Multiple Antennas for MIMO Communications - Basic Theory

Transcription

1 Multiple Antennas for MIMO Communications - Basic Theory 1 Introduction The multiple-input multiple-output (MIMO) technology (Fig. 1) is a breakthrough in wireless communication system design. It uses the spatial dimension (provided by the multiple antennas at the transmitter and the receiver) to combat the multipath fading effect. Fig. 2 shows the dramatic increase in transmission data rate with the increase in the number of transmitting and receiving antennas M of a MIMO system. 1

2 Fig. 1. A 33 MIMO system [1]. 2

3 Fig. 2. The average data rate versus SNR with different number of antennas M in a MIMO system. The channel bandwidth is 100 kz [2]. 3

4 2 Channel and Signal Model Consider a typical MIMO system shown in Fig 3 below. Fig. 3. A typical MIMO system including the signal processing subsystems. 4

5 The wireless channel part is extracted below. y 1 Space-time processor x M y 2 y N Space-time processor h NM Fig. 4. The channel model of a typical MIMO system. The received signal vector y can be expressed in terms of the channel matrix as: 5

6 NUS/ECE 6 y x n received signal vector transmitted signal vector N M y y y x x x y x where the symbols are: (1) (2) (3)

7 7 h11 h12 h1 M h21 h22 h 2M channel matrix (4) hn1 hn2 h NM n1 n 2 n noise vector (5) n M Thereafter, we study the transmit power P T constrained MIMO systems only, i.e., P T C for some fixed C. We can also write:

8 2 2 2 PT x1 x2 xm C x x (6) 2.1 The Covariance Matrices The covariance matrices of the transmitted signals and received signals are: R R xx yy E E E xx yy xx E nn (7) The covariance matrices are important parameters to characterize a MIMO communication system. 8

9 9 The traces of R xx and R yy give the total powers of the transmitted and received signals, respectively. The offdiagonal elements of R xx and R yy give the correlations between the signals at different antenna elements. Consider a symbol period of time T s, for the transmitted signals, it is usually made such that: Then within T s, Rxx E xx IM yy E R E xx nn E xx Rnn R nn (8) (9)

10 where R nn is the noise covariance matrix. In (8) and (9), we have assumed that the channels are stable within T s. Thus over a longer period of time (>> T s ), the average received signal covariance matrix is: Ryy E R nn (10) From (10), it can be seen that the received signal power is determined by the channel covariance matrix E{ } and the noise covariance matrix R nn. As R nn is determined by the environment and cannot be changed, we can manipulate or select an to optimize the channel output SNR (so as the capacity) of the MIMO system. 10

11 3 Channel Capacity 3.1 For SISO Systems 11 The channel capacity C of a single-input single-output (SISO) system is given by [3]: C Blog 1 S N bit/s (11) 2 where B (in z) is the channel bandwidth, S (in Watt) is the signal power, and N (in Watt) is the noise power. Both S and N are measured at the output of the channel. The channel capacity is a measure of the maximum rate that information (in bits) can be transmitted through the channel with an arbitrarily small error after using a certain coding method.

12 Example 1 A black-and-white TV screen picture may be considered as composed of approximately picture elements. Assume that each picture element has 10 brightness levels each being equally likely to occur. TV signals are transmitted at 30 picture frames per second. The signal-to-noise ratio at the TV is required to be at least 30 db. What is the required channel bandwidth for TV broadcast? Solutions Information per picture element = log 2 10 = 3.32 bits Information per picture frame = = bits 12

13 As 30 picture frames are transmitting per second, therefore the maximum information rate, R, for the TV transmission is then: 5 R bit/s This maximum information rate is the channel capacity C for TV broadcast. That is, R C B S N log2 1 Therefore the bandwidth B can be calculated as: B 6 C log 1 log SN Mz 13

14 For MIMO Systems For a MIMO system, the calculation of the capacity is more complicated due to the determination of the signalto-noise ratio S/N. Consider a MIMO system with a channel matrix (NM) as below: y x n (12) By the singular value decomposition (SVD) theorem [4], any N M matrix can be written as: UDV where D is an N M a diagonal matrix with non-negative elements, U is an N N unitary matrix, and V is a M M (13)

15 unitary matrix. That is, UU = U U = I N and VV = V V = I M. The diagonal elements of D are called the singular values of and they are the non-negative square roots of the eigenvalues of the following equation: x x, if N x x, if N M M (14) where x is the N 1 eigenvector associated with. 15

16 Example 2 Find the SVD for the following matrix (with N < M): Solutions

17 The eigenvalues of are: Therefore, 1 = , 2 = , 3 = D By using Matlab with the command: [U,S,V]=svd(), we can find the SVD of as: 17

18 UDV

19 Now putting (13) into (12), we have, y UDV x n Consider the following transformations: y x n U y V x U n Eq. (15) can be transformed as: 19 U y U UDV x U n y DV x n ydxn (15) (16) (17)

