Space-time Diversity Codes for Fading Channels by S03OBzbO

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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
                             Summary
1.    Introduction
2.    Diversity: frequency, time and space
3.    Space diversity and MIMO channels
4.    Maximum likelihood sequence detection for ST codes
5.    Space-time block coding
6.    Capacity of MIMO systems on fading channels
7.    Space-time trellis codes
8.    Code design
9.    System block diagram
10.   Example
11.   BER performance
12.   Conclusions
                            1. Introduction

•   Migration to 3G standards
                                                         2 Mbps indoors
       high data rate communications required
                                                        144 kbps outdoors
       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. 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.
                        S(f)




                                                        f [Hz]
                                       f1          f2

     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 … s 1                 time

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




                                       l/2


                               l/2
                       3. Space Diversity and MIMO channels
Tx 1        c1 (1)c1 (2)c1 (3)               g11                Rx 1
                                                                                   n

                                                    g21                 r1 (t )   g i1 (t )ci (t )  1 (t )
                                       g12
                            g1m                                                   i 1


Tx 2                                          g22               Rx 2              n
            c2 (1)c2 (2)c2 (3)                                         r2 (t )   g i 2 (t )ci (t )  2 (t )
                                                                                 i 1

                                 gn1




                                                                                          …
     ...




                                                          ...
Tx n                                                                              n
                                                                       rm (t )   g im (t )ci (t )  m (t )
            cn (1)cn (2)cn (3)                gnm               Rx m
                                                                                 i 1



 •     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.
        – ηj(t) is an independent Gaussian random variable (AWGN channel)
          3. Space diversity and MIMO channels
                                               n

Taking the equation: rm (t )   gim (t )ci (t )  m (t ) the signals at the
                               i 1
receiving antennas can be expressed in matrix form:

         r1 (t )   g11(t )     g 21(t )         g 31(t )     ... g n1 (t )   c1 (t )  1 (t ) 
         r (t )   g (t )       g 22 (t )        g 32 (t )    ... g n 2 (t )   c2 (t )  2 (t ) 
         2   12                                                                                  
         r3 (t )    g13(t )   g 23(t )         g 33 (t )    ... g n 3 (t )   c3 (t )   3 (t ) 
                                                                                                 
            |   |                   |               |         |       |  |   | 
         rm (t )  g1m (t )
                               g 2 m (t )       g 3m ( t )   ... g nm (t ) cm (t ) m (t )
                                                                                                    

Therefore we can create the MIMO channel matrix:
                         g11(t ) g 21(t ) g 31(t )                ...   g n1 (t ) 
                         g (t ) g (t ) g (t )                     ...   g n 2 (t ) 
                         12         22         32                                  
               G (t )   g13 (t ) g 23 (t ) g 33 (t )             ...   g n 3 (t ) 
                                                                                   
                         |            |         |                  |        | 
                         g1m (t ) g 2 m (t ) g 3m (t )
                                                                  ...   g nm (t ) m xn
                                                                                    
     4. Maximum likelihood sequence detection for ST codes
                                                                   r1 (1) r1 (2) ... r1 ( L) 
                                                                   r (1) r (2) ... r ( L) 
    The probability of receiving a sequence                     R 2       2          2           if the code matrix
                                                                   |          |   |      | 
                                                                                             
   c1 (1) c1 (2) ... c1 ( L)                                    rm (1) rm (2) ... rm ( L)
  c (1) c (2) ... c ( L) 
C 2       2          2         has been transmitted is:
   |         |    |      | 
                             
  cn (1) cn (2) ... cn ( L)                                       n
                                                                                        
                                                                                        2

                                                        rj (l )   g ij (l )ci (l )  
                                  L   m
                                              1                                          
            p ( R | C , G )                    exp            i 1
                                                                                        
                              l 1    j 1   N 0                   2N0               
                                                                                      
                                                                                      
    Taking the likelihood function as the logarithm:
                                                                     n
                                                                                        
                                                                                            2
                                                                                                
                               L m  
                                                           rj (l )   g ij (l )ci (l )        
                                           1                                                 
         ln p( R | C , G )   ln                              i 1
                                                                                               
                                         
                             l 1 j 1   N 0
                                                                     2N0              
                                                                                               
                                                        
                                                                                       
                                                                                               
                                                                                               
