Revision of Lecture Twenty-Eight

Transcription

1 ELEC64 Advanced Wireless Communications Networks and Systems Revision of Lecture Twenty-Eight MIMO classification: roughly three classes create diversity, increase throughput, support multi-users Some classification also includes beamforming Single-user fractional-spaced receiver Baseband continuous-time model, discrete-time multirate model, discrete-time multichannel model SDMA induced MIMOs Adaptive beamforming assisted receiver, and transmit beamforming This lecture carries on MIMO A, B, C 38

2 ELEC64 Advanced Wireless Communications Networks and Systems SDMA Systems Previous lecture considers flat MIMO, we now considers frequency-selective MIMO, requiring space-time processing SDMA induced MIMO system: Assume one transmit antenna and L receiver antennas supporting M users No specific antenna array structure is assumed, so it is most generic Channels are frequency selective, and CIR connecting user m and lth receiver antenna is c l,m = [c 0,l,m c,l,m c nc,l,m] T user user user M Tx Tx Tx Symbol-rate received signal samples x l (k) for l L are given by s (k) s (k) s (k) M n (k) n (k) n (k) L x (k) x (k) x (k) L Multiuser Detector y (k) y (k) y (k) M s ^ (k d) s ^ (k d) s ^ M(k d) x l (k) = M m= n C i=0 c i,l,m s m (k i) n l (k) = x l (k) n l (k) n l (k) is complex-valued AWGN with E[ n l (k) ] = σ n, x l(k) is noise-free part of lth receive antenna s output, s m (k) is kth transmitted symbol of user m (assuming BPSK for simplicity) 38

3 ELEC64 Advanced Wireless Communications Networks and Systems Multiuser supporting capability Multiuser Detection in SDMA Systems CDMA: each user is separated by a unique user-specific spreading code SDMA: each user is associated with a unique user-specific CIR encountered at receiver antennas Unique user-specific CIR plays role of user-specific CDMA signature Owing to non-orthogonal nature of CIRs, effective multiuser detection is required for separating users A bank of M space-time equalisers forms MUD, whose soft outputs are L y m (k) = l= n F i=0 x (k) x (k) x (k) L w * 0,,m w * w*,,m n,,m * * 0,L,m,L,m w* w* 0,,m,,m w* n,,m w w w i,l,m x l(k i), m M w F F * n,l,m F w l,m = [w 0,l,m w,l,m w nf,l,m] T is mth user detector s equaliser weight vector associated with lth receive antenna, STE has order n F and decision delay d y (k) m 383

4 ELEC64 Advanced Wireless Communications Networks and Systems System Model Define n F (n F n C ) CIR matrix associated with user m and lth receive antenna C l,m = 6 4 c 0,l,m c,l,m c nc,l,m c 0,l,m c,l,m c nc,l,m c 0,l,m c,l,m c nc,l,m Introduce overall system CIR convolution matrix C = 6 4 C, C, C,M C, C, C,M... C L, C L, C L,M Then received signal vector x(k) = [x (k) x (k) x L (k)] T can be expressed by x(k) = C s(k) n(k) = x(k) n(k) where x l (k) = [x l (k) x l (k ) x l (k n F )] T for l L, n(k) = [n (k) n (k) n L (k)] T with n l (k) = [n l (k) n l (k ) n l (k n F )] T, and s(k) = [s T (k) st (k) st M (k)]t with s m (k) = [s m (k) s m (k ) s m (k n F n C )] T 384

5 ELEC64 Advanced Wireless Communications Networks and Systems Space-Time Equalisation Output of mth STE detector can be written as y m (k) = L w H l,m x l(k) = w H m x(k) l= where w m = [w T,m wt,m wt L,m ]T With y R m(k) = Re[y m (k)], M user detectors decisions are defined by ŝ m (k d) = sgn (y R m(k)), m M Minimum mean square error solution is defined by closed-form w (MMSE)m = C C H σ n I C (m )(nf n C )(d) for m M, where I denotes Ln F Ln F identity matrix and C i the ith column of C Adaptive implementation using LMS algorithm where ǫ(k) = s m (k d) y m (k) w m (k ) = w m (k) µx(k)ǫ (k) 385

