# CHAPTER 5 DIVERSITY. Xijun Wang

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1 CHAPTER 5 DIVERSITY Xijun Wang

2 WEEKLY READING 1. Goldsmith, Wireless Communications, Chapters 7 2. Tse, Fundamentals of Wireless Communication, Chapter 3 2

3 FADING HURTS THE RELIABILITY n The detection error probability decays exponentially in SNR in the AWGN channel n While it decays only inversely with the SNR in the fading channel. 3

4 FADING HURTS THE RELIABILITY n With only a single signal path There is a significant probability that this path will be in a deep fade. When the path is in a deep fade, any communication scheme will likely suffer from errors. n A natural solution to improve the performance is Let the information symbols pass through multiple signal paths, Each of which fades independently, Reliable communication is possible as long as one of the paths is strong. 4

5 DIVERSITY n Sending signals that carry the same information through different paths n Multiple independently faded replicas of data symbols are obtained at the receiver end n Independent signal paths have a low probability of experiencing deep fades simultaneously n More reliable detection can be achieved by diversity combining. n Diversity can be provided across time, frequency and space. 5

6 DIVERSITY n Time diversity n Frequency diversity Frequency-selective fading channel n Space diversity with multiple transmit or receive antennas spaced sufficiently far enough n Macro diversity In a cellular network, the signal from a mobile can be received at two base-stations. 6

7 TIME DIVERSITY n Time diversity is achieved by averaging the fading of the channel over time Information is coded and the coded symbols are dispersed over time in different coherence periods Different parts of the codewords experience independent fades. n Typically, the channel coherence time is of the order of 10 s to 100 s of symbols 7

8 TIME DIVERSITY n One simple scheme Repetition coding + interleaving 8

9 SIGNAL MODEL IN FLAT FADING CHANNEL n Transmit a codeword x = [x 1,..., x L ] t of length L symbols n The received signal is n Assuming ideal interleaving so that consecutive symbols x l are transmitted sufficiently far apart in time n Assume that the h l s are independent 9

10 REPETITION CODING 10

11 COHERENT DETECTION n The receiver structure is a matched filter and is also called a maximal ratio combiner. it weighs the received signal in each branch in proportion to the signal strength and also aligns the phases of the signals in the summation to maximize the output SNR n Sufficient statistic 11

12 ERROR PROBABILITY n Consider BPSK modulation, with x 1 = ±a. n The error probability conditioned on h n Chi-squared distribution with 2L degrees of freedom. n At high SNR 12

13 DEEP FADES BECOME RARER 13

14 ERROR PROBABILITY L is the diversity order 14

15 EXAMPLE: TIME DIVERSITY IN GSM 15

16 EXAMPLE: TIME DIVERSITY IN GSM n The maximum possible time diversity gain is 8. n Amount of time diversity limited by delay constraint and how fast channel varies. 16

17 EXAMPLE: TIME DIVERSITY IN GSM n In GSM, delay constraint is 40ms (voice). n The coherence time should be less than 5 ms to obtain the maximum diversity. n For fc = 900 MHz, this translates into a mobile speed of at least 30 km/h. n For a walking speed of say 3 km/h, there may be too little time diversity. 17

18 EXAMPLE: TIME DIVERSITY IN GSM n Frequency hopping consecutive frames (each composed of the time slots of the 8 users) can hop from one 200 khz sub-channel to another. With a typical delay spread of about 1μs, the coherence bandwidth is 500 khz The total bandwidth equal to 25 MHz is thus much larger than the typical coherence bandwidth of the channel and the consecutive frames can be expected to fade independently. 18

19 FREQUENCY DIVERSITY n Frequency-selective fading channel n This diversity is achieved by the ability of resolving the multipaths at the receiver. n One simple communication scheme Sending an information symbol every L symbol times. The maximal diversity gain of L can be achieved Only one symbol can be transmitted every delay spread. 19

20 FREQUENCY DIVERSITY n Once one tries to transmit symbols more frequently, inter-symbol interference (ISI) occurs n How to deal with the ISI while at the same time exploiting the inherent frequency diversity in the channel Single-carrier systems with equalization Direct sequence spread spectrum Multi-carrier systems 20

21 SINGLE-CARRIER WITH ISI EQUALIZATION n A sequence of uncoded independent symbols x[1],x[2],... is transmitted over the frequencyselective channel n Assuming that the channel taps do not vary over these N symbol times, the received symbol at time m is n Can we still get the maximum diversity gain of L, even though there is no coding across the transmitted symbols? 21

