MIMO and multiple antennas

Transcription

1 MIMO and multiple antennas Cairo University Faculty of Engineering Department of Electronics and Electrical Communications Dr. Karim Ossama Abbas Fall 2010

2 Contents Brief on diversity Principles of spatial multiplexing (true MIMO) MIMO decoding algorithms Zero Forcing Minimum Mean Square Error Singular Value Decomposition Maximum Likelihood Sphere Decoding The CORDIC processor Implementing MMSE and ZF -> QRD Implementing SVD -> Two-sided Jacobi Case study Squre Root MMSE

3 Receiver and transmitter diversity Classical multiple antenna systems used multiple antennas at either the transmitter or the receiver to improve performance At the receiver, conceptually, multiple receive antennas lead to better performance because if one antenna is in deep fade, it is not very likely the others will also be Transmit diversity follows the same principle but at the transmitter This is a gross oversimplification of course and for further reading refer to Rappaport

4 Spatial multiplexing Spatial Multiplexing or true Multiple Input Multiple Output systems contain multiple antennas at both the transmitter and the receiver. In the classical sense, consider equal numbers Ntx=Nrx y1 = h11x1 + h12x2 + h13x3 + n1 y = h x + h x + h x + n y3 = h31x1 + h32x2 + h33x3 + n3

5 Spatial multiplexing There are enough simultaneous equations to solve for the unknown transmitted signals This ignores the effect of noise The problem can be written in a more general form as a matrix equation We now have a channel matrix, a transmitted vector, an observation vector, and a noise vector instead of scalars y1 h11 h12 h13 x1 n1 y = h h h x + n y 3 h31 h32 h 33 x 3 n 3 y N rx 1 =HNrx N x tx Ntx 1+nNrx 1

6 MIMO capacity Recall the channel has a maximum information-theoretic capacity at any given SNR There is no conceivable way to raise data rate above capacity However, true MIMO raises the capacity itself This result is formalized, and it can be shown that channel capacity increases linearly JUST by raising the number of Nrx, Ntx The main cost is hardware G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, no. 3, pp , Mar

7 The MIMO channel The MIMO channel includes a fading matrix H and a noise vector n The noise vector is simulated as iid Gaussians, thus SNR is usually assumed the same over the antennas due to their proximity H is simulated in its simplest form as iid complex Gaussians, i.e. every element is independent Rayleigh with different parameters How realistic, and how good is this? It is not very realistic because antenna proximity introduces dependence. It is optimistic because it introduces further independence into the paths

8 A good channel for MIMO Multipath leads to fades and fades cause severe degradation to performance (refer to system simulation) However, in MIMO multipath is considered useful In an environment with a lot of reflectors, there will be a lot of multipaths This means that in the end as each ray arrives at each antenna it has followed an independent path This means H has independent elements Which means that the matrix is invertible We are utilizing spatial diversity

9 A bad channel for MIMO An open desert is pretty bad for MIMO Line of sight is a MIMO killer In this case the paths are very correlated This means that two or more vectors in the H matrix are highly correlated The channel will be singular or near singular within your fixed point precision Consider what happens to the inverse as the contribution of the independent vector H3 is reduced. First the inverse rises in value, but also the ratio of the inverse components is reduced, eventually the channel becomes singular once processing precision is reached H H = H2 H H = = H H Hx = H 2 Hx = H1 + H H H Hy = H 2 1 Hy = H1 + H H H1 1 Hz = H 2 Hz = H1 + H H3

10 Practical limitation to capacity As you increase the number of antennas but maintain platform size, you have to crowd the antennas close together This in turn leads to correlation between paths that has nothing to do with the environment This antenna spatial correlation component leads to a limit on MIMO capacity in practice

11 Matrix algebra revision Recall the following for matrices: AB BA 1 AA = I, I = Diag(1) C = A B mxp mxn nxp A T is the transpose where a ij a ji A H is the Hermitian where a ij a * ji

12 Inverting the channel The most basic approach to solving the MIMO problem is to find the inverse matrix of H and multiply it to get an estimate for sent signals This is ideal in a noiseless channel But what if the channel is singular? y = Hx + n xˆ = -1-1 x+h n=h y

