Channel%20Equalization%20To%20Achieve%20High%20Bit%20Rates%20In%20Discrete%20Multitone%20Modulation%20Systems

1
Channel Equalization To Achieve High Bit Rates
In Discrete Multitone Modulation Systems

• Ming Ding
• Ph.D. Defense
• Committee members
• Prof. Ross Baldick
• Prof. Melba M. Crawford
• Prof. Brian L. Evans (Advisor)
• Prof. Robert W. Heath, Jr.
• Prof. Edward J. Powers

April 21, 2004
2
Outline

• Introduction
• Unification of Discrete Multitone (DMT)
  Equalization
• Common Mathematical Framework
• Case Studies
• Contributions in DMT Equalization Methods
• Symmetric Design
• Minimum Intersymbol Interference Methods
• Filter Bank Equalization
• Simulation Results
• Conclusions

3
Multicarrier Modulation

• Divide wideband channel into narrowband
  subchannels
• Subchannel is approximately flat
• DMT is baseband muliticarrier modulation method
• Band partition based on fast Fourier transform
  (FFT)
• Line code for asymmetric digital subscribe line
  (ADSL) and very-high speed digital subscriber
  line standards

4
DMT Transmission

• Quadrature Amplitude Modulation (QAM)
  constellation mapping in each subchannel
• Composed of N/2 complex-valued subsymbols
• Mirror and conjugate subsymbols to obtain
  real-valued inverse FFT output

5
Cyclic Prefix (CP)

• Prepended to each DMT symbol
• Serves as guard time to combat intersymbol
  interference (ISI)
• Converts linear convolution of transmitted symbol
  and channel impulse response into circular
  convolution
• FFT of circular convolution is product of FFTs
• Allows receiver to remove ISI if cyclic prefix
  length 1 is greater than length of channel
  impulse response
• Reduces throughput by a factor of

copy
copy
s y m b o l ( i1)
CP
CP
s y m b o l i
N samples
v samples
6
Bit Loading in DMT

• Number of bits allocated to ith subchannel
• SNRi is SNR in subchannel i
• ?i is SNR gap to channel capacity
• Turn off subchannels that cannotsupport minimum
  number of bits
• Bit rate
• Channels with length longer than cyclic prefix
  cause ISI
• Significantly lowers SNR and bit rate
• Channel equalization essential for combating ISI

?i 9.8 dB in uncoded DMT ADSL/VDSL system
Symbol rate is 4 kHz inDMT ADSL/VDSL system
7
ADSL TransceiverData Transmission Subsystem
N/2 subchannels
N real samples
QAM mapping (Trellis)
mirror data and N-IFFT
add cyclic prefix
P/S
D/A transmit filter
superframe scramble, encode, interleave tone order
ATM
TRANSMITTER
channel
RECEIVER
N real samples
N/2 subchannels
reverse function
time domain equalizer
QAM decisiondevice (Viterbi)
N-FFT and remove mirrored data
S/P
remove cyclic prefix
receive filter A/D
N/2 complex multiply units
8
Conventional Two-Step Equalization

• Channel modeled as finite impulse
• response filter plus additive noise
• Time domain equalizer (TEQ)
• Finite impulse response filter
• Shortens channel impulse responseto be at most n
  1 samples
• Converts linear convolution to circular
• Frequency domain equalizer (FEQ)
• Single division per subchannel (tone)
• Compensate for amplitude/phase distortions
• Design objectives
• High bit rates at fixed bit error rate
• Low implementation complexity

9
Linear Equalizer Structures
up to N/2 FEQs
up to N/2 TEQs
TEQ
Sliding FFT
Goertzel filter bank
N-Point FFT
Per-tone Equalizers
Complex TEQ Filter Bank
Time Domain Equalizer Filter Bank
10
Equalizer Training Complexity

• Periodic 4-QAM training sequence
• No cyclic prefix
• Constant transmit power spectrum Sx
• Receiver monitors additive noise power spectrum Sn

Example ADSL Parameters FFT Size N 512 TEQ
Length Lw 17 Martin, Vanbleu, Ding et al.
2004
Multiplications Additions Memory (Words)
Single TEQ O(Lw3) Lw
TEQ Filter Bank O(Lw2N2) N/2 Lw
Per Tone Equalizer O(Lw2N LwN2) N Lw
Complex Filter Bank O(Lw2N LwN2) N Lw
11
Outline

