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A matrix description

The linear equalization problem is depicted in [link] . A prearranged training sequence s [ k ] is assumed known at the receiver. The goal is to find an FIR filter (called the equalizer ) so that the output of the equalizer is approximatelyequal to the known source, though possibly delayed in time. Thus, the goal is to choose the impulse response f i so that y [ k ] s [ k - δ ] for some specific δ .

The problem of linear equalization is to find a linear system f that undoes the effects of the channel while minimizing the effects of the interferences.
The problem of linear equalization is to find a linear system f that undoes the effects of the channel while minimizing the effects of the interferences.

The input–output behavior of the FIR linear equalizer can be described as the convolution

y [ k ] = j = 0 n f j r [ k - j ] ,

where the lower index on j can be no lower than zero (or else the equalizer is noncausal;that is, it can illogically respond to an input before the input is applied). This convolution is illustrated in [link] as a “direct form FIR” or “tapped delay line.”

The direct form FIR filter of Equation 5 can be pictured as a tapped delay line where each z^-1 block represents a time delay of one symbol period. The impulse response of the filter is f_0, f_1, ..., f_n.
The direct form FIR filter of Equation [link] can be pictured as a tapped delay line where each z - 1 block represents a time delay of one symbol period. The impulse responseof the filter is f 0 , f 1 , . . . , f n .

The summation in [link] can also be written (e.g., for k = n + 1 ) as the inner product of two vectors

y [ n + 1 ] = [ r [ n + 1 ] , r [ n ] , ... , r [ 1 ] ] f 0 f 1 f n .

Note that y [ n + 1 ] is the earliest output that can be formed given no knowledge of r [ i ] for i < 1 . Incrementing the time index in [link] gives

y [ n + 2 ] = [ r [ n + 2 ] , r [ n + 1 ] , ... , r [ 2 ] ] f 0 f 1 f n

and

y [ n + 3 ] = [ r [ n + 3 ] , r [ n + 2 ] , ... , r [ 3 ] ] f 0 f 1 f n .

Observe that each of these uses the same equalizer parameter vector. Concatenating p - n of these measurements into one matrix equation over the availabledata set for i = 1 to p gives

y [ n + 1 ] y [ n + 2 ] y [ n + 3 ] y [ p ] = r [ n + 1 ] r [ n ] r [ 1 ] r [ n + 2 ] r [ n + 1 ] r [ 2 ] r [ n + 3 ] r [ n + 2 ] r [ 3 ] r [ p ] r [ p - 1 ] r [ p - n ] f 0 f 1 f n ,

or, with the appropriate matrix definitions,

Y = R F .

Note that R has a special structure, that the entries along each diagonal are the same. R is known as a Toeplitz matrix and the toeplitz command in M atlab makes it easy to build matrices with this structure.

Source recovery error

The delayed source recovery error is

e [ k ] = s [ k - δ ] - y [ k ]

for a particular δ . This section shows how the source recoveryerror can be used to define a performance function that depends on the unknown parameters f i . Calculating the parameters that minimize this performancefunction provides a good solution to the equalization problem.

Define

S = s [ n + 1 - δ ] s [ n + 2 - δ ] s [ n + 3 - δ ] s [ p - δ ]

and

E = e [ n + 1 ] e [ n + 2 ] e [ n + 3 ] e [ p ] .

Using [link] , write

E = S - Y = S - R F .

As a measure of the performance of the f i in F , consider

J L S = i = n + 1 p e 2 [ i ] .

J L S is nonnegative since it is a sum of squares. Minimizing such a summed squareddelayed source recovery error is a common objective in equalizer design,since the f i that minimize J L S cause the output of the equalizer to become close to thevalues of the (delayed) source.

Given [link] and [link] , J L S in [link] can be written as

J L S = E T E = ( S - R F ) T ( S - R F ) = S T S - ( R F ) T S - S T R F + ( R F ) T R F .

Because J L S is a scalar, ( R F ) T S and S T R F are also scalars. Since the transpose of a scalar is equal to itself, ( R F ) T S = S T R F , and [link] can be rewritten as

J L S = S T S - 2 S T R F + ( R F ) T R F .

The issue is now one of choosing the n + 1 entries of F to make J L S as small as possible.

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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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