Wireless Communication Systems @CS.NCTU Lecture 6: Multiple-Input - PowerPoint PPT Presentation

Wireless Communication Systems @CS.NCTU Lecture 6: Multiple-Input Multiple-Output (MIMO) Instructor: Kate Ching-Ju Lin ( 林靖茹 ) 1

Agenda • Channel model • MIMO decoding • Degrees of freedom • Multiplexing and Diversity 2

MIMO • Each node has multiple antennas ⎻ Capable of transmitting (receiving) multiple streams concurrently ⎻ Exploit antenna diversity to increase the capacity h 11   h 11 h 12 h 13 h 12 h 21 H N × M = h 21 h 22 h 23 h 31   h 22 h 31 h 32 h 33 h 32 h 13 N: number of antennas at Rx h 23 h 33 M: number of antennas at Tx H ij : channel from the j-th Tx antenna to the i-th Rx antenna … … 3

Channel Model (2x2) • Say a 2-antenna transmitter sends 2 streams simultaneously to a 2-antenna receiver h 11 x 1 y 1 h 21 h 12 x 2 y 2 h 22 Equations Matrix form: y = Hx + n y 1 = h 11 x 1 + h 12 x 2 + n 1 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ y 1 h 11 h 12 x 1 n 1 = + y 2 h 21 h 22 x 2 n 2 y 2 = h 21 x 1 + h 22 x 2 + n 2 4

MIMO (MxN) • An M-antenna Tx sends to an N-antenna Rx h 11 h 12 h 21 h 31 h 22 N-antenna M-antenna h 32 h 13 h 23 h 33 y = Hx + n … …         y 1 h 11 h 12 h 1 M x 1 n 1 · · · y 2 h 21 h 22 h 2 M x 2 n 2 · · ·         à . . . . ...          =  + . . . .         . . . .       y N h N 1 h N 2 h NM x M n N · · · 5

Antenna Space (2x2, 3x3) N-antenna node receives in N-dimensional space 3 x 3 2 x 2 ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ y 1 h 11 h 12 n 1 x 1 + x 2 + = y 2 h 21 h 22 n 2 y = ~ h 1 x 1 + ~ h 2 x 2 + ~ y = ~ h 1 x 1 + ~ ~ h 3 x 3 + ~ n ~ h 2 x 2 + ~ n y = ( y 1 , y 2 ) ~ antenna 2 ~ h 2 = ( h 12 , h 22 ) antenna 2 x 2 ~ h 1 = ( h 11 , h 21 ) x 1 antenna 1 antenna 3 antenna 1 6

Agenda • Channel model • MIMO decoding • Degrees of freedom • Multiplexing and Diversity 7

Zero-Forcing (ZF) Decoding • Decode x 1 orthogonal vectors ✓ y 1 ◆ ✓ h 11 ◆ ✓ h 12 ◆ ✓ n 1 ◆ * h 22 x 1 + x 2 + = y 2 h 21 h 22 n 2 * - h 12 + ) y 1 h 22 − y 2 h 12 = ( h 11 h 22 − h 21 h 12 ) x 1 + n 0 y 1 h 22 − y 2 h 12 x 0 1 = h 11 h 22 − h 21 h 12 n 0 = x 1 + h 11 h 22 − h 21 h 12 n 0 = x 1 + ~ h 1 · ~ h ? 2 8

Zero-Forcing (ZF) Decoding • Decode x 2 orthogonal vectors ✓ y 1 ◆ ✓ h 11 ◆ ✓ h 12 ◆ ✓ n 1 ◆ * h 21 x 1 + x 2 + = y 2 h 21 h 22 n 2 * - h 11 + ) y 1 h 21 − y 2 h 11 = ( h 12 h 21 − h 22 h 11 ) x 2 + n 0 y 1 h 21 − y 2 h 11 x 0 2 = h 12 h 21 − h 22 h 11 n 0 = x 2 + h 12 h 21 − h 22 h 11 n 0 = x 2 + ~ h 2 · ~ h ? 1 8