                                                                                                2
                                                                     n
                                                                                      
                                                L m 
                                                          rj (l )   g ij (l )ci (l )
                                 ln N 0                                        
                              mL                                    i 1
                                                                        2
                               2              l 1 j 1                4N 0
  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:

                                                                                                          2
                                                                    L   m
                                                                                     n
                                                                                                      
m(rj (l ), ci (l ) | g ij (l ); i  1,...,n; j  1,...m; l  1...L)   rj (l )   g ij (l )ci (l )
                                                                      l 1 j 1    i 1              

    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.
                5. Space-time block coding
                 C   1          rt1
                     t



  x1                                              R             ˆ
                                                                x1
                                                  E
                         gij                      C             ˆ
                                                                x2
                                                  E
                   n                m    t1      I
                C t            rt                 V
                                                  E
                                                  R
  xk                                                            ˆ
                                                                xk



                                        tm

At time t, the signal rt j, received at antenna j is given by
        n
rt   g ij Cti  t j
  j

       i 1
                      5. Space-time block coding
where the noise samples jt are independent samples of a zero-mean complex Gaussian
random variable with variance n/(2*SNR) 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
                         l   m               n               2

                        r   g C
                        t 1 j 1
                                    t
                                        j

                                            i 1
                                                   ij   t
                                                         i


over all codewords
                         1           1 2     n
                       C1 C12 ...C1nC2C2 ...C2 ...Cl1Cl2 ...Cln
and decides in favour of the codeword that minimizes the sum.
                      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.
                         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.


                x      x2 
         H2   1*
                x 2   x1 
                         *
                           
                x1      x2      x3             x1      x2      x3      x4 
                x      x1      x4            x      x1      x4     x3 
                2                              2                            
                 x3    x4      x1              x3    x4      x1      x2 
                                                                            
                 x      x3     x2              x      x3     x2      x1 
         H 3   *4                       H 4   *4
                x1       *
                         x2      x3 
                                  *              x1       *
                                                          x2       *
                                                                  x3      x4 
                                                                           *

                *                  *           *                        * 
                x 2            x4            x 2            x4
                          *                                *         *
                         x1                               x1              x3 
                x *     *
                         x4      x1 
                                  *              x *     *
                                                          x4       *
                                                                  x1      x2 
                                                                             *
                3                              3                            
                x 4    x3     x2             x 4    x3             x1 
                    *       *     *                  *       *     *       *
                                                               x2           
                                   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
                             m
                                  r j  g s  g s 2  r j  g s*  g s* 2 
                              1 1, j 1 2, j 2 2 1, j 2 1, j 1 
                            j 1                                          
                                                                                                                  (1)

  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
                                                          *
                                                               
             r1 j g1, j s1  r1 j g1, j s1  r1 j g 2, j s2  r1 j g 2, j s2  r2j g1 , j s2  r2j g1, j s2 
                     * *               *              *               *               *
                                                                                                 *        *

              j 1


                                                                  g
                                                                                     2

                                    g
                                                                  m       2
                                     j *
                     r g s  r                s  s1  s2
                      j   *                     *       2     2
                     2    2, j 1    2      2, j 1                             i, j
                                                                  j 1 i 1

        The above metric decomposes in two parts, one of which
                                           5. Space-time block coding
               m


               j 1
                      
               r1 g s  r1
                          j   *    *
                              1, j 1      g   j *
                                                      1, j s1  r g
                                                             2
                                                              j    *
                                                                    s  r
                                                                   2, j 1      g2
                                                                                   j *
                                                                                              s  s1
                                                                                              *
                                                                                         2, j 1                2
                                                                                                                        m   n

                                                                                                                     g i, j
                                                                                                                    j 1 i 1
                                                                                                                                2




 is only a function of s1, and the other one
                                                                                 
                                                                                                                    2

                               g                                g
   m                                                                                         m       2
  r2j g 2, j s2  r1         j *
                                          s  r2j g1, j s2  r     j *
                                                                            s  s2            
          *     *                                  *                          *          2
                                       2, j 2                     2      1, j 2                             i, j
  j 1                                                                                       j 1 i 1

 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

                                                     
                                                             2
              m j *                                                                               
                                        
                                                        m 2

               r1 g1, j  r2 g 2, j   s1    1   g i , j
                                                                                                 2
                              j *
                                                                                                      s1
                                                                                                            2
                                                                                                    
               j 1                                 j 1 i 1                                     
 for detecting s1, and the decision metric

