Interleaving and adaptive equalization of ISI

Interleaving

Interleaving is the process of rearranging coded bits in time before transmission so that burst errors from deep fades are spread into random, isolated errors after de-interleaving, where the channel decoder can correct them.

How equalization offsets ISI

In a multipath channel, delayed copies of a symbol overlap with later symbols, causing inter-symbol interference (ISI). If the channel impulse response is f(t), the received signal is y(t) = x(t) ⊛ f(t) + n(t). An equalizer h_eq(t) is chosen so that the combined response is an impulse:

f(t) ⊛ h_eq(t) = δ(t)
F(f) · H_eq(f) = 1   →  H_eq(f) = 1 / F(f)

So the equalizer is an inverse filter of the channel. Since the mobile channel changes with time and position, the equalizer must be adaptive.

Example: adaptive linear (transversal) equalizer

y_k ─┬─[T]─┬─[T]─┬─ ... ─[T]─┐
     │     │     │           │
   (x)w₀  (x)w₁ (x)w₂ ...  (x)w_N
     └─────┴────(Σ)──────────┘
                 │ d̂_k
                 ├──────►[Decision]──► x̂_k
                 v              │
     x_k ───►(+)(−)◄────────────┘ (tracking)
     (training)  │ e_k
                 v
          [LMS: w ← w + α e_k y_k]

Operation

1. Training mode: the transmitter sends a known training sequence (e.g. 26-bit midamble in GSM). The receiver compares the equalizer output d̂_k with the known symbol x_k to form the error e_k = x_k − d̂_k.
2. Weight update: an adaptive algorithm (e.g. LMS) adjusts the taps to reduce the mean square error:

w_(k+1) = w_k + α e_k y_k

The taps converge near the optimum ŵ = R⁻¹p, so the filter undoes the channel's ISI. 3. Tracking mode: during user data, the receiver's own decisions x̂_k replace the training symbols, so the equalizer keeps following slow channel changes. 4. Retraining: training is repeated periodically (in every TDMA burst), because over a long time the channel can change too much for decision-directed tracking to follow reliably.

Example: a two-path channel with ISI from one earlier symbol (f = [1, 0.5]) needs an inverse filter approximately [1, −0.5, 0.25]; the adaptive equalizer finds these taps automatically from the training sequence.