13 Avoiding singularity issues Matrix singularity results from a highly correlated channel A highly correlated channel will almost certainly fail in MIMO However, to maintain consistent mathematical performance, we calculate the pseudoinverse The pseudoinverse always exists and always yields the inverse for non-singular matrices, but give the next best thing when they are singular Pseudo inverse of H: + H H -1 H = A (AA ) xˆ = + + x+h n=h y

14 Zero forcing (ZF) This very direct approach to channel inversion is a valid form of MIMO decoding ZF essentially nullifies the interference of each antenna on the other by forcing the channel inverse on the received Thus inter-antenna interference is minimum However we have a major problem with small channel values: Yield very large dynamic range on inverse Cause magnification on the channel colored noise term

15 ZF magnifies noise Note that x_hat is the metric upon which the demapper will then work to decide either the sent symbol (hard decision) or soft metrics (soft decision) x_hat includes a noise component n_tilde that is colored by the channel pseudoinverse If the channel has nulls or small values, this noise terms is greatly magnified ZF nullifies effect of antennas on each other only in zero noise and behaves badly with noise ˆ + + x = x+h n=h y xˆ = + x+n=h y

16 MMSE, an optimal (?) estimator MMSE = Minimum Mean Square Error This is a method where we find an estimator that minimizes the MSE In this case we have a set of measurments y, what is the estimator x_hat that minimizes the error from the true random variable x given y? MMSE solves this problem optimally in the presence of noise xˆ = Min E x xˆ 2 x ˆ ( (( ) )) given measurment y of x in noise n

17 MMSE Derivation of the estimator is an interesting estimation and detection problem, but without derivation, the solution is shown below As shown, MMSE does not magnify noise, nor does it try to completely cancel out the channel In terms of implementation we will see that ZF and MMSE are nearly identical So nobody uses plain ZF x ˆ = WH y = H (HH + I ) y SNR * * * -1

18 Sidenote on MMSE The SNR term in MMSE assumes white noise on all antennas and equal SNR on the all antennas If the antennas have different SNR then the term becomes Diag (1/SNRi) If the noise is colored then you will have a general matrix G Whether or not to consider G diagonal is an interesting question in implementation, note however the weight matrix W is symmetric by necessity * * * -1 x ˆ = WH y = H (HH + G) y * W = HH + G

19 Unitary matrices, diagonal matrices, and triangular matrices Notice that we have a lot of matrix inverses along the way Performing this is a big question mark Recognize that some matrices have desirable properties that help in implementation A unitary matrix: Its inverse is its Hermitian A diagonal matrix: Its inverse is the inverse of the diagonal elements A triangular matrix: Also easier than a general matrix Unitary matrix: UU H H = U U = I

20 SVD Singular value decomposition is a matrix decomposition that breaks a general matrix into three matrices, two unitary and one diagonal Because of the desirable properties of unitary and diagonal matrices in inversion, SVD could be useful in MIMO decoding H = USV UU VV S = Diagonal= Diag( λ i ) λ = Channel signular values i H H = I = I

21 Direct implementation of SVD At the receiver, the estimated channel is decomposed, then the inversion is implemented on the different components Unitary inversions and diagonal inversion are fairly simple, but is the decomposition simple? This leads to performance very similar to ZF y = Hx + n y = USVx + n H H H U y = U USVx + U n -1 H -1-1 H S U y =S SVx+S U n H -1 H H H -1 H V S U y=v Vx+V S U n=xˆ

22 SVD and rudimentary beamforming If the V matrix is fedback from the receiver to the transmitter to predistort the signal: This leads to some form of eigenbeamforming It also distributes the computational load H sentsignal = x=v x y = Hx + n y = USVx + n H y = USVV x + n = USx + n -1 H x=s ˆ U y

23 Notes on SVD SVD breaks a matrix down into very basic components The diagonal elements of the S matrix are the singular values In MIMO the effect of SVD is that it reduces the MIMO operation into almost parallel SISO streams The singular values then represent the powers on these streams If this information is sent back to the transmitter, then waterfilling and beamforming can be very effective n provides some basic facilities for a feedback channel and support for SVD channel state information feedback

