Slide 1OutlineIntroductionIntroduction: Rayleigh fadingIntroduction: Information TheoryMathematical modelCapacities: Matrix channel is RayleighCapacities: ContdLower Bound On CapacityCapacity Derivation:Capacity improvement: CCDFCapacity per dimension:Comparison of systemsMin-Max communication systemMin Max performanceSummaryThank YouLimits On Wireless Communication In Fading Environment Using Multiple AntennasPresented By Fabian RozarioECE DepartmentPaper By G.J. Foschini and M.J. GansOutlineIntroduction.Mathematical model.Capacity formulas.Lower bound on capacity.Capacity improvement.Comparison of various systems.Min-Max strategy.Summary.IntroductionWe make the following assumptionsReceiver knows channel characteristics but not transmitter.Ie: fast feedback link required otherwise.We allow changes to propagation environment to be slow in time scale compared to burst rate.Model used is Rayleigh fading.Use information theory to find out increase in bit/cycle compared to no of antennas used.Introduction: Rayleigh fadingModel useful when no LOS path exists.Zero mean Gaussian process.Can be used to model ionospheric and tropospheric scatters.If relative motion exists between TX and RX: fading is correlated and varying in time.We can decorrelate path losses by using antennas separated by λ/2 on a rectangular lattice. This belongs to small scale fading.Introduction: Information TheoryUse Shannon capacity formula.We get capacity in terms of bits/second.In our application we can get the increase in bps/Hz for given no of TX and RX.Roughly for n antennas increase is n bits per 3db increase in SNR.)1(l og*2SNRBC )||.1(log22HCMathematical modelFocus on single point to point channel. whereHow does receiver diversity affect capacityNoise remains same but output signal is linear combination of diff antennas.This is maximal ratio combiner.)(.)ˆ/()()()(*)()(2/1thPPtgtvtstgtr)||.1(log022RniHCCapacities: Matrix channel is RayleighRandom channel model(|H|) is treated as Rayleigh channel model with zero mean, unit variance, complex.H matrix is assumed to be measured at receiver using training preamble.No Diversity case: nt=nr=1|H| replaced by Chi squared variate with 2 degree of freedom.].1[log222CCapacities: ContdReceiver Diversity case: nt=1, nr=nTransmit Diversity case: nt=n, nr=1Combined Transmit and Receiver Diversity: nt=nrCycling using one transmitted at a time:].1[log222nC])./(1[log222nTnCTRTnnnkkTnC)1(222])./(1[logTRniinTnC1222].1[log)./1(Lower Bound On CapacityEmploys unitarily equivalent rectangular matrices, here H is unitarily equivalent to m*n matrix.Where are chi squared variables with j degree freedom.Final result: contribution of the form L+Qwhereand Q is positive, negative term are cancelled out by positive Qhence C>L with probability 1.22,jjyxmnmjjXL)1(22)1(jTjjTjynYxnX22/1222/12)/()/(Capacity Derivation:One spatially cycled transmitting antenna/symbol:Channel capacity defined in terms of mutual information between input and output.Where ε- entropyTniToutcomeinoiseoutcomeioutputnoutputinputI1)]|()|([.1)/(Capacity improvement: CCDF2 antenna case4 antenna caseCapacity per dimension:Comparison of systemsMin-Max communication systemWhen multiple antennas are used the other antennas will add noise.Detectors have optimal combining.Detect 1st signal component using optimal combining and treat 2nd component as noise.After 1st is detected subtract that from received signal vector and extract 2nd signal by optimal combining.2nd component affected by thermal noise as 1st already removed.Same procedure for second detector.Min Max performanceSummaryWe were able to analyze receiver and transmitter diversity.Conclude that increase in bit rate is n bits/cycle for n antennas for each 3db increase in SNR.Compare various combinations of systems with different no of Rx and Tx.See the use of min-max strategy.This application is useful for indoor wireless LAN.Thank