6 ELEC64 Advanced Wireless Communications Networks and Systems Bit Error Rate of Space-Time Equaliser Note transmitted symbol sequence s(k) {s (q), q N s }, where N s = M(n F n C ) Let the element of s (q) corresponding to desired symbol s m (k d) be s (q) m,d Noise-free part of mth detector input signal x(k) assumes values from signal set m = { x (q) = C s (q), q N s } m can be partitioned into two subsets, depending on the value of s m (k d), as follows (±) m = { x(q,±) m : s m (k d) = ±} Similarly, noise-free part of mth detector s output ȳ m (k) assumes values from the scalar set Y m = {ȳ (q) m = wh m x(q), q N s } Thus ȳ R m(k) = Re[ȳ m (k)] can only take the values from the set Y R m = {ȳ (q) Rm = Re[ȳ(q) m ], q N s} Y R m can be divided into the two subsets conditioned on the value of s m (k d) Y (±) Rm = {ȳ(q,±) Rm Y Rm : s m (k d) = ±} 386

7 ELEC64 Advanced Wireless Communications Networks and Systems Bit Error Rate of STE (continue) Conditional PDF of y R m(k) given s m (k d) = is a Gaussian mixture where ȳ (q,) Rm p m (y R ) = N sb N sb q= p πσ n wm Hw e m y R ȳ (q,) «Rm σ n wh m w m Y() Rm and N sb = N s / is the number of points in Y () Rm Thus BER of the mth detector associated with the detector s weight vector w m is given by P E (w m ) = N sb N sb q= Q g (q,) (w m ) where Q(u) = π Z u e v d v and g (q,) (w m ) = sgn(s(q) m,d )ȳ(q,) Rm p σ n w H m w m Note that BER is invariant to a positive scaling of w m Alternatively, the BER may be calculated based on the other subset Y ( ) Rm. 387

8 ELEC64 Advanced Wireless Communications Networks and Systems Minimum Bit Error Rate Solution MBER solution for the mth STE detector is defined as w (MBER)m = arg min wm No closed-form solution, but gradient of P E (w m ) is P E (w m ) P E (w m ) = N sb πσn p w H m w m N sb q= e ȳ (q,) «Rm σ n wh m w msgn s (q) ȳ(q,) Rm w m m,d wm Hw m x (q,)! Gradient optimisation can be applied to obtain a w (MBER)m Adaptive implementation using LBER algorithm w m (k ) = w m (k) µ sgn(s m(k d)) e y Rm (k) ρ n x(k) πρ n where µ is adaptive gain, and ρ n kernel width 388

9 ELEC64 Advanced Wireless Communications Networks and Systems Simulation Results: Stationary System CIRs of 3-user 4-antenna stationary system C l,m (z) m = m = m = 3 l = ( 0.5 j0.4) (0.7 j0.6)z ( 0. j0.) (0.7 j0.6)z ( 0.7 j0.9) (0.6 j0.4)z l = (0.5 j0.4) ( 0.8 j0.3)z ( 0.3 j0.5) ( 0.7 j0.9)z ( 0.6 j0.8) ( 0.6 j0.7)z l = 3 (0.4 j0.4) ( 0.7 j0.8)z ( 0. j0.) (0.7 j0.6)z (0.3 j0.5) (0.9 j0.)z l = 4 (0.5 j0.5) (0.6 j0.9)z ( 0.6 j0.4) (0.9 j0.4)z ( 0.6 j0.6) (0.8 j0.0)z CIR order n C =, STE order n F = 3 and decision delay d = BER comparison of MMSE/MBER and LMS/LBER for three users 0 - LMS() MMSE() LBER() MBER() 0 - LMS() MMSE() LBER() MBER() 0 - LMS(3) MMSE(3) LBER(3) MBER(3) log0(bit Error Rate) log0(bit Error Rate) log0(bit Error Rate) SNR (db) SNR (db) SNR (db) 389

10 ELEC64 Advanced Wireless Communications Networks and Systems Simulation Results: Fading System 3 users, 4 receive antennas, and Rayleigh fading channels with each of CIRs having n C = 3 taps Each channel tap has root mean power of 0.5 j 0.5 Normalised Doppler frequency for simulated system was 0 5, which for a carrier of 900 MHz and a symbol rate of 3 Msymbols/s corresponded to a user velocity of 0 m/s (36 km/h) STE order n F = 5 and decision delay d = Frame structure: 50 training symbols followed by 450 data symbols BER comparison of LMS/LBER for three users 0 0 LMS() LBER() 0 0 LMS() LBER() 0 0 LMS(3) LBER(3) BER BER BER Average SNR (db) Average SNR (db) Average SNR (db) 390

