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# Diversity.

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Diversity

Outline Introduction to Diversity Receiver Diversity
Combining Techniques Performance of Diversity in Fading Selection Combining Maximal Ratio Combining MGF Approach to MRC Performance EGC Transmit Diversity

Introduction to Diversity
Basic Idea Send same bits over independent fading paths Independent fading paths obtained by time, space, frequency, or polarization diversity Combine paths to mitigate fading effects Tb 瑞利衰落和对数正态衰落都会使调制性能产生很大的功率损失。减轻衰落影响的有效技术之一就是对独立的衰落信号进行分集合并。 分集是在独立的衰落路径上发送相同的数据。 由于独立路径在同一时刻经历深衰落的概率很小，因此经过适当的合并后，接收信号的衰落程度会被减小。 Multiple paths unlikely to fade simultaneously t

Realization of independent fading paths
Space diversity Polarization diversity Angle (pattern) diversity Frequency diversity Time diversity

Receiver Diversity System Model
Coherently combine the independent fading paths S(t) multiplied by a random complex amplitude term Suppose no fading so that , noise PSD N0/2 on each branch and pulse shaping such that BTs=1. Then each branch has the same SNR: Set Then the received SNR after combination: Array gain: SNR increase in the absence of fading

Receiver Diversity System Model
With fading More favorable distribution of , increase and Pout Average probability of error Outage probability Diversity gain: change in BER slope The average probability of error Diversity order: M how the slope of average probability of error as a function of average SNR changes with diversity.

Array/Diversity gain Array gain is from noise averaging (AWGN and fading) Diversity gain is change in BER slope (fading)

Receiver Diversity ——Combining Techniques
Selection Combining Fading path with highest gain used Maximal Ratio Combining All paths cophased and summed with optimal weighting to maximize combiner output SNR Equal Gain Combining All paths cophased and summed with equal weighting

Selection Combining Analysis and Performance
CDF of PDF of Differentiating relative to For Rayleigh fading, the average SNR on the ith branch as The SNR distribution: Outage probability for a target on ith branch in Rayleigh fading: Outage probability of SC: If the average SNR for all branches are the same ( , for all i)

Selection Combining Analysis and Performance
CDF of Differentiating relative to Average SNR of combiner output in independent and identically distributed Rayleigh fading: The average SNR gain and corresponding array gain increase with M, but not linearly. Average probability of symbol error , we can have diversity gain.

Selection Combining Analysis and Performance
Selection Combining (SC) Combiner SNR is the maximum of the branch SNRs. CDF easy to obtain, pdf found by differentiating. Diminishing returns with number of antennas. Can get up to about 20 dB of gain. Outage Probability

MRC and its Performance
With MRC, gS=gi for branch SNRs gi Optimal technique to maximize output SNR Yields dB performance gains Distribution of gS hard to obtain Standard average BER calculation Hard to obtain in closed form Integral often diverges MGF Approach

EGC EGQ simpler than MRC Harder to analyze
Performance about 1 dB worse than MRC

Transmit Diversity With channel knowledge, similar to receiver diversity, same array/diversity gain Without channel knowledge, can obtain diversity gain through Alamouti scheme: works over 2 consecutive symbols

Alamouti Alamouti code h11 h12 Transmitter 1, S1 Receiver 1, Y1 h21
Tx1 Tx2 Time k k+1

Single and double symbol detection
Y1=[y1,k y1,k+1]; Y2=[y2,k y2,k+1] Maximal-ratio receiver combining (MRRC)

two-branch transmit diversity scheme with one receiver

two-branch transmit diversity scheme with two receivers

Attached Matlab Program
Function GUI interface Original reference S. M. Alamouti, A Simple Transmit Diversity Technique for Wireless Communications, IEEE Journal on Select Areas in Communications, vol. 16, no. 8, Oct

Alamouti code-Time domain
Before coding Data1 Data2 Data3 Data4 Data5 Data6 Transmitter 1 OFDM symbol After coding and framing Preamble Tx1 -Preamble Tx2* Data 1,1 -Data 2,1* Data 1,n -Data 2,n* Transmitter 1 OFDM symbol Transmitter 2 Preamble Tx2 Preamble Tx1* Data 2,1 Data 1,1* Data 2,n -Data 1,n* OFDM symbol

Total data sub-carrier
Frequency domain Frequency Frequency F 1,1 F 2,1 -F 2,1* F 1,1* F 1,96 F 2,96 -F 2,96* F 1,96* Time Time Total data sub-carrier 192 Total pilots 8 DC 1 FFT size 256

Main Points Diversity overcomes the effects of fading by combining fading paths Diversity typically entails some penalty in terms of rate, bandwidth, complexity, or size. Techniques trade complexity for performance. MRC yields dB gain, SC around 20 dB. Analysis of MRC simplified using MGF approach EGC easier to implement than MRC: hard to analyze. Performance about 1 dB worse than MRC Transmit diversity can obtain diversity gain even without channel information at transmitter.

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