SYSC 4607 – Slide Set 15 Outline

1. SYSC 4607 – Slide Set 15 Outline • Review of Previous Lecture • Diversity Systems - Diversity Combining Techniques - Performance of Diversity in Fading Channels - Transmitter Diversity - CSI at Tx - No CSI at Tx (Alamouti Scheme)

2. Review of Previous Lecture • Diversity overcomes the effects of flat fading by combining multiple independent fading paths • Diversity typically entails some penalty in terms of rate, bandwidth, complexity, or size. • Different combining techniques offer different levels of complexity and performance.

3. Diversity Combining Techniques • Selection Combining (SC) - Strongest signal is selected. Cophasing not required. • Threshold (Switching) Combining - Signal above a given threshold is used. Switching to a different branch if it drops below the threshold. • Maximal Ratio Combining (MRC) - Signals are cophased and summed after optimal weighting proportional to individual SNR’s. Goal is to maximize SNR at the combiner output. • Equal Gain Combining (EGC) - Branch signals are cophased and added (Maximal Ratio with equal weights).

4. Linear Diversity Combining • Individual branches are weighed by αi and summed • Selection and Threshold Combining: all αi = 0, except one. Cophasing not required • Maximal Ratio Combining: αi function of γi. Co-phasing required • Equal Gain: αi = 1. Co-phasing required

5. Linear Diversity Combining • is a random variable with PDF and CDF which depends on the type of fading and the choice of combining • Most often PDF is obtained by differentiating CDF • is the random probability of error for AWGN non-fading channel • Most often closed form solution for CDF, Pout and unavailable. Results based on computer simulation.

6. Array and Diversity Gains • Array Gain - Gain in SNR from coherent addition of signals and non-coherent addition (averaging) of noise over multiple antennas - Gain in both fading and non-fading channels • Diversity Gain - Gain in SNR due to elimination of weak signals (deep fades). Changes slope of probability of error. - Gains in fading channels

7. Selection Combining • Combiner outputs the signal with the highest SNR • The chance that all the branches are in deep fade simultaneously is very low. • Since at each instant only one signal is used co-phasing is not required.

8. Selection Combining (Assuming independent branches) For iid Rayleigh fading (ri Rayleigh, γi exponential):

9. Selection Combining • The average SNR gain (array gain) increases with M, but not linearly.

10. Threshold (Switching) Combining • Branches are scanned sequentially. First one above a given threshold is selected. The signal is used as long as its SNR is above threshold. • Since at each instant only one signal is used, co-phasing is not required.

11. Threshold (Switching) Combining • For two-branch diversity with iid branch statistics: • For iid Rayleigh fading with

12. Maximal Ratio Combining • In the general model set • Then, • Assuming the same noise psd at all branches: • Maximizing by Cauchy-Schwartz inequality:

13. Maximal Ratio Combining • Assuming iid Rayleigh fading in each branch with equal average branch SNR, , resulting has chi-squared distribution with 2M degrees of freedom:

14. BER Performance of MRC • Average Pb for Maximal Ratio Combining with iid Rayleigh Fading

15. Equal Gain Combining • In maximal ratio combining, set • Then, • In general, no closed-form solution for . For iid two-branch Rayleigh channel with same CDF in terms of Q function:

16. Diversity Improvements with M • For Selection Combining, average SNR increases with M. The increase, however, is not linear. Maximum benefit gained when M is increased form 1 to 2. • For Maximal Ratio and Equal Gain Combining diversity gain and array gain both contribute to the performance improvement. For large M, array gain dominates performance improvements.

17. Diversity Improvements with M • For Selection Combining SNR saturates with modest increase in M. Maximal Ratio and Equal Gain do not exhibit saturation, but slope changes • Maximal Ratio and Equal Gain outperform Selection; however, their relative performance is close (within 1 dB). Equal Gain simpler to implement • Improvements in average SNR ( ) for M-Branch compared to one branch. (a) Maximal Ratio, (b) Equal Gain, (c) Selection

18. Transmitter Diversity • Multiple transmit antennas, with power divided among them • Suitable for systems with greater capabilities at the transmit site (example: cellular, downlink) • Implementation depends on channel knowledge - With transmitter channel knowledge (CSIT), performance is similar to receiver diversity (same array/diversity gain) - Without channel knowledge, can obtain diversity gain (but not array gain) through Alamouti scheme: transmission over space and time (2 consecutive symbols)

19. Channel Known at Tx • We assume path gain is known at the transmitter. • s(t), transmitted signal with energy per symbol Es , is multiplied by constant complex gain αi : • Total energy constraint Es requires • Signals transmitted from M antennas are combined in the air:

20. Channel Known at Tx • Similar to MRC at Rx coefficients are found to maximize SNR: • Resulting SNR: • For large SNR: • Similar to MRC full diversity order of M is achieved

21. Channel Unknown at TxAlamouti Scheme • Transmission in space and time: First symbol period; Second symbol period Antenna 1: s1 Antenna 1: -s2* Antenna 2: s2 Antenna 2: s1* Each transmission has energy Es/2

22. The Alamouti Scheme Channel gains: Received signals: y = = HAs + n

23. The Alamouti Scheme Define z = HAH y, where HAH is the conjugate transpose of HA. We have HAH HA =(|h1|2+|h2|2)I2 Thus z = (z1z2)T =(|h1|2+|h2|2)I2 s + , where is zero-mean complex Gaussian with .

24. The Alamouti Scheme So, (Factor 2 represents transmission of half energy Es/2 per symbol per antenna) Observations: • Alamouti scheme achieves a diversity order of 2 although channel knowledge not available at transmitter • Alamouti scheme achieves an array gain of 1

25. Performance of the Alamouti Scheme BER performance of Alamouti Scheme vs. SNR γb: comparison of BPSK with Maximal Ratio and two-branch transmit diversity in Rayleigh fading

26. Main Points • MRC provides best diversity performance • EGC easier to implement compared to MRC - Performance about 1 dB worse than MRC • Performance of Transmit Diversity depends on channel knowledge - With channel knowledge at transmitter (CSIT) same performance as receiver diversity - Without CSIT, the Alamouti scheme provides same diversity gain, but array gain is 3 dB lower.