# Patent

## Title
Fast computation of linear equalizer coefficients from channel estimate

## Abstract

Computing optimal Linear Equalizer (LE) coefficients gopt from a channel estimate h. A channel impulse response h is first estimated based upon either a known training sequence or an unknown sequence. The channel estimate is formulated as a convolution matrix H. The convolution matrix H is then related to the LE coefficients in a matrix format equation, the matrix format equation based upon the structure of the LE, the convolution matrix, and an expected output of the LE. A Fast Transversal Filter (FTF) algorithm is then used to formulate a recursive least squares solution to the matrix format equation. Computing the recursive least squares solution yields the LE coefficients using structured equations.

## Background
CROSS REFERENCES TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. Regular Utility application Ser. No. 10/044,013, filed Oct. 26, 2001, now U.S. Pat. No. 7,027,504 which claims priority to U.S. Provisional Application Ser. No. 60/322,994, filed Sep. 18, 2001, both of which are incorporated herein by reference in their entirety.


FIELD OF THE INVENTION
This invention relates generally to digital communications; and more particularly to linear equalizers.
BACKGROUND OF THE INVENTION
The structure and operation of communication systems is generally known. Many communication systems carry data, e.g., voice, audio, video, file, or other digital data that is sent from a transmitter to a receiver. On the transmitter side, data is first formed into packets. This data may be raw data or encoded data that represents the raw data. Each of these packets also typically includes a header, a known training sequence, and a tail. These packets are then modulated into symbols and the symbols are transmitted by the transmitter and intended for receipt by the receiver. The receiver then receives the symbols and attempts to extract the data that is carried by the symbols.
A “channel” carries the symbols from the transmitter to the receiver. The channel is serviced by a wired, wireless, optical, or another media, depending upon the communication system type. In many communication systems, such as terrestrial based wireless communication systems, satellite based communication systems, cable based communication systems, etc., the channel distorts the transmitted symbols, from the perspective of the receiver, causing interference between a subject symbol and a plurality of symbols surrounding the subject symbol. This type of distortion is referred to as “inter-symbol-interference” and is, generally speaking, the time-dispersed receipt of multiple copies the symbols caused by multipath. The channel also introduces noise into the symbols prior to their receipt. Each of these concepts is well known.
Equalizers are now generally employed in an attempt to remove channel effects from a received symbol stream. Thus, equalizers are essential building blocks of modern receivers, especially in broadband applications where inter-symbol-interference is a critical problem. In a typical equalizer, the channel between the transmitter and the receiver is first estimated based upon the training sequence contained in one or more preambles. Then optimal equalizer coefficients (also referred to as taps and/or tap coefficients for the equalizer) are estimated based upon the channel estimate. The optimal equalizer coefficients are then used by the equalizer in extracting the data from the packet. The optimal equalizer coefficients may also be computed after extraction of the data from the equalized data stream based upon blind channel estimates. Equalizer coefficient generation should be repeated as often as possible, especially in fast varying channel cases, to generate new equalizer coefficients. The received data stream is usually buffered during the period that is required for channel estimation and equalizer coefficient computations. Thus, the preamble (and also actual data) contained in a packet may be used to generate the channel estimate and optimal equalizer coefficients that are employed by the equalizer to extract the data from the packet.
As symbol rates increase and modulation schemes become more complex, equalizers have increasingly greater importance. A critical factor in increasing the effectiveness of these equalizers is the complexity of optimal equalizer coefficient computation. A reduction in this complexity: (1) reduces the memory size required to buffer the received symbol stream sequence during the period required for coefficient computations; (2) allows more frequent uploading of new coefficients thus enabling the equalizer to track fast channel variations; and (3) simplifies the hardware and, resultantly, the die area required for coefficient computation.
FIG. 1A is a block diagram illustrating a discrete time symbol-spaced Decision Feedback Equalizer (DFE) based channel equalization model 100. The channel equalization model 100 includes a channel 102, a Feed Forward Equalizer (FFE) 104, a Decision block 106, and a Feed Back Equalizer (FBE) 108. An input sequence x(n) is complex, independent, and identically distributed with unit power. Additive noise v(n) is white Gaussian with power spectral density σv2. Furthermore, the decisions {tilde over (x)}(n−δ) are assumed to be correct, and hence equal to x(n−δ). This assumption makes the design of the FBE 108 and FFE 104 easier, but at the expense of introducing error propagation due to possibly wrong decisions. The FFE 104 function G(z) has length L. The channel (impulse) response vector of the channel h is given in Equation (1) as:




  
    
      
        h
        ⁢
        
          =
          Δ
        
        ⁢
        
          [
          
            
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
              
                ...
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          1
          )
        
      
    
  




The number of coefficients (taps) M of the FBE 108 function B(z) is assumed greater or equal to the channel memory, i.e., M≧N−1. These modeling assumptions are feasible in practice.

In estimating FFE 104 and FBE 108 equalizer coefficients, the goal is to minimize the mean square error quantity of Equation (2).

ζ=E|x(n−δ)−{circumflex over (x)}(n−δ)|2,  Equation (2)

Where {circumflex over (x)}(n−δ) is the delayed input signal estimate prior to the Decision block 106. By collecting the coefficients of both G(z) and B(z) into vectors, we can express the received signal {circumflex over (x)}(n−δ) in Equation (3) as:

xn=yng−{tilde over (x)}nb  Equation (3)

A channel output model defining yn may be expressed by:

yn=xnH+vn  Equation (4)

where H is the (N+L−1)×L convolution matrix corresponding to the channel response and expressed as:





  
    
      
        H
        =
        
          [
          
            
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                ⋯
              
              
                0
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋰
              
              
                ⋮
              
            
            
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      2
                    
                    )
                  
                
              
              
                ⋰
              
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
            
            
              
                0
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
              
                ⋰
              
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋰
              
              
                ⋮
              
            
            
              
                0
              
              
                0
              
              
                ⋯
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          5
          )
        
      
    
  




In this model, xn is the 1×(N+L−1) input vector,




  
    
      
        
          x
          n
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          [
          
            
              
                
                  x
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      1
                    
                    )
                  
                
              
              
                ...
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      N
                      -
                      L
                      +
                      2
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          6
          )
        
      
    
  




yn is the 1×L input regression vector to the FFE 104,





  
    
      
        
          y
          n
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          [
          
            
              
                
                  y
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
              
                
                  y
                  ⁡
                  
                    (
                    
                      n
                      -
                      1
                    
                    )
                  
                
              
              
                ...
              
              
                
                  x
                  -
                  
                    y
                    ⁡
                    
                      (
                      
                        n
                        -
                        L
                        +
                        1
                      
                      )
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          7
          )
        
      
    
  




{tilde over (x)}n is the 1×M input regression vector to the (strictly causal) FBE 108,





  
    
      
        
          
            x
            ⋓
          
          n
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          [
          
            
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      δ
                      -
                      1
                    
                    )
                  
                
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      δ
                      -
                      2
                    
                    )
                  
                
              
              
                ...
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      δ
                      -
                      M
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          8
          )
        
      
    
  




and vn is the 1×L vector noise process.

The current efficient methods for computing the optimal filter coefficients of a decision feedback equalizer, which optimizes (1), are based on the well-known Cholesky decomposition method (from a finite-dimension problem formulation). Two published papers: (1) N. Al-Dhahir and J. M. Cioffi, “MMSE Decision-Feedback Equalizers: Finite-Length Results,” IEEE Trans. on Information Theory, vol. 41, no. 4, pp. 961-973, July 1995; and (2) N. Al-Dhahir and J. M. Cioffi, “Fast Computation of Channel-Estimate Based Equalizers in Packet Data Transmission,” IEEE Trans. on Signal Processing, vol. 43, no. 11, pp. 2462-2473, November 1995 provide one procedure for computing optimal DFE settings. These equations are referred to hereinafter as “Al-Dhahir's equations.”
Generally speaking, Al-Dhahir's equations, as well as other existent techniques rely on the use of the generalized Schur algorithm for fast Cholesky decomposition of the matrices involved in both the FBE and FFE optimal coefficient setting computations. However, the overall procedures for calculation of the DFE (FBE and FFE) coefficients have the following problems:

    
    
        1. These procedures require the use of nonstructured recursive equations. These equations are difficult to implement from the perspective of integrated circuit design. In particular, the recursive equations used in Al-Dhahir's DFE tap (coefficient) computer often requires the use of a DSP processor. As is generally known, the use of a DSP processor in a real-time communication system application is severely constricts system throughput.
        2. These procedures, in particular Al-Dhahir's equations require a complex tap computer. In other words, they require the use of a relatively large number of complex multiplies.
        3. These prior art DFE coefficient computation techniques can be used only for equalizers that use a fixed equalizer delay (δ), which is set to its maximum value L−1, where L is the length of the FFE 104. The prior art techniques cannot use a different delay δ of the equalizer.
    
