1. Fire Tom Wada Professor, Information Engineering, Univ. of the Ryukyus Chief Scientist at Magna Design Net, Inc wada@ie.u-ryukyu.ac.jp http://www.ie.u-ryukyu.ac.jp/~wada/ 1/10/20161 Fire Tom Wada, Univ. of the Ryukyus

2. Matrix Based OFDM Modeling Channel Matrix diagonalization by Unitary Matrix FFT 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 2

3. SISO Channel 1/10/20163 Transmission Antenna Reception Antenna Single Input and Single Output (SISO) Channel Fire Tom Wada, Univ. of the Ryukyus

4. OFDM Modulator 1/10/20164 MAPMAPMAPMAPS/PIFFT P/S Generate Complex symbol d 0 ~d Ｎ－１ Bit stream Copy to make Guard Interval OFDM symbol (1/f 0 ) TgTgTgTg Fire Tom Wada, Univ. of the Ryukyus

5. Multi-path channel 1/10/20165 OFDM symbol (1/f 0 ) TgTgTgTg TgTgTgTg Fire Tom Wada, Univ. of the Ryukyus

6. OFDM Demodulator 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 6 S/P FFT Equalize DEMAPDEMAPDEMAPDEMAP Bit Stream OFDM symbol (1/f 0 ) TgTgTgTg P/S Noise Remove Guard Interval

7. FFT matrix 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 7

8. IFFT matrix 1/10/20168 Fire Tom Wada, Univ. of the Ryukyus

9. Twiddle Factor W N nk 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 9

10. Multi-path channel in Matrix 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 10 Symbol n Symbol n-1 GI of n GI of n-1

11. If Multi-path delay is small than GI length 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 11 Channel Matrix is Cyclic Matrix by GI.

12. Base Station Receiver Two path Multi path Channel Example 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 12 Channel Impulse Response = [1, 0.5, 0, 0]

13. Two path Multi path Channel Example 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 13 If time domain channel matrix is cyclic, Frequency Domain Channel Matrix is diagonal!

14. Additive Noise 1/10/201614 Fire Tom Wada, Univ. of the Ryukyus

15. How to recover sending signal from receiver signal. - EQUALIZE - 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 15

16. Summary of Matrix model of OFDM 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 16 Transmission Antenna Reception Antenna Channel

17. Important Mathematics 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 17 Cyclic Matrix can be diagonalized by FFT and IFFT. X H is Hermitian of X, that is, complex conjugate and transpose.

18. Unitary Matrix Unitary Matrix U can satisfy following property. 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 18 Eigen value of Channel Cyclic Matrix is Channel Transfer Function as (H(0), H(1), H(2), … ).

19. MIMO Channel Modeling 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 19

20. SISO Channel OFDM makes Multi-path channel simple complex h(k) for freq=k. 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 20 Transmission Antenna Reception Antenna SISO Channel IFFT FFT

21. MIMO Channel - Nr X Nt SISO Channels for Freq=k - 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 21

22. Singular value decomposition of Nr x Nt Matrix H Nr x Nt matrix H can be decomposed as below using Nr x Nr Unitary matrix V and Nt x Nt Unitary matrix U. Σ is Nr x Nt diagonal matrix. 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 22

23. SVD Example by Matlab(1) H = 1 2 3 2 4 5 >> [U,S,V] = svd(H) U = -0.4863 -0.8738 -0.8738 0.4863 S = 7.6756 0 0 0 0.2913 0 V = -0.2910 0.3396 -0.8944 -0.5821 0.6791 0.4472 -0.7593 -0.6508 -0.0000 H = 1 2 2 4 3 5 >> [U,S,V] = svd(H) U = -0.2910 -0.3396 -0.8944 -0.5821 -0.6791 0.4472 -0.7593 0.6508 -0.0000 S = 7.6756 0 0 0.2913 0 0 V = -0.4863 0.8738 -0.8738 -0.4863 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 23

24. SVD Example by Matlab(2) H = 1.0000 + 1.0000i 2.0000 + 1.0000i 1.0000 - 3.0000i 3.0000 - 1.0000i >> [U,S,V] = svd(H) U = -0.4616 - 0.0659i -0.4907 + 0.7361i -0.3956 + 0.7913i -0.2863 - 0.3680i S = 5.0000 0 0 1.4142 V = -0.6594 0.7518 -0.5934 + 0.4616i -0.5205 + 0.4048i >> U*S*V' ans = 1.0000 + 1.0000i 2.0000 + 1.0000i 1.0000 - 3.0000i 3.0000 - 1.0000i >> U*S*V' ans = 1.0000 + 1.0000i 2.0000 + 1.0000i 1.0000 - 3.0000i 3.0000 - 1.0000i >> U'*U ans = 1.0000 0.0000 - 0.0000i 0.0000 + 0.0000i 1.0000 >> U*U' ans = 1.0000 0.0000 - 0.0000i 0.0000 + 0.0000i 1.0000 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 24

25. MIMO communication 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 25 H MIMO Channel

26. Introduce pre-processing and post-processing 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 26 H MIMO Channel Nt x Nt Nr x Nr

27. There are K(=rank(H)) independent channel 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 27

28. SVD-MIMO system 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 28 H =VΣU H MIMO Channel Nt x Nt Nr x Nr

29. Put them altogether MIMO-OFDM system Space Division Multiplexing by MIMO (K stream) Orthogonal Frequency Division Multplxing (OFDM) 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 29 IFFTFFT IFFT FFT Nt x K K x Nr

30. Summary This presentation shows matrix based modeling for both OFDM and MIMO and there are many similarity in mathematics. 1. OFDM realizes many parallel communication channels in frequency domain. 2. OFDM converts multi-path channel to simple one tap channel such as h(k)=a+bj for Frequency=k. 3. Then OFDM-based MIMO system can focus on simple channel matrix. 4. By singular value decomposition (SVD), MIMO channel matrix H can be decomposed to V*Σ*U H. 5. Non-zero elements of Σ (rank of H) indicates parallel communication channel in space. 1/10/2016 Fire Tom Wada, Univ. of the Ryukyus 30