                                                     
                                              2
                                                                2
               r1 g 2, j  r2  g1, j   s 2    1   g i , j  s 2
           m                                              m 2

        
                 j *          j *                                         2
                                                                   
         j 1                                         j 1 i 1  
 for detecting s2.
        Similarly, the decoders for H3 and H4 can be derived.
                                  5. Space-time block coding
          The decoder for H3 minimizes the decision metric
                                                                            
                                                                                           2
 m j *                                                                                                           2
                                                               
                                                                                                m 3

  r1 g1, j  r2 g 2, j  r3 g3, j  r5 g1, j  r6 g 2, j  r7 g3, j           s1    1  2 gi , j
                                                                                                               2
                 j *         j *        j *        j *         j *
                                                                                                                   s1
                                                                                                                  
  j 1                                                                                      j 1 i 1           
for decoding s1. The decision metric

                                                                                
                                                                                               2
 m j *                                                                                                                2
                                                                 
                                                                                           m 3
                                                                                   1  2 gi , j
  r1 g 2, j  r2 g1, j  r4 g 3, j  r5 g 2, j  r6 g1, j  r8 g 3, j   s2  
                                                                                                                    2
                  j *        j *         j *         j *        j *
                                                                                                                         s2
                                                                                                                        
  j 1                                                                                 j 1 i 1                     
for decoding s2. The decision metric

                                                                                  
                                                                                               2
 m j *                                                                                                                    2
                                                                 
                                                                                                        m 3

                                                                                              1  2 g i , j
                                                                                                                        2
                                                                                                                             s3
                                                *            *           *
        r1 g 3, j  r3j g1, j  r4j g 2, j  r5j g 3, j  r7j g1, j  r8j g 2, j
                         *            *
                                                                                       s3                               
  j 1                                                                                              j 1 i 1            
for decoding s3, and the decision metric

                                                                                      
                                                                                                   2
 m                                                                                                                        2
                                                                   
                                                                                                          m 3

   r2 g 3, j  r3 g 2, j  r4 g1, j  r6 g 3, j  r7 g 2, j  r8 g1, j                s4    1  2 g i , j
                                                                                                                        2
        j *         j *         j *        j *         j *         j *
                                                                                                                             s4
                                                                                                                           
  j 1                                                                                                j 1 i 1          
for decoding s4.
                                    5. Space-time block coding
    For decoding H4, the decoder minimizes the decision metric
                                                                                                      
                                                                                                                   2
   m j *                                                                                                                                   2
                                                                                  
                                                                                                                              m 4

    r1 g1, j  r2 g 2, j  r3 g 3, j  r4 g 4, j  r5 g1, j  r6 g 2, j  r7 g 3, j  r8 g 4, j            s1    1  2 gi , j
                                                                                                                                         2
                   j *         j *         j *         j *        j *         j *         j *
                                                                                                                                              s1
                                                                                                                                            
    j 1                                                                                                                  j 1 i 1       
for decoding s1. The decision metric

                                                                                                  
                                                                                                               2
   m j *                                                                                                                               2
                                                                                 
                                                                                                                      m 4

    r1 g 2, j  r2 g1, j  r3 g 4, j  r4 g 3, j  r5 g 2, j  r6 g1, j  r7 g 4, j  r8 g3, j     s2    1  2 gi , j
                                                                                                                                     2
                    j *        j *         j *         j *         j *        j *         j *
                                                                                                                                          s2
                                                                                                                                        
    j 1                                                                                                          j 1 i 1           
for decoding s2. The decision metric

                                                                                                      
                                                                                                                   2
  m j *                                                                                                                                    2
                                                                                 
                                                                                                                          m 4

   r1 g3, j  r2 g 4, j  r3 g1, j  r4 g 2, j  r5 g3, j  r6 g 4, j  r7 g1, j  r8 g 2, j           s3    1  2 gi , j
                                                                                                                                         2
                  j *         j *        j    *      j *        j *         j *        j *
                                                                                                                                              s3
                                                                                                                                            
   j 1                                                                                                               j 1 i 1           
for decoding s3, and the decision metric
                                                                                                          
