Channel Capacity of MIMO Channels

1. Channel Capacity of MIMO Channels 指導教授：黃文傑 老師 學　　生：曾凱霖 學　　號：M9121014 無線通訊實驗室

2. Outline 1、Introduction 2、Shannon capacity of MIMO systems 3、The ”pipe” interpretation 4、To exploit the MIMO channel • BLAST • Space Time Coding 5、Conclusion

3. Why multiple antennas ???? • Frequency and time processing are at limits. • Space processing is interesting because it does not increase bandwidth.

4. Initial Assumptions • Flat fading channel (Bcoh>> 1/ Tsymb) • Slowly fading channel (Tcoh>> Tsymb) • receive and nt transmit antennas • Receiver estimates the channel perfectly • We consider space diversity only

5. SISO Systems x(t): transmitted signal y(t): received signal h(t): channel transfer function n(t): noise (AWGN, 2) h y(t) x(t) y(t) = h •x(t) + n(t) Signal to noise ratio : Capacity : C = log2(1+)

6. = log2[1+(PT/s2)·|H|2] [bit/(Hz·s)] H = [ H11 H21] Capacity increases logarithmically with number of receive antennas... Receive Diversity H11 H21

7. H11 H12 • Capacity increases logarithmically with nt Transmit Diversity / Beamforming Cdiversity = log2(1+(PT/2s2)·|H|2) [bit/(Hz·s)]

8. Interpretation: l1 Receiver Transmitter l2 m=min(nr, nt) parallel channels, equal power allocated to each ”pipe” MIMO Systems H11 H21 H12 H22 Where the i are the eigenvalues to HH† Cdiversity = log2det[I +(PT/2s2 )·HH†]=

9. H known at TX Where the power distribution over ”pipes” are given by a water filling solution l1 p1 l2 p2 l3 p3 l4 p4 MIMO Capacity in General H unknown at TX

10. The Channel Eigenvalues Orthogonal channelsHH† =I,1=2=…= m=1 • Capacity increases linearly with min( nr , nt ) • An equal amount of power PT/nt is allocated • to each ”pipe” Transmitter Receiver

11. Time s1 s1 s1 s1 s1 s1 V-BLAST Antenna s2 s2 s2 s2 s2 s2 s3 s3 s3 s3 s3 s3 s0 s1 s2 s0 s1 s2 D-BLAST s0 s1 s2 s0 s1 s0 s1 s2 s0 To Exploit the MIMO Channel Bell Labs Layered Space Time Architecture • nr  nt required • Symbol by symbol detection. Using nulling and symbol cancellation • V-BLAST implemented -98 by Bell Labs (40 bps/Hz) {G.J.Foschini, Bell Labs Technical Journal 1996 }

12. Space Time Coding • Use parallel channel to obtain diversity not • spectral efficiency as in BLAST • Space-Time trellis codes : coding and diversity gain (require Viterbi detector) • Space-Time block codes : diversity gain • (use outer code to get coding gain) • nr= 1 is possible • Properly designed codes acheive diversity of nr nt

13. Orthogonal Space-time Block Codes Block of T symbols Constellation mapper STBC Data in nt transmit antennas Block of K symbols • K input symbols, T output symbols T K • R=K/T is the code rate • If R=1 the STBC has full rate • If T= ntthe code has minimum delay • Detector is linear !!!

14. STBC for 2 Transmit Antennas Full rate and minimum delay [ c0 c1 ]  Antenna Time Assume 1 RX antenna: Received signal at time 0 Received signal at time 1

15. The MIMO/ MISO system is in fact transformed to an equivalent SISO system with SNR SNReq = ||H ||F2SNR/nt l1+l2 ||H ||F2 = l1+l2 Diagonal matrix due to orthogonality

16. Conclusion • MIMO systems are a promising technique for high data rates. • Their efficiency depends on the channel between the transmitters and the receivers (power and correlation). • Practical issues need to be resolved.