Lecture 7 MIMO Communica2ons

Transcription

1 Wireless Communications Lecture 7 MIMO Communica2ons Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall

2 Outline MIMO Communications (Chapter 10 in Goldsmith s Book) Narrowband MIMO Model Parallel Decomposition of the MIMO Channel MIMO Channel Capacity MIMO Diversity Gain: Beamforming Diversity-Multiplexing Trade-off Space-Time Modulation and Coding 2

3 Circular Complex Gaussian Vectors A complex random vector is of the form where and are real random vectors. Complex Gaussian random vectors are ones in which [x R, x I ] t is a real Gaussian random vector. The distribution is completely specified by the mean and covariance matrix of the real vector [x R, x I ] t. Define x = x R + jx I x R x I where A is the transpose of the matrix A with each element replaced by its complex conjugate, and is just the transpose of A. Note that in general the covariance matrix K of the complex random vector x by itself is not enough to specify the full second-order statistics of x. Indeed, since K is Hermitian, i.e., K = K, the diagonal elements are real and the elements in the lower and upper triangles are complex conjugates of each other. A t 3

4 Circular Complex Gaussian Vectors In wireless communication, we are almost exclusively interested in complex random vectors that have the circular symmetry property: For a circular symmetric complex random vector x, for any ; hence the mean µ =0. Moreover for any ; hence the pseudo-covariance matrix J is also zero. Thus, the covariance matrix K fully specifies the first and second order statistics of a circular symmetric random vector. 4

5 Circular Complex Gaussian Vectors And if the complex random vector is also Gaussian, K in fact specifies its entire statistics. A circular symmetric Gaussian random vector with covariance matrix K is denoted as. Some special cases: A complex Gaussian random variable w = w R + jw I with i.i.d. zero-mean Gaussian real and imaginary components is circular symmetric. In fact, a circular symmetric Gaussian random variable must have i.i.d. zero-mean real and imaginary components. A collection of n i.i.d. CN(0, 1) random variables forms a standard circular symmetric Gaussian random vector w and is denoted by CN(0, I). The density function of w can be explicitly written as Uw has the same distribution as w for any complex orthogonal matrix U (such a matrix is called a unitary matrix and characterized by the property. UU = I 5

6 Narrowband MIMO Model Here we consider a narrowband MIMO channel. A narrowband point-topoint communication system of transmit and receive antennas is shown in the following figure. 6

7 Or simply as Narrowband MIMO Model We assume a channel bandwidth of B and complex Gaussian noise with zero mean and covariance matrix, where typically. For simplicity, given a transmit power constraint P we will assume an equivalent model with a noise power of unity and transmit power, where ρ can be interpreted as the average SNR per receive antenna under unity channel gain. This power constraint implies that the input symbols satisfy (10.1) where is the trace of the input covariance matrix 7

8 Parallel Decomposition of the MIMO Channel When both the transmitter and receiver have multiple antennas, there is another mechanism for performance gain called (spatial) multiplexing gain. The multiplexing gain of an MIMO system results from the fact that a MIMO channel can be decomposed into a number R of parallel independent channels. By multiplexing independent data onto these independent channels, we get an R-fold increase in data rate in comparison to a system with just one antenna at the transmitter and receiver. Consider a MIMO channel with channel gain matrix known to both the transmitter and the receiver. Let denote the rank of H. From matrix theory, for any matrix H we can obtain its singular value decomposition (SVD) as (10.2) where Σ is an diagonal matrix of singular values of H. 8

9 Parallel Decomposition of the MIMO Channel Since cannot exceed the number of columns or rows of H, If H is full rank, which is sometimes referred to as a rich scattering environment, then. The parallel decomposition of the channel is obtained by defining a transformation on the channel input and output x and y through transmit precoding and receiver shaping, as shown in the following figure: The transmit precoding and receiver shaping transform the MIMO channel into RH parallel single-input single-output (SISO) channels with input and output, since from the SVD, we have that 9

