Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Waterfilling at high SNR N 0 λ 2 k ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ������������� ������������� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ������������� ������������� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ������������� ������������� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ������������� ������������� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ������������� ������������� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� P ∗ k ≈ P ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� n ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ... n k 1 2 3 15
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Waterfilling at low SNR ➮ At low SNR, the water is shallow. Then � k = argmax λ 2 P, P ∗ k k = 0 , else � P � � � 1 + P λ 2 λ 2 ≈ · log 2 ( e ) C = log 2 max max N 0 N 0 ➮ MIMO provides an array gain (power gain of λ 2 max ) but no DoF gains. ➮ Channel rank does not matter, only power matters. ➮ Transmit one beam in the direction associated with largest λ k ➮ Knowing H is very important! (to select what beam to use) 16
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Waterfilling at low SNR N 0 λ 2 k 4 = P P ∗ µ ... n k 1 2 3 17
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing In practice ➮ Feedback of channel state information, requires quantization ➮ Potentially, by scheduling only “good” users, one may always operate at high SNR ➮ Selection of modulation scheme — e.g., M -QAM per subchannel, different M — better channel, larger constellation — should be done with outer code in mind ➮ Imperfect CSI ➠ cross-talk! 18
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing MIMO channel models ➮ MIMO channel modeling is a rich research field, with both empirical (measurement) work and theoretical models. ➮ We will explore the main underlying physical phenomena of MIMO propagation and how they connect to the DoF concept. 19
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Line-of-sight SIMO channel ➮ Consider m -ULA at TX and RX, wavelength λ = c/f c , ant. spacing ∆ ∆ RX array φ TX 1 e − j 2 π λ ∆ cos φ ➮ Let u ( φ ) � . . . e − j ( m − 1) 2 π λ ∆ cos φ ➮ Signal from point source impinging on RX array (large TX-RX distance): 20 y = α u ( φ ) · s + e , ( α ∈ C , dep. on distance )
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Line-of-sight MIMO channel φ r RX array φ t TX array H ➮ MIMO channel: y = α · u ( φ r ) · · x + e , u ( φ t ) n = rank( H ) = 1 � �� � � �� � n r × 1 n t × 1 � �� � H ➮ The LoS-MIMO channel has rank one, so no DoF gain! 21
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Lobes and resolvability ➮ Consider unit-power point sources at φ 1 , φ 2 with sign. u ( φ 1 ) , u ( φ 2 ) . How similar do these signatures look? 1 1 s 2 · 1 m � s 1 u ( φ 1 ) − s 2 u ( φ 2 ) � 2 = 2 − 2 Re s ∗ m u H ( φ 1 ) u ( φ 2 ) � �� � |·| = f ( · ) where the lobe pattern f (cos( φ 1 ) − cos( φ 2 )) � 1 m | u H ( φ 1 ) u ( φ 2 ) | . ➮ If f ( · ) < 1 , then φ 1 , φ 2 resolvable. ➮ Resolvability criterion: | cos φ 1 − cos φ 2 | ≥ 2 π A � ( m − 1)∆ A , ➮ Grating lobes avoided if ∆ ≤ λ 2 ⇒ A ≤ ( m − 1) λ 2 . 22
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Two separated point sources and an m -receive-array TX2 φ 1 φ 2 RX array TX1 ➮ Define H = [ h 1 h 2 ] , h i = α i u ( φ i ) � ➮ Condition number κ ( H ) = λ max ( H ) 1 + f (cos( φ 1 ) − cos( φ 2 )) λ min ( H ) = 1 − f (cos( φ 1 ) − cos( φ 2 )) ➮ κ ( H ) is small if f ( · · · ) � = 1 ⇔ φ 1 , φ 2 resolvable ⇔ | cos φ 1 − cos φ 2 | > 2 π A 23
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing MIMO with two plane scatters A RX array B TX array ➮ Here, H TX-RX = H AB-RX · H TX-AB ➮ We have rank ( H TX-RX ) = 2 only if rank ( H AB-RX ) = 2 and rank ( H TX-AB ) = 2 ➮ For H to offer 2 DoF, A and B must be sufficiently separated in angle, 24 as seen both from TX and RX
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Angular decomposition of MIMO channel ➮ For φ 1 , φ 2 , ..., φ m define 1 U � √ m [ u ( φ 1 ) · · · u ( φ m )] ➮ Can show: With cos( φ k ) = k/m , { u ( φ k ) } forms ON-basis. Then U H U = I . ➮ Let U r and U t be the U matrices associated with the TX and RX arrays. Note that U H r U r = I and U H t U t = I ➮ If ∆ = λ/ 2 , then u ( φ i ) correspond to simple, perfectly resolvable beams, with a single mainlobe. ➮ We assume ∆ = λ/ 2 from now on. The case of ∆ � = λ/ 2 is more involved. 25