                                                                                                                       2
m                                                                                                                                              2
                                                                                   
                                                                                                                    m 4
                                                                                                            1  2 gi , j
  r1 g 4, j  r2 g 3, j  r3 g 2, j  r4 g1, j  r5 g 4, j  r6 g 3, j  r7 g 2, j  r8 g1, j   s4  
                                                                                                                                             2
       j *         j *         j *         j *        j *         j *         j *         j *
                                                                                                                                                  s4
                                                                                                                                                 
 j 1                                                                                                           j 1 i 1                     
for decoding s4.
                    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.
      6. Capacity of MIMO systems on fading channels
•   For the single Tx/Rx channel the capacity is given by Shannon’s classical formula:

         C  B log 2 (1  snr  g ) bits/sec
                                   2



    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:

                                 snr  * 
         C  B log 2 det  I m       GG          bits/sec
                                 n      
    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
     6. Capacity of MIMO systems on fading channels
• A particular case is when m = n and G = In (completely uncorrelated
  parallel sub-channels), then:
                       snr  
     C  B log 2 det 1    I n         bits/sec
                       n  
                  snr 
     C  n log 2 1           bits/sec/Hz
                     n 
• Conclusion:
    o Capacity can scale linearly with increasing snr
    o Capacity can increase in almost n more bits/cycle for every 3 dB increase
      in the snr.
                                        6. Capacity of MIMO systems on fading channels
• Average capacity of a MIMO Rayleigh fading channel
                                   60


                                   55


                                   50


                                   45


                                   40
  Average Capacity [bits/sec/Hz]




                                   35


                                   30


                                   25


                                   20


                                   15


                                   10


                                   5


                                   0
                                        0   1   2      3      4   5   6     7   8   9     10   11   12   13    14   15   16   17   18   19    20   21   22     23   24   25
                                                                                                    SNR [dB]

                                                    N=1 M=1       N=2 M=1       N=1 M=2        N=2 M=2         N=2 M=4        N=2 M=6        N=4 M=4         N=8 M=8
      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:
    j   g ij  1
                     2

            i
                     snr 
   C  log 2 det I     R
                      n 
where R is the normalized channel correlation matrix ( rij  1 )
whose components are
                1
   rij 
             i j
                     g  k
                             ki   g kj   g ki g kj
                                    *             *



Therefore
                                                              
                                       1  snr n              
                snr
   C  n log 2 1 
                           2 
                              
                      1  r   log 2                        
                   n                 1  snr 1  r 2
                                      
                                            n
                                                              , Where r = correlation
                                                                coefficient
                                                               
      6. Capacity of MIMO systems on fading channels
•   In the case of n >> 1 and r < 1, we finally obtain
                 snr        2 
    C  n log 2 1    (1  r ) 
                    n          
    When n → ∞
          snr
    C         (1  r )
                     2

          ln 2
    When r = 0 (H = I)
                 snr 
    C  n log 2 1    
                    n 
    and
           snr
    C 
           ln 2
                                            Channel Capacity for 3dB & 7dB

                                                                  3 dB & 7 dB SNR


                        8
                                                                                                                       4 Antennas
                                                                                                                       10 Antennas
                        7                                                                                              20 Antennas
                                                                                                                       50 Antennas
                                                                                                                       4 Antennas
                        6                                                                       7 dB SNR
                                                                                                                       10 Antennas
                                                                                                                       20 Antennas
Capacity (bit/sec/Hz)




                        5                                                                                              50 Antennas



                        4

                                                      3 dB SNR
                        3



                        2



                        1



                        0
                            0   0.1   0.2       0.3         0.4              0.5                0.6        0.7   0.8   0.9           1
                                                                  Correlation Coefficient (r)
                                             Channel Capacity for 5dB & 9dB


                                                             5 dB & 9 dB SNR


                        12
                                                                                                                          4 Antennas
                                                                                                                          10 Antennas
                                                                                                                          20 Antennas
                        10
                                                                                                                          50 Antennas
                                                                                                                          4 Antennas
                                                                                                                          10 Antennas
                         8                                                                                                20 Antennas
Capacity (bit/sec/Hz)




                                                                                                        9 dB SNR          50 Antennas


                         6


                                                                5 dB SNR

                         4




                         2




                         0
                             0   0.1   0.2       0.3   0.4                0.5               0.6   0.7              0.8   0.9            1
                                                              Correlation Coefficient (r)
                                        Channel Capacity for 11dB & 30dB


