Space-time Diversity Codes for Fading Channels

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Space-time Diversity Codes for Fading Channels

• by Professor R. A. Carrasco
• School of Electrical, Electronic and Computing
  Engineering
• University of Newcastle-upon-Tyne

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Summary

• Introduction
• Diversity frequency, time and space
• Space diversity and MIMO channels
• Maximum likelihood sequence detection for ST
  codes
• Space-time block coding
• Capacity of MIMO systems on fading channels
• Space-time trellis codes
• Code design
• System block diagram
• Example
• BER performance
• Conclusions

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1. Introduction

• Migration to 3G standards
• high data rate communications required
• high quality transmission and bandwidth efficient
  communications
• low decoding complexity
• Major obstacles to be solved
• Multipath fading signal is scattered among
  several paths, each path has a different time
  delay.
• Interference ISI in case of channels with
  memory
• multi-user interference

2 Mbps indoors
144 kbps outdoors
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2. Diversity

• Solution of the multipath fading problem, by
  transmission of several redundant replicas that
  undergo different multipath profiles
• Types
• Frequency diversity same information is
  transmitted on different frequency carriers,
  which will face different multipath fading.
• Time diversity replicas of the signal are
  provided in the form of redundancy in the time
  domain by the use of an error control code
  together with a proper interleaver
• s1 s2 s3 s1 time

redundant of
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2. Diversity

• Types (cont.)
• Space diversity redundancy is provided by
  employing an array of antennas, with a minimum
  separation of ?/2 between neighbouring antennas.
  Differently polarized antennas can also be used.

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3. Space Diversity and MIMO channels

• where
• ci(l) is the modulation symbol transmitted by
  antenna i at the time instant l. It is generated
  by a space-time encoder.
• gij is the path gain from Tx antenna i to Rx
  antenna j.
• ?j(t) is an independent Gaussian random variable
  (AWGN channel)

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3. Space diversity and MIMO channels

• Taking the equation
  the signals at the receiving
  antennas can be expressed in matrix form
• Therefore we can create the MIMO channel matrix

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4. Maximum likelihood sequence detection for ST
codes

• The probability of receiving a sequence
  if the code matrix
• Taking the likelihood function as the logarithm

has been transmitted is
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4. Maximum likelihood sequence detection for ST
codes

• When maximising the log-likelihood we eliminate
  the constant term. After this,
• the problem is equivalent to minimising the
  following expression
• This can be easily done with the Viterbi
  algorithm, using the above expression
• as a metric and computing the maximum likelihood
  path through the trellis.

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5. Space-time block coding
At time t, the signal , received at antenna j
is given by
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5. Space-time block coding

• where the noise samples ?jt are independent
  samples of a zero-mean complex Gaussian random
  variable with variance n/(2SNR) per sample
  dimension.
• The average energy of symbols transmitted from
  each antenna is normalised to be one.
• Assuming perfect channel state information is
  available, the receiver computes the decision
  metric
• over all codewords
• and decides in favour of the codeword that
  minimizes the sum.

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5. Space-time block coding

• Encoding algorithm
• A space-time block code is defined by a p x n
  transmission matrix H. The entries of the matrix
  H are linear combinations of the variable x1, x2,
  , xk and their conjugates. The number of
  Transmission antennas is n.
•  
• We assume that transmission at the baseband
  employs a signal constellation A, with 2b
  elements. At time slot 1, Kb bits arrive at the
  encoder and select constellation signals
  s1,,sK, setting xi si for i 1,2,.,K in H,
  we arrive at a matrix C with entries linear
  combinations of s1,s2,..,sK and their
  conjugates. So, while H contains indeterminates
  x1,x2,.,xK C contains specific constellation
  symbols.

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5. Space-time block coding

• Encoding algorithm
•  Examples H2 represents a code that utilizes two
  antennas, H3 represents a code that utilizes
  three antennas and H4 represents a code that
  utilizes four antennas.

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5. Space-time block coding

• Decoding algorithm
• Maximum likelihood decoding of any space-time
  block code can be achieved using linear
  processing at the receiver. Then maximum
  likelihood detection amounts to minimizing the
  decision metric
• over all possible values of s1 and s2.
• We expand the above metric and delete the terms
  that are independent of the code words and
  observe that the above minimization is equivalent
  to minimizing
• The above metric decomposes in two parts, one
  of which

(1)
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5. Space-time block coding

• is only a function of s1, and the other one
• is only a function of s2. Thus the minimization
  of (1) is equivalent to minimizing these two
  parts separately. This in turn is equivalent to
  minimizing the decision metric
• for detecting s1, and the decision metric
• for detecting s2.
• Similarly, the decoders for H3 and H4 can be
  derived.

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5. Space-time block coding

• The decoder for H3 minimizes the decision
  metric
• for decoding s1. The decision metric
• for decoding s2. The decision metric
• for decoding s3, and the decision metric
• for decoding s4.

