noc18-ee31-Lecture 39-Practical Application

[Music] [Music] hello welcome to another module in this massive open online course so we are looking at robust beamforming as an application of convex optimization or the optimization framework that we have seen so far so let us continue our discussion we are looking at robust beamforming or multiple antenna system remember what beamforming does is to focus the wireless signal in a particular direction formal beam in a particular direction and robust beam forming is the paradigm where the knowledge of the channel is not known precisely so there is uncertainty in the channel knowledge and how to design a beam former that is robust to that uncertainty we said the robust beam former can be designed as the solution to the following optimization problem W minimize W bar transpose RW bar W bar is the beam former R is the noise plus interference covariance matrix subject to the constraint that W bar transpose H bar is greater than or equal to 1 for all H bar belongs to this ellipse correct this ellipsoid this is also termed as the uncertainty ellipsoid this is also termed as uncertainty ellipsoid and this ellipsoid is described as follows this ellipsoid is the ellipse which has the center H bar II that is the nominal channel or the estimated channel so H bar E Plus P u bar such that Norm u bar is less than or equal to 1 okay and well now we want to solve this optimization problem to basically determine the optimal beam form okay the optimal robust be for more if you were okay and that solution first of all let's simplify this optimization problem and this is where it can be done in a very interesting fashion as described below so let us look at the constraint and constraint can be simplified as follows okay remember the constraint ensures a minimum gain of unity for all vectors h-bar belonging to the uncertainty ellipsoid so we have what is the constraint the constraint is W bar transpose H bar is greater than equal to V for all H bar belonging to the uncertainty ellipsoid which means now you substitute for H bar W bar transpose H bar s well we have seen that is simply H bar e the estimated channel plus PU bar greater than equal to one for all now for all H bar belong to e now becomes because the equivalent condition is H bar E Plus P u bar for all vectors u bar such that norm u bar is less than or equal to 1 for all norm u bar less than or equal to 1 ok now this is the interesting part now this has to be true for all vectors u bar such that norm u bar is less than equal to 1 this has to be greater than equal to 1 all right which implies basically this also has to hold at that value of u bar where this is the minimum all right so is the minimum of this overall u bar is greater than equal to 1 that automatically implies that it is going to be greater than equal to 1 for all u bar such that norm u bar is less than equal to so this can be written equal until and you can convince yourself this implies that for the minimum of u bar that is you take the minimum over u bar says in norm u bar less or equal to 1 this W bar transpose H bar E Plus P u bar greater than equal to 1 okay if this if this this definitely implies that about the minimum this has to be greater than equal to 1 and this implies now I can simplify this further you take the minimum of norm or norm you bar all you bars is that norm you bar is less than equal to one no I can simplify this as W bar transpose H e bar plus W bar transpose P u bar the minimum has to be okay over this minimum this has to be greater than or equal to now this is a constant W bar transpose H e bar this does not depend on u bar okay so this will come out of the minimization so this implies if you look at this W bar transpose H bar E Plus minimum of norm of u bar less than equal to one W bar transpose P u bar greater than equal to one now what we are going to do we are going to set this let us set this W bar transpose P as W tilde which implies P transpose W bar I am sorry W tilde transpose so P transpose W bar will be W tilde so I will write this as W bar transpose H bar E Plus the minimum or norm for all u bar such that norm u bar less than equal to one minimum or u bars that W tilde transpose u bar greater than equal to one and now this is very interesting now if you observe now you see what is this this is the nothing but the dot product W tilde transpose u bar so we have W tilde and we have this vector u bar correct and we have the dot product now when is the dot product between W tilde and you are minimum remember the dot products maximum when you bar is perfectly aligned with W tilde and the dot product is minimum when the vector is 180 degree that is it's completely in opposite direction and in a direction opposite the data of the material so the dot product is minimum when you bar forms a 180 degree angle with W tilde so this is where so the this is let us say this is star so this is a 180 degree angle so we say W tilde u bar is minimum when u bar is opposite that is forms a 180 degree angle with W tilde and therefore what we say is u