• Introduction
• Unification of DMT Equalization
• Common Mathematical Framework
• Case Studies
• Contributions in DMT Equalization Methods
• Symmetric Design
• Minimum Intersymbol Interference Methods
• Filter Bank Equalization
• Simulation Results
• Conclusions

12
Unification of Equalizer Design Algorithms

• Most algorithms minimizeproduct of
  generalizedRayleigh quotients
• For M 1, solution is generalizedeigenvector of
  the matrix pair(B, A) corresponding to
  smallestgeneralized eigenvalue
• For M gt 1, solution is not well-understood
• Various searching methods exist to find a local
  optimum

13
Single Quotient Cases

• Minimum Mean Square Error Chow Cioffi, 1992
• Minimizes squared error between output of TEQ w
  and output of virtual target impulse response
  filter b
• Maximum Shortening SNR Melsa et al. 1996
• Channel convolution matrix H

nk
yk
Channel
TEQ
ek
Adependson D
xk
w
h


hwin
A and Bdependon D
hwall
14
Single Quotient Cases

• Minimum Intersymbol Interference Arslan et al.
  2000
• Minimum Delay Spread Schur et al. 2001
• Modified Maximum Shortening SNR with distance
  weighting

Generalization of MaximumShortening SNR method
withfrequency weighting
d1
n1
d2
k 0, 1, 2,, N-1 c center of mass
channel taps
15
Multiple Filters (each with a Single Quotient)

• Per-tone equalization Acker et al. 2001
• Generalized eigenvalue problem for each tone i
• Received frame (CP symbol) is y and ith FFT
  coefficient is Yi
• Time domain equalizer bank Milosevic et al.
  2002

16
Product of Quotients

• Bit rate
• Maximum Geometric SNR Al-Dhahir et al. 1995
• Additive white Gaussian Noise (AWGN), Sequential
  Quadratic Programming
• Maximum Bit Rate Arslan et al. 2001
• ISI AWGN, Quasi-Newton algorithm
• Maximum Data Rate Milosevic et al. 2002
• ISI Cross-talk Echo digital noise floor
• Almogy and Levin iteration
• Bitrate Maximizing Vanbleu et al. 2003
• Eventually all possible noises and interference
  resources
• Recursive Gauss-Newton update

17
Outline

• Introduction
• Unification of DMT Equalization
• Common Mathematical Framework
• Case Studies
• Contributions in DMT Equalization Methods
• Symmetric Design
• Minimum Intersymbol Interference Methods
• Filter Bank Equalization
• Simulation Results
• Conclusions

18
Contribution 1Infinite Length TEQ Results

• Eigenvectors of a doubly symmetric matrix
• Maximum Shortening SNR TEQ with unit energy
• A HT DT D H converges asymptotically to doubly
  symmetric HT H
• Minimum Mean Square Error TEQ
• Target impulse response is symmetric/skew
  symmetric
• A becomes a doubly symmetric matrix

symmetric
skew symmetric
19
Contribution 1 Observation of Long TEQ Designs

• Minimum Mean Square Error TEQs
• Target impulse response is
• approximately symmetric
• Maximum Shortening SNR TEQs
• A and B are almost doubly symmetric
• w becomes almost perfectly symmetric
• Minimum Intersymbol Interference TEQs
• Same as Maximum Shortening SNR case
• Can exploit symmetry in TEQ designs
• Force TEQ to be symmetric
• Compute half of TEQ coefficients
• Apply symmetry

20
Contribution 1 Symmetric TEQ design

• Implementation instead of finding eigenvector of
  Lw ? Lw matrix, find eigenvector of
  matrix
• Some matrix operations O(Lw3))
• Phase response of symmetric TEQ is linear
• Phase response fixed when
• given TEQ length
• No amplitude scaling needed
• for 4-QAM
• Enables design of FEQ in parallel

21
Contribution 2 Minimum ISI Method

• Advantages
• Push ISI to unused subchannels or subchannels
  with lower SNR
• Practical real-time implementation on digital
  signal processors
• Disadvantages
• TEQs longer than ? 1 taps
• B is not invertible method fails
• Cholesky decomposition sensitive to
• fixed-point computation
• High computational cost when performing
• delay optimization (A and B depend on ? )

22
Contribution 2 Improving Minimum ISI Method

• Define new cost function
• weighting value for subchannel i
• HT H is always positive definite and invertible
• Suitable for arbitrary length TEQ design
• Reduces computational cost when performing delay
  optimization