20 The system in (17) is called the equivalent MIMO system of (12). Note that: E E Ryy yy U yy U U RyyU E E Rxx xx V xx V V RxxV E E Rnn nn U nn U U RnnU (18) So that: tr tr tr R tr R yy yy Rxx trrxx R trr nn nn (19) 20

21 This means that the equivalent MIMO system has the same total input power, total output power and total noise power as the actual MIMO system in (12). The output SNR of the equivalent MIMO system is thus the same as the actual MIMO system. This in turn means that the channel capacity of the equivalent MIMO system is the same as that of the actual MIMO system because capacity is a function of the output SNR. Now the system in (17) has its channels all decoupled. The N channels are parallel to each other, with channel gains given by the diagonal elements of D, i.e., i, i = 1, 2,, N. 21

22 The number of nonzero eigenvalues of matrix is equal to the rank of matrix, denoted by r. This means that we can expand (17) as: y xn, for i 1,2, r i i i i y0 n, for i r,2, N i i (20) We note that if the MIMO system has more transmitting antennas than the receiving antennas (M > N), than is a horizontal matrix with a maximum rank = N. According to (20), the maximum number of uncoupled equivalent MIMO channels is N (<M). The remaining M-N transmitting antennas will become redundant with no 22

23 receiving antennas. This situation is illustrated below: x 1 y 1 x 2 y 2 x N N y N x N+1 x M Fig. 5. The equivalent MIMO system with M > N. 23

24 On the other hand, if the MIMO system has more receiving antennas than the transmitting antennas (M < N), than is a vertical matrix with a maximum rank = M. According to (20), the maximum number of uncoupled equivalent MIMO channels is M (<N). The remaining N- M receiving antennas will become redundant with no received signals. This is illustrated on next page. In general, for an NM MIMO system, the maximum number of uncoupled equivalent channels is min(n, M). 24

25 x 1 y 1 x 2 y 2 x M M y M y M+1 y N Fig. 6. The equivalent MIMO system with M < N. 25

26 26 As the channels of the equivalent MIMO system in (17) are uncoupled and parallel, the channel capacity of (17) can be calculated by a summation of the individual capacities of the parallel channels. That is, r Py i C B log2 1 2 i1 (21) where B (in z) is the channel bandwidth, Py i (in Watt) is the power received at the ith receiving antenna, 2 (in Watt) is the noise power at the ith receiving antenna, and r is the rank of. In order to related the received power to the channel parameters, we need to classify a MIMO system according to the availability of the channel knowledge to the transmitter or receiver.

27 (A) Channel state information (CSI) known to the receiver only As the transmitter does not know the CSI, its best strategy is to transmit power equally from all its transmitting antennas. For the equivalent MIMO system in (17), this can be done by making all the elements of x to have the same power. Under this situation, the received power is then calculated as: P Py i i M (22) where P is the total transmitting power. 27

28 Therefore, (21) can be written as: r r P P C B log 1 Blog 1 M M 2 i 2 2 i 2 i1 i1 (23) The eigenvalue i in (23) can be expressed in terms of the matrix or in (14) and (23) can be re-written as (see details of derivation in [5], pp. 7-8): C P Blog2 det I N, if 2 N M M (24) P Blog2 det I M, if N M 2 M 28

29 The total transmitting power P in (24) may not be easily known. If the average received powers P r at each of the receiving antennas are the same, we have: P P P r loss (25) where P loss is the average path loss from the transmitter to the receiver. Then (24) can be re-written as: C Pr Blog2 det IN, if N M 2 M P loss Pr Blog2 det IM, if N M 2 M P loss (26) 29

30 Or, in terms of the SNR at the receiving antennas, we have: C Blog2 det IN, if N M M P loss Blog2 det IM, if N M M P loss (27) 30

31 (B) 31 Channel state information (CSI) known to both the transmitter and receiver If the transmitter knows the CSI, i.e., the channel matrix, its best strategy is to transmit more powers along those channels whose channel gains are larger and to transmit less powers or along those channels with a smaller channel gain. This is called the water-filling principle. Under this condition, the transmitting power P i for the ith channel in the equivalent MIMO system in (17) is given by (see details of derivation in [5], pp ): 2 Pi, i 1,2,, r rank( ) (28) i

32 where if P i in (28) is negative, it will be set to zero. The parameter in (28) is determined by satisfying the transmitting power constraint: P r Pi i1 With the transmitting powers in (28), the received powers in (22) is then: 2 P P (30) i y i i i The channel capacity is then obtained as: (29) 1 C B r 2 log i i1 (31) 32

33 3.3 Random channels When the channels are random in nature, the channel capacity is a random number. The most popular random channel model is the Gaussian channel model whose channel matrix elements are all complex Gaussian random numbers with a mean and a variance 2. Note that the channel capacity expression is same as for the deterministic channel case except that C becomes a random number. Because the capacity is a random number, a pdf and cdf of C can be obtained. Instead of finding the instantaneous C, it is more often to find the average channel capacity E{C}. 33