ZF Decoding (antenna space) y = ( y 1 , y 2 ) ~ ~ h 2 = ( h 12 , h 22 ) antenna 2 x 2 ~ h 1 = ( h 11 , h 21 ) x 1 antenna 1 |x’ 1 | ≤ |x 1 | x’ 1 • To decode x 1 , project the received signal y onto the interference-free direction h 2 ⊥ • To decode x 2 , project the received signal y onto the interference-free direction h 1 ⊥ • SNR reduces if the channels h 1 and h 2 are correlated, i.e., not perfect orthogonal (h 1 ⋅ h 2 =0) 10

SNR Loss due to ZF Detection y = ( y 1 , y 2 ) ~ ~ h 2 = ( h 12 , h 22 ) antenna 2 x 2 ~ θ h 1 = ( h 11 , h 21 ) x 1 antenna 1 x’ 1 1 | 2 = | x 1 | 2 cos 2 (90 − θ ) = | x 1 | 2 sin 2 ( θ ) | x � n SNR ZF = SNR SISO x 0 • From equation: 1 = x 1 + ~ h 1 · ~ h ? when h 1 ⊥ h 2 2 2 ) 2 = | x 1 | 2 sin 2 ( � ) | x 1 | 2 SNR � = = SNR ∗ sin 2 ( � ) N 0 / ( � h 1 · � N 0 h � • The more correlated the channels (the smaller angles), the larger SNR reduction 11

When will MIMO Fail? • In the worst case, SNR might drop down to 0 if the channels are strongly correlated to each other, e.g., h 1 ⫽ h 2 in the 2x2 MIMO • To ensure channel independency, should guarantee the full rank of H ⎻ Antenna spacing at the transmitter and receiver must exceed half of the wavelength 12

ZF Decoding – General Eq. • For a N x M MIMO system, y = Hx + n • To solve x , find a decoder W satisfying the constraint WH = I , then x � = Wy = x + Wn à W is the pseudo inverse of H W = ( H ∗ H ) − 1 H ∗ 13

ZF-SIC Decoding • Combine ZF with SIC to improve SNR ⎻ Decode one stream and subtract it from the received signal ⎻ Repeat until all the streams are recovered ⎻ Example: after decoding x 2 , we have y 1 = h 1 x 1 +n 1 à decode x 1 using standard SISO decoder • Why it achieves a higher SNR? ⎻ The streams recovered after SIC can be projected to a smaller subspace à lower SNR reduction ⎻ In the 2x2 example, x 1 can be decoded as usual without ZF à no SNR reduction (though x2 still experience SNR loss) 14

Other Detection Schemes • Maximum-Likelihood (ML) decoding ⎻ Measure the distance between the received signal and all the possible symbol vectors ⎻ Optimal Decoding ⎻ High complexity (exhaustive search) • Minimum Mean Square Error (MMSE) decoding ⎻ Minimize the mean square error ⎻ Bayesian approach: conditional expectation of x given the known observed value of the measurements • ML-SIC, MMSE-SIC 15

Channel Estimation • Estimate N x M matrix H h 11 x 1 y 1 y 1 = h 11 x 1 + h 12 x 2 + n 1 h 21 y 2 = h 21 x 1 + h 22 x 2 + n 2 h 12 x 2 y 2 h 22 Two equations, but four unknowns preamble Stream 1 Antenna 1 at Tx preamble Stream 2 Antenna 2 at Tx Estimate h 11 , h 21 Estimate h 12 , h 22

Agenda • Channel model • MIMO decoding • Degrees of freedom • Multiplexing and Diversity 17

Degree of Freedom For N x M MIMO channel • Degree of Freedom (DoF): min {N,M} ⎻ Can transmit at most DoF streams • Maximum diversity: NM ⎻ There exist NM paths among Tx and Rx

MIMO Gains • Multiplex Gain ⎻ Exploit DoF to deliver multiple streams concurrently • Diversity Gain ⎻ Exploit path diversity to increase the SNR of a single stream ⎻ Receive diversity and transmit diversity

