Linear Phase Impulse Response Shortening for xDSL DMT modems by mek10591

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									      Impulse response shortening for xDSL discrete
        multitone modems with linear phase filters
                            Carlos Ribeiro1,2, Vitor Silva2 and Paulo S. R. Diniz3
           1
              Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria, 2401-951 Leiria, Portugal
  2
      Instituto de Telecomunicações, Departamento de Engenharia Electrotécnica e de Computadores, Universidade de
                                         Coimbra, 3030-290 Coimbra, Portugal
                                 3
                                   COPPE, Universidade Federal do Rio de Janeiro, Brasil
                               cribeiro@estg.iplei.pt; vitor.silva@co.it.pt; diniz@lps.ufrj.br


                         Abstract                               2. Channel shortening with linear phase filters
The problem of shortening the effective channel impulse         Unlike previously proposed method [7], the maximum
response in xDSL is addressed. A new approach is                shortening method [4] explicitly uses the length ν of the
described based on linear phase FIR equaliser filters. The      cyclic prefix (CP) as the desired effective length of the
mathematical framework for the LPIRS method is                  channel impulse response, thus minimising the channel
developed and simulation results are presented for several      Inter Symbolic Interference (ISI). As Melsa et al. referred
VDSL loops.                                                     in [4], there is no direct relation between the achievable bit
For a similar performance, the novel method offers a design     rate and the channel shortening, but this is certainly a good
with considerably lower computational complexity and a          criteria. The coefficients of the optimal shortening filter are
simpler solution, on either DSP or dedicated hardware           obtained by an eigenvector decomposition of the energy
implementations.                                                function SSNR, as described next.
                                                                Consider the effective channel heff in its matrix form as
                     1. Introduction                                                       heff = Hw,                       (1)
Discrete Multitone Modulation (DMT) is a powerful               where H is the convolution matrix built from the channel
modulation scheme [1], adopted by the standardisation           impulse response of length M and w is the equaliser filter of
bodies [2], [3] in the development of recent Digital            length t. By defining the vector hwin as a window of ν+1
Subscriber Line (xDSL) standards. One of the main issues        consecutive samples of heff, starting at sample d, and the
in the xDSL systems is the severe attenuation and distortion    vector hwall containing the remaining samples of heff, vectors
caused by the channel and other services on the same            hwin and hwall are given by (2) and (3), respectively,
bundle. To overcome these channel effects on the
transmitted signals, equalisation techniques are commonly       h win = H win w
used in time [4] - [8] and frequency [9] domains. The                                                              T
equalisation process can be simplified by periodically               =  heff (d ) heff (d + 1) ! heff (d +ν ) 
                                                                                                              
extending each symbol with a convenient number of prefix               h( d )     h(d − 1)          !   h(d − t + 1)   w(0) 
and suffix samples [1]. Hence, circular convolution can be             h(d + 1)    h( d )           ! h(d − t + 2)   w(1) 
used instead of linear convolution, so that the channel              =                                                ×          
equalisation can then be easily performed in the frequency             "             "              #        "          " 
                                                                                                                                 
domain. Unfortunately, this is only possible if the effective          h(d +ν ) h(d +ν + 1)
                                                                                                    ! h(d +ν − t + 1)   w(t − 1) 
                                                                                                                                  
length of the channel impulse response is shorter than the
                                                                                                                                  (2)
guard period (cyclic prefix and suffix). In order to shorten
                                                                h wall = H wall w
the impulse response, several methods have been proposed
                                                                                                                                             T
[4] - [8], based on Finite Impulse Response (FIR)                     =  heff (0) ! heff ( d − 1) heff ( d + ν + 1) ! heff ( M + t − 2) 
                                                                                                                                        
equalisers.
In this paper is proposed a novel method of equalisation for            h(0)             0      !        0         
                                                                                                                   
xDSL DMT modems based on linear phase FIR equalisers.                         "          "      #        "           w(0) 
The next section presents a review of the original Minimum              h(d − 1) h(d − 2) !          h(d − t )   w(1) 
Shortening Signal-to-Noise Ratio (MSSNR) method [4] and               =                                            ×          
its improvements [5]. Also, is described the development of             h( d + ν + 1) h(d + ν ) ! h(d + ν − t + 2)   " 
                                                                              "         #       "        "           w(t − 1) 
                                                                                                                                
