Multiple Antenna Techniques 1 Introduction 2 Introduction In - PowerPoint PPT Presentation

Chapter 5 Multiple Antenna Techniques 1

Introduction 2

Introduction In mobile systems, a key requirement is not to increase the total transmitted power, since more transmitted power generally means more interference. The multiple antenna techniques can increase received power without increasing transmitted power. This achieves through improved power gain. An important concept covered in this chapter is antenna diversity to overcome fading effect. This is because the probability that all antennas fade simultaneously is relatively small if they are separated by sufficient distances, as illustrated by the figure below. No CSIT is required in this case. g average g minimum g 0 required position 3

Introduction We will focus on relatively simple single-input multiple-output (SIMO) and multiple-input single-output (MISO) systems. We will only briefly outline the multiple-input multiple-output (MIMO) principle that has been identified as a key technology for future cellular systems. Chapter outline: Part 1 Antenna combining techniques Part 2 Space-time coding Part 3 MIMO 4

5.1 Antenna combining techniques 5

Selection combining Combing techniques refer to the methods of utilizing the signals from different antennas. A common technique is switched combining, in which the best signal is selected from different antennas. In theory the best signal is the one with maximum SNR. In practice, however, it is difficult to judge and normally the strongest signal is selected. 6

Selection combining Define local mean signal power at antenna = i p g = i  i 2 mean noise power where i =1, 2, … , M for M antennas. We assume that p i is exponentially is a constant. Then g i has an exponential pdf  2 distributed (Rayleigh fading) and g − i − g = g g g  1 p ( ) e , for all 0 single antenna i i The outage is defined as g  g Pr( ) i 0 This is the probability that the signal from one antenna is below a threshold value g 0 . The outage can be calculated using a cumulated distribution function (cdf) given below. g g − 0 0  g = g  g = g g = − g P ( ) Pr( ) p ( ) d 1 e single antenna 0 i 0 i i 0 7

Selection combining The outage probability that the best signal among M antennas falling below a threshold value g 0 is given by the following cdf: M  g  − 0 (5.3) g = g  g = g  g = − g   M P ( ) Pr( ) ( Pr( ) ) 1 e   M antennas 0 max 0 i 0   g It characterizes switching combining with M antennas, average SNR and threshold g 0 We can also get a pdf by taking differentiation to cdf in (5.3). − M 1    g g − − 0 0 g = g g = g g − g    p ( ) dP ( ) / d Me / 1 e    M antennas 0 M antennas 0 0    The average SNR at the combining output is given by − M 1    g g   − − 0 0   g g g = g g g − g g    p ( ) d Me / 1 e d    0 M antennas 0 0 0 0 (5.4)    0 0 M 1  = g i 8 = i 1

Selection combining M  g  − 0 g = g  g = g  g = − g   M P ( ) Pr( ) ( Pr( ) ) 1 e   M antennas 0 max 0 i 0   9

Outage probability of selection combining in Rayleigh fading 10

Gain combining With gain combining, the signal from the i th antenna is scaled by a complex coefficient a i (meaning adjustment on both amplitude and phase) and summed. The received signal for the i th antenna is given in its phasor form as = +  r h d (5.6) i i i where h i is a channel coefficient, d is the transmitted signal and η i is a noise sample with variance  2 . For simplicity, assume the same noise power for every antenna i . The output of the combiner is ( ) M M M    = = + a  (5.7) r a r a h d i i i i i i = = = i 1 i 1 i 1 11

Background: coherent signal adding Consider   +  =   +  +   +  cos(2 ft ) cos(2 ft ) cos(2 ft ) 1 1 2 2 Given  1 ,  2 ,  1 and  2 , we can find  and  using a phasor diagram:  e j    j 2 e 2  e j    1 e j      +  and max(  ) =  1 +  2 at Clearly, 1 2  1 =  2 . We say that the addition is coherent when  1 =  2 . Coherent addition is crucial for power maximization. 12

Adding non-coherent and coherent variables We now discuss more general cases of adding multiple variables M =  (5.8a)  i z , i = 1 where { z i } are phasors: =   +  j 2 ft z e (5.8b) i i i We consider two extreme cases: (i) { z i } are independent random variables with zero means. In this case, from probability theory, ( ) ( ) ( ) M ( ) M     = = 2 2 E = Var Var z E z (5.8c) i i = = i 1 i 1 This is referred to as non-coherent adding. In particular, if all { z i } have the same average power, then ( ) ( )   2 2 E = M E z (5.8d) i 13