22 SINGLE-CARRIER WITH ISI EQUALIZATION n Uncoded transmission combined with maximum likelihood sequence detection (MLSD) achieves full diversity on symbol x[n] using the observations up to time N +L 1, i.e., a delay of L 1 symbol times. n Compared to the scheme in which a symbol is transmitted every L symbol times, the same diversity gain of L is achieved and yet an independent symbol can be transmitted every symbol time. 22

23 THE VITERBI ALGORITHM n Given the received vector y of length n, MLSD requires solving the optimization problem n A brute-force exhaustive search The complexity grows exponentially with the block length n n Viterbi algorithm exploit the structure of the problem recursive in n so that the problem does not have to be solved from scratch for every symbol time 23

24 FINITE STATE MACHINE n The key observation is that the memory in the frequency-selective channel can be captured by a finite state machine n The number of states is M L, where M is the constellation size for each symbol x[m]. 24

25 FINITE STATE MACHINE n A finite state machine when x[m] are ±1 BPSK symbols and L = 2 25

26 TRELLIS GRAPH 26

27 MLSD n The received symbol y[m] is given by n The MLSD problem Conditioned on the state sequence s[1],...,s[n], the received symbols are independent and the loglikelihood ratio breaks into a sum 27

28 MLSD n The optimization problem can be represented as the problem of finding the shortest path through an n-stage trellis n Each state sequence (s[1],..., s[n]) is visualized as a path through the trellis n the cost associated with the mth transition is n Let Vm(s) be the cost of the shortest path to a given state s at stage m. 28

29 TRELLIS GRAPH 29

30 CAPACITY WITH TIME & FREQUENCY DIVERSITY n Parallel channels n Without CSIT, a reasonable strategy is to allocate equal power P to each of the sub-channels. n If the target rate is R bits/s/hz per sub-channel, then outage occurs when 30

31 CAPACITY WITH TIME & FREQUENCY DIVERSITY n Outage probability for uniform power allocation n Outage probability for non-uniform power allocation 31

32 CODING STRATEGY n With CSIT, dynamic rate allocation and separate coding for each sub-channel suffices. n Without CSIT separate coding would mean using a fixed-rate code for each sub-channel and poor diversity results: errors occur whenever one of the sub-channels is bad. coding across the different coherence periods is now necessary: if the channel is in deep fade during one of the coherence periods, the information bits can still be protected if the channel is strong in other periods. 32

33 A GEOMETRIC VIEW Coding across the sub-channels. Separate, non-adaptive code for each sub-channel. 33

34 ANTENNA DIVERSITY n The antennas has to be placed sufficiently far apart. The required antenna separation depends on the local scattering environment as well as on the carrier frequency. For a mobile which is near the ground with many scatterers around, the channel decorrelates over shorter spatial distances, and typical antenna separation of half to one carrier wavelength is sufficient. For base stations on high towers, larger antenna separation of several to 10 s of wavelengths may be required. 34

35 RECEIVE DIVERSITY n A flat fading channel with 1 transmit antenna and L receive antennas If the antennas are spaced sufficiently far apart, then the gains are independent Rayleigh. Optimal reception is via match filtering (receive beamforming). the error probability of BPSK conditioned on the channel gains 35

36 RECEIVE DIVERSITY n Power gain (also called array gain) by having multiple receive antennas and coherent combining at the receiver, the effective total received n Diversity gain by averaging over multiple independent signal paths, the probability that the overall gain is small is decreased. 36

37 RECEIVE DIVERSITY n Diversity gain If the channel gains are fully correlated across all branches, then we only get a power gain but no diversity gain as we increase L. When all the hl are independent there is a diminishing marginal return as L increases n Power gain suffers from no such limitation: a 3 db gain is obtained for every doubling of the number of antennas. 1 37

38 TRANSMIT DIVERSITY WITH CSIT n L transmit antennas and 1 receive antenna n When this gain is known at the transmitter, the system is very similar to receiver diversity with MRC n Transmit beamforming maximizes the received SNR by in-phase addition of signals at the receiver Reduce to scalar channel 38

39 TRANSMIT DIVERSITY W/O CSIT n L transmit antennas and 1 receive antenna n To get a diversity gain of L without CSIT simply transmit the same symbol over the L different antennas during L symbol times. At any one time, only one antenna is turned on and the rest are silent. Simple but wasteful of degrees of freedom 39

40 ALAMOUTI CODE n With flat fading, the two transmit, single receive channel n Transmits two complex symbols u 1 and u 2 over two symbol times time 1: time 2: Assume 40