    


FIG. 1B is a block diagram illustrating a discrete time symbol-spaced Linear Equalizer (LE) channel equalization model 150. The LE channel equalization model 150 includes a channel 152 having length N, a LE 154 having length L, and a decision block 156. An input sequence x(n) is complex, independent, and identically distributed with unit power. Additive noise v(n) is white Gaussian with power spectral density σv2.
A minimization of the Mean Square Error (MSE) across the decision block 156 may be written as:




  
    
      
        
          min
          g
        
        ⁢
        
          
        
        ⁢
        
          E
          ⁢
          
            
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      δ
                    
                    )
                  
                
                -
                
                  
                    y
                    n
                  
                  ·
                  g
                
              
              
            
            2
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          9
          )
        
      
    
  




Denoting Ry as the variance of the input regression vector y, and cross variance Ryx(n−δ), the smoothing problem for this minimization is given by Equation (10) as:

gopt=Ry−1Ryx(n−1δ).  Equation (10)

A convolution matrix H as defined in Equation (5) may also be assumed for the LE channel equalization model 150. With a vector d defined as a vector of all zeros except for the δth element, which is unity, a solution for the coefficients of the LE 154 solution may be written as:

gopt=[σv2I+H*H]−1H*d  Equation (11)

Prior art solutions of Equation (11) require matrix inversion. Such matrix inversion requires the greater of L3 or N3 multiplications and cannot be accomplished timely enough to allow a prior art transceiver to compute LE coefficients based upon a channel estimate. Accordingly, prior art techniques for training LEs included: (1) blindly assigning LE coefficients; (2) adapting the LE coefficients to a received preamble, training sequence, or other data; or (3) a combination of both of these techniques. Typically, convergence in training the LE using an iterative training approach, e.g., Least Squares, required a significant number of iterations. Oftentimes, such convergence in training could not be accomplished to meet the requirements of rapidly changing channels.
Thus, there is a need in the art for a technique to determine LE coefficients based upon a channel estimate.
SUMMARY OF THE INVENTION
In order to overcome the above-cited shortcomings, among others of the prior art, the method and system of the present invention computes optimal Linear Equalizer (LE) coefficients gopt from a channel estimate h. As contrasted to the prior art, the methodology of the present invention relies on iterating simple structured recursions to yield the LE coefficients. Resultantly, the operation according to the present invention produces LE coefficients faster and with less computational requirements than the prior art operations, thus providing significant advantages over prior techniques.
According to the present invention, a channel impulse response h is first estimated based upon either a known training sequence or an unknown sequence. The channel estimate is formulated as a convolution matrix H. The convolution matrix H is then related to the LE coefficients in a matrix format equation, the matrix format equation based upon the structure of the LE, the convolution matrix, and an expected output of the LE. A Fast Transversal Filter (FTF) algorithm is then used to formulate a recursive least squares solution to the matrix format equation. The method concludes with computing the recursive least squares solution to yield the LE coefficients using structured equations.
In one particular embodiment, the recursive least squares solution is formulated as a Kalman gain solution using the FTF algorithm. In such embodiment, the Kalman gain solution may be determined using an Array Form Algorithm. The length of the LE may be chosen to force the Array Form Algorithm to use a lower triangular matrix. Further, the length of the LE may be chosen to force the FTF algorithm to use a lower triangular matrix.
The channel estimate is typically based upon a known preamble sequence of a data packet upon which the LE operates. After the LE coefficients have been determined, the LE may be used to equalize data received via the channel.
The method of the present invention is exact and may be employed to compute LE coefficients to track a rapidly fading channel. Because the method of the present invention relies upon simple structured recursion, the computation period and/or the hardware required for the LE coefficient computations may be implemented in both portable devices and non-portable devices. The method of the preset invention may be efficiently used in many communication systems that require the use of equalizers such as mobile wireless receivers, fixed wireless receivers, cable modem receivers, HDTV receivers, etc. to increase the performance of such communication systems. Other features and advantages of the present invention will become apparent from the following detailed description of the invention made with reference to the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
These and other features, aspects and advantages of the present invention will be more fully understood when considered with respect to the following detailed description, appended claims and accompanying drawings wherein:
FIG. 1A is a block diagram illustrating a discrete time symbol-spaced Decision Feedback Equalizer (DFE) channel equalization model;
FIG. 1B is a block diagram illustrating a discrete time symbol-spaced Linear Equalizer (LE) channel equalization model;
FIG. 2 is a logic diagram generally illustrating operation according to the present invention in determining DFE coefficients and applying such coefficients to a DFE;
FIG. 3 is a logic diagram illustrating operations according to the present invention employed to determine Feed Forward Equalizer (FFE) coefficients for the DFE;
FIG. 4 is a logic diagram illustrating operations according to the present invention employed to determine Feed Back Equalizer (FBE) coefficients for the DFE;
FIG. 5 is a block diagram illustrating a discrete time fractionally-spaced DFE that operates according to the present invention;
FIG. 6 is a block diagram illustrating a multi-channel equivalent of the discrete time fractionally-spaced DFE of FIG. 5;
FIG. 7 is a block diagram illustrating a transceiver having a DFE constructed according to the present invention;
FIG. 8 is a block diagram illustrating a transceiver having a LE constructed according to the present invention;
FIG. 9 is a logic diagram generally illustrating operation according to the present invention in determining LE coefficients and applying such coefficients to a LE; and
FIG. 10 is a logic diagram illustrating operations according to the present invention employed to determine LE coefficients.



DETAILED DESCRIPTION OF THE INVENTION
FIG. 2 is a logic diagram generally illustrating operation according to the present invention in determining Decision Feedback Equalizer (DFE) coefficients and in applying such coefficients to a DFE. The operations of the present invention are performed by a processor, such as a Digital Signal Processor (DSP), or other circuitry present within a receiver that determines DFE coefficients to be applied to a DFE, also resident in the receiver. The DFE operates upon samples of a received signal in an attempt to remove channel effects from the samples so that digital data may be extracted from the samples. The structure and operation of DFEs, one of which was illustrated in FIG. 1A, are generally known and will not be further described herein except as they relate to the present invention.
The operations of the present invention will be first performed by a processor, tap computer, DSP, or other receiver device, to determine initial DFE coefficients to be used in subsequent operations by the receiver. Thus, during startup or reset, a channel corresponding to the receiver is estimated (step 202). According to one embodiment of the present invention, the channel is estimated based upon a known preamble sequence. However, in other embodiments, the channel could also be estimated based upon unknown received data. In either case, channel estimation operations are generally well known and are not described further herein except as it relates to the present invention.
With the channel estimated, Feed Forward Equalizer (FFE) coefficients are determined based upon the channel estimate (step 206). Then, Feed Back Equalizer (FBE) coefficients are determined based upon the FFE coefficients and the channel estimate (step 208). The manner in which the FFE and FBE coefficients are generated is step 206 and step 208 will be described in detail herein with reference to FIGS. 3-7.
With the FFE and FBE coefficients determined, they are applied to the DFE (step 208) and are used in equalizing samples of a received signal to remove channel effects. These DFE coefficients are continually updated (step 210) using a known technique. Periodically, upon the receipt of a next packet for example, upon an indication that redetermination is required, or upon another triggering event, the DFE coefficients are redetermined (step 212). In this event, another channel estimate may be obtained (step 214). Then, the DFE coefficients are again determined according to the present invention and applied to the DFE. The operations of FIG. 2 continue until the receiver is turned off, placed in a sleep mode, or otherwise inactivated.
FIG. 3 is a logic diagram illustrating operations according to the present invention employed to determine Feed Forward Equalizer (FFE) coefficients for the DFE. In a first operation of FIG. 3, a DFE delay is selected (step 302). In one embodiment, this delay is selected as the channel length. In such case, the DFE delay corresponds to the length of the FFE.
Next, the DFE solution is formulated into a least squares solution (step 304). By collecting the FFE coefficients g and the FBE coefficients b into a single vector w, the minimization of the Mean Square Error may be written as:




  
    
      
        
          min
          w
        
        ⁢
        
          
        
        ⁢
        
          E
          ⁢
          
            
              
              
                
                  x
                  ⁡
                  
                    (
                    
                      n
                      -
                      δ
                    
                    )
                  
                
                -
                
                  
                    
