7. Inter-Symbol Interference and Equalization // Digital Communications // bhaswanth

Send a square pulse through any band-limited channel and it stretches in time. Two pulses sent close together blur into one another at the receiver. That is inter-symbol interference (ISI): the energy of one symbol leaks into the next.

7.1 Nyquist's pulse-shaping criterion

Nyquist asked: is there a pulse shape $p(t)$ such that, sampled at the symbol rate $1/T$ , only one sample is non-zero per symbol? If yes, ISI vanishes at the sampling instants. He proved: a pulse satisfies the zero-ISI condition if and only if its spectrum $P(f)$ satisfies

$\sum_{k} P\!\left(f - \tfrac{k}{T}\right) = T \text{ for all } f.$

In words, the periodically-summed spectrum is flat. The sinc pulse $p(t) = \text{sinc}(t/T)$ is the limiting case (with brick-wall spectrum), but its long oscillating tails make it useless in practice (any timing error is catastrophic).

7.2 The raised-cosine pulse

The practical solution is the raised-cosine pulse, with spectrum

$P_\text{RC}(f) = \begin{cases} T & |f| \le \frac{1-\beta}{2T} \\ \frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left(|f| - \frac{1-\beta}{2T}\right)\right)\right] & \frac{1-\beta}{2T} \le |f| \le \frac{1+\beta}{2T} \\ 0 & \text{otherwise} \end{cases}$

The roll-off factor $\beta$ is between 0 (sinc, brick wall) and 1 (full bandwidth doubled). Larger $\beta$ uses more bandwidth but gives shorter, less timing-sensitive time-domain pulses. The total occupied bandwidth is $(1 + \beta)/T$ .

Practically, the pulse is split into a square-root raised-cosine (SRRC, RRC) at the transmitter and an identical SRRC at the receiver. Their cascade is full raised-cosine, satisfying Nyquist. The receiver SRRC also doubles as the matched filter, which is wonderfully tidy. DVB-S, DVB-S2, LTE, and most satellite links use SRRC pulse shaping.

7.3 Partial-response signalling

What if you accept a little controlled ISI in exchange for narrower bandwidth? Partial-response signalling does exactly that. Class IV ( $1 + D$ , "duobinary") and Class I ( $1 - D^2$ ) shapes deliberately let one symbol of ISI through, knowing the receiver can subtract it out. Magnetic disk read channels used this for decades.

7.4 Equalization

Real channels are worse than band-limited; they have multipath, dispersion, frequency-dependent loss. The receiver must equalize: undo the channel's distortion as best it can.

Linear equalizer. A finite-impulse-response filter at the receiver whose taps are chosen to invert the channel response (zero-forcing) or minimize mean square error. Cheap and often sufficient on mild channels.
Decision-feedback equalizer (DFE). A linear feedforward stage plus a feedback stage that subtracts ISI from already-decided symbols. Avoids amplifying noise at deep channel nulls. Used in 10/100GBASE-T Ethernet, DSL, and many cable modems.
Adaptive equalizer. Tap weights updated continuously by gradient algorithms (LMS, RLS) using either a known training sequence or blind decision-directed feedback. Required for time-varying channels (mobile radio, indoor Wi-Fi).
Maximum-likelihood sequence estimation (MLSE). A Viterbi-like search over all possible transmitted sequences, picking the one most likely given the received samples. Optimal but expensive; used in GSM (5-tap MLSE) and many radar systems.

A modern OFDM receiver, however, mostly sidesteps equalization. Each subcarrier is narrow enough that its channel is roughly flat; a single-tap equalizer per subcarrier (one complex multiply) corrects amplitude and phase. That is part of OFDM's appeal.