41 ALAMOUTI CODE n Rewrite the received signal n Define the new vector n Then, we have n Decouples due to the diagonal nature of z 41

42 ALAMOUTI CODE n A linear receiver allows us to decouple the two symbols sent over the two transmit antennas in two time slots n The received SNR thus corresponds to the SNR for z i n Alamouti scheme Achieves a diversity order of 2, despite the fact that channel knowledge is not available at the transmitter. Achieves an array gain of 1, since s i is transmitted using half the total symbol energy. n Transmit diversity with CSIT Achieves an array gain and a diversity gain of 2. 42

43 TRANSMIT AND RECEIVE DIVERSITY n A MIMO channel with two transmit and two receive antennas n Both the transmit antennas and the receive antennas are spaced sufficiently far apart n The maximum diversity gain that can be achieved is 4 43

44 REPETITION CODE n Transmit the same symbol over the two antennas in two consecutive symbol times Time 1 Time 2 n Performing maximal-ratio combining of the four received symbols yields a 4-fold diversity gain Effective channel gain 44

45 ALAMOUTI CODE n Transmitter n Receiver The received signal in the first time slot is The received signal in the second time slot is 45

46 ALAMOUTI CODE n Combination n Detection 46

47 ALAMOUTI CODE n Diversity order of 4 47

48 MORE DEGREE OF FREEDOM n Spatial Multiplexing transmit independent uncoded symbols over the different antennas as well as over the different symbol times (V-BLAST) diversity gain of 2, since there is no coding across the transmit antennas full use of the spatial degrees of freedom should allow a more efficient packing of bits joint detection of the two symbols is required 48

49 2 2 MIMO SCHEMES 49

50 CAPACITY n Linear Time-Invariant Gaussian Channels known to both the transmitter and the receiver optimal codes for these channels can be constructed directly from an optimal code for the basic AWGN channel. n Fading Channels Slow fading 50

51 SIMO AND MISO CHANNEL n Receive beamforming in SIMO channels the projection of the L- dimensional received signal on to h n Transmit beamforming in MISO channels send information only in the direction of the channel vector h n Capacity (only power gain) 51

52 SLOW FADING SIMO CHANNEL n Receive diversity Outage probability At high SNR Outage capacity At low SNR and small ε 52

53 SLOW FADING MISO CHANNEL n Transmit diversity with CSIT Outage probability with CSIT same as the corresponding SIMO system achievable only if the transmitter knew the phases and magnitudes of the gains perform transmit beamforming, i.e., allocate more power to the stronger antennas and arrange the signals from the different antennas to align in phase at the receiver 53

54 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT Outage probability with 2 antennas using Alamouti Scheme 3 db loss in received SNR In the Alamouti scheme, the symbols sent at the two transmit antennas in each time are independent. Each of them has power P/2. In transmit beamforming, the symbols transmitted at the two antennas are completely correlated in such a way that the signals add up in phase at the receive antenna 54

55 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT Outage probability with 2 antennas using Alamouti Scheme Alamouti scheme has the best outage probability among all schemes which radiates energy isotropically. If h1, h2 are i.i.d. Rayleigh, that correlation never improves the outage performance the Alamouti scheme has the optimal outage performance for the i.i.d. Rayleigh fading channel. 55

56 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT Outage probability with L antennas 56

57 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT Outage probability with L antennas using repetition scheme The same symbol is transmitted over the L different antennas over L symbol periods, using only one antenna at a time to transmit. to achieve the same outage probability for the same target rate R 57

58 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT to achieve the same outage probability with Alamouti scheme for the same target rate R, the SNR has to be increased by a factor of For a fixed R, the performance loss increases with L: the repetition scheme becomes increasingly inefficient in using the degrees of freedom of the channel. For a fixed L, the performance loss increases with the target rate R. 58

59 SLOW FADING MISO CHANNEL n Transmit diversity without CSIT to achieve the same outage probability with Alamouti scheme for the same target rate R, the SNR has to be increased by a factor of For a small R, we have the repetition scheme is very sub-optimal in the high SNR regime where the target rate can be high it is nearly optimal in the low SNR regime. 59

60 MAIN POINTS n Fading hurts the reliability of the system (we saw 30dB possible power penalty). n Reliability is increased by providing more signal paths that fade independently. n Diversity can be provided across time, frequency and space. n Macro-diversity vs. Micro-diversity Macro-diversity: Combat shadowing Micro-diversity: Combat small scale fading (Rayleigh) 60

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