                                                                                                                               
the proposed Linear Phase Impulse Response Shortening
                                                                       
(LPIRS) method and the mathematical framework, which                   
                                                                              0         !       0    h( M − 1)    
leads to an optimal solution for every type of linear phase                                                                              (3)
filter.                                                         The energy of these vectors can be expressed as
Simulation results are presented in section 3, which clearly
show that the achieved shortened signal noise ratios                      hT h wall = w T H T H wall w = w T Aw
                                                                           wall              wall                                        (4)
(SSNR) are similar to yielded by other methods [4], [5], at a                   win  =w
                                                                              h T h win        T
                                                                                                      = w Bw ,
                                                                                                   H T H win w
                                                                                                     win
                                                                                                                       T
                                                                                                                     (5)
considerably lower computational complexity.                    where A and B are presumably positive definite matrices.
Finally, conclusions about the new method are drawn in          The solution to the shortening problem can be found by
section 4, where it is discussed the influence of some design   either minimising (4) or maximising (5).
parameters, e.g. the filter type (symmetry) and length.

                                                                                                                                   1/4
Originally, the optimal solution was found by minimising                      Since linear phase filters can provide a prescribed
the energy of hwall [4]. To avoid an unbounded solution, a                    magnitude frequency selectivity with lower computational
unit energy constraint was imposed on hwin [6], given by                      complexity than a more general nonlinear phase filters, we
                               2                                              propose to constraint the equaliser filter w to have linear
                       h win = 1 .                  (6)
                                                                              phase (LPIRS method), as following described. Define
Assuming that matrix B is positive definite, we can use                                                w L = Lx ,                        (16)
Cholesky decomposition to obtain
                                                                              where wL is the linear phase equaliser filter of length t, L is
                       (      )(            )
                                                T             T
         B = QDQT = Q D Q D                         = B B ,             (7)   an extension matrix (see Table 1 and 2) and x is a generic
                                                                              vector.
where D is the diagonal matrix formed from the
eigenvalues of B, and the columns of Q are the orthonormal
eigenvectors of B. Defining the following matrix                                       Symmetric                      Anti-symmetric
                                                                                       1 0 0 !         0            1 0 0!                0
           (     ) A ( DQ ) = ( B )
                                                                  −1
                                                         A B  ,
                  −1        T −1                    −1       T
      C= Q D                                                          (8)            0 1 0           "            0 10                   "
                                                                                     0 0 #                        0 0 #                   
the optimal solution is given as                                                                                                            
                                                                                             0 1       0                    0 1           0
                                       −1
                                                                                       "               1            "                      1
                   w opt =  B  l min ,
                               T                                                              0 0                              0 0
                                                     (9)                          L=                            L=
                                                                                           0 0       1                    0 0          − 1
                                                                                             0 1       0                    0 −1          0
where lmin is the unit-length eigenvector corresponding to                             0 0 $                        0 0 $                   
the minimum eigenvalue λmin of C.                                                                                                           
The optimal shortening SNR is defined as                                               0 1 0           "            0 −1 0                 "
                                                                                       1 0 0 !
                                                                                                       0
                                                                                                                    − 1 0 0 !
                                                                                                                                            0
                      w T Bw opt 
     SSNRopt = 10 log T
                         opt        = −10 log(λ ) , (10)                       Table 1. Extension matrices L (t×t 2 ) for even length vectors.
                      w Aw                    min
                        opt   opt 

which depends on the minimum eigenvalue λmin of the                                   Symmetric                        Anti-symmetric
composite matrix C. This approach is valid as long as B is                            1 0 0 !          0              1 0 0!             0
non-singular and B exists, which is true if the length of                             0 1 0            "              0 10               "
the equaliser t is shorter than ν [5].                                                0 0 #                           0 0 #               
                                                                                                                                          
The solution to the shortening problem was found in [5], by                                 0 1        0                     0 1         0
maximising (5) and imposing a unit energy constraint on                             L="     0 0        1           L= "      0 0         0
hwall,                                                                                      0 1        0                     0 −1        0
                               2                                                      0 0 $                           0 0 $               
                      h wall = 1 ,                                                    0 1 0            "              0 −1 0             "
                                                (11)
                                                                                                                                          
                                                                                                                       − 1 0 0 !
under the assumption that matrix A is non-singular.
                                                                                      1 0 0 !          0                                  0
Therefore, matrix A can be decomposed using Cholesky
decomposition as                                                               Table 2. Extension matrices L (t×(t −1) 2 ) for odd length vectors.

