1. MIMO (Multiple Input Multiple Output)

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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 𝑛 1 ℎ ℎ 𝑛 2 ℎ ℎ 𝑛 3 ℎ ℎ 𝑛 4 ℎ ℎ 5 2 ( 𝑛 5 ℎ 5 ) ℎ ℎ ℎ ℎ ℎ 5 2
Effective Noise
MRC achieves the best combining with minimal effective noise

9. 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 ℎ ℎ ℎ ℎ 5 2

10. SIMO – Maximal Ratio Combining (MRC)
ℎ 𝐻 ℎ=
ℎ ℎ ℎ ℎ ℎ 5 2
(ℎ 𝐻 ℎ) −1 =
1 ℎ ℎ ℎ ℎ 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 ℎ ℎ ℎ ℎ 5 2

11. 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 ∗ =

12. 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 ℎ 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 ℎ 2
ℎ 2 ∗ − ℎ 1 ∗
ℎ ℎ
ℎ ℎ 2 2
𝐻 𝐻 𝐻 = =
(𝐻 𝐻 𝐻) −1 =
1 ℎ ℎ
ℎ ℎ 2 2

13. 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 ℎ ℎ 31
ℎ ℎ ℎ 32
ℎ ℎ 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

14. 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 ℎ ℎ 31
𝑛 1
𝑛 2
𝑛 3
𝑥 1
𝑥 2
𝑥 3
ℎ ℎ ℎ 32 = +
ℎ ℎ ℎ 33
ℎ ℎ ℎ 34
𝑦 = 𝐻𝑥 𝑛
𝑦 ≈ 𝐻𝑥
𝑥 𝑒𝑠𝑡 ≈ 𝐻 −1 𝑦
𝑥 𝑒𝑠𝑡 ≈ (𝐻 𝐻 𝐻) −1 𝐻 𝐻 𝑦

15. 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

16. 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

17. 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

18. 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 𝑑
𝑑 3 𝑑 𝑑
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)