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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) 4/8/2015 11 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. 4/8/2015 21 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). 4/8/2015 31 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. Cophasing required 4/8/2015 41 Linear Diversity Combining is a random variable with PDF p ( ) and CDF P ( ) which depends on the type of fading and the choice of combining Most often PDF is obtained by differentiating CDF Pout p( 0 ) 0 p ( )d P ( 0 ) 0 Ps Ps ( ) p ( )d 0 Ps ( ) is the random probability of error for AWGN non-fading channel Most often closed form solution for CDF, Pout and Ps unavailable. Results based on computer simulation. 4/8/2015 51 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 4/8/2015 61 Selection Combining Combiner outputs the signal with the 2 r highest SNR i / N i 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. 4/8/2015 71 Selection Combining P ( ) p( ) p(max[ 1 ,, M ] ) M p( 1 M ) p( i ) i 1 (Assuming independent branches) For iid Rayleigh fading (ri Rayleigh, γi exponential): M P ( ) (1 e / i ) (1 e / ) M i 1 E( i ), same for all branches p ( ) 4/8/2015 dP ( ) d M (1 e / ) M 1 e / 81 Selection Combining M 1 p ( )d i 1 i 0 Pout ( 0 ) p( 0 ) P ( 0 ) (1 e 0 / ) M The average SNR gain (array gain) increases with M, but not linearly. 4/8/2015 91 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. 4/8/2015 101 Threshold (Switching) Combining For two-branch diversity with iid branch statistics: P 1 ( T ) P 2 ( ) T P ( ) p( T 2 ) P 1 ( T ) P 2 ( ) T For iid Rayleigh fading with E( i ) 1 e T / e / e ( T ) / P ( ) ( T ) / / 1 2 e e 4/8/2015 T T 111 Maximal Ratio Combining In the general model set i ai e j M M Then, r i ri e j ai ri i i 1 i i 1 Assuming the same noise psd at all branches: M r2 1 N tot N 0 ( ai ri ) 2 i 1 M 2 a i i 1 Maximizing by Cauchy-Schwartz inequality: 2 2 a r i / N0 4/8/2015 i 121 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: 1 M 2 M ri i N 0 i 1 i 1 M 1e / p ( ) M , 0 (M 1)! 4/8/2015 131 BER Performance of MRC Average Pb for Maximal Ratio Combining with iid Rayleigh Fading 4/8/2015 141 Equal Gain Combining In maximal ratio combining, set ai 1, i.e., i e -j 1 ri N 0 M i 1 M 2 Then, In general, no closed-form solution for P ( ) . For iid two-branch Rayleigh channel with same CDF in terms of Q function: P ( ) 1 e 2 / / e / [1 2Q( 2 / )] , 0 4/8/2015 151 i 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. 4/8/2015 161 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 (10log10 ( / )) for M-Branch compared to one branch. (a) Maximal Ratio, (b) Equal Gain, (c) Selection 4/8/2015 171 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) 4/8/2015 181 Channel Known at Tx We assume path gain ri e j is known at the transmitter. i s(t), transmitted signal with energy per symbol Es , is multiplied by constant complex gain αi : i ai e j , 0 ai 1 M 2 a Total energy constraint Es requires i 1 i i 1 Signals transmitted from M antennas are M combined in the air: r (t ) ai ri s(t ) 4/8/2015 i 1 191 Channel Known at Tx Similar to MRC at Rx coefficients are found to maximize SNR: ai ri M r i 1 2 i Es Resulting SNR: N0 M r i 1 Ps M Q( M ) M e M 1 Ps M i 1 1 M i / 2 i M 2 i i 1 M / 2 M M M exp M i / 2 i 1 M For large SNR: Ps M Similar to MRC full diversity order of M is achieved 2 4/8/2015 201 Channel Unknown at Tx Alamouti 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 4/8/2015 211 The Alamouti Scheme Channel gains: h1 r1e j , h2 r2 e j 1 Received signals: y= 4/8/2015 y1 h1 * * y 2 h2 2 y1 h1 s1 h2 s 2 n1 * * y h s h s 1 2 2 1 n2 2 h2 s1 n1 * * h1 s 2 n2 = HAs + n 221 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 = (z1 z2)T =(|h1|2+|h2|2)I2 ~ n s+ , where n~ H AH n is zero-mean complex Gaussian 2 2 H ~ ~ with E (n n ) ( h1 h2 ) N 0 I 2 . 4/8/2015 231 The Alamouti Scheme ( h1 h2 ) E s 2 So, z i ( h1 h2 ) si n~i , i 1, 2, and 2 2 i 2 2N 0 (Factor 2 represents transmission of half energy Es/2 per symbol per antenna) Observations: 1. Alamouti scheme achieves a diversity order of 2 although channel knowledge not available at transmitter 2. Alamouti scheme achieves an array gain of 1 4/8/2015 241 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 4/8/2015 251 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. 4/8/2015 261