1. Lecture 9,10: Beam forming Transmit diversity Aliazam Abbasfar

2. Outline Beam forming Transmit Diversity Space-time codes

3. Diversity gain – Power gain SNR = ‖ h ‖ 2 SNR avg = L SNR avg ‖ h ‖ 2 /L Diversity gain ‖ h ‖ 2 /L E[ ‖ h ‖ 2 /L ]  1 Less likely to fade deeply Power gain : L SNR avg Array gain

4. Antenna array Antenna arrays Combining waves linearly (TX or RX) Radiation pattern Gain = maximum radiation pattern / isotropic radiation intensity Main lobe and side lobes Beam width 3-db (Half power) beam-width Changes angular radiation intensity Total radiated power is constant Side lobes cannot be ignored Phased array Combine phase-shifted signals

5. Beam forming Choice of coefficients makes the beam Direction of the beam can change Adaptive beam forming Tracking Coefficients can be chosen to null out interference Antenna pattern has nulls at certain directions

6. Array transfer function φ m = 2πD(m-1)sin()/ If G m = exp(-jφ m ), the beam point to the direction  Unity gain and linear phase shift (linear phased array) Array is a spatial filter Combine arriving signals with different weights Place nulls in the direction of interfering signals M-antenna array can place M-1 nulls in the beam pattern

7. Some Array patterns

8. Frequency response of array Array response is a function of direction and frequency ( dependence) A phased array has a bandwidth Nulls have limited bandwidth too The bandwidth can be increased using delay lines instead of phase shifters Using filters in each branch

9. Multi-beam forming A single array can be used to form different beams for two signals Superposition law TX or RX

10. Adaptive beam forming The direction of main lobe or the nulls can change by changing the weights Smart antenna changes the weights adaptively to track a target or minimize a cost function Minimize mean-square-error (MSE) Use adaptive filter algorithms such as LMS and RLS

11. Transmit diversity If channel response is known at the TX SC, MRC, and EGC can be used MRC is optimum, Why? The same diversity gain Less power gain vs RX diversity The total power should be divided among branches Channel response is measured in the receiver Sent to TX using a back channel Use downlink channel response in TDD systems

12. Space-time codes Can we achieve TX diversity if the channel response is not known? YES Add temporal encoding Repetition code + antenna muxing Coded system

13. Alamouti scheme Very simple 2-antenna transmit diversity No rate reduction T1 T2 Antenna 1 : x 1 x 2 * Antenna 2 : x 2 -x 1 * Encodes x 1 and x 2 into two orthogonal vectors Decouples data detections SNR = ‖ h ‖ 2 SNR 0 /2 Similar to MRC (half power gain) Full diversity order (2) Extension to more antennas (OSTBC) Real symbols ( for any M T ) Lower rate ( ¾ for M T =3,4, ½ for any M T )

14. Pair-wise error probability (PEP) P( X A  X B | h) = Q ( ‖ (X A -X B )h ‖ /2 n )

15. Space-time code design Diversity order (L) Rank criterion : L = min [rank(X A -X B )] L <= N and M T Shortest error event in coded systems Full diversity order (L = M T ) Coding gain Determinant criterion Squared distance products in coded systems

16. MIMO diversity If there are M R antenna at the RX P( X A  X B | H) = Q ( ‖ (X A -X B )H ‖ /2 n ) Diversity order = L M R <= M T M R Alamouti scheme = 2 M R

17. Ch. 7 Goldsmith Ch. 3.3 Tse Reading