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5. Space-time block coding

• For decoding H4, the decoder minimizes the
  decision metric
• for decoding s1. The decision metric
• for decoding s2. The decision metric
• for decoding s3, and the decision metric
• for decoding s4.

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5. Space-time block coding

• There are two attractions in providing transmit
  diversity via orthogonal designs.
• There is no loss in bandwidth, in the sense that
  orthogonal designs provide the maximum possible
  transmission rate at full diversity.
• There is an extremely simple maximum-likelihood
  decoding algorithm which only uses linear
  combining at the receiver. The simplicity of the
  algorithm comes from the orthogonality of the
  columns of the orthogonal design.

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6. Capacity of MIMO systems on fading channels

• For the single Tx/Rx channel the capacity is
  given by Shannons classical formula
• where B is the bandwidth
• g is the fading gain (the realization of a
  complex Gaussian random variable)
• For a MIMO channel of n inputs and m outputs, the
  capacity is now given by
• where Im is the identity matrix of order m
• snr is the signal-to-noise ratio per receive
  antenna
• G is the MIMO channel matrix
• denotes the transpose conjugate

bits/sec
bits/sec
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6. Capacity of MIMO systems on fading channels

• A particular case is when m n and G In
  (completely uncorrelated parallel sub-channels),
  then
• Conclusion
• Capacity can scale linearly with increasing snr
• Capacity can increase in almost n more bits/cycle
  for every 3 dB increase in the snr.

bits/sec
bits/sec/Hz
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6. Capacity of MIMO systems on fading channels

• Average capacity of a MIMO Rayleigh fading channel

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6. Capacity of MIMO systems on fading channels

• Channel correlation influence in the MIMO channel
  capacity
• Assume that all the received powers are equal.
  In this case we define
• where R is the normalized channel correlation
  matrix ( )
• whose components are
• Therefore
• , Where r correlation
  coefficient

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6. Capacity of MIMO systems on fading channels

• In the case of n gtgt 1 and r lt 1, we finally
  obtain
• When n ? 8
• When r 0 (H I)
• and

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Channel Capacity for 3dB 7dB
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Channel Capacity for 5dB 9dB
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Channel Capacity for 11dB 30dB
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7. Space-time trellis codes

• The matrix C is called the code matrix, whose
  element ci(l) is the symbol transmitted by
  antenna i at the instant l. and l 1, , L
• The system model is

..
Ant 1
?
..
Ant 2
.
.
.
..
Ant n
, where Es is the average symbol energy
?j(t) is an independent sample of a complex
Gaussian random variable with variance No/2 per
dimension.
gij(t) is a complex Gaussian random variable with
variance 0.5 per dimension.
Signal to noise ratio per receive antenna
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7. Space-time trellis codes

• The probability of decoding erroneously the code
  matrix C and choosing instead another code matrix
  E, assuming ideal channel state information, is
  given by
• where
• and the distance between codewords C and E is
  given by
• where ci element of
  matrix C
• and ei element of
  matrix E
• after some manipulation we rewrite the distance
  as

with
and
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7. Space-time trellis codes

• A(l) is an Hermitian matrix, therefore there
  exists a unitary matrix U and a diagonal matrix D
  such that
• where ?i(l) eigenvalues of
  matrix A(l)
• Let , so
• Considering the Chernoff bound of the error
  probability
• Now we must distinguish between two cases.

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7. Space-time trellis codes

• Quasi-static fading
• gij(l) is constant within a frame of length L and
  changes randomly from one frame to another.
• where r(A) is the rank of matrix A.
• Time-varying fading
• where ?(C,E) is the set of indexes of the all
  zero columns of the difference matrix C-E.
• where D C-E is the difference matrix
• r(D) is its rank
• O(D) is the set of column indexes that differ
  from zero.

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• Error Probability for fading channels.
• Single Input/Single Output (SISO)
• Multi-antenna (MIMO), from the Chernoff bound of
  the error probability.

(coherent binary PSK, Rayleigh fading)
(coherent orthogonal, Rayleigh fading)
(orthogonal, noncoherent, Rayleigh fading)
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• The output noise power of the branch K can be
  written as
• Assume that the total energy of a block is
  limited to Etot K.E0
• (E0 is the transmitting energy for each source
  symbol).

Where
Signal-to-Noise (SNR)
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• The total energy can also be expressed as
• Assume the constellation at the receiver
  satisfies ER ?.dR2 , where dR is the minimum
  distance of the constellation, and ? is a
  constant depending on the different
  constellations.
• Using minimum distance sphere bound, the instant
  symbol error rate bound is

Thus
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• We assume ?i,j are independent samples of zero
  mean complex Gaussian random variables having a
  variance of 0.5 per dimension. Thus
  are independent Rayleigh distributed with a PDF
  of
• Thus, the average symbol error rate bound is
  given by

Where
or
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Probability of error for different numbers of Tx
Rx antennas
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Probability of error for large number of Tx
antennas (n??)
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8. Code design

