Lecture 9,10: Beam forming Transmit diversity Aliazam Abbasfar

Outline Beam forming Transmit Diversity Space-time codes

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

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

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

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

Some Array patterns

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

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

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

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

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

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 )

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

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

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

Ch. 7 Goldsmith Ch. 3.3 Tse Reading