24 True optimality: Maximum Likelihood (ML) Consider the following argument: If we calculate every possible version of sent vector distorted by the channel matrix Then for each version we calculate the vector distance between this version and the observation vector y If we look at the minimum Distance from within these choices Then it is the most likely solution Hx all possible vectors x y - Hx x all possible vectors

25 Maximum Likelihood ML solution is performed by calculating vector norms which are fairly simple However the number of points over which the search space extends grows tremendously with constellation, and especially with antenna count If we have a constellation of size k and Nrx antennas then the number of search candidates is: k N rx Thus for SISO QPSK there are 4 points, SISO 64 QAM 64 points. But for 4x4 MIMO with 64 QAM there are possible points

26 ML in SISO Note that in the SISO mode, ML reduces to drawing the boundaries and measuring the closest point Except we are distorting the constellation instead of inverting the channel on the received points And also we are dealing with matrix-vector problems so we have vector norms

27 MMSE or ML? ML has the best performance of ANY MIMO decoding algorithm ML is the theoretical best performance for a memoryless channel (which a wireless channel is) Why is MMSE a minimum error estimator then? MMSE reduces the average error and gives an optimal estimate IF the value being estimated is continuous Note that in ZF, MMSE, SVD we have to slice the symbols after inversion ML gives the optimal solution if the candidates are fixed discrete points in space

28 ML to SD ML is an impossible solution in most MIMO systems However, is there a way to get ML or close to ML performance while reducing complexity? This is where sphere decoding comes in Sphere decoding depends on drawing a sphere around the received point, then searching only candidates within this hypersphere, recursively cancelling a large number of candidates at each step If the radius of the sphere is large we are more likely to get the true solution but will search more points, if the radius is small we will search less points but will likely miss the correct solution We will come back to SD later

29 Implementation issues -> General complex matrix inversion The central issue in MIMO decoding is matrix inversion This is explicit in ZF and MMSE, and implicit in other algorithms Trying to invert a general complex matrix is counterproductive We try to decompose the matrices (as in SVD) into more invertible forms

30 Unitary matrices and transformations As a reminder unitary matrices are matrices whose inverse is their Hermitian transpose A unitary transformation is the multiplication of a general matrix by a unitary matrix Thus multiplying y by U in SVD is a unitary transformation

31 Givens rotations A Givens rotation is an operation on a matrix or vector of the form: cosθ sinθ sinθ cosθ Now consider its operation on the following matrix if theta is carefully calculated from the matrix as follows: cosθ sinθ A B E F C sinθ cosθ C D = 0 G A Note that in reality Givens operate on vectors 1 where θ = tan ( )

32 QR Decomposition A QR decomposition is the decomposition of a matrix into a unitary matrix Q, and an upper triangular matrix R If you observe the Givens rotation operation in the previous slide, you will notice it performed QRD on the 2x2 matrix cosθ sinθ A B E F C θ sinθ cosθ C D = 0 G A A B cosθ sinθ E F C D = QR = sinθ cosθ 0 G 1 where = tan ( ) Note that so far we are operating only on real matrices

33 Complex Givens rotation Note that each Givens operation calculates its phase from a leading pair. In complex matrices we are likely to need more than one rotation angle In the first step, the leading element is made real jφa jφa jφb' jφb e 0 Ae Be A Be jφ jφ = jφ jφ 0 1 Ce De Ce De c' d ' c d In the second step, we need a two angle Givens rotation jφ jφ jφ b f cosθ sinθe A Be E Fe C φ φ θ jφ jφc jφ = = = d jφg sinθ cosθe Ce De 0 Ge A 1 c, tan ( )

34 Givens rotations on larger matrices Note that the effect of a Givens rotation is that it nulls the second element in the leading pair In a larger matrix, we have a sparse rotation matrix and the result is nulling only one element For example in a 4x4 matrix To null element 3,1 we can anchor on element 1,1 We then calculate the necessary angles And the rotation matrix will be very sparse, having the shape (considering a pure real example): Note the effect of Givens cosθ 0 sinθ 0 is not only nulling the element, but rotating the rest of the matrix sinθ 0 cosθ