                      
                        [
                        
                          
                            y
                            n
                          
                          -
                          
                            
                              x
                              ⋓
                            
                            n
                          
                        
                        ]
                      
                      }
                    
                    u
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        [
                        
                          
                            
                              g
                            
                          
                          
                            
                              b
                            
                          
                        
                        ]
                      
                      }
                    
                    w
                  
                
              
              
            
            2
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          12
          )
        
      
    
  




Now, denoting Ru the variance of the augmented input regression vector u, and cross variance Rux(n−δ), the well-known solution to this smoothing problem is given by Equation (13) as:

wopt=Ru−1Rux(n−δ).  Equation (13)

where





  
    
      
        
          R
          u
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            
              E
              ⁡
              
                [
                
                  
                    
                      
                        y
                        n
                        *
                      
                    
                  
                  
                    
                      
                        -
                        
                          
                            x
                            ⋓
                          
                          n
                          *
                        
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  y
                  n
                
                -
                
                  
                    x
                    ⋓
                  
                  n
                
              
              ]
            
          
          =
          
            [
            
              
                
                  
                    R
                    y
                  
                
                
                  
                    -
                    
                      R
                      
                        y
                        ⁢
                        
                          x
                          ⋓
                        
                      
                    
                  
                
              
              
                
                  
                    -
                    
                      R
                      
                        
                          x
                          ⋓
                        
                        ⁢
                        y
                      
                    
                  
                
                
                  
                    R
                    
                      x
                      ⋓
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          14
          )
        
      
    
  




and





  
    
      
        
          R
          
            ux
            ⁡
            
              (
              
                n
                -
                δ
              
              )
            
          
        
        =
        
          
            [
            
              
                
                  
                    E
                    ⁢
                    
                      
                    
                    ⁢
                    
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          =
          
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                        ⋓
                      
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                          (
                          
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            ]
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          15
          )
        
      
    
  




Using the channel output model of Equation (4), and the fact that x(n) is individually identically distributed (i.i.d.), the following closed form expressions for {Ry, Ry{tilde over (x)}, R{tilde over (x)}, Ryx(n−δ), R{tilde over (x)}x(n−δ)} are determined:

Ry=Eyn*yn=σv2I+H*H  Equation (16)





  
    
      
        
          
            R
            
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                x
                ⋓
              
            
          
          =
          
            
              
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                *
              
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                    n
                    *
                  
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            =
            
              
                
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                  *
                
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                  [
                  
                    
                      
                        
                          0
                          
                            
                              (
                              
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                                +
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                            M
                          
                        
                      
                    
                    
                      
                        
                          I
                          M
                        
                      
                    
                    
                      
                        
                          0
                          
                            
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                                +
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              ⁢
              
                =
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              ⁢
              
                
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                  _
                
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        ,
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          17
          )
        
      
    
  




R{tilde over (x)}=IM  Equation (18)





  
    
      
        
          R
          
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            ⁡
            
              (
              
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                -
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              )
            
          
        
        =
        
          
            
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              *
            
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              *
            
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              ⁡
              
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              T
            
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              =
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              *
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          19
          )
        
      
    
  
  
    
      
        
          R
          
            
              x
              ⋓
            
            ⁢
            
              
            
            ⁢
            
              x
              ⁡
              
                (
                
                  n
                  -
                  δ
                
                )
              
            
          
        
        =
        0
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          20
          )
        
      
    
  




where H is a submatrix of H as set forth in Equation (5),





  
    
      
        H
        =
        
          [
          
            
              
                
                  H
                  1
                
              
            
            
              
                
                  H
                  _
                
              
            
            
              
                
                  H
                  2
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          21
          )
        
      
    
  




H1 is defined as the (δ+1)×L submatrix of H, consisting of the first (δ+1) rows of H. Note that for the case of colored noise, the matrix σv2I should be replaced by the autocorrelation matrix of the noise Rv. Extending the derivation to the colored noise case is straightforward.

Now, with the quantities defined above, Equation (13) becomes:




  
    
      
        
          w
          opt
        
        =
        
          
            
              [
              
                
                  
                    
                      
                        
                          σ
                          v
                          2
                        
                        ⁢
                        I
                      
                      +
                      
                        
                          H
                          *
                        
                        ⁢
                        H
                      
                    
                  
                  
                    
                      -
                      
                        
                          H
                          _
                        
                        *
                      
                    
                  
                
                
                  
                    
                      -
                      
                        H
                        _
                      
                    
                  
                  
                    I
                  
                
              
              ]
            
            
              -
              1
            
          
          [
          
            
              
                
                  h
                  *
                
              
            
            
              
                0
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          22
          )
        
      
    
  




Using the well known inverse formula of block matrices, wopt may be rewritten as:





  
    
      
        
          w
          opt
        
        =
        
          
            (
            
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      I
                    
                  
                
                ]
              
              +
              
                
                  [
                  
                    
                      
                        I
                      
                    
                    
                      
                        
                          H
                          _
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                    
                      (
                      
                        
                          
                            σ
                            v
                            2
                          
                          ⁢
                          I
                        
                        +
                        
                          
                            H
                            *
                          
                          ⁢
                          H
                        
                        -
                        
                          
                            
                              H
                              _
                            
                            *
                          
                          ⁢
                          
                            H
                            _
                          
                        
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          I
                        
                        
                          
                            H
                            _
                          
                        
                      
                    
                    ]
                  
                
              
            
            )
          
          [
          
            
              
                
                  h
                  *
                
              
            
            
              
                0
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          23
          )
        
      
    
  




and as:





  
    
      
        =
        
          
            [
            
              
                
                  I
                
              
              
                
                  
                    H
                    _
                  
                
              
            
            ]
          
          ⁢
          
            
              (
              
                
                  
                    σ
                    v
                    2
                  
                  ⁢
                  I
                
                +
                
                  
                    H
                    1
                    *
                  
                  ⁢
                  
                    H
                    1
                  
                
                +
                
                  
                    H
                    2
                    *
                  
                  ⁢
                  
                    H
                    2
                  
                
              
              )
            
            
              -
              1
            
          
          ⁢
          
            h
            *
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          24
          )
        
      
    
  




which may be written as:





  
    
      
        
          [
          
            
              
                
                  g
                  opt
                
              
            
            
              
                
                  b
                  opt
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  I
                
              
              
                
                  
                    H
                    _
                  
                
              
            
            ]
          
          ⁢
          
            
              (
              
                
                  
                    σ
                    v
                    2
                  
                  ⁢
                  I
                
                +
                
                  
                    [
                    
                      
                        H
                        1
                        *
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        H
                        2
                        *
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            H
                            1
                          
                        
                      
                      
                        
                          
                            H
                            2
                          
                        
                      
                    
                    ]
                  
                
              
              )
            
            
              -
              1
            
          
          ⁢
          
            h
            *
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          25
          )
        
      
    
  




Although each of the two matrices H1 and H2 has a shift structure, the augmented matrix




   
  
    [
    
      
        
          
            H
            1
          
        
      
      
        
          
            H
            2
          
        
      
    
    ]
  




does not have a shift structure.

In selecting the length of the FBE (DFE delay) to be M≧N−1, the quantities hδ, H1 and H2 are such that:




  
    
      
        
          [
          
            
              H
              1
            
            
              H
              2
            
          
          ]
        
        =
        
          [
          
            
              
                H
                δ
              
              ⁢
              
                
              
              ⁢
              0
            
            
              0
              ⁢
              
                
              
              ⁢
              
                H
                ~
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          26
          )
        
      
    
  




and

h=[hδ0]  Equation (27)

This implies that:





  
    
      
        
          
            
              σ
              v
              2
            
            ⁢
            I
          
          +
          
            
              H
              1
              *
            
            ⁢
            
              H
              1
            
          
          +
          
            
              H
              2
              *
            
            ⁢
            
              H
              2
            
          
        
        =
        
          [
          
            
              
                
                  
                    
                      σ
                      v
                      2
                    
                    ⁢
                    I
                  
                  +
                  
                    
                      H
                      δ
                      *
                    
                    ⁢
                    
                      H
                      δ
                    
                  
                
              
              
                0
              
            
            
              
                0
              
              
                
                  
                    
                      σ
                      v
                      2
                    
                    ⁢
                    I
                  
                  +
                  
                    
                      
                        H
                        ~
                      
                      *
                    
                    ⁢
                    
                      H
                      ~
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          28
          )
        
      
    
  




is block diagonal. In this case, the expressions for gopt and bopt decouple into a simple form.