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Angular decomposition, cont. 5 4 5 3 4 2 3 RX 1 2 1 26 TX
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Angular decomposition, cont. ➮ Now define H a � U H r HU t ⇒ H a, ( k,l ) = u H ( φ r k ) Hu ( φ t l ) �� � u H ( φ r α i u ( φ ′ r i ) u H ( φ ′ t u ( φ t k ) i ) l ) i � �� � physical model � u H ( φ r k ) u ( φ ′ r u H ( φ ′ t i ) u ( φ t · = α i i ) l ) � �� � � �� � i =0 unless φ ′ r i falls in lobe φ r =0 unless φ ′ t i falls in lobe φ t k l ➮ Elements of H a correspond to different propagation paths ➮ H a, ( k,l ) =gain of ray going out in TX lobe l and arriving in RX lobe k 27
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Angular decomposition, key points ➮ “Rich scattering” if all angular bins filled ( H a has “no zeros”) ➮ “Diversity order” = measure of error resilience = number of propagation paths = number of nonzero elements in H a ➮ Number of DoF = rank( H ) = rank( H a ) ➮ If H a, ( k,l ) are i.i.d. then H k,l are i.i.d. ➮ With i.i.d. H a and many terms in � , then we get i.i.d. Rayleigh fading. 28
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Fast fading, no CSI at TX ➮ Each codeword spans ∞ number of H ➮ The V-BLAST architecture is optimal here Note: Reminiscent of architecture for slow fading and full CSI@TX ➮ Transmit vectors x = Q ˜ x where ˜ x 1 , ..., ˜ x n are independent streams with powers P k and rates R k 29
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing V-BLAST architecture e 1 x 1 y 1 x 1 ˜ x 2 x 2 ˜ y 2 optimal Q H receiver e n r x n t y n r x n ˜ 30
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing P 1 0 · · · 0 . ... . 0 P 2 . Q H ➮ Transmit covariance: K x � cov ( x ) = Q . ... ... . . 0 · · · 0 0 P n ➮ Achievable rate, for fixed H : � � � I + 1 � � HK x H H R = log 2 � � N 0 � ➮ Intuition: Volume of noise ball is | N 0 I | N . Volume of signal ball is | HK x H H + N o I | N . 31
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing ➮ Fast fading, coding over ∞ number of H matrices gives ergodic capacity � � � � � I + 1 � � HK x H H C = E log 2 � � N 0 � ➮ Choose Q and P k to � � � � � I + 1 � � HK x H H max E log 2 � � N 0 � K x , Tr ( K x ) ≤ P ➮ Optimal K x depends on the statistics of H 32
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing ➮ In i.i.d. Rayleigh fading, K ∗ x = P n t I (i.i.d. streams) and � � � � n � � �� � I + P 1 1 + SNR � � � HH H λ 2 C = E log 2 = E log 2 � � k N 0 n t n t � k =1 where n = rank ( H ) = min( n r , n t ) SNR � P N 0 { λ k } are the singular values of H ➠ Antennas then transmit separate streams. ➠ Coding across antennas is unimportant. 33
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Some special cases ➮ SISO: n t = n r = 1 � � log 2 (1 + SNR | h | 2 ) C = E At high SNR, the loss is -0.83 bpcu relative to AWGN channel ➮ SIMO: n t = 1 (power gain relative to SISO) � � �� n r � | h k | 2 C = E log 2 1 + SNR k =1 ➮ MISO: n r = 1 (no power gain relative to SISO) � � �� n t 1 + SNR � | h k | 2 C = E log 2 n t k =1 34
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Large arrays (infinite apertures) ➮ Large MISO ( n t TX, 1 RX) becomes AWGN channel: � � �� n t 1 + SNR � | h k | 2 → log 2 (1 + SNR ) C = E log 2 n t k =1 ➮ Large SIMO (1 TX, n r RX) � � �� n r � | h k | 2 C = E log 2 1 + SNR ≈ log 2 ( n r SNR ) = log 2 ( n r )+log 2 ( SNR ) k =1 ➮ Large square MIMO ( n t TX, n r RX, n r = n t = n ): Linear incr. with n : � � � � � � 4 1 1 t − 1 C ≈ n · log 2 (1 + t · SNR ) dt 35 π 4 0
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Fast fading, no CSI at TX, high SNR ➮ Here n � � �� 1 + SNR � λ 2 C = E log 2 ≈ n log 2 ( SNR ) + const k n t k =1 ➮ Both n r and n t must be large to provide DoF gain ➮ “Capacity grows as min ( n r , n t ) ” 36
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Fast fading, no CSI at TX, low SNR ➮ Here n n � � �� 1 + SNR ≈ log 2 ( e ) · SNR � � λ 2 E [ λ 2 C = E log 2 · k ] k n t n t k =1 k =1 = log 2 ( e ) · SNR · E [ || H || 2 ] = log 2 ( e ) · n r · SNR n t � �� � = n r n t ➮ Capacity independent of n t ! ➮ No DoF gain. All what matters here is power ➮ Relative to SISO, a power gain of n r (array/beamforming gain) ➮ Multiple TX antennas do not help here 37 (but with CSI at TX, things are very different)