Does not depend on ?
23
Contribution 2 Quantized Frequency Weighting

• Min-ISI weighting in each subchannel is
  
• On-off quantization
• Compare noise power with threshold
  
• Choose zero weights in subchannels with
  larger-than-threshold noise power
• Choose unit weights in other subchannels
• Choose threshold as noise power forsupporting 2
  bits in subchannel

ADSL fixes Sx -40 dBm/Hz ?gap 9.8 dB During
training
24
Contribution 2 Iterative Minimum ISI Method

• Obtain weighting values for subchannel i
• Pre-compute
  and
• Choose step size ?
• Start with non-zero initial guess w0, and
  iteratively calculate wk, using deterministic
  gradient search

Chatterjee, et. al 1997
Division-free iteration
Method avoids Cholesky decomposition and directly
calculates generalized eigenvector associated
with minimum eigenvalue
25
Contribution 3 Complex Filter Bank Equalization

• Move all FEQ operations to time domain
• Combine with TEQ to obtain multi-tap
  complex-valued FIR filter bank

26
Contribution 3 Design of Filter Bank

• For each subchannel, define at
  FEQ output
• Classical MMSE solution for TEQ for each
  subchannel
• Quadratic cost function leads to iterative
  implementation use deterministic steepest descent
  search
• Different delays can be introduced on each
  subchannel
• Introduce different TEQ length to each subchannel
• Upper bound on achievable bit rate performance

27
Contribution 3 Dual-path TEQ

• Each path exploits a different TEQ aiming at
  optimize over a different subset of data-carrying
  subchannels
• Advantages
• Less frequency selectivity makes equalization
  easier
• Achieve higher data rates than conventional
  structure at relatively low implementation cost
  Examples

PFFT Partial FFT
28
Outline

• Introduction
• Unification of DMT Equalization
• Common Mathematical Framework
• Case Studies
• Contributions in DMT Equalization Methods
• Symmetric Design
• Minimum Intersymbol Interference Methods
• Filter Bank Equalization
• Simulation Results
• Conclusions

29
Proposed Dual-Path andComplex TEQ Filter Bank
Equalizers

• Simulation Parameters
• TEQ length 17
• Cyclic prefix 32 samples
• FFT size (N) 512 samples
• Coding gain 5 dB
• Margin 6 dB
• Input power 23 dBm
• Noise PSD -140 dBm/Hz
• Crosstalk noise 5 ISDN
• RF interference 6 AM stations
• Channels Carrier Serving
  Area Loops 1-8
• Testing 1000 symbols

30
Proposed Symmetric TEQ Design Methods

• Simulation Parameters
• TEQ length 17
• Cyclic prefix 32 samples
• FFT size (N) 512 samples
• Coding gain 5 dB
• Margin 6 dB
• Input power 23 dBm
• Noise PSD -140 dBm/Hz
• Crosstalk noise 5 ISDN
• RF interference 6 AM stations
• Channels Carrier Serving
  Area Loops 1-8
• Testing 1000 symbols

31
Proposed Iterative Minimum ISI Method
Simulation Parameters TEQ length
3-32 Cyclic prefix 32 samples FFT size (N)
512 samples Coding gain 5 dB Margin
6 dB Input power 23 dBm Noise PSD
-140 dBm/Hz Crosstalk noise 24 HDSL RF
interference none Channels Carrier
Serving Area Loop
average Testing 1000 symbols
Mbps
32
Conclusions

• Unification and evaluation of existing methods
• Design methods for conventional equalizer
  structures
• Symmetric methods reduce complexity by order of
  magnitude
• Modified Minimum ISI method simplifies delay
  optimization
• Iterative Minimum ISI method applicable to any
  generalized eigendecomposition method and
  suitable for fixed-point realization
• Filter bank equalization structures
• Complex filter bank benchmarks achievable bit
  rate
• Dual path achieves best tradeoff of bit rate vs.
  training complexity and allows VLSI design reuse
  of a conventional equalizer
• Deliverables
• MATLAB discrete multitone equalization toolbox
• Analysis of Advanced Signal Technology ADSL
  measurements

33
Future topics

• Effect of channel estimation error on bit rate
  performance
• Channel estimation based on frequency domain
  zero-forcing
• Perturbation bounds on generalized eigenvector
  computation
• Minimum phase equalizer design
• Minimum group delay, energy delay and phase lag
• Reduced TEQ length compare to linear phase design
• Efficient designs use a linear phase design as a
  start point
• Upstream transmission
• Equalization in multi-input multi-output case
• Multiple lines are grouped in cable
• Future DSL systems deployed with central unit