Therefore, the optimal FFE and FBE coefficients are represented as:

gopt=(σv2I+Hδ*Hδ)−1hδ*  Equation (29)

bopt= Hgopt  Equation (30)

The above expressions are valid for all values of the DFE delay δ. In general, the optimal value for the DFE delay δ is within the range L−1≦δopt≦N+L−2. In the special case of the choice δ=L−1, the matrices involved in the above expressions are given by




  
    
      
        
          
            H
            δ
          
          =
          
            [
            
              
                
                  
                    h
                    ⁡
                    
                      (
                      0
                      )
                    
                  
                
                
                  0
                
                
                  ...
                
                
                  0
                
              
              
                
                  
                    h
                    ⁡
                    
                      (
                      1
                      )
                    
                  
                
                
                  
                    h
                    ⁡
                    
                      (
                      0
                      )
                    
                  
                
                
                  ...
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋰
                
                
                  ⋮
                
              
              
                
                  
                    h
                    ⁡
                    
                      (
                      
                        N
                        -
                        1
                      
                      )
                    
                  
                
                
                  
                    h
                    ⁡
                    
                      (
                      
                        N
                        -
                        2
                      
                      )
                    
                  
                
                
                  ⋰
                
                
                  
                    h
                    ⁡
                    
                      (
                      0
                      )
                    
                  
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        
          
            h
            δ
            *
          
          =
          
            [
            
              
                
                  
                    
                      h
                      *
                    
                    ⁡
                    
                      (
                      
                        N
                        -
                        1
                      
                      )
                    
                  
                
              
              
                
                  
                    
                      h
                      *
                    
                    ⁡
                    
                      (
                      
                        N
                        -
                        2
                      
                      )
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    
                      h
                      *
                    
                    ⁡
                    
                      (
                      0
                      )
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          31
          )
        
      
    
  




and





  
    
      
        
          H
          _
        
        =
        
          [
          
            
              
                0
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
              
                ⋰
              
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
            
            
              
                0
              
              
                0
              
              
                ⋰
              
              
                
                  h
                  ⁡
                  
                    (
                    2
                    )
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋰
              
              
                ⋮
              
            
            
              
                0
              
              
                0
              
              
                ⋯
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          32
          )
        
      
    
  




Note that this solution is equivalent to the solution of Al-Dhahir's equations, due to the uniqueness of wopt when Equation (2) is minimized. However, the expressions obtained above for Equations (29) and (30) provide alternative methods to compute the FFE and FBE coefficients gopt and bopt in a simpler and more efficient way than the prior art techniques.
In calculating gopt, a coefficient matrix is first defined in Equation (33) as:




  
    
      
        
          P
          δ
        
        ⁢
        
          =
          Δ
        
        ⁢
        
          
            (
            
              
                
                  σ
                  v
                  2
                
                ⁢
                I
              
              +
              
                
                  H
                  δ
                  *
                
                ⁢
                
                  H
                  δ
                
              
            
            )
          
          
            -
            1
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          33
          )
        
      
    
  




so that the optimal solution for the FFE coefficients gopt is given by

gopt=Pδhδ*.  Equation (34)

The expression for gopt corresponds exactly to the definition of the Kalman gain vector, used to update the optimal weights in a certain regularized Recursive Least Squares (RLS) problem. More specifically, given a (n+1)×L data matrix Hn and the corresponding coefficient matrix Pn, the Kalman gain gn=Pnhn* can be time-updated according to the following recursions:

γ−1(n)=1+hnPn−1hn*,  Equation (35)

gn=Pn−1hn*γ(n),  Equation (36)

Pn=Pn−1−gnγ−1(n)gn*,  Equation (37)

where P−1=σv−2I (the term P−1 is the initial condition for Pn) and g0=0. The computation of the Kalman gain vector gn+1 in the above solution relies on the propagation of the Riccati variable Pn. This method of computation requires O(L2) operations per iteration.

Well known fast RLS schemes avoid the propagation of Pn and compute the gain gn in a more efficient way. In this case, the computational complexity required is only of O(L) per iteration. This implies that the overall complexity needed to calculate the FFE coefficients can be on the order of O(L2) thus yielding an efficient method for computing the FFE coefficients. Therefore, fast RLS filters are used to determine gopt. In such case, fast transversal computations are employed to determine the Kalman gain for the RLS solution (step 306). Here we also note that it is straightforward to extend this method to use other fast RLS algorithms, e.g., array form fast RLS algorithms.
Faster recursions may be obtained by selecting the FBE length. Such selection eliminates certain variables that remain constant and equal to zero during adaptations in the special solution for gopt. Fast RLS recursion in its explicit form propagates gn in an efficient manner, see, e.g.: [1] Ljung, M. Morf, and D. Falconer, “Fast calculation of gain matrices for recursive estimation schemes,” Int. J. Contr. vol. 27, pp. 1-19, January 1978); [2] G. Carayannis, D. Manolakis, and N. Kalouptsidis, “A fast sequential algorithm for least squares filtering and prediction,” IEEE Trans. on Acoustic., Speech, Signal Proc., vol. ASSP-31, pp. 1394-1402, December 1983; and [3] J. Cioffi and T. Kailath, “Fast recursive-least-squares transversal filters for adaptive filtering,” IEEE Trans. on Acoust., Speech Signal Processing, vol. ASSP-32, pp. 304-337, April 1984.
Table 1 lists the fast recursions employed in one embodiment for computing the normalized Kalman gain. The additional index included in kL,n and γL(n) stems from the fact that these quantities admit an order-update relation, instead of time-update.







TABLE 1





Fast Transversal Computation of the Kalman gain.

















Initialization



ζf(−1) = ζh(0) = σv2



w−1f = w0b = k0 = 0



γL(0) = 1



For n = 0 to δ repeat operations (1) to (13):



(1) αL(n − 1) = h(n) − hn−1wn−2f



(2) f(n − 1) = γL(n − 1)α(n − 1)



 







  
    
      (
      3
      )
    
    ⁢
    
      
    
    ⁢
    
      k
      
        
          L
          +
          1
        
        ,
        
          n
          -
          1
        
      
    
  
  =
  
    
      [
      
        
          
            0
          
        
        
          
            
              k
              
                L
                ,
                
                  n
                  -
                  1
                
              
            
          
        
      
      ]
    
    +
    
      
        
          α
          *
          
            (
            
              n
              -
              1
            
            )
          
        
        
          
            ζ
            f
          
          ⁡
          
            (
            
              n
              -
              2
            
            )
          
        
      
      ⁡
      
        [
        
          
            
              1
            
          
          
            
              
                -
                
                  w
                  
                    n
                    -
                    2
                  
                  f
                
              
            
          
        
        ]
      
    
  







 



(4) ζf(n − 1) = ζf(n − 2) + α*(n − 1)f(n − 1)



(5) wn−1f = wn−2f + kL,n−1f(n − 1)



 







  
    
      (
      6
      )
    
    ⁢
    
      
    
    ⁢
    
      
        γ
        
          L
          +
          1
        
      
      ⁡
      
        (
        n
        )
      
    
  
  =
  
    
      
        γ
        L
      
      ⁡
      
        (
        
          n
          -
          1
        
        )
      
    
    ⁢
    
      
        
          ζ
          f
        
        ⁡
        
          (
          
            n
            -
            2
          
          )
        
      
      
        
          ζ
          f
        
        ⁡
        
          (
          
            n
            -
            1
          
          )
        
      
    
  







 



(7) v(n) = (last entry of kL+1,n−1)



(8) kL,n = kL+1,n−1(1:L) + v(n)wn−1b



(9) β(n) = ζb(n−1)v*(n)



 







  
    
      (
      10
      )
    
    ⁢
    
      
    
    ⁢
    
      
        γ
        L
      
      ⁡
      
        (
        n
        )
      
    
  
  =
  
    
      
        γ
        
          L
          +
          1
        
      
      ⁡
      
        (
        n
        )
      
    
    
      1
      -
      
        
          
            γ
            
              L
              +
              1
            
          
          ⁡
          
            (
            n
            )
          
        
        ⁢
        
          
            β
            L
          
          ⁡
          
            (
            n
            )
          
        
        ⁢
        
          
            ν
            L
          
          ⁡
          
            (
            n
            )
          
        
      
    
  







 



(11) b(n) = γL(n)βL(n)



(12) ζb(n) = ζb(n − 1) + β*(n)b(n)



(13) wnb = wn−1b + kL,nb(n)



Set gopt = kL,δγL(δ).