34 34 Example 3 Find the channel capacity of a MIMO system with N = M = 1 and = h = 1. Assume that the total transmitting power = P and the noise power at the receiver = 2. The transmitter has no knowledge of the channels. Solutions Without CSI, the transmitter transmits power equally over all transmitting antennas. r = rank () = 1, 1 = 1. Therefore, P Py 1 1 P M r Py P i C B log2 1 Blog i1

35 Example 4 Find the channel capacity of a MIMO system with N = M = 4 and h ij = 1, (i = 1,2,,4, j = 1,2,,4). Assume that the total transmitting power = P and the noise power at the receiver = 2. The transmitter has no knowledge of the channels. Solutions , r = rank( ) 1,

36 Without CSI, the transmitter transmits power equally over all transmitting antennas. Therefore, P P M y P r Py 4 i P C B log2 1 Blog i1 36

37 37 Example 5 The conditions are same as those in Example 4 but the transmitter now knows the channel matrix perfectly. Find the channel capacity. Solutions , r = rank( ) 1, With knowledge of, the transmitter can transmit power along only one channel, i.e., the channel with eigenvalue 1.

38 The received power will then be: P Py P 1 The capacity will then be: r Py 16 i P C B log2 1 Blog i1 Note the capacity in this example is much larger than the one in Example 4, due to the availability of the CSI, i.e,. 38

39 Example 6 Find the channel capacity of a MIMO system with N = M = 4 and Assume that the total transmitting power = P and the noise power at the receiver = 2. The transmitter has no knowledge of the channels. 39

40 Solutions C r = rank( ) P Py 1 1 Py 2 Py P 3 y4 M r Py i Blog2 1 2 i 1 P 4Blog P 4 Without channel knowledge, the transmitter transmits equally over all transmitting antennas. Therefore,

41 Example 7 Find the channel capacity of a MIMO system with N = 4, M = 1, and Assume that the total transmitting power = P and the noise power at the receiver = 2. The transmitter has no knowledge of the channels. 41

42 Solutions r = rank( ) 1, 4 1 Without channel knowledge, P P M y P C r P Blog 1 i1 4P Blog yi

43 43 Example 8 Find the channel capacity of a MIMO system with N = 1, M = 4, and Assume that the total transmitting power = P and the noise power at the receiver = 2. The transmitter has no knowledge of the channels. Solutions r yi C Blog i1 r = rank( ) 1, 4 P Py 1 1 P M P P Blog 2 1 2

44 44 Example 9 Find the average channel capacity of a MIMO system with N = M = 4 and the channel matrix is a random matrix with r11 r12 r13 r14 rij aij jbij( i, j 1,,4) where aij, bij r21 r22 r23 r 24 are real random Gaussian numbers r31 r32 r33 r34 with Ea ij E bij 0, r41 r42 r43 r 44 and Vara ij Var bij 1 2 r ij (i, j = 1,, 4) are random complex numbers with a mean equal to zero and a variance equal to one. Assume that the transmitter has no knowledge of the channels. The SNR at the receiving antennas is = 20 db and there is no path loss such that P loss = 1.

45 Solutions r a jb (, i j 1,,4), a, b 0,12 ij ij ij ij ij E r E a je b ij ij ij ij ij ij ij ij Using (27) with N = M = 4, = 20 db, P loss = 1, we have C 100 log2det I 4 bits/s/z 4 0 Var r E r E r E a E b Using Matlab, we can find 2 4 Normal distribution EC Elog det I bits/s/z Normalized by bandwidth B 45

46 We can further plot the cdf of C as follows: cdf (C) C (bits/s/z) 46

47 The Matlab codes are shown below (filename: mimo_iid.m): clear all; M=4; % number of transmitting antennas N=4; % number of receiving antennas snrdb=20; % SNR snr=10^(snrdb/10); % SNR in numerical value for n=1:5000; % number of runs =sqrt(0.5)*(randn(n,m)+1j*randn(n,m)); % channel matrix C(n)=log2(real(det(eye(N)+snr/M*( *)))); % random capacity end; cdfplot(c) Average_capacity=mean(C) 47

48 References: [1] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, From theory to practice: an overview of MIMO space time coded wireless systems, IEEE Journal on Selected Areas in Communications, vol. 21, no. 3, pp , [2] E. Biglieri, R. Calderban, A. Constantinides, A. Goldsmith, A. Paulraj, and. V. Poor, MIMO Wireless Communications, Cambridge University Press, [3] F. G. Stremler, Introduction to Communication Systems, Addison-Wesley, [4] R. orn and C. Johnson, Matrix Analysis, Cambridge University Press, [5] Branka Vucetic and Jinhong Yuan, Space-Time Coding, John Wiley & Sons Ltd,