Short note: Adaptive equalization

Adaptive equalization is equalization in which the filter coefficients are adjusted automatically to track a time-varying channel, so that inter-symbol interference (ISI) is removed even as the mobile moves.

Why adaptive: the mobile radio channel is unknown and changes with time (Doppler, moving scatterers), so a fixed equalizer cannot stay matched to it.

 x(k)   +-------------------+ d^(k)   +----------+
 ------>| Transversal filter|-------->| Decision |--> d~(k)
        | taps w0..wN       |    |    +----------+   |
        +-------------------+    |                   |
               ^                 v                   |
               |     e(k)=d(k)-d^(k)  <-- d(k) ------+
               +---- adaptive algorithm (LMS/RLS)
     d(k): training sequence, or decided symbol later

Two modes of operation

1. Training mode – The transmitter sends a known, fixed-length training sequence (e.g. a PN sequence or the GSM midamble). The receiver knows d(k), computes the error e(k) = d(k) − d̂(k), and the algorithm sets the tap weights to near optimum.
2. Tracking (decision-directed) mode – After training, user data follows. The detector's own decisions d̃(k) are used as the reference to keep updating the taps as the channel slowly changes.

Training is repeated periodically (for TDMA, in every burst/time slot).

Algorithms

• Zero forcing (ZF) – forces ISI to zero; amplifies noise at spectral nulls.
• LMS – w(k+1) = w(k) + μ·e(k)·x(k); simple (2N+1 operations per iteration), slow convergence.
• RLS – faster convergence, more computation.

Structures: linear transversal equalizer, decision feedback equalizer (DFE), and MLSE (Viterbi) equalizer.

Performance factors: rate of convergence, misadjustment, computational complexity and numerical stability.