0.12 Linear equalization  (Page 4/17)

A matrix description

The linear equalization problem is depicted in [link] . A prearranged training sequence $s\left[k\right]$ 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 $y\left[k\right]\approx s[k-\delta ]$ for some specific $\delta$ .

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

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

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

and

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

or, with the appropriate matrix definitions,

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

for a particular $\delta$ . 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

and

Using [link] , write

As a measure of the performance of the $f_i$ in $F$ , consider

$J_{LS}$ 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 $J_{LS}$ cause the output of the equalizer to become close to thevalues of the (delayed) source.

Given [link] and [link] , $J_{LS}$ in [link] can be written as

Because $J_{LS}$ is a scalar, ${\left(RF\right)}^{T}S$ and $S^{T}RF$ are also scalars. Since the transpose of a scalar is equal to itself, ${\left(RF\right)}^{T}S=S^{T}RF$ , and [link] can be rewritten as

The issue is now one of choosing the $n+1$ entries of $F$ to make $J_{LS}$ as small as possible.