Equalization fundamentals and RAKE receiver

Fundamentals of equalization

Equalization is a receiver technique that removes inter-symbol interference (ISI) caused by multipath delay spread or band-limited channels. ISI appears when the delay spread is comparable to the symbol period, so symbols overlap and the error rate becomes high.

• The equalizer is a filter whose response is the inverse of the channel: F(f)·H_eq(f) = 1, i.e. f(t) * h_eq(t) = δ(t), giving an overall flat, ISI-free response.
• Since the mobile channel is unknown and time-varying, the equalizer is adaptive:
  1. Training mode: a known training sequence is sent; the error e(k) = d(k) − d̂(k) is used to set the taps.
  2. Tracking mode: detected data symbols are used as the reference to follow channel changes.
• Basic structure: a transversal (tapped delay line) filter, d̂(k) = Σ wₙ·y(k − n).
• Types: linear (transversal, lattice) and non-linear (decision feedback equalizer, MLSE).
• Algorithms: zero forcing, LMS, RLS.

 y(k)->[T]->[T]->...->[T]
   |    |    |          |
  w0   w1   w2   ...   wN   (adapted by LMS/RLS)
   +----+----+----------+--(Σ)--> d^(k) -> decision

RAKE receiver

A RAKE receiver is used in CDMA to collect the energy of several multipath copies instead of suffering from them. Paths that arrive more than one chip apart are resolvable because the PN code has low autocorrelation away from zero shift.

             +------------+   +----+   w1
        +--->| Correlator |-->|Z1  |--(x)--+
        |    |  code c(t) |   +----+       |
        |    +------------+                |
 r(t)   |    +------------+   +----+   w2  v   +-------+
 ------>+--->| Correlator |-->|Z2  |--(x)-(+)->|Integr.|
        |    | c(t - τ2)  |   +----+       ^   | & dec.|
        |    +------------+                |   +-------+
        |    +------------+   +----+   wM  |      |
        +--->| Correlator |-->|ZM  |--(x)--+      v
             | c(t - τM)  |   +----+          data bits
             +------------+
      (one "finger" per resolvable multipath)

Function

1. M correlators ("fingers") each despread the received signal using the PN code delayed to match one strong multipath delay τₖ.
2. A searcher finds the path delays and amplitudes and assigns fingers.
3. Finger outputs Zₖ are weighted by wₖ (proportional to path strength) and summed: Z′ = Σ wₖZₖ (maximal-ratio combining).
4. The sum is integrated over the bit period and a decision gives the data.

So multipath, which causes ISI in narrowband systems, is turned into time diversity, raising SNR and reducing fading in IS-95 and WCDMA.