Multiplexing-Diversity Tradeoff • Tradeoff between the diversity gain and the multiplex gain • Say we have a N x N system ⎻ Degree of freedom: N ⎻ The transmitter can send k streams concurrently, where k ≤ N ⎻ If k < N, leverage partial multiplexing gains, while each stream gets some diversity ⎻ The optimal value of k maximizing the capacity should be determined by the tradeoff between the diversity gain and multiplex gain

Agenda • Channel model • MIMO decoding • Degrees of freedom • Multiplexing and Diversity 21

Receive Diversity • 1 x 2 example h 1 x y 1 y 1 = h 1 x + n 1 h 2 y 2 = h 2 x + n 2 y 2 ⎻ Uncorrelated whit Gaussian noise with zero mean ⎻ Packet can be delivered through at least one of the many diverse paths

Theoretical SNR of Receive Diversity • 1 x 2 example h 1 x y 1 Increase SNR by 3dB • h 2 Especially beneficial for • the low SNR link y 2 P (2 X ) SNR = P ( n 1 + n 2 ) , where P refers to the power = E [(2 X ) 2 ] E [ n 2 1 + n 2 2 ] = 4 E [ X 2 ] , where σ is the variance of AWGN 2 σ = 2 ∗ SNR single antenna

Maximal Ratio Combining (MRC) • Extract receive diversity via MRC decoding • Multiply each y with the conjugate of the channel 1 y 1 = | h 1 | 2 x + h ∗ ⇒ h ∗ y 1 = h 1 x + n 1 1 n 1 = 2 y 2 = | h 2 | 2 x + h ∗ y 2 = h 2 x + n 2 h ∗ 2 n 2 • Combine two signals constructively 2 y 2 = ( | h 1 | 2 + | h 2 | 2 ) x + ( h ∗ h ∗ 1 y 1 + h ∗ 1 + h ∗ 2 ) n • Decode using the standard SISO decoder h ⇤ 1 y 1 + h ⇤ 2 y 2 x 0 = ( | h 1 | 2 + | h 2 | 2 ) + n 0 24

Achievable SNR of MRC 2 y 2 = ( | h 1 | 2 + | h 2 | 2 ) x + ( h ∗ h ∗ 1 y 1 + h ∗ 1 + h ∗ 2 ) n SNR MRC = E [(( | h 1 | 2 + | h 2 | 2 ) X ) 2 ] SNR single = E [ | h 1 | 2 X 2 ] ( h ∗ 1 + h ∗ 2 ) 2 n 2 n 2 = ( | h 1 | 2 + | h 2 | 2 ) 2 E [ X 2 ] = | h 1 | 2 E [ X 2 ] ( | h 1 | 2 + | h 2 | 2 ) σ 2 σ 2 = ( | h 1 | 2 + | h 2 | 2 ) E [ X 2 ] σ 2 gain = | h 1 | 2 + | h 2 | 2 • | h 1 | 2 ~2x gain if |h 1 |~=|h 2 | • 25

Transmit Diversity h 1 y x x h 2 • Signals go through two diverse paths • Theoretical SNR gain: similar to receive diversity • How to extract the SNR gain? ⎻ Simply transmit from two antennas simultaneous? ⎻ No! Again, h 1 and h 2 might be destructive

Transmit Diversity: Repetitive Code t+1 t h 1 y(t) = h 1 x x 0 y(t+1) = h 2 x 0 x h 2 • Deliver a symbol twice in two consecutive time slots • Repetitive code time Diversity: 2 • � x � 0 Data rate: 1/2 symbols/s/Hz • X = 0 space x • Decode and extract the diversity gain via MRC • Improve SNR, but reduce the data rate!!

Transmit Diversity: Alamouti Code t+1 t h 1 y(t) = h 1 x 1 +h 2 x 2 ＊ + n x 1 -x 2 * y(t+1) = h 2 x 1 * - h 1 x 2 + n x 2 x 1 * h 2 • Deliver 2 symbols in two consecutive time slots, but switch the antennas • Alamouti code (space-time block code) time ✓ x 1 ◆ Diversity: 2 • − x 2 x = Data rate: 1 symbols/s/Hz x ∗ x ∗ • space 2 1 • Improve SNR, while, meanwhile, maintain the data rate