                                                     11 dB & 30 dB SNR


                        250                                                                               4 Antennas
                                                                                                          10 Antennas
                                                                                                          20 Antennas
                                                                                                          50 Antennas
                                                                                                          4 Antennas
                        200
                                                                                                          10 Antennas
                                                                                                          20 Antennas
                                                                                                          50 Antennas
Capacity (bit/sec/Hz)




                        150
                                                                                                                30 dB SNR



                        100




                         50
                                                                            11 dB SNR



                          0
                              0   0.1   0.2   0.3   0.4               0.5               0.6   0.7   0.8   0.9               1
                                                          Correlation Coefficient (r)
                              7. Space-time trellis codes
Ant 1                   ..…   c1 (1)c1 (2)....c1 ( L)          c1 (1) c1 (2) c1 (3) ... c1 ( L) 
                                                              c (1) c (2) c (3) ... c ( L) 
Ant 2                   ..…   c2 (1)c2 (2)....c2 ( L)      ⇒C 2       2      2          2      
                                                               |         |      |           | 
                                                                                                
Ant n                   ..…   cn (1)cn (2)....cn ( L)          cn (1) cn (2) cn (3) ... cn ( L)

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:
                    n
   rj (t )  E s  g ij (t )ci (t )   j (t )          , where Es is the average symbol energy
                   i 1
  ηj(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:
          nE s
    snr 
          N0
                                 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:
                  d 2 (C , E )E                              
P(C  E | G )  Q               s                            
                       2N0                                   
                                                             
                     1                   
                                     
                                                   x2 / 2
where       Q( x )                           e              dx
                     2               x

and the distance between codewords C and E is given by
                          L      m        n                                2

      d 2 (C , E )    g ij (l )ci (l )  ei (l )
                                                                                     where ci = element of matrix C
                                                                                       and ei = element of matrix E
                         l 1 j 1 i 1
after some manipulation we rewrite the distance as:
                          L      m
      d (C , E )   v j (l ) A(l )v j (l )
        2

                         l 1 j 1
                  g1 j (l )                     (c1 (l )  e1 (l ))(c1 (l )  e1 (l )) ... (c1 (l )  e1 (l ))(cn (l )  en (l )) 
                  g (l )                                                                                                          
                                                   (c2 (l )  e2 (l ))(c1 (l )  e1 (l )) ...
                                     and A(l )                                                                                     
                                                                                                                 |
with: v j (l )   2 j 
                  |                                                |                    |                     |                   
                                                                                                                                  
                 
                  g nj (l ) 
                                                (cn (l )  en (l ))(c1 (l )  e1 (l )) ... (cn (l )  en (l ))(cn (l )  en (l ))
                                                                                                                                    
                          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
                            l1 (l )       0 
                                                
          UA(l )U  D  
                       *
                                       .              where li(l) = eigenvalues of matrix A(l)
                            0                   
                                         ln (l ) 
          h1 j (l )      
         h (l )                                                   n                   2

    Let  2 j   v j (l )U *
          |                                                
                                    , so v j (l ) A(l )v j (l ) 
                                                                  i 1
                                                                       li (l ) hij (l )
                    
          hnj (l ) 
                    
    Considering the Chernoff bound of the error probability:
                                d 2 (C , E )E s 
         P(C  E | G )  exp  
                                                
                                                 
                                     4N0        
                                  Es n                         2
                         exp  
                                  4N         li (l ) hij (l ) 
                                                                 
                          j ,l           0 i 1                 
 Now we must distinguish between two cases.
                            7. Space-time trellis codes
A.       Quasi-static fading
gij(l) is constant within a frame of length L and changes randomly from one frame to another.
                                    m              r ( A) m
                             
                          r ( A)
                                          Es 
       P(C  E | G )    li 
                                       
                                          4N 
                        i 1              0 

where r(A) is the rank of matrix A.

