Presentation on theme: "MIMO Communication Systems"— Presentation transcript:
1 MIMO Communication Systems So far in this courseWe have considered wireless communications with only one transmit and receive antenna- SISOHowever there are a lot advantages to be had if we extend that to multiple transmit and multiple receive antennasMIMOMIMO is short for multiple input multiple output systemsThe “multiple” refers to multiple transmit and receiver antennasAllows huge increases in capacity and performanceMIMO became a “hot” area in 1998 and remains hot
2 Motivation Current wireless systems Increasing demand Cellular mobile phone systems, WLAN, Bluetooth, Mobile LEO satellite systems, …Increasing demandHigher data rate ( > 100Mbps) IEEE802.11nHigher transmission reliability (comparable to wire lines)4GPhysical limitations in wireless systemsMultipath fadingLimited spectrum resourcesLimited battery life of mobile devices…Video transmission, mobile game, internet browsingMobile streaming
3 Multiple-Antenna Wireless Systems Time and frequency processing hardly meet new requirementsMultiple antennas open a new signaling dimension: spaceCreate a MIMO channelHigher transmission rateHigher link reliabilityWider coverage
4 General IdeasDigital transmission over Multi-Input Multi-Output (MIMO) wireless channelObjective: develop Space-Time (ST) techniques with low error probability, high spectral efficiency, and low complexity (mutually conflicting)
5 Possible Gains: Multiplexing Multiple antennas at both Tx and RxCan create multiple parallel channelsMultiplexing order = min(M, N), where M =Tx, N =RxTransmission rate increases linearlyTxTxRxRxSpatial Channel 1Spatial Channel 2
6 Possible Gains: Diversity Multiple Tx or multiple Rx or bothCan create multiple independently faded branchesDiversity order = MNLink reliability improved exponentiallyRxTxRxTxFading Channel 1Fading Channel 2Fading Channel 3Fading Channel 4
7 Key Notation- Channels Assume flat fading for nowAllows MIMO channel to be written as a matrix HGeneralize to arbitrary number of inputs M and outputs N so H becomes a NxM matrix of complex zero mean Gaussian random variables of unity varianceCan understand that each output is a mixture of all the different inputs- interferenceWe assume UNCORRELATED channel elementsTxRx
8 Key Decomposition- SVD SVD- singular value decompositionAllows H of NxM to be decomposed into parallel channels as followsWhere S is a NxM diagonal matrix with elements only along the diagonal m=n that are real and non-negativeU is a unitary N x N matrix and V is a unitary M x M matrixThe superscript H denotes Hermitian and means complex transposeA Matrix is Unitary if AH=A-1 so that AHA = IThe rank k of H is the number of singular valuesThe first k left singular vectors form an orthonormal basis for the range space of HThe last right N-k right singular vectors of V form an orthogonal basis for the null space of HWhat does SVD mean?
