# 0.12 Linear equalization  (Page 2/17)

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This chapter suggests several different ways that the coefficients of the equalizer can be chosen.The first procedure, in "A Matrix Description" , minimizes the square of thesymbol recovery error This is the error between the equalizer output and the transmitted symbol,and is known whenever there is a training sequence. over a block of data, which can be doneusing a matrix pseudoinversion. Minimizing the (square of the)error between the received data values and the transmitted values can also be achieved usingan adaptive element, as detailed in "An Adaptive Approach to Trained Equalization" . When there is no training sequence, other performance functionsare appropriate, and these lead to equalizers such as the decision-directed approach in "Decision-Directed Linear Equalization" and the dispersion minimization method in "Dispersion-Minimizing Linear Equalization" . The adaptive methods considered here are only modestlycomplex to implement, and they can potentially track time variations in the channel model,assuming the changes are sufficiently slow.

## Multipath interference

The villains of this chapter are multipath and other additive interferers. Both should be familiar from [link] .

The distortion caused by an analog wireless channel can be thought of as a combination of scaled and delayedreflections of the original transmitted signal. These reflections occur when there are differentpaths from the transmitting antenna to the receiving antenna. Between two microwave towers, for instance, the paths may include one alongthe line-of-sight, reflections from nearby hills, and bounces from a field or lake betweenthe towers. For indoor digital TV reception, there are many (local)time-varying reflectors, including people in the receiving room, and nearby vehicles.The strength of the reflections depends on the physical properties of the reflecting objects, while the delay of thereflections is primarily determined by the length of the transmission path. Let $u\left(t\right)$ be the transmitted signal. If $N$ delays are represented by ${\Delta }_{1},\phantom{\rule{4pt}{0ex}}{\Delta }_{2},...,\phantom{\rule{4pt}{0ex}}{\Delta }_{N}$ , and the strength of the reflections is ${\alpha }_{1},\phantom{\rule{4pt}{0ex}}{\alpha }_{2},...,\phantom{\rule{4pt}{0ex}}{\alpha }_{N}$ , then the received signal is

$y\left(t\right)={\alpha }_{1}u\left(t-{\Delta }_{1}\right)+{\alpha }_{2}u\left(t-{\Delta }_{2}\right)+...+{\alpha }_{N}u\left(t-{\Delta }_{N}\right)+\eta \left(t\right),$

where $\eta \left(t\right)$ represents additive interferences. This model of the transmission channelhas the form of a finite impulse response filter, and the total length of time ${\Delta }_{N}-{\Delta }_{1}$ over which the impulse response is nonzero is called the delay spread of the physical medium.

This transmission channel is typically modelled digitally assuming a fixed sampling period ${T}_{s}$ . Thus, [link] is approximated by

$\begin{array}{ccc}\hfill y\left(k{T}_{s}\right)={a}_{1}u\left(k{T}_{s}\right)& +& {a}_{2}u\left(\left(k-1\right){T}_{s}\right)+...\hfill \\ & +& {a}_{n}u\left(\left(k-n\right){T}_{s}\right)+\eta \left(k{T}_{s}\right).\hfill \end{array}$

In order for the model [link] to closely represent the system [link] , the total time over which the impulse response is nonzero (the time $n{T}_{s}$ ) must be at least as large as the maximum delay ${\Delta }_{N}$ . Since the delay is not a function of the symbol period ${T}_{s}$ , smaller ${T}_{s}$ require more terms in the filter (i.e., larger $n$ ).

For example, consider a sampling interval of ${T}_{s}\approx 40$ nanoseconds (i.e., a transmission rate of 25 MHz).A delay spread of approximately 4 microseconds would correspond to one hundred taps in the model [link] . Thus, at any time instant, the received signalwould be a combination of (up to) one hundred data values.If ${T}_{s}$ were increased to 0.4 microsecond (i.e., 2.5 MHz), only 10 terms wouldbe needed, and there would be interference with only the 10 nearest data values.If ${T}_{s}$ were larger than 4 microseconds (i.e., 0.25 MHz), only one term wouldbe needed in the discrete-time impulse response. In this case, adjacent sampled symbols would notinterfere. Such finite duration impulse response models as [link] can also be used to represent the frequency-selective dynamicsthat occur in the wired local end-loop in telephony, and other (approximately) linear, finite-delay-spread channels.

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scanning tunneling microscope
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