MIMO (Multiple Input Multiple Output)

MIMO (multiple input, multiple output) Tx Rx Multiple antennas at both transmitters and receivers

SISO (single input single output) Wireless channel Noise Tx Rx 𝑦 𝑡 = h t ∗ x t + n t 𝑦 𝑡 𝑦 𝑓 = h f x f + n f 𝑥 𝑡 𝑦=ℎ x+𝑛 𝑥 𝑒𝑠𝑡 = 𝑦 ℎ =x+ 𝑛 ℎ True Symbol Noise corruption Goal of a communication system is to minimize noise corruption in estimated symbol

SIMO (Single Input Multiple Output) 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 𝑥

SIMO – Selection Combining (SC) 𝑥 𝑒𝑠𝑡 = 𝑦 1 ℎ 1 =x+ 𝑛 1 ℎ 1 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 𝑥 𝑒𝑠𝑡 = 𝑦 2 ℎ 2 =x+ 𝑛 2 ℎ 2 𝑥 𝑥 𝑒𝑠𝑡 = 𝑦 3 ℎ 3 =x+ 𝑛 3 ℎ 3 𝑥 𝑒𝑠𝑡 = 𝑦 4 ℎ 4 =x+ 𝑛 4 ℎ 4 𝑥 𝑒𝑠𝑡 = 𝑦 5 ℎ 5 =x+ 𝑛 5 ℎ 5 SC chooses the antenna with lowest noise component among 𝑛 1 ℎ 1 , 𝑛 2 ℎ 2 , 𝑛 3 ℎ 3 , 𝑛 4 ℎ 4 , 𝑛 5 ℎ 5 Equivalent to choosing antenna with maximum h among h 1 , h 2 , h 3 , h 4 , h 5

SIMO – Selection Combining 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 𝑖=max⁡( ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 ,| ℎ 5 |) 𝑥 𝑒𝑠𝑡 = 𝑦 𝑖 ℎ 𝑖 =𝑥+ 𝑛 𝑖 ℎ 𝑖 𝑥 Selection combining picks the least attenuated antenna output

SIMO – Equal Gain Combining 𝑥 𝑒𝑠 𝑡 𝑖 = 𝑦 𝑖 ℎ 𝑖 =𝑥+ 𝑛 ℎ 𝑖 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 𝑥 𝑠𝑐 = 𝑥 𝑒𝑠 𝑡 1 + 𝑥 𝑒𝑠 𝑡 2 + 𝑥 𝑒𝑠 𝑡 3 + 𝑥 𝑒𝑠 𝑡 4 + 𝑥 𝑒𝑠 𝑡 5 5 𝑥 𝑥 𝑠𝑐 =𝑥 + 𝑛 ℎ 1 + 𝑛 ℎ 2 + 𝑛 ℎ 3 + 𝑛 ℎ 4 + 𝑛 ℎ 5 5 Effective Noise The effective noise is much lower than individual noises at antennas due to the smoothening effect of averaging

SIMO – Maximal Ratio Combining (MRC) 𝑥 𝑒𝑠𝑡 = 𝑎 1 𝑥 𝑒𝑠𝑡 1 + 𝑎 2 𝑥 𝑒𝑠 𝑡 2 + 𝑎 3 𝑥 𝑒𝑠 𝑡 3 + 𝑎 4 𝑥 𝑒𝑠𝑡 4 + 𝑎 5 𝑥 𝑒𝑠𝑡 5 𝑎 1 + 𝑎 2 + 𝑎 3 + 𝑎 4 + 𝑎 5 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 For optimal combining, 𝑎 𝑖 = ℎ 𝑖 2 𝑥 𝑥 𝑒𝑠𝑡 =𝑥 + ℎ 1 2 𝑛 1 ℎ 1 + ℎ 2 2 𝑛 2 ℎ 2 + ℎ 3 2 𝑛 3 ℎ 3 + ℎ 4 2 𝑛 4 ℎ 4 + ℎ 5 2 ( 𝑛 5 ℎ 5 ) ℎ 1 2 + ℎ 2 2 + ℎ 3 2 + ℎ 4 2 + ℎ 5 2 Effective Noise MRC achieves the best combining with minimal effective noise

SIMO – Maximal Ratio Combining (MRC) 𝑦 1 𝑦 2 𝑦 3 𝑦 4 𝑦 5 ℎ 1 ℎ 2 ℎ 3 ℎ 4 ℎ 5 𝑛 1 𝑛 2 𝑛 3 𝑛 4 𝑛 5 = 𝑥 + 𝑦 1 = ℎ 1 𝑥+ 𝑛 1 𝑦 2 = ℎ 2 𝑥+ 𝑛 2 𝑦 3 = ℎ 3 𝑥+ 𝑛 3 𝑦 4 = ℎ 4 𝑥+ 𝑛 4 𝑦 5 = ℎ 5 𝑥+ 𝑛 5 𝑦 = ℎ𝑥 + 𝑛 𝑦 ≈ ℎ𝑥 𝑥 𝑥 𝑒𝑠𝑡 ≈ (ℎ 𝐻 ℎ) −1 ℎ ℎ 𝑦 ℎ 𝐻 ℎ = ℎ 1 ℎ 2 ℎ 3 ℎ 4 ℎ 5 [ℎ 1 ∗ ℎ 2 ∗ ℎ 3 ∗ ℎ 4 ∗ ℎ 5 ∗ ] = ℎ 1 2 + ℎ 2 2 + ℎ 3 2 + ℎ 4 2 + ℎ 5 2