11 ELEC64 Advanced Wireless Communications Networks and Systems Diversity We now consider diversity gain aspect of MIMO Transmit diversity: assume Two transmit antennas, which are sufficiently apart One receive antenna Two channel estimates are available at transmitter Receive diversity: assume One transmit antenna Two receive antennas, which are sufficiently apart Two channel estimates are available at receiver channel estimate x ML detector x h * h * h n Transmit Diversity y h channel estimate n h channel estimate Receive Diversity h x y h ML detector h * * Transmit diversity order of two: two transmit signals are h x and h x, and receive signal is y = h h x h h x n = ` h h x n Receive diversity order of two: optimal combined signal of two receive signals is y = h `h x n h `h x n = ` h h x n n channel estimate 39

12 ELEC64 Advanced Wireless Communications Networks and Systems G Space-Time Block Code Alamouti s G space-time block code uses two transmitter antennas and one receiver antenna In time slot (one symbol period), two symbols (x,x ) are transmitted While in time slot, transformed (x,x ), i.e. ( x,x ), are transmitted Assume narrowband channels with channel, h = h e jα and channel, h = h e jα Antenna spacing is sufficiently large, e.g. 0 wavelengths, so two channels are independently faded Fading is sufficiently slow so during two time slots channels h, h are unchanged n n x x x * x h * Linear Combiner ~ x x~ Maximum Likelihood Detector h y =h x h x n y = h x * h x * n h h Time slot x^ Time slot ^x Channel Esimator 39

13 ELEC64 Advanced Wireless Communications Networks and Systems G STBC (continue) Received signals at two time slots are respectively y = h x h x n y = h x h x n Assume perfect channel estimate h,h, linear combiner s outputs are x = h y h y = ( h h )x h n h n x = h y h y = ( h h )x h n h n Maximum likelihood decoding involves minimising decision metric x ( h h )x for decoding x and minimising decision metric x ( h h )x for decoding x 393

14 ELEC64 Advanced Wireless Communications Networks and Systems Space-Time Block Codes Encoding: generic STBC is defined by n p transmission matrix 3 g g g p x, x, x,p G = 6 g g g p = 6 x, x, x,p 4... g n g n g np x n, x n, x n,p Each entry g ij = x i,j is a linear combination of k input symbols x, x, x k and their conjugates Number of rows n is equal to number of time slots, and number of columns is equal to number of transmit antennas During time slot i, encoded symbols x i,, x i,,, x i,p are transmitted simultaneously from transmit antennas,,, p, respectively Code rate is obviously R = k/n Assume L receiver antennas, and channel connecting jth transmit antenna and lth receiver antenna is h j,l, then received signal arriving at receiver l during time slot i is p y i,l = h j,l x i,j n j,l where n j,l is AWGN for j, l-th channel ML detector or suboptimal low-complexity detector can be employed j=

15 ELEC64 Advanced Wireless Communications Networks and Systems Space-Time Block Codes (continue) Decoding: assuming perfect channel estimate, maximum likelihood decoding decides in favour of specific entry x i,j, i n, j p, that minimises the decision metric n i= L l= yi,l p h j,l x i,j j= An alternative is maximum a posteriori probability decoding, for details see relevant reference STBC examples (transmit antennas p =, 3, 4) G =» x x x x, G 3 = 6 4 x x x 3 x x x 4 x 3 x 4 x x 4 x 3 x x x x 3 x x x 4 x 3 x 4 x x 4 x 3 x 3, G 4 = x x x 3 x 4 x x x 4 x 3 x 3 x 4 x x x 4 x 3 x x x x x 3 x 4 x x x 4 x 3 x 3 x 4 x x x 4 x 3 x x G has time slots n =, G 3 and G 4 have time slots n = 8 395

16 ELEC64 Advanced Wireless Communications Networks and Systems STBC Examples (continue) STBC examples (transmit antennas p = 3, 4) H 3 = 6 4 x x x 3 x x x 3 x 3 x 3 x x x x x 3 x 3 x x x x 3, H 4 = x x x 3 x 3 x x x 3 x 3 x x x x x 3 x 3 x x x x x 3 x 3 x x x x x x x x H 3 and H 4 have time slots n = 4 Parameters of space-time block codes space-time code rate number of number of number of block code R transmitters p input symbol k time slots n G G 3 / G 4 / H 3 3/ H 4 3/

17 ELEC64 Advanced Wireless Communications Networks and Systems Summary Multiuser capacity of SDMA systems Space-time equalisation assisted multiuser detection for SDMA systems MMSE design and MBER design, adaptive implementation Diversity order, and space-time block codes 397