Signal equalization

Electrical computers: arithmetic processing and calculating – Electrical digital calculating computer – Particular function performed

Reexamination Certificate

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C708S323000

Reexamination Certificate

active

06341298

ABSTRACT:

FIELD OF THE INVENTION
The present invention relates to the transmission and reception of signals, and more particularly, to methods and apparatus for optimizing the equalization of a signal that has been transmitted through a distorting medium.
BACKGROUND OF THE INVENTION
Modern modems, and in particular ADSL modems with a DMT line code, generally employ block-based modulation to transmit data across a twisted pair. In such systems equalization of the received signal is necessary to ensure that as much of the energy of the overall impulse response as possible is contained in a fixed number of symbol periods, called the cyclic prefix of the system. The equalization process is typically performed in two stages, the initial stage and the steady state. In the initial stage, a predetermined signal is transmitted in order to study the impulse response of the transmission channel and determine the equalizing coefficients. In the steady state, any signal may be transmitted, and the equalization coefficients are applied as a finite impulse response (FIR) filter to the signal received by the receiver.
While steady state operation is straightforward, determining the equalization coefficients, also referred to as “equalizing the channel,” is less so. Proposed methods for determining the equalizing coefficients in a DMT system are either computationally complex, and therefore require an enormous amount of computer memory and processing time in order to implement them, or they suffer from inaccuracies due to non-optimized algorithms or due to computational roundoff errors that arise when performing operations such as division on a DSP.
Methods of the prior art generally comprise two major parts:
Preparing the parameters for the algorithm, including computation of auto-correlation and cross-correlation matrices; and
Iteratively performing computations on a certain range of delays in order to find an optimum delay such that the overall signal contained in the corresponding response region will have maximal energy.
Most of the computational effort is generally required for the iterative computations as the preparation is performed only once.
U.S. Pat. No. 5,461,640 describes a method where the auto-correlations and cross-correlations are computed during the parameter preparation part. U.S. Pat. No. 5,461,640 is disadvantageous in that it presents computational complexities, discussed hereinbelow, that might be avoided during the iterative computation part were another method used during the parameter preparation part.
To understand the equalization problem we may denote an input signal x
k
, an output signal y
k
, a channel response by h, and equalizer taps w. Thus the overall response of the channel may be expressed as:
{tilde over (h)}=h*w,
We may further denote {tilde over (b)} as the ideal overall response which is equivalent to the overall response {tilde over (h)} in the response region, and zero outside the response region, and denote {tilde over (b)} as the filter which is the non-zero portion of the ideal overall response {tilde over (b)}.
Two prior art approaches are now described.
1. In one approach, methods used for optimization of decision feedback equalization (DFE) are employed, where the equalizer taps w is the forward filter, and the filter b is the feedback filter. The only difference is that in the DMT case b does not have to be monic and causal. That is, the first tap does not have to be constrained to unity. The following algorithm is proposed:
a) Estimate the cross-correlation between the channel input x
k
, and the channel output y
k
, and the auto-correlation of the output y
k
.
b) For delay &Dgr;=&Dgr;
min
to &Dgr;
max
for i=1 to v
(i) Constrain the i-th tap of b to be unity
(ii)
minimize



&LeftDoubleBracketingBar;
b
~
-
h
~
&RightDoubleBracketingBar;
2


 using the DFE minimization techniques.
c) Choose the minimum of all the solutions found.
2. Another approach, such as is described in U.S. Pat. No. 5,461,640, tries to minimize the ratio of the energy in the output of the ideal response filter to the energy in the output of the overall response filter that is outside the response region. This is accomplished by applying a constraint of the form:
∥b∥
2
=1,
where b has a length v, and minimizing
z
T
R
uu
z
where R
uu
is a matrix comprising the auto-correlations of the inputs x
k
, the outputs of the channel y
k
, and the cross-correlations between them, and z is a vector comprising the filter b and the equalizer taps w. The computations that follow require solving an eigenvalue problem for a matrix whose dimensions are (m+v) by (m+v), where m is the number of taps in the equalizer, and v the length of the time region allowed for the response region (in DMT systems v is the cyclic prefix+1). This eigenvalue problem is solved for each possible delay, and after performing the computation for all possible delays the global minimum is chosen.
Some straightforward manipulations transform the problem from an (m+v) by (m+v) size eigenvalue problem to a v by v size problem by performing a matrix inversion and multiplication of matrices of sizes m by m and m by v at each stage, and, moreover, by introducing division at each stage.
In the first approach 1) described above, each iteration of step b) requires the formation of two new matrices and the performing of a matrix inversion, usually leading to an enormous number of calculations for each &Dgr; tried.
The second approach 2) greatly relaxes the computation load by performing only one matrix inversion and solving one maximal eigenvalue problem for each &Dgr;. However, as stated above, the matrices involved are still quite large, and even after simplification methods are used a matrix inversion of size m by m must be performed at each stage. The formation of a v by v size matrix at each step involves computations with larger size m by m matrices if v<=m, and an eigenvalue problem of size v by v (or perhaps a generalized eigenvalue problem) must be solved, which is very inefficient where v>m.
Relevant methods useful in equalization are discussed in “Fundamentals of Matrix Computations” by D. S. Watkins, John Wiley & Sons, 1991.
U.S. Pat. No. 5,285,474 is also believed to be representative of the art.
The disclosures of all U.S. Patents and publications mentioned in this specification and of U.S. patents and publications cited therein are hereby incorporated by reference.
SUMMARY OF THE INVENTION
The present invention seeks to provide novel apparatus and methods for determining equalizer coefficients which overcomes disadvantages of the prior art as discussed above. Other objects and advantages of the invention will become apparent from the description and claims which follow taken together with the appended drawings.
In accordance with the present invention there is provided a method for calculating a vector A from a set comprising at least one pair of subspaces of R
N
wherein N is the dimension of said vector space, each of the at least one pair comprises a first subspace and a second subspace of the vector space R
N
, and a set of maximal projections and their corresponding unit vectors, each of said maximal projections is associated with one of the pairs of subspaces, wherein for each of the pairs of subspaces the first subspace is the column space of a N by m matrix M and the second subspace is a function of a predefined set of indices between 1 and N, comprising:
for each of the pairs of subspaces:
determining a unit vector C in one of said subspaces so that a projection of the unit vector upon another of the vector subspaces is maximal for all unit vectors in one of the vector subspaces;
applying a predefined optimizing function to the set of maximal projections and unit vectors, thereby selecting a maximal projection and its corresponding unit vector and pair of first and second subspaces;
determining a vector A having length m, so that the product of vector A and matrix M is a unit vector B and the projection o

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