9 Key Decomposition- SVD What does it mean?Implies that UHHV=S is a diagonal matrixTherefore if we pre-process the signals by V at the transmitter and then post-process them with UH we will produce an equivalent diagonal matrixThis is a channel without any interference and channel gains s11 and s22 for exampleSpatial Channel 1Spatial Channel 2TxRx
10 Key Decomposition- SVD What are the singular values?You can remember eigenvalues and eigenvectorsIf A is any square matrix then it can diagonalized using E-1AE = D where E is the matrix of eigenvectorsNote we can generate a square M x M matrix as HHH= (USVH)H(USVH)=V(SHS)VHLetting A=HHH so that E-1AE = D = VHHHHV= SHSAlternatively we can generate a square N x N matrix as HHH= (USVH)(USVH)H= UH (SSH)UTherefore we can see that the square of the singular values are the eigenvalues of HHHAlso note that V is the matrix of Eigenvectors of HHHSimilarly U is the matrix of eigenvectors of HHH
11 Capacity For a SISO channel capacity C is given by where is the SNR at a receiver antenna and h is the normalized channel gainFor a MIMO channel we can make use of SVD to produce multiple parallel channels so thatWhere are the eigenvalues of W
12 Capacity We can also alternatively write the MIMO capacity It can be demonstrated for Rayleigh fading channels that if N<M then the average capacity grows linearly with N asThis is an impressive result because now we can arbitrarily increase the capacity of the wireless channel just by adding more antennas with no further power or spectrum requiredIn these calculations it is also assumed the transmitter has no knowledge about the channel
13 Note on SNRThe definition of the SNR used previously is simply the receiver SNR at each receiver antennaIn this definition the channel must be normalized rather than be the actual measured channelThis approach is used since it is more usual to specify things in terms of received SNRHowever in calculations it is perhaps easier to think of total transmit power, un-normalized channel G and the received noise power per receive antenna, a, so capacity becomes
16 Special CasesTake M=N and H =In and assume noise has cross-correlation In thenLet Hij = 1 so that there is only one singular value given by and assume noise has cross-correlation InThe first column of U and V isThusEach transmitter is sending a power P/M and each is sending the same signal HxThese M signals coherently add at each receiver to give power PThere are N receivers so the total power is NPGiven the noise has correlation In SNR is also NP
17 Example Consider the following six wireless channels Determine the capacity of each of the six channels above, assuming the transmit power is uniformly distributed over the transmit antennas and the total transmit power is 1W while the noise per receive antenna is 0.1W.Note which channels are SIMO and MISO
21 CapacityThese capacity results are however the theoretical best that can be achievedThe problem is how do we create receivers and transmitters that can achieve close to these capacitiesThere are a number of methods that have been suggestedZero-forcingMLDBLASTS-T coding
22 MIMO DectectionConsider a MIMO system with M transmit and N receive antennas (M,N)wherex is the Mx1 transmit vector with constellation QH is a NxM channel matrixy is Nx1 received vectorn is a Nx1 white Gaussian complex noise vectorEnergy per bit per transmit antenna isOur basic requirement is to be able to detect or receive our MIMO signals x
23 MIMO MLDLets first consider optimum receivers in the sense of maximum liklihood detection (MLD)In MLD we wish to maximize the probability of p(y|x)To calculate p(y|x) we observe that the distribution must be jointly Gaussian and we can use previous results from M-ary to write it as
24 MIMO MLDThat is we need to find an x from the set of all possible transmit vectors that minimizesIf we have Q-ary modulation and M transmit antennas then we will have to search through QM combinations of transmitted signals for each transmit vector and perform N QM multiplicationsBecause of the exponent M the complexity can get quite high and sub-optimal schemes with less complexity are desired
25 MIMO Zero-Forcing In zero-forcing we use the idea of minimizing However instead of minimizing only over the constellation points of x we minimize over all possible complex numbers (this is why it is sub-optimum)We then quantize the complex number to the nearest constellation point of xThe solution then becomes a matrix inverse when N=M and we force to zero (zero-forcing)What about when M does not equal N?
