SYSC 4607 – Lecture 14 Outline

1. SYSC 4607 – Lecture 14 Outline • Review of Previous Lecture • Diversity Systems • Methods for Obtaining Diversity Branches • Diversity Combining Techniques • Performance of Diversity in Fading Channels

2. Review of Previous Lecture • Average Ps in fast fading: - Averaged over fast fading distribution - Good metric when Tc~Ts • Combined outage and average Ps - Used for combined fast and slow fading: average over fast fading, outage relative to slow fading • Irreducible error floor due to Doppler - Differential modulation has a poor phase reference due to phase decorrelation - Little impact on high speed systems • Irreducible error floor due to ISI - Self-interference due to delay spread - Without compensation, severely limits data rates

3. Diversity Systems: Basic Principles and Classifications • Basic Concept - Same Information is Sent over Independent Fading Paths - Signals are Combined to Mitigate the Effects of Fading • Design Issues - Methods to Obtain Diversity Branches - Diversity Combining Methods • Different Classifications - Receiver versus Transmitter - Predetection versus Postdetection - Microscopic versus Macroscopic

4. Methods to Obtain Diversity Branches • Space - Multiple antenna elements spaced apart by decorrelation distance. Theoretical decorrelation λ/2. Most common form of diversity. No additional power or bandwidth. • Frequency - Multiple narrowband channels separated by channel coherence bandwidth. Less often used. Wasteful of scarce spectrum. • Polarization - Two antennas (one horizontally, the other verticallypolarized) are used. Orthogonal polarization in wireless channels exhibit uncorrelated fading. Only two-branch diversity possible. Not common.

5. Methods to Obtain Diversity Branches • Angle of Arrival -Directional antennas facing widely different directions. Scattered signal from different directions having approximately independent fading. • Time - Transmission of the same information in time slots separated by channel coherence time. Inefficient for high-speed transmissions. Useless for stationary users. • Multipath - Same as Time-diversity, except that branches are provided by channel through multipath. Takes advantage of channel provided usually undesirable multipath echoes. Principle of Rake Receivers.

6. 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).

7. 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

8. 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.

9. 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

10. 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.

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

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

13. 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.

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

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

16. 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:

17. 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:

18. Main Points • 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.