine m 3 pi over m is it 3 pi over m 3 pi yeah 3 pi over m but there is also a 2 right so 3 pi over 8. no no way this is 2 pi over m 4 pi over m half of that right and here it's going to be sine 3 pi over m but by 2 so this is this so look here m cancels what is sine 3 pi over 2 anybody what yeah yeah here right so m cancer sine 3 pi by 2 is minus 1 right sine 3 by 8 so square of that is 1 so this turns out to be 1 over sine 3 pi over 2 m and there is an m squared so i'm going to expand that quantity so this if you expand you will get one um sine expansion is what 3 pi by 2 2m minus what is it 1 3 factorial 3 pi cubed over 2 m cubed etcetera right squared so look here this m cancels with this so you get this quantity to be 1 over 3 pi by 2 minus 1 3 3 pi cubed over 2 cubed m squared then the next term will be 1 over m 4 right some constant but the whole square so you can see as m becomes large these terms become negligible and this whole thing goes to what is it 2 over 3 pi square that's a constant so this is so this is a famous result that's a minus 13.2 db so if you use simple beamforming whatever you do you can't reduce the peak side low beyond a certain level by increasing more number of sensors the peak lobe is not going you'll get an error or beam width but the peak side lobe will be at the best 13 point so there's a saturation right so then the only thing you could do is you have to bring in additional weights but then you bring the net this becomes a window design business so you bring in any any other weights instead of remember it's called beam forming because you are a face facing array you are only changing the faces so if you put any other weights of course as you know the main beam width will decrease increase but you can have a better side lobes this is where the chebyshev weights tailor weights etc comes in and so you can look into that so you can go in that direction but we will take a different direction in the next class

Channel: Probability, Stochastic Processes - Random Videos

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Id: O9NscnDukaA

Length: 43min 31sec (2611 seconds)

Published: Fri Dec 04 2020