bar star this will be equal to minus W tilde because the vector that is exactly opposite to W tilde is minus W tilde however we need u bar to be normally bar to be less than equal to 1 therefore we normalize this with norm of the method that is so u bar is the unit norm vector that is opposite to W tilde and this is for which this is precisely the U bar for which you have minimum such that norm u bar less than equal to one W tilde transpose u by okay this is where the minimum occurs that is when u bar is a unit norm vector it is exactly opposite in direction to W tilde therefore the inner product is basically negative number Lindy this cosine 180 is minus one okay so this implies that now the minimum will be W bar transpose H bar E Plus the minimum over norm u bar u bar such that norm u bar less or equal to one occurs when u bar equals minus W when u bar equals minus W tilde divided by Norm W tilde therefore and that will be W tilde transpose into and I am now substituting for that value of u bar which is minus W tilde divided by W tilde and this has to be greater than or equal to one if this is greater than equal to one then it's going to be greater than equal to one for all by implication by a so by following this argument it's going to be greater than equal to one for all vectors H bar belonging to that else material and now you see this is mine W tilde transpose into W tilde is nothing but norm W tilde square divided by W tilde so that is norm W tilde so this implies W bar transpose H bar B minus norm W tilde greater than or equal to one and now we substitute for not W tilde W tilde is nothing but we have seen earlier W tilde is basically you are a P transpose W bar so this implies W bar transpose H bar II minus norm of B bar transpose W bar this has to be greater than or equal to one this is the equivalent constraint or this you can say is the simplified constraint simplified constrained and you can also write this as now W a transpose H bar e minus norm of P transpose W bar greater than equal to one this can also be simplified as follows norm of P transpose W bar less than or equal to W bar transpose H bar e minus one okay and now if you look at this is very interesting you can recall that this is a norm and this is your a fine this is the affine portion so we have no less than equal to something that is affine so this is basically you can recall and you can look the notes this is a Konak constraint in fact this pair this constraint represents a conic region or this is basically a cone or this is known as a this is let us say this is a cone also known as a conic constraint it's a very interesting constraint it reduces to a cone or a conic constraint and therefore now the equivalent optimization problem to find the robust beam former okay that can be formulated as even an optimization problem for robust beamforming that will be minimum of minimum or W bar transpose R W bar such that W bar transpose P norm is less than or equal to W bar transpose H were e- okay this is a coning constraint this is a second-order quadratic optimization so this is a second-order objective this is a conic constraint so this is known as the second-order cone problem so this is basically this is known as an SOC P equals second-order cone so this is a very interesting aspect so the robust beamforming problem reduces to a very interesting optimization for more either a very interesting or belongs to a very interesting class of optimization problems turned as second-order cone programs with the objective function is second-order all right our objective function and the constraint is a conic constraint alright so this is known as an SOC p problem and it's a very interesting problem and it can be solved and i will demonstrate it separately because it is a little involved so robust beamforming problem is n s of CP let us note that does the robust beamforming problem does the robust to inform a problem is in SOC P and the robust beam former W bar it can be shown that robust beam former is minus of lambda in fact I will show this in a subsequent module plus R Q inverse of H bar e this is the robust this is the robust B former this is the worst being former and lambda this is the Lagrange multiplier lambda equals LaGrange multiplier and this has to be determined suitably and the matrix Q depends on R and P basically Q equals I'm sorry depends on P and H bar e so PP transpose minus H bar e age bar of course you see notice PP is the matrix corresponding to the uncertainty ellipsoid and H bar e is the estimate the nominal estimate of the channel and this is the solution to the robust beamforming problem that is a blue bar equals you can say this is w bar star W R star equals minus lambda R plus lambda Q inverse into h bar e alright so I will conclude this module with this alright so there's a very interesting problem the robust beamforming problem which can be shown to be an associate a second order code problem and this is the solution it's slightly involved which I will in illustrate in a separate one you thank you very much [Music] [Music]