                        (     )(
         A = QDQT = Q D Q D = A A           )
                                            T                 T
                                                                       (12)
                                                                              There are four possible L matrices depending on the length
and the new composite matrix C may be defined as                              of vector wL and type of symmetry adopted.
           (     ) B ( DQ ) = ( A )
                                                                  −1
                                                         B A  . (13)
                  −1        T −1                    −1       T                As in [4] and [5], we can rewrite (4) and (5) as
      C= Q D                                                  
                                                                                      h T h wall = w L T H T H wall w L = x T A L x
                                                                                          wall               wall                             (17)
The new solution is given in [5] as
                                       −1
                                                                                       h T h win = w L T H T H win w L = x T B L x ,
                                                                                         win               win                                (18)
                    w opt =  A  l max ,
                                   T
                                                       (14)                 where AL and BL are given by
                                  
                                                                                                     A L = LT AL ,                            (19)
where lmax is the unit-length eigenvector corresponding to
the maximum eigenvalue λmax of C.                                                                     B L = L BL .
                                                                                                                 T
                                                                                                                                 (20)
The new shortening SNR is then defined as                                     The solution proposed in [5] was the one adopted in this
                          w T Bw opt                                        work, since it was shown to be valid regardless of the
        SSNRopt = 10 log T
                              opt        = 10 log(λ ) . (15)                 lengths of either the CP or the equaliser.
                          w Aw                    max
                             opt   opt                                      Assuming that a proper delay d is chosen so that A is
This new solution is valid for any value of t and yielding a                  positive definite, we can show that AL is also positive
similar performance [5] as the original method [4].                           definite since
A common and key issue is to guarantee the non-singularity                     w L T Aw L = (Lx)T A (Lx) = x T (LT AL)x = x T A L x > 0 , (21)
                                                                              and the matrices L are full rank ( t 2 or (t − 1) 2 ,
of both A and B matrices. By ensuring that there is at least
one non-zero sample of heff outside hwin, it guarantees, by
definition, that A is positive definitive. The same principle                 respectively, for even or odd length filters).
can be applied to matrix B.                                                   As in [5], Cholesky decomposition is applied to matrix AL,

Linear phase is an important property in digital signal
                                                                                                  (
                                                                                           AL = Q D Q D  )(      )T
                                                                                                                      = AL AL .
                                                                                                                                   T
                                                                                                                                              (22)
                                                                              Using a similar development as in [5], we define
processing. It ensures the absence of phase distortion on the
signal (only an integer delay).

                                                                                                                                                 2/4
                                   (A )
                                                                    −1                    Fig. 3 shows the normalized impulse response of FSAN
                             BL  AL 
                                       T       −1
                           CL =         L             (23)                              Cabinet #3 test loop, with MSSNR and LPIRS (symmetric)
                                        
                                                                                          equalisation.
and the optimum solution for the equaliser wL is given by
                                                            −1                                    1

                         w L opt = L A L  l L max ,
                                          T
                                                             (24)
                                                                                              0.8

where l Lmax        is the unit-length eigenvector corresponding to
                                                                                                0.6
the maximum eigenvalue λ L max of CL.
                                                                                                0.4
It is important to highlight that the dimensions of matrices
AL and BL are t 2 × t 2 or (t − 1) 2 × (t − 1) 2 , thus resulting                               0.2

in a substantial complexity reduction on the Cholesky
decomposition and eigen analysis, as compared to [4] and                                          0

[5].
                                                                                                -0.2


                            3. Simulation results                                               -0.4
                                                                                                       0   500   1000    1500   2000    2500   3000   3500    4000
To evaluate the performance of the proposed method,
several experiments were carried out on different VDSL                                    Fig. 3. FSAN Cabinet #3 MSSNR (with dashed line) and LPIRS
test loops, as described in [3] and [10].                                                 equalised impulse response.
In Fig. 1 is shown the original impulse response of the
FSAN Cabinet #3 test loop, which is one of the most                                       For performance comparison purposes, several simulations
difficult VDSL test loops. Fig. 2 shows the normalized                                    results are presented in Table 3 and 4, using the same target
impulse response of the same loop after MSSNR                                             impulse response and number of carriers, as in the previous
equalisation.                                                                             simulations. In Table 3, an equaliser of length t = 10 is
The same input parameters were used for all simulations                                   used, whereas in Table 4, an equaliser of length t = 19 is
presented, i.e. 4096 carries, a target impulse response                                   used instead.
length of 64 samples and an equaliser w with length t = 10 .                              Considering the short equaliser filter lengths (low
                  -4
                                                                                          complexity implementation), the results in Tables 3 and 4
             x 10                                                                         show, from a practical point of view, a similar performance
        3
                                                                                          between the MSSNR and LPIRS methods, taking in
      2.5                                                                                 consideration the high levels of obtained SSNR (apart from
                                                                                          the highly demanding FSAN Cabinet #3 loop).
        2

