ISI, RAKE and linear transversal equalizer

Signal processing to minimize ISI

Equalization is the signal processing operation that minimizes ISI. An equalizer is a filter at the receiver whose response is roughly the inverse of the channel, so the combined channel + equalizer response is flat. Because the mobile channel changes with time, the equalizer is adaptive and tracks the channel using training sequences and decision feedback.

RAKE receiver

A RAKE receiver is a multi-correlator CDMA receiver. Each correlator ("finger") locks to one multipath component delayed by at least one chip, and the finger outputs are weighted and combined. It turns multipath into diversity gain instead of ISI.

Linear transversal equalizer (LTE)

The most common equalizer structure is a tapped delay line (FIR) filter. The received signal passes through a chain of delay elements of one symbol period T; each tap output is multiplied by an adjustable complex weight c_n, and all products are summed.

y_k ─┬─[T]─┬─[T]─┬─[T]─┬─ ... ─[T]─┐
     │     │     │     │           │
   (x)c₋N₁ (x)  (x)   (x)  ...   (x)c_N₂
     │     │     │     │           │
     └─────┴─────┴──(Σ)┴───────────┘
                     │
                     v d̂_k ─►[Decision]─► x̂_k
                     │
        x_k (training) ─►(−)─► e_k ─► [Adapt taps]

Output:

d̂_k = Σ_(n = −N₁)^(N₂) c_n* y_(k−n)

• N₁ = number of taps for future (precursor) samples, N₂ for past (postcursor) samples.
• The error e_k = x_k − d̂_k (or decision − d̂_k in tracking mode) is used by an algorithm such as LMS to adjust the taps.
• The minimum MSE of an LTE is:

E[|e|²] = (T/2π) ∫_(−π/T)^(π/T) N₀ dω / (|F(e^jωT)|² + N₀)

Merits: simple, stable, easy to adapt. Limitation: in channels with deep spectral nulls it boosts noise a lot, so non-linear equalizers (DFE, MLSE) are preferred for severe multipath.