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing V-BLAST in practice ➮ Transmitter architecture “simple” but the receiver must separate the streams ➠ major challenge ➮ Problems are conceptually similar to uplink MUD in CDMA and to equalization for ISI channels ➮ Stream-by-stream receivers: Successive-interference-cancellation ➠ MMSE-SIC is theoretically optimal but suffers from error propagation ➠ Rate allocation necessary ➮ Iterative architectures ➠ Iteration between outer code and demodulator ➠ Demodulator design is major problem 38 ➮ Receivers for MIMO to be discussed more in lecture 3
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Fast fading, full CSI at TX ➮ The transmitter can do waterfilling over both space and time ➮ Parallel channels: ˜ y k [ m ] = λ k [ m ]˜ x k [ m ] + ˜ e k [ m ] ➠ Waterfilling over space ( k ) and time ( m ). ➠ Optimal powers P ∗ k [ m ] ➠ Capacity n � � �� 1 + P ∗ ( λ k ) λ 2 � k C = E log 2 N 0 k =1 39
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing ➮ High SNR: P ∗ ( λ k ) ≈ P n (equal power) n � � �� 1 + SNR � n λ 2 C ≈ E log 2 , n D.o.F. k k =1 An SNR gain (compared to no CSI) of n t min( n t , n r ) = n t n t if n t ≥ n r n = , n r ➮ Low SNR: Even larger gain, so here multiple antennas do help! 40
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Slow fading, no CSI at TX � N 0 HK x H H � � 1 � ➮ Reliable communication for fixed H if log 2 � I + � > R � N 0 HK x H H � � � � 1 � ➮ Outage probability, for fixed R : P out = P log 2 � I + � < R ➮ Optimal K x as function of H ’s statistics: � � � � � I + 1 � � K ∗ HK x H H x = argmin P log 2 � < R � � N 0 K x , Tr K x ≤ P For H i.i.d. Rayleigh fading: ➠ K ∗ x = P n t I optimal at large SNR n ′ diag { 1 , ..., 1 , 0 , ..., 0 } at low SNR ( n ′ < n t ) ➠ K ∗ x = P 41
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing ➮ Notion of diversity : P out behaves as SNR − d where d =diversity order ➮ Maximal diversity: d = n r n t ➮ To achieve diversity, we need coding across streams ➮ V-BLAST does not work here. Each stream has diversity at most n r , while the channel offers n r n t ➮ Architectures for slow fading: ➠ Theoretically, D-BLAST is optimal ➠ Pragmatic approaches include STBC combined with FEC 42
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Example: Outage probability at rate R = 2 bpcu n t =1, n r =1 (SISO) n t =2, n r =1 n t =1, n r =2 −1 10 n t =2, n r =2 −2 10 FER −3 10 −4 10 −5 0 5 10 15 20 25 30 35 40 43 SNR
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing D-BLAST architecture epl x B (1) x B (2) x A (1) x A (2) x A (3) ➮ Decoding in steps: 1. Decode x A (1) 2. Decode x B (1) , suppressing x A (2) via MMSE 3. Strip off x B (1) , and decode x A (2) 4. Decode x B (2) , suppressing x A (3) via MMSE ➮ One codeword: x ( i ) = [ x A ( i ) x B ( i )] ➮ Requires appropriate rate allocation among x A ( i ) , x B ( i ) ➮ In practice, error propagation and rate loss due to initialization 44
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Le 2: Low-complexity MIMO 45
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Antenna diversity basics ➮ Recall transmission model (single time interval): y 1 h 1 , 1 · · · h 1 ,n t x 1 e 1 . . . . . . . . . . . . . . . = + y n r h n r , 1 · · · h n r ,n t x n t e n r � �� � � �� � � �� � � �� � y (RX data) H (channel) x (TX data) e (noise) ➮ Transmission model ( N time intervals): � y 1 , 1 � h 1 , 1 � x 1 , 1 � e 1 , 1 � � � � · · · x 1 ,N · · · y 1 ,N · · · h 1 ,nt · · · e 1 ,N . . . . . . . . = + . . . . . . . . . . . . . . . . xnt, 1 · · · xnt,N ynr, 1 · · · ynr,N hnr, 1 · · · hnr,nt enr, 1 · · · enr,N � �� � � �� � � �� � � �� � Y H E X ∈X “code matrix” 46
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Introduction and preliminaries ➮ Transmitting with low error probability at fixed rate requires N large. ➮ For practical systems, it is often of interest to design short space-time blocks (small N ) with good error probability performance. Outer FEC can then be used over these blocks. ➮ Throughout, we will assume Gaussian noise, e ∼ N (0 , N 0 I ) Usually, we assume i.i.d. Rayleigh fading, H i,j i.i.d. N (0 , 1) Sometimes, for SIMO/MISO, we take h ∼ N ( 0 , Q ) 47
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Receive diversity ( n t = 1 ) ➮ Suppose s transmitted, and h known at RX. ➮ Receive: y = h s + e ➮ Detection of s via maximum-likelihood (in AWGN): s � h H y � � 2 � y − h s � 2 = ... = � h � 2 · � � � s − ˆ s + const. , where ˆ � � h � 2 ➮ MRC+scalar detection problem! ➮ Distribution of ˆ s determines performance: � � s � h H y SNR | h = � h � 2 s, N 0 = � h � 2 · P � | s | 2 � � h � 2 ∼ N · E ˆ , � h � 2 N 0 N 0 � �� � ���� = P SNR 48