B.    Time-varying fading
                                                                      m
                                     n        2 Es 
       P(C  E | G )     ci (l )  ei (l )
                                                   
                      l ( C , E )  i 1      4N0 
                                                    
where Ν(C,E) is the set of indexes of the all zero columns of the difference matrix C-E.
                                                                 m              m r( D )
                                         n
                                                            2         Es 
        P(C  E | G )                  ci (l )  ei (l ) 
                                  )  i 1                           
                                                                       4N 
                               l ( D                                  0 

where D = C-E is the difference matrix
      r(D) is its rank
      Ω(D) is the set of column indexes that differ from zero.
• Error Probability for fading channels.
   Single Input/Single Output (SISO)
           1
   Pb                 (coherent binary PSK, Rayleigh fading)
        4 Eb N 0
            1
   Pb             (coherent orthogonal, Rayleigh fading)
        2 Eb N 0
              1
   Pb 
        2  Eb N 0 
                      (orthogonal, noncoherent, Rayleigh fading)

               L'
                               1
   Pe (a, c)  
                           Es 2
               k 1
                      1      d k ( a, c )
                           N0


   Multi-antenna (MIMO), from the Chernoff bound of the error probability.
The output noise power of the branch K can be written as:


        m   P                                        m   n
 Pn   t , K  gj 2 gj        N 
                                       gi, j N0
                               2                2

      j 1 t 1
                
                       t ,K k  0
                                     j 1 i 1

             P                                   n           2

             g               g tj, K  k    g i , j
                    j     2                2
Where
             
             
            t 1
                   t ,K                      i 1
                                                                                                                2
                                                                                           m n                 
                                                                                            g i , j
                                                                                                            2
                                                                                                                 ES
 Signal-to-Noise (SNR)                SNRrev     
                                                   ER
                                                                m
                                                                     E yK
                                                                       n
                                                                              2         
                                                                                         
                                                                                            j 1 i 1
                                                                                             m     n
                                                                                                                
                                                                                                                

                                                                                          
                                                   PN                               2                        2
                                                                             gi, j N 0                   gi, j N 0
                                                                 j 1 i 1                   j 1 i 1

                      m       n
        SNRrev   gi , j
                                       2   ES
                      j 1 i 1
                                                N0

Assume that the total energy of a block is limited to Etot = K.E0
(E0 is the transmitting energy for each source symbol).

Using the orthogonal martix H to transmit the sequence (x1,x2,…..,xK)
The total energy can also be expressed as:

           P n i 2     n P i 2                           n K        2
 Etot  E   Ct   E  Ct                           E  X k   n.k .ES
           t 1 t 1   t 1 t 1                          t 1 k 1  
 Thus            E0
          ES 
                 n
                                   2                           2
                  m   n
                           2                   m       n
                                                          2  E0
                 gi , j  Es
          ER                                  gi , j 
                                             
                                                            n
                j 1 t 1                    j 1 t 1   

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:

                            dR            E       
 Pe , instg i , j   exp  
                               2

                            4P      exp   R
                                           4P     
                                                     
                              N               N   
              n m                   
              
                                2
                           g ij      
               i 1 j 1             exp   A             
                                                     n m
    Pe  exp                     E0        
                                                            exp  A m n
                    4 nN 0                       i 1 j 1   
             
                                    
                                     
                                                                             1 
                                                                                      mn

                                          e A  1  A             Pe          
                                                                            1  A 
                       E0
    Where A 
                    4 nN 0
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:

                                  
         P g i , j  2 g i , j exp  g i , j
                                               2
                                                   exp g 
                                                         i, j
                                                                2



Thus, the average symbol error rate bound is given by

                               nm
                  
                  
    Pe           
              1
              Eb                                                    
           1
          4 N n                        or
                                                                   E
                                                      Pe  exp   b m ;    ( n  )
                0                                             4 N 
                                                                    0 
                  Probability of error for different numbers of Tx & Rx
                                         antennas

     1.E+00



     1.E-01



     1.E-02



     1.E-03
                        Pe (2 Tx 2 Rx)
                        Pe (2 Tx 4 Rx)
Pe




     1.E-04             Pe (4 Tx 2 Rx)
                        Pe (4 Tx 4 Rx)
                        Pe (4 Tx 8 Rx)
     1.E-05
                        Pe (8 Tx 4 Rx)
                        Pe (8 Tx 8 Rx)

     1.E-06



     1.E-07



     1.E-08
              1     2      3             4   5   6              7   8   9   10   11   12
                                                     SNR (dB)
              Probability of error for large number of Tx antennas (n)