Adding non-coherent and coherent variables The phases of { z i } are aligned, i.e.,  1 =  2 = …=  M . In this case, ( ) 2 M   2 (5.8e) = z i = i 1 This is referred to as coherent adding. In particular, if all { z i } have the same power, then   2 2 2 = M z (5.8f) i From (5.8d) and (5.8f), coherent adding has a power gain of M times than non- coherent adding. The advantage of coherent adding can also be seen from the following general inequality: ) ( 2 2 M M    z z (5.8g) i i = = i 1 i 1 The above shows the optimality of coherent adding. 14

Phase alignment We now consider a set of variables { z i = a i h i }. Clearly, to achieve coherent adding, we can properly choose { a i } to align the phases of { a i h i }. Example: Let { a i = ± 2} and { h i = ± 2}. We have two cases: Non-coherent adding: All { a i } and { h i } are independent. Then   = ( ) 2 M M ( )   =   = 2 2 E  a h  E a h M 2 2 16 M  i i  i i = = i 1 i 1 Coherent adding: We choose a i = h i for every i . Then ) (     2 2 ( ) M M ( )   = =   2 = 2 E  a h  E  a h  M 2 2 16 M  i i   i i  = = i 1 i 1 Note: In the above, non-coherent adding leads to “ power adding ” while coherent adding leads to “ magnitude adding ” . The former may involve signal cancelation since the signs of { a i h i } can be different, while the latter always involve signal enhancement since the signs of {| a i h i |} are all non-negative. 15

Equal gain combining (EGC) Let = for i =1, 2, …, M .  j h h e i i i With EGC, the combining coefficients are = for i =1, 2, …, M . −  j a e i i Clearly, for i =1, 2, …, M. | a i |=1 The combining output is ( ) M M M    = = +   − j r a r h d e i i i i i = = = i 1 i 1 i 1 The above operation is referred to as “phase alignment” or “co - phasing” . It −  j results in coherent adding. In practice, each e is realized by a time delay i circuit of  i . We will come back to this later. 16

Power gain with EGC EGC improves SNR statistically. To see this, define SNR at receiver output (after combinging) = effective power gain SNR at the transmitter Let P = | d | 2 . Then SNR at the transmitter is P /  2 . ( ) 2 =  M signal power after combiner h P i = i 1   2 M  =   =  − j 2 n oise power after combine r E  i e  M i  i  = 1 Note that  2 is actually the noise power at the receiver. Thus P /  2 is only a reference value that is useful in measuring the effects of the channel and receiver. ( ) ( )  2 M  M 2 SNR after the combiner: i h P / = 1 i ( )  2 M SNR gain: i h / M = 1 i 17

Example: EGC is not optimal. Let r 1 =1+  1 and r 2 =-2+  2 . Compare 3 methods: Method 1: a 1 =1, a 2 =1, r = r 1 + r 2 + (  1 +  2 )= -1 + (  1 +  2 ), SNR = (-1) 2 /2  2 =1/2  2 . Method 2(EGC): a 1 =1, a 2 =-1, r = r 1 - r 2 + (  1 -  2 )= 3 + (  1 -  2 ), SNR = 9/2  2 =4.5/  2 . Method 3: (MRC) a 1 =1, a 2 =-2, r = r 1 -2 r 2 + (  1 -2  2 ) = 5 + (  1 -2  2 ), SNR = 25/5  2 =5/  2 . 18

Time domain derivation for the co-phase operation We now summarize two different approaches to the derivation of the co-phase operation. Time domain approach : Consider a transmitted cosine signal (ignoring noise) =  s t ( ) d cos(2 f t ) c Let the received signal on antenna i be ( ) =  +  r t ( ) h d cos 2 f t i i c i where is the amplitude gain and  i is the phase change due to delay. The co- h i phasing operation adds a phase of -  i to r i ( t ) and ( ) M ( )  =  +  −  r t ( ) h d cos 2 f t (5.10) i c i i = i 1 This is to align the phases of different r i ( t ). 19