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Diversity order ( n × 1 fading vector h ) �� � SNR · � h � 2 ➮ P ( e | h ) = Q and h ∼ N ( 0 , Q ) (SNR up to a constant) � � n − 1 � � − 1 � I + SNR 1 + SNR � � � ➮ Then P ( e ) = E [ P ( e | h )] ≤ 2 Q = 2 λ k ( Q ) � � � k =1 � SNR � − rank( Q ) 1 ➮ As SNR → ∞ , P ( e ) ≤ · � rank( Q ) 2 λ k ( Q ) k =1 ➮ Diversity order d � − log P ( e ) log SNR = rank( Q ) � n � 1 /n n ≤ 1 λ k = 1 n Tr { Q } = 1 � � nE [ � h � 2 ] ➮ Note that λ k n k =1 k =1 49 with eq. if λ 1 = · · · = λ n so Q ∝ I minimizes the bound on P ( e )
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Diversity order P ( e ) d = 1 10 − 1 d = 2 d = 3 10 − 2 10 − 3 10 − 4 10 − 5 0 10 20 30 40 50 SNR 50
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Transmit diversity, H known at transmitter ➮ Try transmit w · s where w is function of H ! (as we did in Le 1) ➮ RX data is y = Hw s + e and optimal decision minimizes the ML metric: � y − Hw s � 2 = ... = � Hw � 2 · | s − ˆ s | 2 + const. s � w H H H y � � N 0 � Hw � 2 ∼ N where ˆ s, � Hw � 2 ➮ The SNR | H in ˆ s is max for w =normalized dominant RSV of H � H � 2 � | s | 2 � � | s | 2 � N 0 λ max ( H H H ) 1 1 ➮ Resulting SNR | H = · E ≥ · E . N 0 n t � �� � ≥� H � 2 /n t ➮ Diversity order: d = n r n t � h � so SNR | h = � h � 2 � | s | 2 � h ∗ 51 ➮ For n r = 1 take w opt = N 0 E (same as for RX-d)
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Transmit diversity, H unknown at transmitter ➮ From now on, TX does not know H ! ➮ Consider Y = HX + E . Optimal receiver in AWGN ( H known at RX): X � Y − HX � 2 X P ( X | Y , H ) ⇔ min max �� � � H ( X 0 − X ) � 2 ➮ Pairwise error probability P ( X 0 → X | H ) = Q 2 N 0 ➮ Consider P ( X 0 → X ) = E [ P ( X 0 → X | H )] . For i.i.d. Rayleigh fading, � ( X 0 − X )( X 0 − X ) H � 1 − n r � � P ( X 0 → X ) ≤ � I + � 4 N 0 � �� � ” − d − d “ 1 ∼ SNR ∼ N 0 d =“diversity order”. Note: d ≤ n r n t and d = n r n t if X 0 − X full rank 52
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Linear space-time block codes (STBC) ➮ STBC maps n s complex symbols onto n t × N matrix X : { s 1 , . . . , s n s } → X ➮ Linear STBC: n s � X = (¯ s n A n + i ˜ s n B n ) n =1 where { A n , B n } are fixed matrices ➮ Typically N small. Need N ≥ n t for max diversity (why?) ➮ Rate: R � N bits/channel use n s 53
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing STBC with a single symbol ➮ Transmit one symbol s during N time intervals, weighted by W : X = W · s, Y = HX + E = HW s + E ➮ Average error probability in Rayleigh fading: � 1 � − n r n t P ( s 0 → s ) ≤ | W W H | − n r | s − s 0 | − 2 n r n t 4 N 0 ➮ What is the optimum W ? Try to maximize: | W W H | max W = � W � 2 ≤ 1 � W W H � s.t. Tr (power constraint) ➮ Solution: W W H = 1 n t I , (antenna cycling). Diversity but rate 1 /N ! 54
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Alamouti scheme for n t = 2 Time 1 Time 2 � � √ √ s ∗ s 1 1 2 s ∗ ➮ X = . That is: Ant 1 s 1 / 2 2 / 2 √ − s ∗ √ √ s 2 2 1 − s ∗ Ant 2 s 2 2 1 / 2 � y 1 � � h 1 s 1 + h 2 s 2 � � e 1 � 1 ➮ RX data: = + √ h 1 s ∗ 2 − h 2 s ∗ y 2 e 2 2 1 � � � � � � � � � � � � h 1 s 1 + h 2 s 2 s 1 y 1 e 1 h 1 h 2 e 1 1 1 ➮ Consider = + = + √ √ h ∗ 1 s 2 − h ∗ − h ∗ h ∗ y ∗ e ∗ e ∗ 2 s 1 s 2 2 2 2 2 2 1 2 ➮ ML detector � � 2 � � � � � � � � � � � � − 1 s 1 y 1 h 1 h 2 � � √ min − h ∗ h ∗ y ∗ � � s 2 s 1 ,s 2 2 � 2 2 1 � � �� � � �� � ���� � � y s � � � G 55
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing � � � � h H − h T h 1 h 2 = � h 1 � 2 + � h 2 � 2 ➮ Observation: G H G = 1 1 2 I − h ∗ h ∗ h H h T 2 2 2 1 2 1 G H y ➮ Hence min � y − Gs � 2 ⇔ min � ˆ s − s � 2 , ˆ s = 2 � h 1 � 2 + � h 2 � 2 ➮ Distribution of ˆ s : � � � h 1 � 2 + � h 2 � 2 = 2 G H ( Gs + e ) G H y s , 2 N 0 ˆ s = 2 � h 1 � 2 + � h 2 � 2 ∼ N � H � 2 I ➮ SNR | H = � H � 2 For 2 × 1 system, 3 dB less than 1 × 2 with MRC . 2 N 0 ➮ Diversity order: 2 n r 56
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Overview of 2-antenna systems Method SNR rate TX knows h 1 , h 2 | h 1 | 2 + | h 2 | 2 1 TX, 2 RX, MRC 1 no N 0 | h 1 | 2 + | h 2 | 2 2 TX, 1 RX, BF 1 yes N 0 | h 1 | 2 + | h 2 | 2 2 TX, 1 RX, ant. cycl. 1/2 no 2 N 0 | h 1 | 2 + | h 2 | 2 2 TX, 1 RX, Alamouti 1 no 2 N 0 ≥ | h 1 | 2 + | h 2 | 2 2 TX, 1 RX, Ant. sel. 1 partly 2 N 0 ➮ For antenna selection, note that | h 1 | 2 + | h 2 | 2 max | h n | 2 ≥ 1 N 0 2 N 0 57