• Diversity advantage (DA) is the exponent in the
  error probability bound.
• In order to improve the performance of ST codes,
  the diversity advantage must be
• maximised by maximising the rank of the
  difference matrix.
• 1st Design criteria the minimum of the ranks of
  all possible matrices D C-E must
• be maximised. To achieve the full rank n all
  matrices D must have full rank.
• Coding gain (CG) is the term independent of SNR
  in the upper error bound.
• It is the product of eigenvalues of the
  difference matrix or of euclidean distances.
• 2nd Design criteria in order to maximise the
  coding gain, the minimum of the
• products of euclidean distance (or equivalently
  the eigenvalues) taken over all pairs of
• codes C and E must be maximised.

for quasi-static fading
for time-varying fading
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9. System block diagram
I. Encoder
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9. System block diagram
II. Decoder
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10. Example

• Delay diversity code
• QPSK modulation
• code rate ½
• 2 Tx antennas
• Encoder structure
• Example
• x 1 3 2 0 1
• c1 1 3 2 0 1
• c2 0 1 3 2 0

1 symbol delay
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10. Example

• The space-time decoding branch metric
  calculation is performed using the following
  equation
• For QPSK, received signal consists of I and Q
  components which are produced separately by the
  demodulator, so the magnitude of this signal must
  be taken. Making the above equation

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10. Example

• For two transmit antennas (n2), the equation
  becomes
• For two receive antennas (m2), the equation
  becomes

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10. Example

• The trellis structure for the Modulo 4, 4-state
  21/3 ST-Ring TCM code is

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10. Example

• From a computer simulation, with 3dB S/N and a
  Rayleigh fading variance set to 0.5 per
  dimension, the following metric is calculated
•  
• r1I 0.998 g11 1.983 c1 0.707 g21 0.554c2
  -0.707
• r1Q -2.027 g11 1.983 c1 0.707 g21 0.554
  c2 0.707
• r2I -0.734 g12 0.552 c1 0.707 g22 1.412
  c2 -0.707
• r2Q -1.586 g12 0.552 c1 0.707 g22 1.412
  c2 0.707
• 0
• 14.597
• 0.016
• 8.848
•  
• 0 14.579 0.016 8.848 23.461

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10. Example

• This value (23.461) is then used as a branch
  metric value (BMV) and assigned to the following
  trellis branch
• All other branch metric values are calculated in
  the same way (16 values per codespace pair for a
  4-state
• code). This procedure is then repeated for each
  received symbol pair through the trellis (from
  left to right)
• until the whole frame has been calculated.

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10. Example

• The path metric values (one value per code
  state) are then calculated in the following
  manner
•  
• At the start of the trellis, the PMV values are
  the same as the BMV values leading into that
  node
• For the repetitive trellis sections, there are
  four (in the case of a Modulo-4 code) paths
  leading into each node, three competitor paths
  and one survivor path.

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10. Example

• The second and further sets (deeper into the
  trellis) are calculated by adding the BMV value
  of a path entering a node to the PMV value from
  the previous node connected to that path to form
  a competitor path, this procedure is then
  repeated for all four paths and the smallest of
  these four calculated values is used to form the
  new node PMV value, if there is more than one
  value that is the smallest of the four, then one
  is chosen arbitrarily.
•  
• Competitor 1 28.6 2.9 31.5
• Competitor 2 0.1 16.0 16.1 Survivor
• Competitor 3 11.7 21.4 33.1
• Competitor 4 18.4 8.3 26.7
•  
• This procedure is then repeated for each node
  throughout the trellis, working from left to
  right.

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10. Example

• Once these values have been calculated, the
  traceback operation is performed which consists
  of identifying the most likely path through the
  trellis based on low PMV values. This operation
  is performed from the end of the trellis to the
  start (i.e. right to left) and so can only be
  performed when an entire frame has been received.
  The operation begins with the selection of the
  smallest PMV for the end (deepest) set. The path
  backwards (the overall survivor path) to the
  start of the trellis is calculated based on the
  selection of the smallest of the four PMVs
  joining the current node, if there exist more
  than one PMV with the smallest value then one is
  chosen arbitrarily.
• This path is then stored as the modulo-4 value
  of the systematic MCE output corresponding to the
  selected path. The survivor path can be seen in
  the trellis diagram above (in red). This stored
  encoder output sequence is then reversed in order
  (symbol by symbol) and becomes the decoded data.

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11. BER performance on time-varying Rayleigh
fading channels
, (Tx2), QPSK, ideal CSI
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12. Conclusions

• The use of space-time diversity techniques for
  transmission over fading channels offers a
  promising increase in the capacity. The capacity
  of MIMO channels can even increase linearly with
  the number of transmit antennas.
• In particular, space-time codes provide a
  significantly better performance than single
  antenna systems.
• The design criteria for space-time codes has been
  presented. These takes into account two factors
  the diversity advantage and the coding gain.
• Maximum-likelihood detection techniques can be
  applied in the decoding process without an
  increase on the decoder complexity.
• Computer simulations avail these statements by
  showing a smaller BER at a fixed SNR.

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Thank you