CHAPTER 5 DIVERSITY Xijun Wang

WEEKLY READING

1. Goldsmith, “Wireless Communications”, Chapters 72. Tse, “Fundamentals of Wireless Communication”,

Chapter 3

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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.

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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.

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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 bydiversity combining.

n Diversity can be provided across time, frequency and space.

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DIVERSITY

n Time diversityn 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.

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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

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TIME DIVERSITY

n One simple scheme¨ Repetition coding + interleaving

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SIGNAL MODEL IN FLAT FADING CHANNEL

n Transmit a codeword x = [x1, . . . , xL]t of length L symbols

n The received signal is

n Assuming ideal interleaving so that consecutive symbols xl are transmitted sufficiently far apart in time

n Assume that the hl’s are independent

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REPETITION CODING

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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

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ERROR PROBABILITY

n Consider BPSK modulation, with x1 = ±a.n The error probability conditioned on h

n Chi-squared distribution with 2L degrees of freedom.

n At high SNR

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DEEP FADES BECOME RARER

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ERROR PROBABILITY

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L is the diversity order

EXAMPLE: TIME DIVERSITY IN GSM

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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.

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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.

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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.

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FREQUENCY DIVERSITY

n Frequency-selective fading channeln 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.

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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

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SINGLE-CARRIER WITH ISI EQUALIZATION

n A sequence of uncoded independent symbols x[1],x[2],... is transmitted over the frequency-selective 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?

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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.

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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

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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 ML, where M is the constellation size for each symbol x[m].

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FINITE STATE MACHINE

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

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TRELLIS GRAPH

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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 log-likelihood ratio breaks into a sum

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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.

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TRELLIS GRAPH

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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

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CAPACITY WITH TIME & FREQUENCY DIVERSITY

n Outage probability for uniform power allocation

n Outage probability for non-uniform powerallocation

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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.

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A GEOMETRIC VIEW

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

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.

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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

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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.

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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.

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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

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Reduce to scalar channel

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

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ALAMOUTI CODE

n With flat fading, the two transmit, single receive channel

n Transmits two complex symbols u1 and u2 over two symbol times ¨ time 1:¨ time 2:¨ Assume

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ALAMOUTI CODE

n Rewrite the received signal

n Define the new vector

n Then, we haven Decouples due to the diagonal nature of z

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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 zi

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 si is transmitted using half

the total symbol energy.n Transmit diversity with CSIT

¨ Achieves an array gain and a diversity gain of 2.

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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

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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

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Effective channel gain

ALAMOUTI CODE

n Transmitter

n Receiver¨ The received signal in the first time slot is

¨ The received signal in the second time slot is

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ALAMOUTI CODE

n Combination

n Detection

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ALAMOUTI CODE

n Diversity order of 4

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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

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2 × 2 MIMO SCHEMES

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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

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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)

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SLOW FADING SIMO CHANNEL

n Receive diversity¨ Outage probability¨ At high SNR¨ Outage capacity

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At low SNR and small ε

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

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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

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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.

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SLOW FADING MISO CHANNEL

n Transmit diversity without CSIT¨ Outage probability with L antennas

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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 an-tenna at a time to transmit.

¨ to achieve the same outage probability for the same target rate R

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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.

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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.

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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)

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