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing 2-antenna systems, cont P ( e ) 1 RX, 1 TX 10 − 1 1 RX, 2 TX, Alamouti 1 RX, 2 TX, antenna selection 10 − 2 2 RX, 1 TX, MRC or 1 TX, 2 RX, BF 10 − 3 10 − 4 10 − 5 0 10 20 30 40 50 58 SNR
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Orthogonal STBC (OSTBC) ➮ Important special case of linear STBC: n s n s � � | s n | 2 · I = � s � 2 · I XX H = X = (¯ s n A n + i ˜ s n B n ) for which n =1 n =1 Notation: ¯ ( · ) =real part, ˜ ( · ) =imaginary part ➮ This is equivalent to requiring for n = 1 , . . . , n s , p = 1 , . . . , n s A n A H n = I , B n B H n = I A n A H p = − A p A H B n B H p = − B p B H n , n , n � = p A n B H p = B p A H n 59
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Proof ➮ To prove ⇒ ), expand: n s n s � � XX H = s p B p ) H (¯ s n A n + i ˜ s n B n )(¯ s p A p + i ˜ n =1 p =1 n s � s 2 n A n A H s 2 n B n B H = (¯ n + ˜ n ) n =1 n s n s � � � � s p ( A n A H p + A p A H s p ( B n B H p + B p B H + s n ¯ ¯ n ) + ˜ s n ˜ n ) n =1 p =1 ,p>n n s n s � � s p ( B n A H p − A p B H + i s n ¯ ˜ n ) n =1 p =1 ➮ Proof of ⇐ ), see e.g., EL&PS book. 60
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Some properties of OSTBC ➮ Manifests the intuition that unitary matrices are good √ ➮ Alamouti code is an OSTBC (up to 1 / 2 normalization) ➮ Pros ➠ Diversity of order n r n t ➠ Detection of { s n } is decoupled ➠ Converts space-time channel into n s AWGN channels ➠ Combination with outer coding is straightforward ➮ Cons ➠ Rate loss for n t > 2 , i.e., n t > 2 ⇒ R = n s N < 1 ➠ Information loss except for when n t = 2 , n r = 1 61
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Diversity order of OSTBC n } n s ➮ Suppose { s 0 n =1 are true symbols and { s n } are any other symbols. Then n s � � � s 0 s 0 X − X 0 = (¯ s n − ¯ n ) A n + i (˜ s n − ˜ n ) B n n =1 n s � ( X − X 0 )( X − X 0 ) H = n | 2 · I | s n − s 0 ⇒ n =1 ➮ Full rank ➠ full diversity for i.i.d. Gaussian channel 62
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Derivation of decoupled detection ➮ Write the ML metric as � Y − HX � 2 = � Y � 2 − 2 ReTr � � Y H HX + � HX � 2 n s n s � � = � Y � 2 − 2 � � � � Y H HA n Y H HB n ReTr s n + 2 ¯ Im Tr s n ˜ n =1 n =1 + � H � 2 · � s � 2 n s � s n + | s n | 2 � H � 2 � � � � � � Y H HA n Y H HB n − 2 ReTr = s n + 2 Im Tr ¯ ˜ n =1 + const. � � 2 � � � � n s Y H HA n Y H HB n − i Im Tr � s n − ReTr � � � = � H � 2 · � � + const. � � � H � 2 63 � n =1
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Decoupled detection, again s ′ � [¯ y = F s ′ + e , s T ] T s T ➮ Linearity: Y = HX + E ⇔ ˜ ➮ Theorem: X is an OSTBC if and only if = � H � 2 · I � � F H F ∀ Re H ➮ ML metric: � y − F s ′ � 2 = � y � 2 − 2 Re � y H F s ′ � � s ′ T F H F s ′ � + Re = � y � 2 − 2 Re + � H � 2 · � s ′ � 2 � y H F s ′ � = � H � 2 · � s ′ − ˆ s ′ � 2 + const. �� � � � � ˆ � � F H y = Re ¯ , N 0 / 2 s s ¯ s ′ � ∼ N where ˆ � H � 2 I ˆ s ˜ ˜ � H � 2 s 64 ➮ F is a “Spatial/temporal (code) matched filter”
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Interpretation of decoupled detection ➮ Space-time channel decouples into n s AWGN channels AWGN signal 1 AWGN signal 2 AWGN signal ns (a) n t × n r space-time channel (b) n s independent AWGN channels · � H � 2 ➮ SNR per subchannel: SNR | H = N · P 65 n s n t N 0
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Example: Alamouti’s code is an OSTBC ➮ Consider the Alamouti code (re-normalized): � � s ∗ s 1 XX H = ( | s 1 | 2 + | s 2 | 2 ) I 2 X = , − s ∗ s 2 1 ➮ Identification of A n and B n gives � � � � 1 0 0 1 A 1 = A 2 = 0 − 1 1 0 � � � � 1 0 0 − 1 B 1 = B 2 = 0 1 1 0 66
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Examples of OSTBC ➮ Best known OSTBC for n t = 3 , N = 4 , n s = 3 : − s 3 s 1 0 s 2 s ∗ s ∗ 0 s 1 X = 3 2 − s ∗ s ∗ − s 3 0 2 1 Code rate: 3/4 bpcu ➮ For n t = 4 , N = 4 , n s = 3 : s 1 0 s 2 − s 3 s ∗ s ∗ 0 s 1 3 2 X = − s ∗ s ∗ − s 3 0 2 1 s ∗ s ∗ − s 2 0 3 1 Rate is 3 / 4 bpcu. 67
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Summary of OSTBC Relations n s � XX H = | s n | 2 · I = � s � 2 · I n =1 � A n A H B n B H = , = I I n n A n A H − A p A H B n B H − B p B H n � = p = , = n , p n p A n B H B p A H = p n � � � = � H � 2 · I F H F Re where F is such that � ¯ � s vec ( Y ) = F · + vec ( E ) ˜ s 68