The purpose of computing kn in the well-known FTF algorithm is to use it in the computation of the corresponding optimal least squared solution for the FFE coefficients. Because we are interested only in kn, the filtering part of the FTF algorithm is not necessary, and does not appear in the algorithm listing.
The quantities {Wnf,wnb} are known as the least-squares solutions of the forward and backward prediction problems of order L, with corresponding residual errors {f(n),b(n)}. Now, because during the first L iterations, the desired signal for the backward prediction problem is equal to zero, the backward least squares solution wnb will be equal to zero. This means that all quantities associated with the backward prediction problems will remain nulls for the first L iterations. Since in our case the optimal solution is achieved exactly at the L-th iteration, we can simply rule out the computation of these quantities from Table 1. These operations correspond to operations (7) and (9) through (13) of Table 1. Note that operation (10) of Table 1 is also eliminated, since β(n)=0, which implies γL(n)=γL+1(n). This means that we can replace operation (6) of Table 1 with:




  
    
      
        
          
            
              γ
              L
            
            ⁡
            
              (
              n
              )
            
          
          =
          
            
              
                γ
                L
              
              ⁡
              
                (
                
                  n
                  -
                  1
                
                )
              
            
            ⁢
            
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    n
                    -
                    2
                  
                  )
                
              
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    n
                    -
                    1
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          38
          )
        
      
    
  




which can be further simplified since





  
    
      
        
          
            γ
            L
          
          ⁡
          
            (
            n
            )
          
        
        =
        
          
            
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    -
                    1
                  
                  )
                
              
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  0
                  )
                
              
            
            ⁢
            
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  0
                  )
                
              
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  1
                  )
                
              
            
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              
            
            ⁢
            
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    n
                    -
                    2
                  
                  )
                
              
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    n
                    -
                    1
                  
                  )
                
              
            
          
          =
          
            
              
                σ
                v
                2
              
              
                
                  ζ
                  f
                
                ⁡
                
                  (
                  
                    n
                    -
                    1
                  
                  )
                
              
            
            .
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          39
          )
        
      
    
  




Moreover, operation (8) of Table 1 becomes simply

kL,n=kL+1,n−1(1:L).  Equation (40)

where (1:L) denotes the first L entries of kL+1,n−1 

Table 2 illustrates a simplified fast recursion for computing the optimal FFE coefficients gopt.







TABLE 2





Fast Transversal Computation of the FFE coefficients.

















Initialization



ζf(−1) = σv2



w−1f = kL,0 = 0



γ(0) = 1



For n = 0 to δ repeat operations (1) through (7):



(1) αL(n − 1) = h(n) − hn−1wn−2f



(2) f(n − 1) = γ(n − 1)α(n − 1)



 







  
    
      (
      3
      )
    
    ⁢
    
      
    
    ⁢
    
      k
      
        
          L
          +
          1
        
        ,
        
          n
          -
          1
        
      
    
  
  =
  
    
      [
      
        
          
            0
          
        
        
          
            
              k
              
                L
                ,
                
                  n
                  -
                  1
                
              
            
          
        
      
      ]
    
    +
    
      
        
          f
          *
          
            (
            
              n
              -
              1
            
            )
          
        
        
          σ
          v
          2
        
      
      ⁡
      
        [
        
          
            
              1
            
          
          
            
              
                -
                
                  w
                  
                    n
                    -
                    2
                  
                  f
                
              
            
          
        
        ]
      
    
  







 



(4) ζf(n − 1) = ζf(n − 2) + α*(n − 1)f(n − 1)



(5) wn−1f = wn−2f + kL,n−1f(n − 1)



 







  
    
      (
      6
      )
    
    ⁢
    
      
    
    ⁢
    
      γ
      ⁡
      
        (
        n
        )
      
    
  
  =
  
    
      σ
      v
      2
    
    
      
        ζ
        f
      
      ⁡
      
        (
        
          n
          -
          1
        
        )
      
    
  







 



(7) kL,n = kL+1,n−1(1:L)



Set gopt = kL,δγ(δ)










With the Kalman gain determined, the FFE coefficients are then determined (step 308). In the recursions of Table 2, the FFE coefficients are determined by setting gopt=kL,δγ(δ) when the number of iterations, n, is equal to the DFE delay.
Unlike the conventional weighted FTF algorithm, the above recursions do not face finite precision difficulties for the following reasons. First, by ruling out the equations associated with the backward prediction problem, we are automatically eliminating many of the recursive loops that are responsible for the finite precision difficulties of the full FTF algorithm. Second, these simplified fast recursions have a forgetting factor λ=1, which yields finite precision stability. Third, the simplified algorithm deals with a finite set of data (δ+1 iterations). This algorithm is then reset, which avoids the accumulation of finite precision errors.
FIG. 4 is a logic diagram illustrating operations according to the present invention employed to determine Feed Back Equalizer (FBE) coefficients for the DFE. The FBE coefficients bopt are determined according to the matrix-vector product of Equation (30). The computation of the feedback filter coefficients simply amounts to convolution of the channel impulse response with gopt. The convolution operation that defines the optimal FBE coefficients in Equation (30) can be computed directly with LM/2 multiplications. Alternatively, the operations may also be efficiently performed using well-known fast FFT convolution techniques.
To illustrate such operations, Equation (32) is first rewritten in Equation (41) as:




  
    
      
        
          
            b
            opt
          
          =
          
            
              
                [
                
                  
                    I
                    M
                  
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        H
                        _
                      
                    
                    
                      
                        H
                        3
                      
                    
                  
                  
                    
                      
                        H
                        4
                      
                    
                    
                      
                        H
                        5
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      g
                      opt
                    
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          41
          )
        
      
    
  




The above equality holds true regardless of the values of H3, H4 or H5. Now, if these values are chosen such that the matrix C, defined by:





  
    
      
        C
        =
        
          
            [
            
              
                
                  
                    H
                    _
                  
                
                
                  
                    H
                    3
                  
                
              
              
                
                  
                    H
                    4
                  
                
                
                  
                    H
                    5
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          42
          )
        
      
    
  




is a square (Q×Q) circulant matrix, where Q is the smallest power-of-two integer larger than or equal M+L. In this case, the matrix C is rewritten in Equation (43) as:

C=F*ΛF,  Equation (43)

where F is a (Q×Q) FFT matrix and A is a diagonal matrix that contains the elements of the FFT of the first row of C. The solution for bopt becomes:





  
    
      
        
          
            b
            opt
          
          =
          
            
              [
              
                
                  
                    
                      I
                      M
                    
                  
                  
                    0
                  
                
              
              ]
            
            ⁢
            F
            *
            Λ
            ⁢
            
              
            
            ⁢
            
              F
              ⁡
              
                [
                
                  
                    
                      
                        g
                        opt
                      
                    
                  
                  
                    
                      0
                    
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          44
          )
        
      
    
  




The complexity of this method is the complexity of obtaining the FFT of two vectors of Q elements of each, the inverse FFT of another Q element vector, and Q complex multiples. Thus the overall complexity is

Q+3Q log2(Q)  Equation (45)

For the case of a power-of-two channel estimate N, the complexity is

2M+6M log2(2M).  Equation (46)

Thus, referring again to FIG. 4, in determining the FBE coefficients, the convolutional matrix H is first determined (step 402). As was previously described, the matrix H may be determined as part of the channel estimation. Then, the convolutional matrix H is extended to form the bigger circulant matrix C (step 404). The bigger circulant matrix C is then converted to the Frequency domain (step 406) as are the FFE coefficients (step 408). With both the bigger circulant matrix C and the FFE coefficients in the frequency domain, they are convolved in the frequency domain by simple matrix multiplication (step 410) to produce the FBE coefficients bopt. Finally, the resultant FBE coefficients bopt are then converted to the time domain using inverse FFT operations to produce FBE coefficients bopt in the time domain (step 412). The FBE coefficients bopt are then applied with the FFE coefficients gopt to the DFE (as previously described with reference to FIG. 2.
FIG. 5 is a block diagram illustrating a discrete time fractionally-spaced DFE that operates according to the present invention. FIG. 6 is a block diagram illustrating a multi-channel equivalent of the discrete time fractionally-spaced DFE of FIG. 5. The approach used for the symbol-spaced model can be easily extended to fractionally spaced models. In this section, we shall derive a fast algorithm for a T/2-spaced equalizer. The fast solution for the general case of faster oversampling factors, say T/3, T/4, . . . , etc. will then follow simply by inspection of the arguments in this section.
FIGS. 5 and 6 illustrate two equivalent representations for a discrete time model of a T/2-spaced equalizer. FIG. 5 illustrates a DFE 500 in terms of an down sampler and up sampler device, while FIG. 6 illustrates a DFE 600 is a corresponding multichannel representation. Thus, as contrasted to FIG. 1A, the DFE 500 of FIG. 5 includes a down sampler 502 and an up sampler 504 that alter the sampling frequency. Further, the DFE 600 of FIG. 6 includes a first channel function 602 having channel estimate h0 and corresponding noise function v0(n) and FFE 606 coefficients g0. Likewise, the DFE 600 of FIG. 6 also includes a second channel function 604 having channel estimate h1 and corresponding noise function v1(n) and FFE 608 coefficients {g0,g1}.
The equivalence between the structures of FIG. 5 and FIG. 6 can be easily verified by writing each of the quantities {h.g} and v(n) in terms of their polyphase components, and interchanging their position with the down sampler and the up sampler (see, e.g., P. P. Vaidyanathan, “Multirate Systems and Filter Banks,” Prentice Hall, N.J., 1993 and J. R. Treichler, I. Fijalkow, C. R. Johnson, Jr., “Fractionally spaced equalizers,” IEEE Signal Processing Magazine, vol. 13, no. 3, May 1996 for details). The quantities {h0,h1} and {g0,g1} are the so-called polyphase components of the channel and FFEs, and are denoted by N and L size vectors {h0,h1} and {g0,g1} (see FIG. 6). The noise sequences {v0(n),v1(n)} are also the even and odd samples of v(n), with powers 2σv2.
A model for the multichannel representation, just like the one for the symbol spaced case in equation (4) may be rewritten by collecting {g0,g1} into a single vector g′, as:

g′=[g1g0]  Equation (47)

with the output of the multichannel system given by:

└y0,ny1,n┘g′  Equation (48)

Thus, the following model of Equation (49) works for the input to g′:

└y0,ny1,n┘=xn[H0H1]+└v0,nv1,n┘  Equation (49)

where {H0,H1} are the convolution matrices associated with the subchannels {h0,h1}. It is more convenient, however, to express the inner product in Equation (45) as a function of the original FFE coefficients vector g, by reordering the entries of g′. In this case, Equation (46) is replaced by:

y′n=xnH′+v′n  Equation (50)

where





  
    
      
        
          H
          ′
        
        =
        
          [
          
            
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
            
            
              
                
                  h
                  ⁡
                  
                    (
                    3
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    2
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                
                  h
                  ⁡
                  
                    (
                    3
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    2
                    )
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
            
            
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      1
                    
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      2
                    
                    )
                  
                
              
              
                ⋮
              
              
                ⋮
              
              
                ⋰
              
              
                
                  h
                  ⁡
                  
                    (
                    0
                    )
                  
                
              
            
            
              
                
                  
                
              
              
                
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      1
                    
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      2
                    
                    )
                  
                
              
              
                
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    2
                    )
                  
                
              
            
            
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                ⋰
              
              
                ⋮
              
            
            
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      1
                    
                    )
                  
                
              
              
                
                  h
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        N
                      
                      -
                      2
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          51
          )
        
      
    
  




Given this convolution matrix and the noise variance σv2, the solution for the fractionally space problem is simply given by Equations (29) and (30), with {Hδ, H,σv2} replaced by {H′δ, H′,σv2} (similarly to the symbol-spaced case, here we assume M>N−1).

For the symbol-spaced DFE, the shift structure of Hδ implies that kL,n=kL+1,n−1(1:L). That is, the normalized gain kL,n can be computed fast, by performing the order update of kL,n−1 to kL+1,n−1, and then retaining the first L entries of kL+1,n−1 as kL,n. The procedure to derive a fast recursion in the T/2-spaced case follows the same principle. The only difference here is that the data matrix H′δ has now a double shift structure, in the sense that each row is formed by shifting two samples of the channel at a time. Thus, two successive order updates from kL,n−1 to kL+2,n−1 are performed and then the first L entries of kL+2,n−1 are retained:

kL,n−1→kL+1kn−1→kL+2,n−1  Equation (52)

so that

kL,n=kL+2,n−1(1:L)  Equation (53)

In other words, the resulting algorithm is given by two consecutive forward prediction problems similar to operations (1) through (6) of Table 2 of orders L and L+1, respectively. Table 3 lists the resulting algorithm for the T/2-spaced case.








TABLE 3





Fast Transversal Computation of


FFE coefficients for the T/2-spaced equalizer.

















Initialization



ζLf(−1) = ζL+1f(−1) = 2σv2



wL,−1 f = wL+1,−1f = kL,0 = kL+1,0 = 0



γL(0) = γL+1(0) = 1



For n = 0 to δ repeat operations (I) to (II):



(I) For q = L to L + 1 repeat operations (1) to (6):



(1) αq(n − 1) = h(2n + q − L + 1) − hn−1wq,n−2f



(2) fq(n − 1) = γq(n − 1)αq(n − 1)



 







  
    
      (
      3
      )
    
    ⁢
    
      
    
    ⁢
    
      k
      
        
          q
          +
          1
        
        ,
        
          n
          -
          1
        
      
    
  
  =
  
    
      [
      
        
          
            0
          
        
        
          
            
              k
              
                q
                ,
                
                  n
                  -
                  1
                
              
            
          
        
      
      ]
    
    +
    
      
        
          
            α
            q
            *
          
          ⁡
          
            (
            
              n
              -
              1
            
            )
          
        
        
          
            ζ
            q
            f
          
          ⁡
          
            (
            
              n
              -
              2
            
            )
          
        
      
      ⁡
      
        [
        
          
            
              1
            
          
          
            
              
                -
                
                  w
                  
                    q
                    ,
                    
                      n
                      -
                      2
                    
                  
                  f
                
              
            
          
        
        ]
      
    
  







 



(4) ζqf(n − 1) = ζqf(n − 2) + αq*(n − 1)fq(n − 1)



(5) wq,n−1f = wq,n−2f + kq,n−1fq(n − 1)



 







  
    
      (
      6
      )
    
    ⁢
    
      
    
    ⁢
    
      
        γ
        
          q
          +
          1
        
      
      ⁡
      
        (
        n
        )
      
    
  
  =
  
    
      
        γ
        q
      
      ⁡
      
        (
        n
        )
      
    
    ⁢
    
      
        
          ζ
          q
          f
        
        ⁡
        
          (
          
            n
            -
            2
          
          )
        
      
      
        
          ζ
          q
          f
        
        ⁡
        
          (
          
            n
            -
            1
          
          )
        
      
    
  







 



(II) kL,n = kL+1,n−1(1:L)



Set gopt = kL,δγL+2(δ)










In the general case of a T/S spaced equalizer, the fast DFE tap computation is a straightforward extension of the above algorithm, where q is iterated from L to (L+S−1). However, in the fractionally spaced equalizer, the RLS problem is formulated as a multi-channel problem. In such case, a multi-channel Kalman gain is calculated for the multi-channel RLS problem and the FFE taps are determined therefrom. Note that in the expressions of Table 3, successive order updates are performed.
Now, the optimal FBE taps could now be computed as




  
    
      
        
          b
          opt
        
        =
        
          
            [
            
              
                
                  
                    I
                    M
                  
                
                
                  0
                
              
            
            ]
          
          ⁢
          F
          *
          
            
              Λ
              ⁢
              
                
              
            
            S
          
          ⁢
          
            
              (
              
                F
                ⊗
                
                  I
                  S
                
              
              )
            
            ⁡
            
              [
              
                
                  
                    
                      g
                      opt
                    
                  
                
                
                  
                    0
                  
                
              
              ]
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          54
          )
        
      
    
  




where Λs is a block diagonal matrix, which satisfies

CS=FS*ΛSFS  Equation (55)

where FS=F⊕IS and CS is a block circulant matrix, which is formed by extending the matrix H in the same manner as in Equation (42). As was the case with the symbol spaced operations described above, the convolution may be performed in a transformed domain, e.g., Frequency Transformation domain, Discrete Cosine Transformation domain and the Discrete Hadamard Transformation domain, among others. In such case a Croniker product may be employed in conjunction with such domain transformation in performing multi-channel convolution in the selected transformed domain.