35 QRD by Givens rotations To perform QRD on a larger matrix we need to successively null off-diagonal elements The remaining matrix is R 1 2 N The product of the givens rotation matrices is the Q matrix One must follow a specific order because if you don t then the matrix may regenerate off diagonal elements One such order for 4x4: 2,1 3,1 4,1 3,2 4,2 4,3 (Q Q...Q )H = R

36 QRD and inversion in hardware Once the matrix is decomposed, finding its inverse is much simpler The inversion of Q is its Hermitian, which is extremely inexpensive Finding the inverse of R is done recursively by first calculating the inverse of R44 then going up the vectors one by one r11 r12 r13 v11 v12 v r22 r23 0 v22 v = 0 0 r v r33v33 = 1, r22v22 = 1, etc.

37 Sphere decoding Having understood QRD we can go back to SD H = QR Condition to lie in a hypershpere y - Hx < QRx - y Norm is invariant to a unitary transform H Rx - Q y 2 2 H Define: p(x) = Rx - Q y r < r < r

38 SD continued H Define: p(x) = Rx - Q y N rx i= 1 p Where r 2 2 i p i is the ith row of p(x) r11 r12 r13 r14 x1 q11 q12 q13 q14 y1 0 r r r x q q q q y 0 0 r33 r 34 x 3 q31 q32 q33 q 34 y r44 x4 q41 q42 q43 q44 y x1 y1 x 2 y ' ' ' ' 2 p1 = [ r11 r12 r13 r14 ] + q11 q21 q31 q 41 x 3 y 3 x y 4 4 * 2 r

39 N rx i= 1 p SD continued 2 2 i In 4x4 leads to: Which will not be satisfied unless: Which in turn is only satisfied if: Again satisfied only if we also have: r p + p + p + p r p + p + p r p + p r p r The SD algorithm depends on: Choosing a value for r that reduces candidates Performing the last condition first, this is easier to calculate, and involves permutations over only x4, if this fails for any x4, we can skip all combinations of the rest of the p s for this p4 This tree pruning operation is the basis of SD

40 SD continued SD thus has two main components: A QRD, now becoming the limiting factor A search algorithm Some search algorithms will search all the tree, but will prune early thus reducing the number of candidates, these algorithms have ML performance but variable throughput Some search algorithms consider only a fixed number of points at every level and thus have constant throughput but suboptimal performance to ML

41 SVD using Jacobi algorithm How do we perform a Singular Value Decomposition The two-sided Jacobi-algorithm is a hardware friendly algorithm with affinity to the QRD Jacobi is multi-step and uses multiples angles to perform the rotation, but it nulls two off-diagonal elements The first step is removing the phase of the leading element jφa jφa jφb jφb1 e 0 A B A Be jφ jφ = jφ jφ 0 1 C D Ce D c d c1 d1

42 2-sided Jacobi continued This is then followed by a Givens rotation to null the lower off-diagonal element jφ jφb1 cosθ sinθe A Be jφ jφ jφ sinθ cosθe Ce De c1 d1 d 2 1 Then the phase of the other diagonal element is removed jφb2 jφb2 1 0 A1 Be 1 A1 Be 1 jφd 2 j d 2 0 e φ = 0 De 1 0 D1 A unitary transform exchanges phases across diagonal, leading to a pure real matrix jφb 2 /2 jφb 2 A Be e jφ /2 0 e 0 D 0 D 1 b 2 1 = A B A 0 Be De jφb 2 = 1 1 jφ

43 2-sided Jacobi continued At this point the real Jacobi algorithm can be used to calculate t singular values Real Jacobi requires the calculation of two fairly complex angles cosθr sinθr A1 B1 cosθl sinθl S1 0 = sinθ cosθ 0 D sinθ cosθ 0 S r r 1 l l 2 B θ θ θ θ θ θ sum = ( r + l ) = tan ( ) diff = ( r l ) = tan ( ) D1 A1 D1+ A1 B

44 SVD of larger matrices SVD of larger matrices involves nulling two off-diagonal elements in a step as opposed to QRD where one element is nulled There is no way to avoid the nulling of an element leading to the regeneration of a previously nulled element in Jacobi Thus Jacobi for any matrix of size larger than 2x2 requires multiple iterations to converge Jacobi is the choice algorithm for hardware implementations due to suitability to parallelism

45 Cost of matrix operations The cost of complex matrix arithmetic extends logically from the cost of complex scalars, in ascending order of complexity: Complex addition Complex multiplication Division Givens rotation (and other unitary transformations) involve trigonometric functions, multiplications, and additions By trial you will find they are roughly equivalent to real division What does unitary nature say about dynamic performance?