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Mutual Information Properties of OSTBC ➮ Average transmitted energy per antenna and time interval = 1 /n t ➮ Channel mutual information, with i.i.d. streams of power 1 /n t : � � HH H � � � I + 1 � � C MIMO ( H ) = log � � n t N 0 � ➮ Mutual information of OSTBC coded system: � � � H � 2 C OSTBC ( H ) = n s 1 + N N log n s n t N 0 ➮ Theorem: C MIMO ≥ C OSTBC , equality only for n t = 2 , n r = 1 69
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Capacity comparison (1% outage) 1 10 Capacity [bits/sec/Hz] 0 10 Average capacity, 1 TX, 1 RX Outage capacity, 1 TX, 1 RX Average capacity, 2 TX, 2 RX Average capacity, 2 TX, 2 RX − OSTBC Outage capacity, 2 TX, 2 RX Outage capacity, 2 TX, 2 RX − OSTBC −1 10 0 5 10 15 20 25 30 35 40 SNR 70
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Non-orthogonal linear STBC ➮ Also called linear dispersion codes ➮ Different approaches: ➠ Optimization of mutual information between the TX & RX: � � �� � max 1 � I + 2 � F H F � � 2 E H log 2 Re � N 0 (no explicit guarantee for full diversity here) ➠ Quasi-orthogonal codes ➠ Codes based on linear constellation (complex-field) precoding s ′ = Φ s 71
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Example: A non-OSTBC ➮ Consider the following diagonal code , where | s n | = 1 : � � s 1 0 X = 0 s 2 ➮ Then � � � � � � � � 1 0 0 0 1 0 0 0 A 1 = , A 2 = , B 1 = B 2 = 0 0 0 1 0 0 0 1 ➮ ML metric for symbol detection: � Y − HX � 2 = � Y � 2 − 2 ReTr � � X H H H Y + � H � 2 � � � � [ H H Y ] 1 , 1 · s 1 [ H H Y ] 2 , 2 · s 2 = − 2 Re − 2 Re + const. 72 ➮ Decoupled detection, but not OSTBC, and not full diversity
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing More examples of linear but not orthogonal STBC ➮ Alamouti code with forgotten conjugates � � s 1 s 2 X = s 2 − s 1 ➮ “Spatial multiplexing” ( R = n t , N = 1 , n s = n t , d = n r ). For n t = 2 : � � � � 1 0 A 1 = , A 2 = 0 1 � � � � 1 0 B 1 = , B 2 = 0 1 73
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Linearly precoded STBC ➮ Transmit W X where W ∈ { W 1 , . . . , W K } . Data model: Y = HW X + E ➮ Fact: If rank { W k } = n t then same diversity order as but without W ➮ Consider correlated fading: R = E [ hh H ] = R T t ⊗ R r , h = vec( H ) ➮ Error probability: ˛ I + 1 − nr ( X 0 − X )( X 0 − X ) H · W H R t W ˛ ˛ E H [ P ( X 0 → X )] ≤ const. · ˛ ˛ N 0 ˛ � � � I + n s ➮ For OSTBC, ( X 0 − X )( X 0 − X ) H ∝ I . Hence, min W H R t W � � � N 0 W 74
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Ex. OSTBC with One-Bit Feedback for n t = 2 ➮ One bit used to choose between �� � | a | � � 0 1 − | a | 2 0 W 1 = , W 2 = � 1 − | a | 2 0 0 | a | � �� � � �� � if � h 1 � > � h 2 � if � h 2 � > � h 1 � ➮ Let P c be the probability that the feedback bit is correct ➮ For P c = 1 (reliable feedback), a = 1 is optimal ➠ antenna selection W may be multiplied with fixed unitary matrix ➠ grid of beams ➮ For P c < 1 (erroneous feedback), � P c | a | 2 + 1 − P c � 2 E H [ P ( X 0 → X )] ≤ SNR 2 · (for n r = 1 ) 1 − | a | 2 75
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing −1 10 Unweighted OSTBC. Optimal weighting (No feedback error). Optimal weighting with feedback error. Error tolerant weighting (No feedback error), Error tolerant weighting with feedback error −2 10 BER −3 10 −4 10 −5 10 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 SNR 76
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing MIMO with feedback - optimized transmission n y x s ˜ s W ( I ) U H Encoder Decoder ˜ I H I ( ˜ H ) ➮ Here ➠ U depends on long-term feedback ➠ I depends on short-term (few bits) feedback ➮ State-of-the art designs rely on vector quantization techniques 77
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Frequency-selective channels ➮ Maximum diversity order (with ML detection) will be n r n t L where L =length of CIR ➮ Variety of techniques to achieve maximum diversity ➮ Most widely used transmission technique is MIMO-OFDM ➠ coding across multiple OFDM symbols ➠ coding across subcarriers within one OFDM symbol ➮ Basic model per subcarrier is Y n = H n X n + E n 78
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Le 3: MIMO receivers 79