FIG. 7 is a block diagram illustrating a transceiver constructed according to the present invention. The components of the transceiver 700 are resident in a communications device and are illustrated only generally to show how the operations of the present invention would be accomplished in such a transceiver. The transceiver 700 includes a receiver section 702 and a transmitter section 704 and further includes either a wireless transceiver 706 or a wired transceiver, depending upon the system in which the transceiver 700 is implemented. In the case of a cellular device, RF device, satellite system device, or another wireless implementation, the transceiver 700 includes a wireless transceiver. However, in the case of a cable modem, a LAN device, a home networking device, or another device that couples to a physical media, the transceiver 700 includes a wired transceiver 708. Further, if the present invention is implemented in a Serializer/Deserializer (SERDES) or similar application, the receiver section 702 and transmitter section 704 may couple to a media without a wired transceiver 708.
Further discussion of the transmitter section 704 and receiver section 702 are in the context of baseband processing. In such case, the receiver section 702 receives a baseband signal from the wireless transceiver 706 (or wired transceiver 708) that is baseband modulated and operates upon the baseband signal to extract data. These operations include determining DFE coefficients according to the present invention and operating upon the baseband signal using the determined DFE coefficients.
The transmitter section 704 receives digital data to be transmitted from a host, codes the digital data into a baseband signal, and passes the baseband signal to the RF transceiver 706. The RF transceiver 706 couples the baseband signal to an RF carrier to create an RF signal and transmits the RF signal to a receiving device across a wireless link.
The receiver section 702 receives a baseband signal that carries coded data from the RF transceiver 706. A Programmable Gain Amplifier (PGA) 712 adjusts the gain of the baseband signal and then provides the gain-adjusted baseband signal to an Analog-to-Digital Converter (ADC) 714 for sampling. The ADC 208 samples the gain adjusted baseband signal at a particular sampling frequency, fS(that is the symbol clock frequency), to produce samples thereof.
A processor 710 couples to the output of the ADC 714 and analyzes a preamble sequence contained in each received physical layer frame. Based upon the preamble sequence, the processor 710 determines a gain to be applied to portions of the baseband signal corresponding to the data carrying portions of the frame and provides this gain to the PGA 712. Further, the processor 710 may also interact with the optional timing compensation section 716 to compensate for symbol timing and RF carrier mismatches.
The processor 710, based upon the preamble sequence (and based upon actual extracted data in some operations), also determines FFE 104 and FBE 108 coefficients. The manner in which these coefficients are determined was previously described in detail herein. Further, and as was also previously described, the processor 710 may estimate a channel and calculate DFE coefficients based upon unknown but assumed data content. After the processor 710 determines these coefficients, they are applied to the FFE 104 and FBE 108 for subsequent use in extracting data from the baseband signal.
The structure described in FIG. 7 may be embodied using various types of circuits formed using various manufacturing processes. For example, in one particular embodiment, the RF transceiver 706 (or wired transceiver 708) is embodied in a first integrated circuit that is coupled to a second integrated circuit that includes the transmitter section 704 and the receiver section 702, among other circuits. In another embodiment, the RF transceiver 706, the transmitter section 704 and the receiver section 702 are all formed on a single monolithic integrated circuit. These integrated circuits may be constructed in CMOS or another semiconductor technology, e.g., PMOS, NMOS, Bipolar, etc.
Further, the receiver section 702 of FIG. 7 may be constructed using various circuit elements/combinations. In one embodiment, all structures past the ADC 714 in the receiver section 702 are embodied using a Digital Signal Processor (DSP) or similar processing device. In another embodiment, dedicated signal path circuitry embodies each of the structural components of the receiver section 702, including the processor 710. While a DSP implementation would provide more flexibility, dedicated signal path circuitry would typically provide higher performance at a lower cost and with lower power consumption.
FIG. 8 is a block diagram illustrating a transceiver having a LE constructed according to the present invention. The components of the transceiver 800 are similar to or the same as are similarly named components in FIG. 7. The transceiver 800 includes a receiver section 802 and a transmitter section 804 and further includes either a wireless transceiver 806 or a wired transceiver, depending upon the system in which the transceiver 800 is implemented. In the case of a cellular device, RF device, satellite system device, or another wireless implementation, the transceiver 800 includes a wireless transceiver. However, in the case of a cable modem, a LAN device, a home networking device, or another device that couples to a physical media, the transceiver 800 includes a wired transceiver 808. Further, if the present invention is implemented in a Serializer/Deserializer (SERDES) or similar application, the receiver section 802 and transmitter section 804 may couple to a media without a wired transceiver 808.
The receiver section 802 receives a baseband signal that carries coded data from the RF transceiver 806. A Programmable Gain Amplifier (PGA) 812 adjusts the gain of the baseband signal and then provides the gain-adjusted baseband signal to an Analog-to-Digital Converter (ADC) 814 for sampling. The ADC 208 samples the gain adjusted baseband signal at a particular sampling frequency, fS(that is the symbol clock frequency), to produce samples thereof.
A processor 810 couples to the output of the ADC 814 and analyzes a preamble sequence contained in each received physical layer frame. The processor 810 may be a LE coefficients processor, a general processor such as a microprocessor, Digital Signal Processor (DSP), a custom processor, or another processing device that performs the LE coefficient processing operations, among other operations. Based upon the preamble sequence, the processor 810 determines a gain to be applied to portions of the baseband signal corresponding to the data carrying portions of the frame and provides this gain to the PGA 812. Further, the processor 810 may also interact with the optional timing compensation section 816 to compensate for symbol timing and RF carrier mismatches.
The processor 810, based upon the preamble sequence, the LE 154 structure, and an expected preamble sequence, determines LE 154 coefficients. The manner in which these coefficients are determined will be described with reference to FIGS. 9 and 10. After the processor 810 determines the LE coefficients, they are applied to the LE 154 for subsequent use in extracting data from the baseband signal.
The structure described in FIG. 8 may be embodied using various types of circuits formed using various manufacturing processes. For example, in one particular embodiment, the RF transceiver 806 (or wired transceiver 808) is embodied in a first integrated circuit that is coupled to a second integrated circuit that includes the transmitter section 804 and the receiver section 802, among other circuits. In another embodiment, the RF transceiver 806, the transmitter section 804 and the receiver section 802 are all formed on a single monolithic integrated circuit. These integrated circuits may be constructed in CMOS or another semiconductor technology, e.g., PMOS, NMOS, Bipolar, etc.
Further, the receiver section 802 of FIG. 8 may be constructed using various circuit elements/combinations. In one embodiment, all structures past the ADC 814 in the receiver section 802 are embodied using a DSP or similar processing device. In another embodiment, dedicated signal path circuitry embodies each of the structural components of the receiver section 802, including the processor 810. While a DSP implementation would provide more flexibility, dedicated signal path circuitry would typically provide higher performance at a lower cost and with lower power consumption.
FIG. 9 is a logic diagram generally illustrating operation according to the present invention in determining LE coefficients and applying such coefficients to a LE. The operations of the present invention are performed by a processor, such as a DSP, or other circuitry present within a receiver that determines LE coefficients to be applied to a LE, also resident in the receiver. The LE operates upon samples of a received signal in an attempt to remove channel effects from the samples so that digital data may be extracted from the samples. The structure and operation of LEs, one of which was illustrated in FIG. 1B, are generally known and will not be further described herein except as they relate to the present invention.
The operations of the present invention will be first performed by a processor, tap computer, DSP, or other receiver device, to determine initial DFE coefficients to be used in subsequent operations by the receiver. Thus, during startup or reset, a channel corresponding to the receiver is estimated (step 902). According to one embodiment of the present invention, the channel is estimated based upon a known preamble sequence. With the channel estimated, LE coefficients are determined based upon the channel estimate (step 904). The LE coefficients are then applied to the LE (step 906) and are used in equalizing samples of a received signal to remove channel effects. These LE coefficients are continually updated (step 910) using a known technique. Periodically, upon the receipt of a next packet for example, upon an indication that redetermination is required, or upon another triggering event, the LE coefficients are redetermined (step 912). In this event, another channel estimate may be obtained (step 914). Then, the LE coefficients are again determined according to the present invention and applied to the LE. The operations of FIG. 9 continue until the receiver is turned off, placed in a sleep mode, or otherwise inactivated.
FIG. 10 is a logic diagram illustrating operations according to the present invention employed to determine LE coefficients. Operation commences with formulating the channel estimate as a convolution matrix H (step 1002). Operation continues with relating the convolution matrix H to the LE coefficients in a matrix format equation, the matrix format equation based upon the structure of the LE, the convolution matrix, and an expected output of the LE (step 1004). Operation continues with using a Fast Transversal Filter (FTF) algorithm to formulate a recursive least squares solution to the matrix format equation (step 1006). Operation concludes with computing the recursive least squares solution to yield the LE coefficients using structured equations (step 1008).
In one particular embodiment, the recursive least squares solution is formulated as a Kalman gain solution using the FTF algorithm. In such embodiment, the Kalman gain solution may be determined using an Array Form Algorithm. The length of the LE may be chosen to force the Array Form Algorithm to use a lower triangular matrix. Further, the length of the LE may be chosen to force the FTF algorithm to use a lower triangular matrix.
The channel estimate is typically based upon a known preamble sequence of a data packet upon which the LE operates. After the LE coefficients have been determined, the LE may be used to equalize data received via the channel.
The method of the present invention is exact and may be employed to compute LE coefficients to track a rapidly fading channel. Because the method of the present invention relies upon simple structured recursion, the computation period and/or the hardware required for the LE coefficient computations may be implemented in both portable devices and non-portable devices. The method of the preset invention may be efficiently used in many communication systems that require the use of equalizers such as mobile wireless receivers, fixed wireless receivers, cable modem receivers, HDTV receivers, etc. to increase the performance of such communication systems.
In one particular embodiment, the channel estimate is formulated as a convolution matrix H according to Equation (5). With this embodiment, the matrix format equation is formulated at step (1004) as:

gopt=[σv2I+H*H]−1H*d  Equatoin (56)

Where:
H is the convolution matrix;
gopt is a vector of the LE coefficients;
σv2 is the power spectral density of the channel estimate;
I is the identity matrix; and
d as a vector of all zeros except for the δth element, which is unity, δ representing a system delay.
With this embodiment, using the FTF algorithm to formulate the recursive least squares solution to the matrix format equation (at step 1006) produces the operations illustrated in Table 4.