46 Givens operations cosθ sinθ A B E F C θ sinθ cosθ C D = 0 G A 1-Finding the angle: 1 C θ = tan ( ) A 2-Rotating a coordinate pair: G = B sinθ + D cosθ The two main operations are: Finding a rotation angle Rotating a coordinate pair with this angle 1 where = tan ( ) Note that for the pair that generates the angle, this is equivalent to a Cartesian to polar transform: θ C A ( AC, ) ( E, ) = ( A + C, tan ( ))

47 Visualizing Givens We are either finding the angle of a coordinate pair to the x-axis (and rotating the pair by the same angle) Or we are finding the result of rotation of a coordinate pair by an angle derived elsewhere

48 A direct implementation of Givens A direct implementation requires a divider, three lookup tables, two multipliers, and one adder One major issue is how all these elements should be sized C/A has a finite range G will have the same magnitude as the coordinate pair (B,D) Thus a direct application of rules for noiseless multiplication is too conservative 1 C θ = tan ( ) A G = B sinθ + D cosθ

49 The CORDIC algorithm [1] CORDIC stands for COordinate Rotation DIgital Computer At its heart the CORDIC algorithm is a way to implement a Givens rotation in a hardware friendly (binary based) fashion It can also be used to implement other functions and unitary transformations The CORDIC uses adders only It allows wordlength to grow only at its natural pace In general, CORDIC was the unit of choice in hardware implementations, however this is starting to change

50 The CORDIC algorithm This is just a restatement of the equation Consider the geometric interpretation of the operation again The rotation of the pair can be approached by a number of microrotations of successfully smaller angle z = x cosθ y sinθ Taking the cos term common z = cos θ( x y tan θ)

51 The CORDIC algorithm Thus at each stage of the microrotation we are performing a Givens rotation z = i 1 cos θi( x + i y i tan θi) If the angles of rotation are restricted so that their tan is a negative power of 2, then the rotations become 1 i z i + 1 = ( x 2 i ± y 2 ) i i 1+ 2 Thus this microrotation becomes a shift-add operation, the major decision at each step is the sign (i.e. Whether we add or subtract) Note because the common factor is cos, it doesn t matter which direction we are rotating

52 Combining microrotations Now any angle can be broken down so that the tans of its components are powers of 2 Thus to form a specific angle we rotate in clockwise and counterclockwise directions until the original angle is reformed At each step a decision is made on whether to add or subtract depending on the direction we need to rotate

53 The CORDIC algorithm formally Begin with the angle you need to rotate by θ, and the original coordinates x o, y then perform N iterations of: o i x = ( x ay 2 ) i + 1 i i i i y = ( y + ax 2 ) θ a i + 1 i i i i = ( θ a arctan(2 )) i + 1 i i i = sign( θ ) End up with x = K ( x cosθ N N o i 2i sin θ ) y = K ( y cosθ + x sin θ ) where K = 1+ 2 o o o N N o o o o N N y o

54 What about the common term The common term K can be a problem Calculating it requires successive multiplications However, if the number of multiplications is constant, this scaling value is constant Thus you don t need to calculate it Also applying it only requires a constant coefficient multiplier As the number of iterations grows the scaling factor approaches a certain number asymptotically

55 Types of CORDIC: Vectoring, rotation The CORDIC algorithm described above performs a rotation of a coordinate pair Another type of CORDIC calculates the angle of a coordinate pair to the x-axis and rotates them there (calculating amplitude) i x = ( x ay 2 ) i + 1 i i i i y = ( y + ax 2 ) i + 1 i i i i θi + 1 = ( θi ai arctan(2 )) ai = sign( y i) End up with x = K x + y y θ 2 2 N N o o N where K 0 = ( θ + arctan( y / x )) N o o o N = = 1+ 2 N 2i