34
Publications in DMT

• Journal Papers
• M. Ding, B. L. Evans, Effect of Channel
  Estimation Error on Bit Rate Performance in a
  Multicarrier Transceiver, IEEE Transactions on
  Signal Processing, to be submitted.
• R. K. Martin, K. Vanbleu, M. Ding, G. Ysebaert,
  M. Milosevic, B. L. Evans, M. Moonen, and C. R.
  Johnson, Jr., Multicarrier Equalization
  Unification and Evaluation. Part I Optimal
  Designs'', IEEE Transactions on Signal
  Processing, submitted.
• R. K. Martin, K. Vanbleu, M. Ding, G. Ysebaert,
  M. Milosevic, B. L. Evans, M. Moonen, and C. R.
  Johnson, Jr., Multicarrier Equalization
  Unification and Evaluation. Part II
  Implementation Issues and Performance
  Comparisons'', IEEE Transactions on Signal
  Processing, submitted.
• R. K. Martin, M. Ding, B. L. Evans, and C. R.
  Johnson, Jr, Infinite Length Results and Design
  Implications for Time-Domain Equalizers'', IEEE
  Trans. on Signal Processing, vol. 52, no. 1, pp.
  297-301, Jan. 2004.
• R. K. Martin, M. Ding, B. L. Evans, and C. R.
  Johnson, Jr, Efficient Channel Shortening
  Equalizer Design '', EURASIP Journal on Applied
  Signal Processing, vol. 2003, no. 13, pp.
  1279-1290, Dec. 1, 2003.
• B. Farhang-Boroujeny and M. Ding, Design
  Methods for Time Domain Equalizer in DMT
  Transceivers'', IEEE Transactions on
  Communications, vol. 49 Issue 3, pp. 554 -562,
  March 2001.  

35
Publications in DMT

• Conference Papers
• M. Ding, Z. Shen, B. L. Evans, An Achievable
  Performance Bound for Discrete Multitone Systems
  Proc. IEEE Globecom Conf., Nov. 29 - Dec. 3,
  2004, Dallas, USA, submitted.
• M. Ding, B. L. Evans, R. K. Martin, and C. R.
  Johnson, Jr, Minimum Intersymbol Interference
  Methods for Time Domain Equalizer Design'', Proc.
  IEEE Globecom Conf., Dec. 1-5 2003, vol. 4, pp.
  2146-2150, San Francisco, CA, USA.
• R. K. Martin, C. R. Johnson, Jr, M. Ding, and B.
  L. Evans, Infinite Length Results for Channel
  Shortening Equalizers '', Proc. IEEE Int. Work.
  on Signal Processing Advances in Wireless
  Communications, June 15-18, 2003, Rome, Italy,
  accepted for publication.
• R. K. Martin, C. R. Johnson, Jr, M. Ding, and B.
  L. Evans, Exploiting Symmetry in Channel
  Shortening Equalizers '', Proc. IEEE Int. Conf.
  on Acoustics, Speech and Signal Processing, April
  6-10, 2003, vol. V, pp. 97-100, Hong Kong, China.
  
• M. Ding, A. J. Redfern, and B. L. Evans, A
  Dual-path TEQ Structure for DMT-ADSL Systems'',
  Proc. IEEE Int. Conf. on Acoustics, Speech and
  Signal Processing, May 13-17, 2002, vol. III, pp.
  2573-2576, Orlando, FL.

36
Backup Slides
37
Overview of ADSL Technology
38
Bi-directional Transmission in ADSL

• ADSL modems divide the available bandwidth in one
  of two ways -- Frequency Division Multiplexing
  (FDM) or Echo Cancellation.
• FDM assigns one band for upstream data and
  another band for downstream data.
• Echo Cancellation assigns the upstream band to
  over-lap the downstream, and separates the two by
  means of local echo cancellation.