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Summary of MIMO receivers ➮ Optimal architectures (from Le 1): y = U H y , separates streams ➠ CSI@TX (any fading): linear processing, ˜ ➠ no CSI@TX, fast fading: V-BLAST, optimal receiver is more involved - linear receiver (channel inversion) is grossly suboptimal - successive interference cancellation (SIC) - using soft MIMO demodulator + decoder, possibly iterative ➠ no CSI@TX, slow fading: D-BLAST ➮ Architectures with STBC+outer FEC (from Le 2) ➠ With OSTBC, decoupled detection and thing are simple: � s 1 | 2 + | s 2 − ˆ s 2 | 2 � min � Y − HX � ∼ min | s 1 − ˆ ➠ With non-OSTBC, min � Y − HX � does not decouple - problem similar to for V-BLAST 80
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Theoretically optimal V-BLAST receiver based on SIC decoded stream 1 decoder 1 MMSE 1 y = Hs + e decoded stream 2 decoder 2 MMSE 2 decoded stream 3 decoder 3 MMSE 3 decoded stream n t decoder n t (MMSE n t ) ➮ Optimality only for fast fading. Requires rate allocation on streams. ➮ Major drawback: Requires long codewords. Prone to error propagation. 81
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Theoretically optimal D-BLAST receiver based on SIC x B (1) x B (2) x A (1) x A (2) x A (3) ➮ One codeword split as x ( i ) = [ x A ( i ) x B ( i )] , with rate allocation ➮ Decoding in steps: 1. Decode x A (1) 2. Decode x B (1) , suppressing x A (2) via MMSE 3. Strip off x B (1) , and decode x A (2) 4. Decode x B (2) , suppressing x A (3) via MMSE ➮ Drawbacks: ➠ error propagation ➠ rate loss due to initialization ➠ requires long codewords 82
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Receivers for linear STBC architectures Training Data 1 Data 2 ➮ Received block: Y = HX + E . ➮ X linear in { s 1 , ..., s n s } so with appropriate F , the ML metric is � � vec ( ¯ � � ¯ �� 2 Y ) s � Y − HX � 2 = � � − F � � vec ( ˜ ˜ Y ) s � � ➮ For OSTBC, F T F = � H � 2 I so detection decouples ➮ Spatial multiplexing (V-BLAST) can be seen a degenerated special case of this architecture, with s 1 . . X = . s n t 83
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Demodulator+decoder architectures soft output P ( b i | y ) y = Hs + e MIMO demodulator channel decoder a priori information P ( b i ) ➮ Demodulator computes P ( b i | y ) given a priori information P ( b i ) ➮ Decoder adds knowledge of what codewords are valid ➮ Added knowledge in decoder is fed back to demodulator as a priori ➮ Iteration until convergence (a few iterations, normally) 84
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing MIMO demodulation (hard) ➮ General transmission model, with G i,j ∈ R = G · + e , s k ∈ S y s ���� ���� ���� ���� m × n n × 1 m × 1 m × 1 ➮ Models V-BLAST architectures, and (non-O)STBC architectures ➮ Other applications: multiuser detection, ISI, crosstalk in cables, ... ➮ Typically, m ≥ n and G is full rank and has no structure. 85
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing The problem ➮ If e ∼ N ( 0 , σ I ) then the problem is to detect s from y s ∈S n � y − Gs � 2 , y ∈ R m , G ∈ R m × n min � is orthonormal ( Q T Q = I ) Q ∈ R m × n ➮ Let G = QL where L ∈ R n × n is lower triangular � � � � 2 2 Then � y − Gs � 2 = � QQ T ( y − Gs ) � ( I − QQ T )( y − Gs ) � � � � + � � � � � � 2 2 � Q T y − Ls � ( I − QQ T ) y � � � � = + � � s ∈S n � y − Gs � 2 y � Q T y ⇔ s ∈S n � ˜ y − Ls � so min min where ˜ 86
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Some remarks ➮ Integer-constrained least-squares problem, known to be NP hard ➮ Brute force complexity O ( |S n | ) ➮ Typical dimension of problem: n ∼ 8 − 16 , so |S| ∼ 2 – 8 , |S n | ∼ 256 – 10 14 ➮ Needs be solved ➠ in real time ➠ once per received vector y ➠ in power-efficient hardware (beware of heavy matrix algebra) ➠ possibly fixed-point arithmetics ➠ preferably, in a parallel architecture ➮ In communications, we can accept a suboptimal algorithm that finds the correct solution quickly, with high probability 87
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Some remarks, cont. ➮ For G ∝ orthogonal (OSTBC), the problem is trivial. ➮ Our focus is on unstructured G ➮ If G has structure (e.g., Toeplitz) then use algorithm that exploits this ➮ Generally, slow fading (no time diversity) is the hard case 88
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Zero-Forcing ➮ Let y − Ls � = L − 1 ˜ s � arg min ˜ s ∈ R n � y − Gs � = arg min s ∈ R n � ˜ y E.g., Gaussian elimination: ˜ s 1 = ˜ y 1 /L 1 , 1 y 2 − ˜ s 2 = (˜ ˜ s 1 L 2 , 1 ) /L 2 , 2 . . . s k ] � arg min ➮ Then project onto S : ˆ s k = [˜ s k ∈S | s k − ˜ s k | ➮ This works very poorly. Why? Note that s = s + L − 1 Q T e = s + ˜ e )= σ · ( L T L ) − 1 ˜ e , where cov( ˜ ZF neglects the correlation between the elements of ˜ e 89