     1.E+00
                                                                           Pe (2 Rx)
                                                                           Pe (4 Rx)
                                                                           Pe (6 Rx)
     1.E-01                                                                Pe (8 Rx)
                                                                           Pe (10 Rx)
                                                                           Pe (15 Rx)
     1.E-02                                                                Pe (20 Rx)
                                                                           Pe (25 Rx)
                                                                           Pe (30 Rx)

     1.E-03                                                                Pe (40 Rx)
                                                                           Pe (50 Rx)
Pe




     1.E-04




     1.E-05




     1.E-06




     1.E-07
              1    2     3     4    5     6              7   8   9   10   11            12
                                              SNR (dB)
                                   8. Code design
A.    Diversity advantage (DA) is the exponent in the error probability bound.
       DA  r ( A)m         for quasi-static fading
       DA  r ( D)m         for time-varying fading
      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.

B.    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.
                          9. System block diagram
I. Encoder
                                                                             P1
                                                                                  ~
                                                                                  c1               S1(t)


                                     c1
                                                        ~
                                                        c1    Frame
                                          Interleaver        Building                  Modulator


               x
                                     c2                 ~
                                                        c2
  Inf
                   Space-Time Ring                            Frame
         Map                              Interleaver        Building                  Modulator
                       Encoder                                                                             S2(t)




                                                                                                                   …
  b0
binary
                                             …




                                                                                          …
                                                                …
                                     cn                 ~
                                                        cn
                                                              Frame
                                          Interleaver        Building                  Modulator
                                                                                                           Sn(t)




                                                                        Pn
                                                                             ~
                                                                             cn
                        9. System block diagram
 II. Decoder

 r1(t)




         Detection &       r1(k)
         Demodulation
                                                 rj      Deinterleaver

 r2(t)
                                                                                                 x  Z4
                                                                                                 ˆ
                                                                                                           binary
                                                                                Space-Time                 output
                                                                               Viterbi Decoder
                                                                                                     Inv
         Detection &    r2(k)       Channel                                                          Map
                                                                                                                    ˆ
                                                                                                                    b0
         Demodulation                                                            (Sequence
                                   Estimation
                                                                                 Detection)

                                                         Deinterleaver
                                                 ~
                                                 gi, j

rm(t)    Detection &       rm(k)
         Demodulation


                                                In case of iterative channel estimation
                                                 10. Example
1.    Delay diversity code
           QPSK modulation
           code rate ½
           2 Tx antennas
                                          x ∈ ℤ4
     Encoder structure:                                                                                    c1
                                                                                                                output
                                                                                                                antenna 1
                                            input

                                                                                                                output
                                                                     D                                     c2   antenna 2



     Example:             State
                           0         0/00
                                                        0                           0/00                   0                0           0          0
     x =13201                                                        1/10                  0/01
                                         1/10                                                                                               1/10
     c1 = 1 3 2 0 1                                         3/30

                                                                             2/20                                                0/02
                                          2/20          1
     c2 = 0 1 3 2 0                                                                                1/11    1                1           1          1
                                                                    2/21                          1/12
                                  3/30

                                                            0/02


                                                                   2/22                                    2
                                                    2                                                                       2           2          2
         1 symbol delay                                                      1/13    3/31
                                                                                                                2/23
                                                            0/03
                                                                                      3/32
                                                                          2/23

                                                    3                               3/33                   3                3           3          3

                                                                                                    transition label: x /c 1c2
                                 10. Example
The space-time decoding branch metric calculation is performed using the following
equation:
                             m              n                            2

                             r (l )   g
                             j 1
                                    j
                                           i 1
                                                      ij   (l )ci (l )

    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:


                                                                                        2
          m                                 2                                   2
                           n
                                                             n
                                                                               
            rjI (l )   gij (l )ci (l )    rjQ (l )   gij (l )ci (l )  
         j 1           i 1                             i 1               
                                                                                 

                                                  2                                 2
             m
                            n
                                                              n
                                                                                
             rjI (l )   gij (l )ci (l )    rjQ (l )   gij (l )ci (l ) 
            j 1          i 1                             i 1              
                                          10. Example