10 Parallel Decomposition of the MIMO Channel 10

11 Parallel Decomposition of the MIMO Channel This parallel decomposition is shown in the following figure: 11

12 MIMO Channel Capacity The capacity of a MIMO channel is an extension of the mutual information formula for a SISO channel given in Lecture 3 to a matrix channel. Specifically, the capacity is given in terms of the mutual information between the channel input vector x and output vector y as (10.5) The definition of entropy yields that H(Y X) = H(N), the entropy in the noise. Since this noise n has fixed entropy independent of the channel input, maximizing mutual information is equivalent to maximizing the entropy in y. The mutual information of y depends on its covariance matrix, which for the narrowband MIMO model is given by (10.6) 12

13 MIMO Channel Capacity where R x is the covariance of the MIMO channel input. The mutual information can be shown as (10.7) The MIMO capacity is achieved by maximizing the mutual information over all input covariance matrices satisfying the power constraint: (10.8) where denotes the determinant of the matrix A. Now let us consider the case of Channel Known at Transmitter 13

14 MIMO Channel Capacity Substituting the matrix SVD of H into C and using properties of unitary matrices we get the MIMO capacity with CSIT and CSIR as (10.9) Since = P/ n 2, the capacity (10.9) can also be expressed in terms of the power allocation to the ith parallel channel as P i (10.10) where i = P i / n 2 and i = i 2P/ n 2 is the SNR associated with the ith channel at full power. Solving the optimization leads to a water-filling power allocation for the MIMO channel: (10.11) 14

15 The resulting capacity is then MIMO Channel Capacity (10.12) Now consider the case of Channel Unknown at Transmitter (Uniform Power Allocation) Suppose now that the receiver knows the channel but the transmitter does not. Without channel information, the transmitter cannot optimize its power allocation or input covariance structure across antennas. If the distribution of H follows the ZMSW channel gain model, there is no bias in terms of the mean or covariance of H. Thus, it seems intuitive that the best strategy should be to allocate equal power to each transmit antenna, resulting in an input covariance matrix equal to the scaled identity matrix: 15

16 MIMO Channel Capacity It is shown that under these assumptions this input covariance matrix indeed maximizes the mutual information of the channel. For an Mt transmit, Mr-receive antenna system, this yields mutual information given by (10.13) Using the SVD of H, we can express this as where and is the number of nonzero singular values of H. The mutual information of the MIMO channel (10.13) depends on the specific realization of the matrix H, in particular its singular values { i }. 16

17 MIMO Channel Capacity In capacity with outage the transmitter fixes a transmission rate C, and the outage probability associated with C is the probability that the transmitted data will not be received correctly or, equivalently, the probability that the channel H has mutual information less than C. This probability is given by (10.14) Note that for fixed M r, under the ZMSW model the law of large numbers implies that (10.15) Substituting this into (10.13) yields that the mutual information in the asymptotic limit of large becomes a constant equal to M t 17

18 MIMO Channel Capacity We can have two important observations from the results in (10.14) and (10.15) As SNR grows large, capacity also grows linearly with for any and. At very low SNRs transmit antennas are not beneficial: Capacity only scales with the number of receive antennas independent of the number of transmit antennas. M t M r M =min{m t,m r } Fading Channels Channel Known at Transmitter: Water-Filling (10.16) 18

19 MIMO Channel Capacity A less restrictive constraint is a long-term power constraint, where we can use different powers for different channel realizations subject to the average power constraint over all channel realizations. The ergodic capacity in this case is (10.17) Channel Unknown at Transmitter: Ergodic Capacity and Capacity with Outage Consider now a time-varying channel with random matrix H known at the receiver but not the transmitter. The transmitter assumes a ZMSW distribution for H. The two relevant capacity definitions in this case are ergodic capacity and capacity with outage. Ergodic capacity defines the maximum rate, averaged over all channel realizations, that can be transmitted over the channel for a transmission strategy based only on the distribution of H. 19

20 MIMO Channel Capacity This leads to the transmitter optimization problem - i.e., finding the optimum input covariance matrix to maximize ergodic capacity subject to the transmit power constraint. Mathematically, the problem is to characterize the optimum R x to maximize (10.18) where the expectation is with respect to the distribution on the channel matrix H, which for the ZMSW model is i.i.d. zero-mean circularly symmetric unit variance. As in the case of scalar channels, the optimum input covariance matrix that maximizes ergodic capacity for the ZMSW model is the scaled identity matrix. Thus the ergodic capacity is given by: M t I Mt (10.19) 20