                                                                                             Test Loops             MSSNR                       LPIRS
      1.5                                                                                                                              Symmetric      Anti-symmetric
                                                                                          ANSI VDSL Loop 3              70.00           69.94            69.90
        1                                                                                 ANSI VDSL Loop 4              38.61           38.38            36.96
                                                                                          ANSI VDSL Loop 7              47.18           39.17            44.79
      0.5                                                                                  FSAN FTTEx #3                73.65           68.36            67.47
                                                                                            FSAN Cab #3                 11.79           7.94              8.01
        0
             0             2000       4000             6000         8000          10000          Table 3. Results in dB for an even length equaliser.

                 Fig. 1. FSAN Cabinet #3 impulse response.                                                                                       LPIRS
                                                                                             Test Loops             MSSNR
                                                                                                                                       Symmetric      Anti-symmetric
                                                                                          ANSI VDSL Loop 3              71.72            71.72               71.62
       1.2
                                                                                          ANSI VDSL Loop 4              40.33            39.92               39.45
        1                                                                                 ANSI VDSL Loop 7              52.90            47.93               48.23
                                                                                           FSAN FTTEx #3                81.53            78.11               77.18
       0.8
                                                                                            FSAN Cab #3                 12.05            8.17                 9.06
       0.6
                                                                                                 Table 4. Results in dB for an odd length equaliser.

       0.4                                                                                                          4. Conclusions
                                                                                          A new approach to the problem of shortening the effective
       0.2
                                                                                          channel impulse response in xDSL has been described,
        0
                                                                                          using linear phase FIR equaliser filters. The use of the full
                                                                                          rank extension matrices is the crucial point in this novel
      -0.2                                                                                method. The resulting target functions are defined with half
             0      1000    2000   3000       4000   5000    6000   7000   8000    9000
                                                                                          size of the original matrices, leading to substantial
Fig. 2. FSAN Cabinet #3 original and MSSNR equalised impulse                              complexity reduction in obtaining the optimal solution.
response (with dashed line).                                                              Hitherto, all efforts leading to simplified solutions for the
                                                                                          implementation of VDSL modems are crucial, as neither


                                                                                                                                                                     3/4
DSP nor dedicated hardware solutions are able to
implement the full VDSL standards.
As shown in section IV, the equaliser performance attained
by the LPIRS method is similar to that yielded by the
reference method proposed by Melsa et al, used as a
reference.

                      References
 [1] John A. C. Bingham, “ADSL, VDSL and
      Multicarrier     Modulation,     Wiley     Series    in
      Telecommunications and Signal Processing, 2000.
 [2] ETSI TM, “VDSL – Part 2: Transceiver
      Specification”, TS 101 270-2 v.1.1.1, Feb 2001.
 [3] T1E1.4, “VDSL Metallic Interface – Part 1:
      Functional       Requirements       and       Common
      Specification”, Draft Trial-Use Standard, Feb. 2001.
 [4] Peter Melsa, Richard Younce and Charles Rohrs,
      “Impulse response shortening for discrete multitone
      transceivers”, IEEE Trans. on Communications, vol.
      44, nº 12, pp. 1662-1672, December 1996.
 [5] Changchuan Yin and Guangxin Yue, “Optimal
      impulse response shortening for discrete multitone
      transceivers”, Electronics Letters, vol. 34, pp. 35-36,
      January 1998.
 [6] D. Daly, C. Heneghan and A. Fagan, “A minimum
      mean-squared error interpretation of residual ISI
      channel shortening for discrete multitone
      transceivers”, Proc. IEEE International Conference
      on Acoustic Speech and Signal Processing
      (ICASSP), 2001.
 [7] Naofal Al-Dhahir, John Cioffi, “Optimum finite-
      length equalization for multicarrier transceivers”,
      Proc. IEEE Transactions on Communications, vol.
      44, nº1, pp 56-64, January 1996.
 [8] Romed Schur, Joachim Speidel, Ralf Angerbauer,
      “Reduction of guard interval by impulse
      compression for DMT modulation on twisted pair
      cables”,     IEEE     Global     Telecommunications
      Conference, pp. 1632-1636, November 2000.
 [9] Katleen Van Acker, Geert Leus, Marc Moonen,
      Olivier Van de Wiel and Thierry Pollet, “Per tone
      equalization for DMT-based systems”, Proc. IEEE
      Trans. on Communications, vol. 49, nº.1, pp. 109-
      119, January 2001.
 [10] Full Service Access Networks group (FSAN),
      www.fsanet.net.




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