       For two transmit antennas (n=2), the equation becomes:



r    jI (l )  g1 j (l )c1 (l )  g 2 j (l )c2 (l )  rjQ (l )  g1 j (l )c1 (l )  g 2 j (l )c2 (l ) 
 m
                                                       2                                                      2

j 1



     For two receive antennas (m=2), the equation becomes:


r1I (l )  g11(l )c1 (l )  g 21(l )c2 (l ) 2  r1Q (l )  g11(l )c1 (l )  g 21(l )c2 (l ) 2 
r2 I (l )  g12 (l )c1 (l )  g 22 (l )c2 (l ) 2  r2Q (l )  g12 (l )c1 (l )  g 22 (l )c2 (l ) 2
                                  10. Example
      The trellis structure for the Modulo 4, 4-state 21/3 ST-Ring TCM code is:

State = 0          00              00                00                00               00

              32              32        33      32        33      32        33               33




State = 1                                 21                21                21
                             13           10   13           10   13           10


                        20 22             20   22           20 22             20 22

State = 2                                 02                02                02
                                  03 01             03 01             03 01
                        12   11           12   11           12   11           12   11

                              31        30      31        30      31        30

State = 3                          23                23                23


       Systematic Symbol (c1)           Parity Symbol (c2)
                                      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

r1I (l )  g11(l )c1 (l )  g21(l )c2 (l )2  0.998  1.983  0.707  0.554  0.7072
          =0
r
 1Q   (l )  g11(l )c1 (l )  g21(l )c2 (l )   2.027  1.983  0.707  0.554  0.707
                                         2                                                      2


          = 14.597
r2 I (l )  g12 (l )c1 (l )  g22 (l )c2 (l )2   0.734  0.552  0.707  1.412  0.7072
          = 0.016
r
 2Q   (l )  g12 (l )c1 (l )  g22 (l )c2 (l )   1.586  0.552  0.707  1.412  0.707
                                         2                                                      2


          = 8.848

          0 + 14.579 + 0.016 + 8.848 = 23.461
                                              10. Example
        This value (23.461) is then used as a branch metric value (BMV) and assigned to the following
        trellis branch:
                       State = 0       28.6

                                     0.1                    5.2



                                                                                        31.7
                       State = 1                       14.1
                                                                               29.3
                                                     16.0

                                              11.7                8.3
                                                                                      15.3
                       State = 2                                        10.2
                                                        23.5
                                              18.4
                                                                                      5.9
                                                                          11.0

                       State = 3                                                      5.9

                                              Calculated Metric


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.
                                           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:

                                                         [28.6]
                                                28.6

                                              0.1


                                                           [0.1]


                                                    11.7




                                                         [11.7]
                                                       18.4




                                                         [18.4]




      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.
                                  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.
                                                                    [28.6]           [16.1]
                                                                               2.9



Competitor 1:   28.6 + 2.9 = 31.5                                            16.0
Competitor 2:   0.1 + 16.0 = 16.1 = Survivor                         [0.1]

Competitor 3:   11.7 + 21.4 = 33.1
Competitor 4:   18.4 + 8.3 = 26.7
                                                                       21.4


                                                                    [11.7]
                                                                         8.3




                                                                    [18.4]



This procedure is then repeated for each node throughout the trellis, working from left to right.
                                  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.
                                  [28.6]     [16.1]      [2.0]     [16.7]     [2.6]
                State = 0




                                   [0.1]     [14.8]     [12.7]      [2.5]
                State = 1




                State = 2
                                  [11.7]      [8.7]      [9.9]     [10.1]




                State = 3
                                  [18.4]      [1.4]     [14.8]     [15.4]


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.
                11. BER performance on time-varying Rayleigh fading
                                     channels
            0
      10


           -1

      10



       -2

      10



       -3
      10
BER




       -4

      10



       -5

      10



           -6

      10



       -7
      10         0    1    2    3   4     5     6    7    8    9    10    11   12   13     14   15   16   17   18      19   20   21   22   23   24   25   26
                                                                                SNR [dB]


                                        BER Performance of 4-state ST-RTCM codes with one, two and four receive antennas , (Tx=2), QPSK, ideal CSI



                  Uncoded QPSK                Delay Diversity m=1        Baro et al. m=1             ST-RTCM m=1                  Delay Diversity m=2
                  Baro et al. m=2             ST-RTCM m=2                Delay Diversity m=4         Baro et al. m=4              ST-RTCM m=4
                              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.
Thank you

								
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