26 Key Theorem- Psuedoinverse When H is square one way to find the transmitted symbols x from Hx = y is by using inverse.What happens when H is not square? Need psuedo-inverseNote that HHH is a square matrix which has an inverseTherefore HHH x =HH y so that (HHH)-1HHH x = (HHH)-1HH y and the psuedo-inverse is defined as H+ = (HHH)-1HHThe psuedo-inverse provides the least squares best fit solution to the minimization of ||Hx-y||2 with respect to x
27 ExampleIf we use a zero-forcing receiver in the previous example what is the receiver processing matrix we need for each of the 6 channels?G1- none neededG2-Inverse not possible- not neededG3- [1,1]G4- Inverse not possible- just MRC weightsG5-G6-
28 Performance analysis of ZF The zero-forcing estimate of the transmitted signal can be written as:where (with elements ) is known as the pseudo-inverse of the channel H and the superscript H denotes conjugate transposeSubstituting :the ith row element of Gn is equal to a zero mean Gaussian random variable with variance:
29 Performance analysis of ZF The noise power is scaled by which is the square 2-norm of the ith row of GThe diagonal elements of GG’ however are the square 2-norm of the rows of GIn addition we can show thatWhich is equal to the the diagonal element of
30 Performance analysis of ZF Since all are all identically distributed so we drop its subscriptw follow the reciprocal of a Chi-Square random variable with 2(N-M+1) degrees of freedomThe probability density function (PDF) of wwhere D=N-M
31 Why Chi-Square? Check out For a N=M it is easy to show as follows H. Winters, J. Salz and R. D. Gitlin, “The Impact of antenna Diversity on the Capacity of Wireless Communication Systems”, IEEE Trans. Commun., VolCOM-42, pp , Feb./March/April.24. J. H. Winters, J. Salz and R. D. Gitlin, “The capacity increase of wireless systems with antenna diversity”, in Proc Conf, Inform. Science Syst., Princeton, NJ, Mar , 1992For a N=M it is easy to show as followsMatrix G, the inverse of H can then be written asWhere Aij is the sub-matrix of H without row i and column j
32 Why Chi-Square?The square of the 2-norm for the i row of G is therefore equal toNoticing that the equation above becomesSince |Aji| is independent of hij we can condition on it so the equation can be further simplifiedRemember hij are random variables (like noise so independent and add up)
33 Why Chi-Square?The square of the 2-norm for the i row of G is therefore equal toWhere h’ is a random variable following the same distribution as hijCanceling common terms we get
34 Why Chi-Square?h’ is a random variable with the same distribution as hijThe weights, w are therefore distributed as the reciprocal of the sum of the square of two Gaussian random variables with zero mean and variance α/2That is the weights are distributed as the reciprocal of a chi-squared random variable with 2 degrees of freedomThis turns out to be the reciprocal of a Rayleigh fading variable for this special case
35 Performance analysis of ZF To obtain the error probabilities when w is random, we must average the probability of error over the probability density function ,where is the probability of error in AWGN channel with depend on the signal constellation.
36 Performance of BPSK and QPSK For BPSK and QPSKPerforming the integral and define as the SNR per bit per channel (see Proakis 4th ed, p825)where
37 Performance of BPSK and QPSK using ZF Exact BER expression for QPSK compared with Monte Carlo simulations
40 Comparison with simulation (ZF) 2468101214161820-6-5-4-3-2-1SNR per bit per channel (dB)BER(3,3)(4,6)(8,12)16-PSK (analysis)16-PSK (simulation)16-QAM (analysis)16-QAM (simulation)BER approximations for 16-PSK and 16-QAM compared with Monte Carlo simulations for (3,3), (4,6) and (8,12) antenna configurations.
41 Comparison with simulation BER approximations for 64-PSK and 64-QAM compared with Monte Carlo simulations for (8,12) antenna configurations.
42 Performance of MLDBER of zero-forcing and MLD for a (4,6) system using 4-QAM.
43 Performance of MLDBER of zero-forcing and MLD for a (3,3) system using 8-QAM and 16-QAM.
44 Performance of MLDBER of zero-forcing and MLD for a (3,3) system using 8-QAM and 16-QAM.
45 MIMO V-BLASTIt turns out the performance of ZF is not good enough while the complexity of MLD is too largeMotivate different sub-optimum approachesBLAST is one well known on (Bell Laboratories Layered Space Time)Based on interference cancellationA key idea is that when we perform ZF we detect all the transmitted bit streams at once
46 MIMO V-BLASTGenerally we would expect some of these bit streams to be of better quality than the othersWe select the best bit stream and output its result using ZFWe then also use it to remove its interference from the other received signalsWe then detect the best of the remaining signals and continue until all signals are detectedIt is a non-linear process because the best signal is always selected from the current group of signals
47 MIMO V-BLAST Basically layers of interference cancellation ) Linear DetectorInterferenceCancellationStage 1Stage (M-1)Stage
48 Performance of V-BLAST BER of zero-forcing, V-BLAST and MLD for a (4,6) system using 4-QAM.