SIMO – Maximal Ratio Combining (MRC) ℎ 𝐻 ℎ= ℎ 1 2 + ℎ 2 2 + ℎ 3 2 + ℎ 4 2 + ℎ 5 2 (ℎ 𝐻 ℎ) −1 = 1 ℎ 1 2 + ℎ 2 2 + ℎ 3 2 + ℎ 4 2 + ℎ 5 2 Optimal noise smoothening matrix 𝑥 𝑒𝑠𝑡 ≈ (ℎ 𝐻 ℎ) −1 ℎ ℎ 𝑦 𝑥 𝑒𝑠𝑡 ≈ (ℎ 𝐻 ℎ) −1 ℎ ℎ (ℎ𝑥+𝑛) 𝑥 𝑒𝑠𝑡 ≈ (ℎ 𝐻 ℎ) −1 ℎ ℎ ℎ𝑥+ (ℎ 𝐻 ℎ) −1 ℎ ℎ 𝑛 𝑛 1 𝑛 2 𝑛 3 𝑛 4 𝑛 5 𝑥 𝑒𝑠𝑡 ≈ 𝑥+ (ℎ 𝐻 ℎ) −1 [ℎ 1 ∗ ℎ 2 ∗ ℎ 3 ∗ ℎ 4 ∗ ℎ 5 ∗ ] 𝑥 𝑒𝑠𝑡 ≈ 𝑥+ (ℎ 𝐻 ℎ) −1 (ℎ 1 ∗ 𝑛 1 + ℎ 2 ∗ 𝑛 2 + ℎ 3 ∗ 𝑛 3 + ℎ 4 ∗ 𝑛 4 + ℎ 5 ∗ 𝑛 5 ) 𝑥 𝑒𝑠𝑡 ≈ 𝑥+ (ℎ 1 ∗ 𝑛 1 + ℎ 2 ∗ 𝑛 2 + ℎ 3 ∗ 𝑛 3 + ℎ 4 ∗ 𝑛 4 + ℎ 5 ∗ 𝑛 5 ) ℎ 1 2 + ℎ 2 2 + ℎ 3 2 + ℎ 4 2 + ℎ 5 2

MISO (Multi Input Single Output) Alamouti STBC (Space time block codes) slot1 slot2 Antennas (Space) 1 𝑥 1 −𝑥 2 ∗ 2 𝑥 2 𝑥 1 ∗ 𝑦 1 𝑦 2 𝑥 1 𝑦= ℎ 1 𝑥 1 + ℎ 2 𝑥 2 +𝑛 𝑦 1 = ℎ 1 𝑥 1 + ℎ 2 𝑥 2 +𝑛 = [ℎ 1 ℎ 2 ] 𝑥 1 𝑥 2 + 𝑛 1 𝑥 2 𝑦 2 = −ℎ 1 𝑥 2 ∗ + ℎ 2 𝑥 1 ∗ +𝑛 = [ℎ 1 ℎ 2 ] −𝑥 2 ∗ 𝑥 1 ∗ + 𝑛 2 [ℎ 1 ∗ ℎ 2 ∗ ]− 𝑥 2 𝑥 1 + 𝑛 2 ∗ 𝑦 2 ∗ = [−ℎ 2 ∗ ℎ 1 ∗ ] 𝑥 1 𝑥 2 + 𝑛 2 ∗ 𝑦 2 ∗ =

MISO – Alamouti STBC ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦 𝑥 𝑒𝑠 𝑡 1 𝑥 𝑒𝑠 𝑡 2 ℎ 1 ℎ 2 𝑥 1 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦 𝑥 𝑒𝑠 𝑡 1 𝑥 𝑒𝑠 𝑡 2 ℎ 1 ℎ 2 𝑥 1 ℎ 2 ∗ − ℎ 1 ∗ 𝑥 2 + 𝑛 1 𝑛 2 ∗ 𝑦 1 𝑦 2 ∗ = ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 (𝐻 + ) 𝑥 1 𝑥 2 𝑛 1 𝑛 2 ∗ 𝑦 = 𝐻𝑥 + 𝑛 𝑦 ≈ 𝐻𝑥 ≈ 𝑥 1 𝑥 2 + 𝑛 1 𝑛 2 ∗ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 Optimal noise smoothening matrix ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦 𝑥 𝑒𝑠 𝑡 1 𝑥 𝑒𝑠 𝑡 2 𝑥 𝑒𝑠𝑡 = ℎ 1 ∗ ℎ 2 ℎ 2 ∗ − ℎ 1 ℎ 1 ℎ 2 ℎ 2 ∗ − ℎ 1 ∗ ℎ 1 2 + ℎ 2 2 0 0 ℎ 1 2 + ℎ 2 2 𝐻 𝐻 𝐻 = = (𝐻 𝐻 𝐻) −1 = 1 ℎ 1 2 + ℎ 2 2 0 0 1 ℎ 1 2 + ℎ 2 2