39
ADSL Specifications

• T1E1.4 group developed ANSI Standard T1.413-1995
• June 1999, ITU-T SG 15 approves G.992.1 (G.dmt)
  standard for full rate ADSL
• Tone subchannel
• Value format Downstream/Upstream
• BER Bit Error Rate

item value item value
No. of Tones 256/32 FFT size (N) 512/64
CP Length (?) 32/4 Tone Width 4.3 KHz
Symbol Rate 4 kHz Sampling Rate 2.208 MHz
Target BER 10-7 SNR Gap (?) 9.8 dB
40
Conventional Channel Shortening Methods

• Design single finite impulse response (FIR)
  filter to convolve with the channel such that
  the combined impulse response has only ? 1
  non-zero values
• This filter is called time domain equalizer (TEQ)
• Major TEQ design methods implemented in real-time
  fixed-point DSP
• Minimum Mean-Squared Error design (MMSE)
  Stanford 1992
• Maximum Shortening SNR design (MSSNR)
  Tellabs 1997
• Minimum Intersymbol Interference design (Min-ISI
  UT 1999

41
MBR TEQ DesignsArslan, Evans Kiaei, 2000

• A subchannel SNR definition
• Maximize nonlinear function to obtain the optimal
  TEQ

42
Pertone EqualizerAcker, Leus, Moonen, van de
Wiel Pollet, 2001

• Output of conventional equalizer structure for
  tone i
• Zi Di rowi(QN ) R w
• Di is the complex value of one-tap FEQ for tone i
• QN is the N ? N complex-valued DFT matrix
• R is an N ? T real-valued Toeplitz matrix of
  received samples
• w is a T ? 1 column vector of real-valued TEQ
  taps
• Rearrange computation of output for tone i
• Zi Di rowi(QN ) R w rowi(QN R) ( w Di )
• A multi-tap FEQ for tone i combines TEQ and FEQ
  operations. The output is
• Zi rowi(QN R) wi

43
Performance Comparison for TEQ vs Pertone

• The performance gap
• for any single tone is
• not universally wide.
• In tones associated
• with higher SNR, the
• improvement of per
• tone tends to be
• significant. For other
• tones, the improvement
• is insignificant.

44
Min-ISI Revisited

• Min-ISI method minimizes the ratio of a weighted
  sum of the ISI power over the sum of desired
  signal power within a target window.
• Solution under the condition that Y is invertible
• A practical solution using Cholesky decomposition
  under a stronger condition Y is positive
  definite

qmin is the eigenvector corresponding to minimum
eigenvalue of C
45
Alternative Solution of Min-ISI

• Under the condition Y is not invertible, but X is
  invertible
• The optimum Min-ISI TEQ is the eigenvector
  corresponding to the maximum eigenvalue of
• Practical Solution
• Use Power Method to iteratively compute the
  dominant eigenvalue and eigenvector of

46
Delay Optimization in Min-ISI design

• Min-ISI needs to perform delay optimization to
  find the optimum transmission delay ? to
  maximizes the bit rate performance.
• Exhaustive searching over all possible ?? is
  required since no other approaches available.
• For each ?, we should solve the Min-ISI problem
  to find the optimum TEQ. To save the computation
  cost
• A fast algorithm to implement matrix
  multiplication Wu, Arslan, Evans 2000
• An efficient algorithm to minimize the redundant
  computations between successive ?s. Martin,
  Ding, Evans Johnson 2003

47
Matrices Definitions
48
Invertibility of X
is obviously a rank 1 matrix.
Conclusion X is invertible if and only if all
?is are non-zero.
49
Goertzel Filters

• The N-point DFT of a length N sequence x(l)
• Define
• Noticed
• A recursive DFT computation scheme

50
More definitions
51
Second Order Conditions of J

• Hessian

is positive-semidefinite.
All ?is are non-negative
52
Constrained Minimization of Iterative Min-ISI

• Use the Lagrange multipliers
• Iterative updates
• where

Noted here X is Hermitian and Y is symmetric.
53
Optimum Complex Filter Bank Solution

• The cost function is
• Take conjugate derivative of the cost function
  and equate to zero
• The optimum solution is

54
MSSNR TEQ Design

• Maximum Shortening SNR (MSSNR) TEQ Choose w to
  minimize energy outside window of desired length
• Design problem
• Disadvantages
• Doesnt consider noise
• Doesnt maximize subchannel SNR
• Longer TEQ start killing subcarriers

55
Min-ISI TEQ Design

• Generalize MSSNR with frequency weighting
• Y is the same matrix as B in MSSNR design
• Convert to a constrained minimization problem
• Optimum Solution is generalized eigenvector of
  matrix pencil (X,Y) corresponding to the minimum
  eigenvalue. In practice we need Cholesky
  decomposition to solve it.