21 MIMO Channel Capacity The ergodic capacity of a 4x4 MIMO system with i.i.d. complex Gaussian channel gains is shown in Figure

22 MIMO Channel Capacity Capacity with outage is defined similar to the definition for static channels described in previous, although now capacity with outage applies to a slowly-varying channel where the channel matrix H is constant over a relatively long transmission time, then changes to a new value. As in the static channel case, the channel realization and corresponding channel capacity is not known at the transmitter, yet the transmitter must still fix a transmission rate to send data over the channel. For any choice of this rate C, there will be an outage probability associated with C, which defines the probability that the transmitted data will not be received correctly. The outage capacity can sometimes be improved by not allocating power to one or more of the transmit antennas, especially when the outage probability is high. This is because outage capacity depends on the tail of the probability distribution. With fewer antennas, less averaging takes place and the spread of the tail increases. 22

23 MIMO Channel Capacity The capacity with outage of a 4 4 MIMO system with i.i.d. complex Gaussian channel gains is shown in Figure

24 MIMO Channel Capacity 24

25 MIMO Diversity Gain: Beamforming The multiple antennas at the transmitter and receiver can be used to obtain diversity gain instead of capacity gain. In this setting, the same symbol, weighted by a complex scale factor, is sent over each transmit antenna, so that the input covariance matrix has unit rank. This scheme is also referred to as MIMO beamforming. A beamforming strategy corresponds to the precoding and receiver matrices described in previous being just column vectors: V = v and U = u, as shown in Figure In the figure, the transmit symbol x is sent over the ith antenna with weight v i. On the receive side, the signal received on the ith antenna is weighted by u i. Both transmit and receive weight vectors are normalized so that kuk = kvk =1 The resulting received signal is given by where if n =(n 1,,n Mr ) has i.i.d. elements. (10.20) 25

26 MIMO Diversity Gain: Beamforming Beamforming provides diversity gain by coherent combining of the multiple signal paths. Channel knowledge at the receiver is typically assumed since this is required for coherent combining. 26

27 MIMO Diversity Gain: Beamforming The diversity gain then depends on whether or not the channel is known at the transmitter. When the channel matrix H is known, the received SNR is optimized by choosing u and v as the principal left and right singular vectors of the channel matrix H. The corresponding received SNR can be shown to equal = max, where max is the largest eigenvalue of the Wishart matrix W = H H H. The resulting capacity is C = B log 2 (1 + max ), corresponding to the capacity of a SISO channel with channel power gain max. When the channel is not known at the transmitter, the transmit antenna weights are all equal, so the received SNR equals = khu k, where u is chosen to maximize γ. Clearly the lack of transmitter CSI will result in a lower SNR and capacity than with optimal transmit weighting. 27

28 Diversity-Multiplexing Tradeoffs So far, we already knew that there are two mechanisms for utilizing multiple antennas to improve wireless system performance. One option is to obtain capacity gain by decomposing the MIMO channel into parallel channels and multiplexing different data streams onto these channels. This capacity gain is also referred to as a multiplexing gain. It is not necessary to use the antennas purely for multiplexing or diversity. Some of the space-time dimensions can be used for diversity gain, and the remaining dimensions used for multiplexing gain. This gives rise to a fundamental design question in MIMO systems: should the antennas be used for diversity gain, multiplexing gain, or both? The diversity/multiplexing tradeoff or, more generally, the tradeoff between data rate, probability of error, and complexity for MIMO systems has been extensively studied in the literature, from both a theoretical perspective and in terms of practical space-time code designs. This work has primarily focused on block fading channels with receiver CSI only since when both transmitter and receiver know the channel the tradeoff is relatively straightforward. 28

29 Diversity-Multiplexing Tradeoffs Antenna subsets can first be grouped for diversity gain and then the multiplexing gain corresponds to the new channel with reduced dimension due to the grouping. For finite blocklengths it is not possible to achieve full diversity and full multiplexing gain simultaneously, in which case there is a tradeoff between these gains. A transmission scheme is said to achieve multiplexing gain r and diversity gain d if the data rate (bps) per unit Hertz R(SNR) and probability of error P e (SNR) as functions of SNR satisfy (10.21) (10.22) 29