MIMO (Multiple Input Multiple Output) Tx Rx 𝑥 1 𝑥 2 𝑥 3 𝑦 1 = h 11 x 1 + h 21 x 2 + h 31 x 3 + n 1 𝑦 2 = h 12 x 1 + h 22 x 2 + h 32 x 3 + n 2 𝑦 3 = h 13 x 1 + h 23 x 2 + h 33 x 3 + n 3 MIMO channel matrix ℎ 11 ℎ 21 ℎ 31 ℎ 12 ℎ 22 ℎ 32 ℎ 13 ℎ 23 ℎ 33 𝑥 1 𝑥 2 𝑥 3 𝑦 1 𝑦 2 𝑦 3 = + 𝑛 1 𝑛 2 𝑛 3 𝑦 = 𝐻𝑥 + 𝑛 𝑦 ≈ 𝐻𝑥 𝑥 𝑒𝑠𝑡 ≈ 𝐻 −1 𝑦

MIMO (Multiple Input Multiple Output) Tx Rx 𝑥 1 𝑦 1 = h 11 x 1 + h 21 x 2 + h 31 x 3 + n 1 Rectangular MIMO channel matrix 𝑥 2 𝑦 2 = h 12 x 1 + h 22 x 2 + h 32 x 3 + n 2 𝑥 3 𝑦 3 = h 13 x 1 + h 23 x 2 + h 33 x 3 + n 3 𝑦 4 = h 14 x 1 + h 24 x 2 + h 34 x 3 + n 4 𝑦 1 𝑦 2 𝑦 3 𝑦 4 ℎ 11 ℎ 21 ℎ 31 𝑛 1 𝑛 2 𝑛 3 𝑥 1 𝑥 2 𝑥 3 ℎ 12 ℎ 22 ℎ 32 = + ℎ 13 ℎ 23 ℎ 33 ℎ 14 ℎ 24 ℎ 34 𝑦 = 𝐻𝑥 + 𝑛 𝑦 ≈ 𝐻𝑥 𝑥 𝑒𝑠𝑡 ≈ 𝐻 −1 𝑦 𝑥 𝑒𝑠𝑡 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦

Multi user MIMO (MU-MIMO) Rx 𝑥 1 𝑦 1 = h 11 x 1 + h 21 x 2 + h 31 x 3 + n 1 𝑥 2 𝑦 2 = h 12 x 1 + h 22 x 2 + h 32 x 3 + n 2 𝑥 3 𝑦 3 = h 13 x 1 + h 23 x 2 + h 33 x 3 + n 3

Multi user MIMO (MU-MIMO) Rx1 𝑥 1 𝑦 1 = h 11 x 1 + h 21 x 2 + h 31 x 3 + n 1 𝑥 2 𝑥 3 Rx2 𝑦 2 = h 12 x 1 + h 22 x 2 + h 32 x 3 + n 2 𝑥 𝑒𝑠𝑡 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦 Rx3 𝑦 3 = h 13 x 1 + h 23 x 2 + h 33 x 3 + n 3 𝑦 1 𝑦 2 𝑦 3 𝑥 𝑒𝑠𝑡 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻

𝑥 𝑒𝑠𝑡 is a linear combination of 𝑦 1 𝑦 2 𝑦 3 . 𝑥 𝑒𝑠𝑡 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑥 𝑒𝑠𝑡 is a linear combination of 𝑦 1 𝑦 2 𝑦 3 . Infeasible to take such a linear combination since a given receiver has only one of 𝑦 1 𝑦 2 𝑦 3 , but not all three of them

Multi user MIMO (MU-MIMO) 𝑦 = 𝐻𝑥 + 𝑛 Pre multiply input vector x by matrix P, and transmit Px instead of x 𝑦 = 𝐻𝑃𝑥 + 𝑛 Choose P such that HP is a diagonal matrix One possibility 𝑃=𝐻 −1 𝑦 = 𝑥 + 𝑛 𝑑 1 0 0 0 𝑑 2 0 0 0 𝑑 3 𝑑 1 0 0 0 𝑑 2 0 0 0 𝑑 3 𝑥 1 𝑥 2 𝑥 3 𝑦 1 𝑦 2 𝑦 3 = 𝑛 1 𝑛 2 𝑛 3 + 𝑦 𝑖 = 𝑑 𝑖 𝑥 𝑖 + 𝑛 𝑖 Each 𝑦 𝑖 is only a function of 𝑥 𝑖 , thus it can be decoded without needing to know other 𝑦 𝑗 (𝑗≠𝑖)