# 0.12 Linear equalization

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The revolution in data communications technology can be dated from the invention of automatic and adaptive channelequalization in the late 1960s.

—Gitlin, Hayes, and Weinstein, Data Communication Principles , 1992

When all is well in the receiver, there is no interaction between successive symbols; each symbol arrivesand is decoded independently of all others. But when symbols interact, when the waveformof one symbol corrupts the value of a nearby symbol, then the received signal becomes distorted. It isdifficult to decipher the message from such a received signal. This impairment is called “intersymbol interference” and was discussed inChapter  [link] in terms of non-Nyquist pulse shapes overlapping in time. This chapter considersanother source of interference between symbols that is caused by multipath reflections(or frequency-selective dispersion) in the channel.

When there is no intersymbol interference (from a multipath channel, from imperfectpulse shaping, or from imperfect timing), the impulse response of thesystem from the source to the recovered message has a single nonzero term. The amplitude ofthis single “spike” depends on the transmission losses, and the delayis determined by the transmission time. When there is intersymbol interference caused by a multipath channel,this single spike is “scattered,” duplicated once for each path in the channel.The number of nonzero terms in the impulse response increases. The channel can be modeled as afinite-impulse-response, linear filter $C$ , and the delay spread is the total time interval during which reflections with significant energy arrive.The idea of the equalizer is to build(another) filter in the receiver that counteracts the effect of the channel. In essence, the equalizermust “unscatter” the impulse response. This can be stated as the goal of designing the equalizer $E$ so that the impulse response of the combined channel and equalizer $CE$ has a single spike. This can be cast as an optimization problem,and can be solved using techniques familiar from Chapters [link] , [link] , and [link] .

The transmission path may also be corrupted by additive interferences such as those caused byother users. These noise components are usually presumed to beuncorrelated with the source sequence and they may be broadband or narrowband, in band or out of bandrelative to the bandlimited spectrum of the source signal. Like the multipath channel interference,they cannot be known to the system designer in advance. The second job of the equalizer is to reject such additivenarrowband interferers by designing appropriate linear notch filters “on-the-fly” based on the received signal.At the same time, it is important that the equalizer not unduly enhance the broadband noise.

The signal path of a baseband digital communication system is shown in [link] , which emphasizes the role of the equalizer in trying to counteract theeffects of the multipath channel and the additive interference. As in previous chapters, all of the inner parts ofthe system are assumed to operate precisely: thus, the upconversion and downconversion,the timing recovery, and the carrier synchronization (all those parts of the receiver that are not shown in [link] ) are assumed to be flawless and unchanging.Modelling the channel as a time-invariant FIR filter, the next section focuses on the taskof selecting the coefficients in the block labelled “linear digital equalizer,” with the goalof removing the intersymbol interference and attenuating the additive interferences.These coefficients are to be chosen based on the sampled received signal sequenceand (possibly) knowledge of a prearranged “training sequence.” While the channel may actually be time varying, thevariations are often much slower than the data rate, and the channel can be viewed as (effectively) time invariant oversmall time scales.

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