56 CORDIC in brief CORDIC can be implemented with only one lookup table (optional, needed only to calculate angle in radians), and a bunch of shift-adds It does this by breaking down rotations into microrotations

57 CORDIC and transcendental functions The CORDIC algorithm can perform the following: Calculate magnitude of a vector Transform from polar to Cartesian Transform from Cartesian to polar General rotation Calculate arctangent Calculate sine and cosine

58 CORDIC and Givens rotations In its very definition, CORDIC performs both types of operations necessary for Givens without using general multipliers cosθ sinθ A B E F C sinθ cosθ C D = 0 G A 1 where θ = tan ( )

59 Complex CORDIC from real CORDIC Complex vectoring, the leading element must be real Complex rotation

60 Types of CORDIC: Serial and parallel These are the two types of CORDIC classically mentioned in literature Bit parallel CORDIC has as many adders as there are iterations It is fully pipelined (Latency=N, throughput=1, adders=n) Bit serial CORDIC may have only one adder and thus takes N cycles to produce an output (latency=throughput=n, adders=1)

61 Systolic arrays A systolic array is a regular connection of identical processors If the array is fully pipelined, each clock data flows as with blood pumping Systolic arrays are: Regular Easy to design Usually faster than needed

62 Systolic array implementation of QRD Consider the following implementation of QRD

63 Systolic R With proper pipelining this should output the R matrix h h h h h h h h h h h h h h h h

64 Collapsed systolic arrays A major issue with systolic arrays is that they are often too fast for their applications One solution to this is to design a linear array in which only one row exists and is reused The disadvantage is that we have much reduced throughput Also we have underuse of the processor

65 Ex1: Systolic array implementation of MMSE [2] Find document attached Describes a systolic array implementation of a typical MMSE [2] JingmingWang, QR-MMSE Algorithm and its systolic array implementation, March 2005

66 Ex2: Non-systolic implementation of Matrix decomposition [3] Find paper attached This paper describes an implementation of a special case of MMSE Contains no inversion Does not use Givens rotations to implement QRD Very suitable for FPGA implementation [3] Hun-Seok Kim, et. al, A Practical, Hardware Friendly MMSE Detector for MIMO-OFDM-Based Systems, EURASIP journal on advances in signal processing, 2008

67 Assignment 3 (cancelled) Write a one page summary comparing the two implementations in [2] and [3] Single space Times New Roman 11 pt.

68 Assignment 4 Simulate (floating point) a wireless comm system with the following: BPSK modulation No coding 2X2 or 4X4 128 subcarrier OFDM Flat faded, quasi-static channels Perfect channel estimation at receiver Plot BER-SNR for the 2 cases (2X2 and 4X4) downto BER 10e-5 for: ZF MMSE ML

69 Assignment 5 Simulate (floating or fixed) a CORDIC unit, the following should be parameters: Vectoring or rotation Number of iterations Maximum integer value of inputs With the x-asis the number of iterations: Plot the SNR (versus number of iterations) of the calculation of absolute value using the vectoring mode This must be done using a large number of inputs so you can get a proper average

70 Comparison of MIMO decoders Algorithm Main adv Main Disadv Implementation ZF Simple Amplifies noise Rarely in practice, but some BLAST implementations MMSE SQRT MMSE Simple, handles colored noise No inversion in hardware Non-optimal White noise-only Common in practice, approaches hard decision ML with soft decision Uncommon ML Optimal Impractically complex Brute force possible in SISO, nonexistent in MIMO SVD SD Practical Still complex, trades off inconstant throughput and optimality Gives info on channel Hardware is complex, performance in AWGN suboptimal In fashion, many hardware architectures to handle search, initial decomposition becoming limiting Fashionable, but rarely practically used

71 Algorithm cognition -> another dimension A MIMO decoder that can change the algorithm according to channel conditions may offer a boost to a cognitive radio beyond mode selection: SVD in interference rich channels More antennas if the channel is scattering-rich SD in more severe channels MMSE in more favorite channels This is particularly applicable if it can be combined with channel decoders, and if the MIMO decoder is not a simple union of decoders This concept drives the implementation of reprogrammable engines in baseband radios as a research area