TABLE 4





Fast Transversal Computation of the LE taps.

















Initialization



ζf(−1) = ζb(0) = σv2



w−1 = w−1f = w−1b = kL,0 = 0



γM(0) = 1



For n = 0 to N + L − 1 repeat operations (1) to (16):



(1) αL(n − 1) = h(n) − hn−1wn−1f



(2) f(n − 1) = γL(n − 1)α(n − 1)



 







  
    
      (
      3
      )
    
    ⁢
    
      
    
    ⁢
    
      
        k
        _
      
      
        
          L
          +
          1
        
        ,
        n
      
    
  
  =
  
    
      [
      
        
          
            0
          
        
        
          
            
              k
              
                L
                ,
                
                  n
                  -
                  1
                
              
            
          
        
      
      ]
    
    +
    
      
        
          α
          *
          
            (
            
              n
              -
              1
            
            )
          
        
        
          
            ζ
            f
          
          ⁡
          
            (
            
              n
              -
              2
            
            )
          
        
      
      ⁡
      
        [
        
          
            
              1
            
          
          
            
              
                -
                
                  w
                  
                    n
                    -
                    1
                  
                  f
                
              
            
          
        
        ]
      
    
  







 



(4) ζf(n − 1) = ζf(n − 2) + α*(n − 1)f(n − 1)



(5) wnf = wn−1f + kL,n−1f(n − 1)



 







  
    
      (
      6
      )
    
    ⁢
    
      
    
    ⁢
    
      
        γ
        
          L
          +
          1
        
      
      ⁡
      
        (
        n
        )
      
    
  
  =
  
    
      
        γ
        L
      
      ⁡
      
        (
        
          n
          -
          1
        
        )
      
    
    ⁢
    
      
        
          ζ
          f
        
        ⁡
        
          (
          
            n
            -
            2
          
          )
        
      
      
        
          ζ
          f
        
        ⁡
        
          (
          
            n
            -
            1
          
          )
        
      
    
  







 



(7) v(n) = (last entry of kL+1,n)



(8) kL,n = k1:L,n + v(n)wn−1b



(9) β(n) = ζb(n − 1)v*(n)



 







  
    
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(11) b(n) = γL(n)βL(n)



(12) ζb(n) = ζb(n − 1) + β*(n)b(n)



(13) wnb = wn−1b + kL,nb(n)



(14) ea(n) = d(n) − hn−1wn−1



(15) ep(n) = γL(n)ea(n)



(16) wn = wn−1 + ep(n)kL,n



If n = N + L − 1 set gopt = wn










With this embodiment, for the values of n≦L−1, the variables corresponding to the backward prediction problem are all equal to zero, while the forward prediction variables are not.
For the values of L−1<n<N−1, the backward prediction variables are non-zero as well. For N−1<n<N+L−1, the forward prediction variables are equal to zero. Exploiting this fact, the complexity (C) of the algorithm (in terms of total number of multiplies) is

C=7NL−2L2, for N>L  Equation (57)

C=7NL−2L2, for N<L 

Where N and L are the lengths of the channel and the LE, respectively. Such complexity is orders of magnitude lower than direct matrix inversion, which requires the greater of N3 or L3 multiplies, whichever is greater.
The structure and operation of the present invention may be employed in satellite communication systems, satellite television systems, HDTV systems, fixed wireless communication systems, mobile wireless communication systems, cable modem/television communication systems, home networking systems, wireless local area networking systems, wired local area networking systems, and many other types of communication systems. The present invention applies to all types of communication devices in which equalizers are employed to operate upon received signals.
The invention disclosed herein is susceptible to various modifications and alternative forms. Specific embodiments therefore have been shown by way of example in the drawings and detailed description. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the present invention as defined by the claims.

## Claims

1. A method for directly computing Linear Equalizer (LE) coefficients from a channel estimate, the method comprising:
formulating the channel estimate as a convolution matrix H;
relating the convolution matrix H to the LE coefficients in a matrix format equation, the matrix format equation based upon the structure of the LE, the convolution matrix, and an expected output of the LE;
using a Fast Transversal Filter (FTF) algorithm to formulate a recursive least squares solution to the matrix format equation, wherein a length of the LE is chosen to force the FTF algorithm to use a lower triangular matrix; and
computing the recursive least squares solution to yield the LE coefficients using structured equations.

2. The method of claim 1, wherein the recursive least squares solution is formulated as a Kalman gain solution using the FTF algorithm.

3. The method of claim 2, wherein the Kalman gain solution is determined using an Array Form Algorithm.

4. The method of claim 3, wherein a length of the LE is chosen to force the Array Form Algorithm to use a lower triangular matrix.

5. The method of claim 1, wherein the matrix format equation comprises:


gopt=[σv2I+H*H]−1H*d

where:
H is the convolution matrix;
gopt is a vector of the LE coefficients;
σv2 is the power spectral density of the channel estimate;
I is the identity matrix; and
d as a vector of all zeros except for the δth element, which is unity, δ representing a system delay.

6. The method of claim 5, wherein the system delay is a combination of a length of the channel and a length of the LE.

7. The method of claim 1, wherein the channel estimate is based upon a known preamble sequence of a data packet upon which the LE operates.

8. The method of claim 1, further comprising the LE using the LE coefficients to equalize received data.

9. The method of claim 7, wherein the LE services at least one a receiver used in a High Definition Television, a receiver used in a Wireless Local Area Network, and a receiver used in a Cellular Telephone.

10. A Linear Equalizer System comprising:
a Liner Equalizer (LE) having an input that receives an uncompensated signal and an output; and
a LE coefficients processor that is operable to:
formulate the channel estimate as a convolution matrix H;
relate the convolution matrix H to the LE coefficients in a matrix format equation, the matrix format equation based upon the structure of the LE, the convolution matrix, and an expected output of the LE;
use a Fast Transversal Filter (FTF) algorithm to formulate a recursive least squares solution to the matrix format equation;
choose a length of the LE to force the FTF algorithm to use a lower triangular matrix; and
solve the recursive least squares solution to yield the LE coefficients.

11. The Linear Equalizer System of claim 10, wherein the LE coefficients processor is operable to formulate the recursive least squares solution as a Kalman gain solution using the FTF algorithm.

12. The Linear Equalizer System of claim 11, wherein the LE coefficients processor is operable to determine the Kalman gain solution using an Array Form Algorithm.

13. The Linear Equalizer System of claim 12, the LE coefficients processor is operable to choose a length of the LE to force the Array Form Algorithm to use a lower triangular matrix.

14. The Linear Equalizer System of claim 10, wherein the matrix format equation comprises:


gopt=[σv2I+H*H]−1H*d

where:
H is the convolution matrix;
gopt is a vector of the LE coefficients;
σv2 is the power spectral density of the channel estimate;
I is the identity matrix; and
d as a vector of all zeros except for the δth element, which is unity, δ representing a system delay.

15. The Linear Equalizer System of claim 14, wherein the system delay is a combination of a length of the channel and a length of the LE.

16. The Linear Equalizer System of claim 10, wherein the channel estimate is based upon a known preamble sequence of a data packet upon which the LE operates.

17. The Linear Equalizer System of claim 10, wherein the LE is operable to equalize received data using the LE coefficients.

18. The Linear Equalizer System of claim 16, wherein the LE is operable to service at least one of a receiver used in a High Definition Television, a receiver used in a Wireless Local Area Network, and a receiver used in a Cellular Telephone.