56
Per-tone Equalizers

• Move the TEQ operations to the frequency domain
  and combine with the FEQ to obtain a multitap FEQ
  for each subchannel

TEQ-FEQ Stucture
y
N-FFT
TEQ
zi
FEQi
z1
zN/2
Pertone Structure
57
Real Dual TEQ Implementations

• Make good ones better
• Path 1 TEQ optimizes some measure of performance
  over the entire bandwidth
• Path 2 TEQ optimizes the subchannels within a
  preset window of frequencies (with highest SNRs)
• Generally those subchannels have higher potential
  to be improved
• Guarantee a higher bit rate than single TEQ case
• Make dead ones alive Warke, Redfern, Sestok
  Ali 2002
• In some cases, good subchannels are killed due to
  receiver operations (such as subcarriers close to
  the transition band)
• TEQ 1 takes care of the transition band
  subcarrier 30 - 40
• TEQ 2 addresses the upstream bandwidth
  subcarrier gt40

58
New SNR model after FEQ

• Define SNR at the ith FEQ output as
• Transmitted power of ith subchannel
• Transmitted complex-valued QAM symbol on
  ith subchannel
• Output of ith FEQ, estimated QAM symbol
  on ith subchannel
• The practical ADSL systems use flat subchannel
  power allocation

59
Optimization based on the proposed SNR model

• A cost function based on the SNR model
• We know how to solve the same Rayleigh Quotient
  minimization problem!
• Adaptive algorithm based on stochastic gradient
  method can be applied to each tone to design the
  filter bank!

60
Time-Domain Per Tone TEQ Filter Bank

• Find optimum TEQ to maximize SNR after FFT for
  every subchannel in use
• Pick the best one as data rate maximum TEQ

61
Frobenius norm

• The Frobenius norm, is matrix norm of an matrix
  defined as the square root of the sum of the
  absolute squares of its elements,
• It is also equal to the square root of the matrix
  trace of

62
Methods with Frequency Control

• ADSL Transmission are partially
  bandwidth-occupied
• Frequency Division Multiplexing
• Unused subcarriers (Bad SNR or coexistence of
  other applications)
• Many methods are targeted to full bandwidth only
• MMSE, MSSNR, MDS, etc
• May not be optimum for partially bandwidth
  occupied case
• Methods with frequency control are suitable
• Multi-tones partition
• Optimum design Min-ISI, MBR, MGSNR, MDR, BM, etc
• Sub-optimum Tones grouping
• Tone-wise Per-tone, Filter bank

63
MMSE TEQ Design

• MMSE TEQ minimizes the squared error between TEQ
  output and the output of a virtual target impulse
  response (TIR) filter.
• Disadvantages
• Doesnt maximize subchannel SNR
• Longer TEQ start killing subcarriers

Poor bit rate performance
64
SNR Gap

• Channel capacity in bits per 2-dimensional symbol
• SNR gap excessive SNR needed to achieve capacity

65
Cholesky Decomposition

• If A is a symmetric (Hermitian) positive definite
  matrix, there exists a non-singular lower
  triangular L with positive real diagonal entries
  such that
• Cholesky Decomposition can be used to convert a
  generalized eigenvalue problem into a normal one

66
Alternative Structure

• The demodulated signal at the FEQ output
• Design freedom is limited in the TEQ-FEQ
  structure
• All tones share same TEQ w
• All taps of TEQ share same complex multiplier Di
  per tone
• Time domain filter bank plus FEQ
• Per-tone equalizer
• Complex time domain filter bank

67
Contribution 1Infinite Length TEQ Results

• TIR for a MMSE TEQ has all zeros on the unit
  circle
• A becomes a symmetric Toeplitz matrix
• Eigenvector of A has all zeros on the unit circle
• TIR for a MMSE TEQ will be symmetric/skew
  symmetric
• A also becomes a doubly symmetric matrix
• Eigenvectors of A will be either symmetric or
  skew symmetric
• A MSSNR TEQ will be symmetric/skew symmetric
• is doubly symmetric
• Infinite length case A converges to
  asymptotically
• Can exploit symmetry in TEQ designs

68
Dual Path TEQ performance

• Simulation Parameters
• TEQ length 17
• AWGN PSD -140 dBm/Hz
• Crosstalk noise 24 ISDN
• FDM filter 5th order IIR
• Test Loop ANSI-13
• Second path only optimizes tones
• 55-85
• Achieved Bit Rate
• Path 1 2.5080 Mbps
• Dual Path 2.6020 Mbps
• 4 improvement in bit rate

69
Carrier Serving Area Loops

• Served by a digital loop carrier, which
  multiplexes hundreds of analog lines into one
  high-speed digital trunk
• Limited to 12000 feet

70
Symmetric design

• Even length TEQ
• Odd length TEQ