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Decision tree view { f 1 ( s 1 ) + f 2 ( s 1 , s 2 ) + · · · + f n ( s 1 , . . . , s n ) } min { s 1 ,...,sn } s k ∈S � 2 � k � where f k ( s 1 , ..., s k ) � y k − ˜ L k,l s l l =1 root node s 1 = − 1 s 1 = +1 f 1 (1) = 5 f 1 ( − 1) = 1 1 5 s 2 = − 1 s 2 = +1 s 2 = − 1 s 2 = +1 f 2 ( − 1 , 1) = 1 f 2 (1 , 1) = 3 f 2 ( − 1 , − 1) = 2 f 2 (1 , − 1) = 2 3 2 7 8 s 3 = − 1 s 3 = +1 s 3 = − 1 s 3 = +1 s 3 = − 1 s 3 = +1 s 3 = − 1 s 3 = +1 f 3 ( · · · ) = 1 3 3 1 f 3 ( · · · ) = 4 4 1 9 leaves 7 4 5 6 10 8 9 17 90 { 1 , − 1 , − 1 } {− 1 , − 1 , − 1 } {− 1 , − 1 , 1 } {− 1 , 1 , − 1 } {− 1 , 1 , 1 } { 1 , − 1 , 1 } { 1 , 1 , − 1 } { 1 , 1 , 1 }
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Zero-Forcing with Decision Feedback (ZF-DF) ➮ Consider the following improvement � ˜ � y 1 i) Detect s 1 via: ˆ s 1 = = arg min s 1 ∈S f 1 ( s 1 ) L 1 , 1 � ˜ � y 2 − ˆ s 1 L 2 , 1 ii) Consider s 1 known and set ˆ s 2 = = arg min s 2 ∈S f 2 (ˆ s 1 , s 2 ) L 2 , 2 iii) Continue for k = 3 , ..., n : � � y k − � k − 1 ˜ l =1 L k,l ˆ s l s k = ˆ = arg min s k ∈S f k (ˆ s 1 , ..., ˆ s k − 1 , s k ) L k,k ➮ This also works poorly. Why? Error propagation. Incorrect decision on s i ➠ most of the following s k wrong as well. ➮ Optimized detection order (start with the best) does not help much. 91
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Zero-Forcing with Decision Feedback (ZF-DF) r1 5 1 1 5 1 3 2 2 3 2 7 8 1 3 3 1 4 1 9 4 6 10 9 17 7 4 5 8 92
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Sphere decoding (SD) ➮ Select a sphere radius, R . Then traverse the tree, but once encountering a node with cumulative metric > R , do not follow it down y − Ls � 2 ≤ R ➮ Enumerates all leaf nodes which lie inside the sphere � ˜ ➮ Improvements: ➠ Pruning: At each leaf, update R according to R := min( R, M ) ➠ Improvements: optimal ordering of s k ➠ Branch enumeration (e.g., s k = {− 5 , − 3 , − 1 , − 1 , 3 , 5 } vs. s k = {− 1 , 1 , − 3 , 3 , − 5 , 5 } ) ➮ Known facts: ➠ The algorithm solves the problem, if allowed to finish ➠ Runtime is random and algorithm cannot be parallelized ➠ Under relevant circumstances, average runtime is O (2 αn ) for α > 0 93
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing SD, without pruning, R = 6 r2 5 1 r2a 1 5 1 3 2 2 3 2 7 8 1 3 3 1 4 1 9 4 7 4 5 6 10 8 9 17 94
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing SD, with pruning, R = ∞ r3 5 1 r3a 1 5 1 3 2 2 3 2 7 8 1 3 3 1 4 1 9 4 7 4 5 6 10 8 9 17 95
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing “Fixed complexity” sphere decoding (FCSD) ➮ Select a user parameter r , 0 ≤ r ≤ n ➮ For each node on layer r , consider { s 1 , ..., s r } fixed and solve ( ∗ ) min { f r +1 ( s 1 , ..., s r +1 ) + · · · + f n ( s 1 , ..., s n ) } { sr +1 ,...,sn } s k ∈S ➮ Subproblem (*) solved using |S| r times ➮ Low-complexity approximation (e.g. ZF-DF) can be used. Why? (*) is overdetermined (equivalent G is tall) ➮ Order can be optimized: start with the “worst” ➮ Fixed runtime, fully parallel structure 96
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing FCSD, r = 1 r4 5 1 r4a 1 5 1 3 2 2 3 2 7 8 1 3 3 1 4 1 9 4 7 4 5 6 10 8 9 17 97
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Semidefinite relaxation (for s k ∈ {± 1 } ) − L T ˜ � � � � � � � L T L s s y � s T = s T s � S � ¯ Ψ � ➮ Let ¯ , s ¯ 1 , y T L 1 1 − ˜ 0 Then y − Ls � 2 = ¯ y � 2 = Trace { Ψ S } + � ˜ y � 2 s T Ψ ¯ � ˜ s + � ˜ so the problem is to min Trace { Ψ S } diag { S } = { 1 ,..., 1 } rank { S } =1 s n +1 =1 ¯ ➮ SDR proceeds by relaxing rank { S } = 1 to S positive semidefinite ➮ Interior point methods used to find S ➮ s recovered, e.g., by taking dominant eigenvector and project onto S n 98
Link¨ oping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing Lattice reduction ➮ Extend S n to lattice. For example, if S = {− 3 , − 1 , 1 , 3 } , S n = { . . . , − 3 , − 1 , 1 , 3 , . . . } × · · · × { . . . , − 3 , − 1 , 1 , 3 , . . . } . then ¯ ➮ Decide on orthogonal integer matrix T ∈ R n × n that maps ¯ S n onto itself: T s ∈ ¯ ∀ s ∈ ¯ S n S n T k,l ∈ Z , | T | = 1 , and ➮ Find one such T for which LT ∝ I s ′ � arg min s = T − 1 ˆ s ′ y − ( LT ) s ′ � 2 , and set ˆ ➮ Then solve ˆ S n � ˜ s ′ ∈ ¯ ➮ Critical steps: ➠ Find suitable T (computationally costly, but amortize over many y ) ∈ S n in general, so clipping is necessary s ∈ ¯ S n , but ˆ ➠ ˆ s / 99
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