30 Diversity-Multiplexing Tradeoffs For each r the optimal diversity gain d opt (r) is the maximum the diversity gain that can be achieved by any scheme. It is shown that if the fading block length exceeds the total number of antennas at the transmitter and receiver, then The function (10.23) is plotted in Fig (10.23) 30

31 Space-Time Modulation and Coding Since a MIMO channel has input-output relationship y = Hx + n, the symbol transmitted over the channel each symbol time is a vector rather than a scalar, as in traditional modulation for the SISO channel. Moreover, when the signal design extends over both space (via the multiple antennas) and time (via multiple symbol times), it is typically referred to as a space-time code. Most space-time codes are designed for quasi-static channels where the channel is constant over a block of T symbol times, and the channel is assumed unknown at the transmitter. Let X =[x 1,...,x T ] denote the M t T channel input matrix with ith column x i equal to the vector channel input over the ith transmission time. Let Y =[y 1,...,y T ] denote the M t T channel output matrix with ith column y i equal to the vector channel output over the ith transmission time. Let N =[n 1,...,n T ] denote the M r T noise matrix with ith column n i equal to the receiver noise vector on the ith transmission time. 31

32 ML Detection and Pairwise Error Probability With this matrix representation the input-output relationship over all T blocks becomes (10.24) Assume a space-time code where the receiver has knowledge of the channel matrix H. Under ML detection and given received matrix Y, the ML transmit matrix satisfies ˆX (10.25) where kak F denotes the Frobenius norm of the matrix A and the minimization is taken over all possible space time input matrices. The pairwise error probability for mistaking a transmit matrix X for another matrix ˆX, denoted as p( ˆX! X), depends only on the distance between the two matrices after transmission through the channel and the noise power, i.e. X T 32

33 ML Detection and Pairwise Error Probability (10.26) Let D x = X ˆX denote the difference matrix between X and ˆX. Applying the Chernoff bound to (10.26) yields (10.27) Let h i denote the ith row of H, i =1,...,M r. Then (10.28) Let H =vec(h T ) T where vec(a) is defined as the vector that results from stacking the columns of matrix A on top of each other to form a vector. 33

34 ML Detection and Pairwise Error Probability So H T is a vector of length M t M r. Also define D X = I Mr D x, where denotes the Kronecker product. With these definitions, khd x k 2 F = khd x k 2 = HD x D H x H H (10.29) Substituting (10.29) into (10.27) and taking the expectation relative to all possible channel realizations yields apple p(x! ˆX) apple det I Mt M r Dx H H H HD x (10.30) n Suppose that the channel matrix H is random and spatially white, so that its entries are i.i.d. zero-mean unit variance complex Gaussian random variables. Then taking the expectation yields where = 1 P DH x D x p(x! ˆX) apple det apple I Mt + P 4 2 n Mr (10.31) 34

35 ML Detection and Pairwise Error Probability p(x! ˆX) apple NY k= k ( )/4 Mr (10.32) = P 2 n 4 N Mr (10.33) Rank and Determinant Criterion: The pairwise error probability in d (10.33) indicates that the probability of error decreases as for d = N M r Thus, N M r is the diversity gain of the space-time code. The maximum diversity gain possible through coherent combining of M t transmit and receive antennas is. M r M t 35 M r

36 ML Detection and Pairwise Error Probability To obtain this maximum diversity gain, the space-time code must be designed such that the M t M t difference matrix Δ between any two code words has full rank equal to M t. This design criterion is referred to as the rank criterion. The coding gain associated with the pairwise error probability in (10.33) depends on the first term A high coding gain is achieved by maximizing the minimum of the determinant of Δ over all input matrix pairs X and ˆX. This criterion is referred to as the determinant criterion. 36

37 Spatial Multiplexing and BLAST Architectures In order to get full diversity order an encoded bit stream must be transmitted over all M t transmit antennas. This can be done through a serial encoding, illustrated in Figure

38 Spatial Multiplexing and BLAST Architectures A simpler method to achieve spatial multiplexing, pioneered at Bell Laboratories as one of the Bell Labs Layered Space Time (BLAST) architectures for MIMO channels, is parallel encoding, illustrated in Figure

39 Spatial Multiplexing and BLAST Architectures 39

40 Spatial Multiplexing and BLAST Architectures 40