MLSE and time diversity

Maximum Likelihood Sequence Estimation (MLSE)

MLSE equalization is a non-linear equalizer that, instead of deciding each symbol separately, chooses the whole sequence of symbols that most probably produced the received samples. It is the optimum detector for a channel with ISI and additive white Gaussian noise.

 r(t)  +---------+  y(k)  +-----------------+   a^(k)
 ----->| Matched |------->| MLSE (Viterbi   |-------->
       | filter  |        | algorithm)      |  data
       +---------+        +-----------------+
            ^                    ^
            |                    |
       +------------------------------+
       | Channel estimator (CIR from  |
       | training sequence)           |
       +------------------------------+

Working

1. The received signal passes through a matched filter and is sampled once per symbol, giving y(k).
2. A channel estimator finds the discrete channel impulse response (CIR) h₀, h₁, …, h_L, usually by correlating with a known training sequence (e.g. the GSM midamble).
3. The channel with memory L is a finite-state machine with Mᴸ states (M = alphabet size). The possible sequences form a trellis.
4. For each candidate sequence the receiver predicts the noiseless output ŷ(k) = Σ hⱼ·a(k−j) and computes a metric, the squared Euclidean distance Σ |y(k) − ŷ(k)|².
5. The Viterbi algorithm searches the trellis efficiently: at each state it keeps only the survivor path with the smallest metric, so complexity grows linearly with sequence length rather than exponentially.
6. The survivor with minimum metric is the maximum-likelihood sequence; its symbols are output.

Simple illustration: for BPSK (M = 2) and a two-tap channel h = [h₀, h₁] (L = 1), the trellis has 2¹ = 2 states (previous symbol +1 or −1). Each new symbol gives two branches per state, and at each step only the better of the two paths entering each state survives. After the whole burst, one survivor remains, and that is the decided sequence.

Comparison with other equalizers

Features: minimum probability of sequence error; complexity grows as Mᴸ, so it is practical only for short channel memory. GSM receivers commonly use MLSE (Viterbi) equalizers.

Time diversity

Time diversity is obtained by transmitting the same information repeatedly at time intervals greater than the coherence time of the channel, so that the copies experience independent fading. (Coherence time T_c ≈ 0.423/f_d, where f_d is the maximum Doppler shift.)

Two implementations of time diversity

1. Channel coding with interleaving

• Redundancy is added by an error-correcting code (convolutional, block, turbo).
• An interleaver rearranges coded bits so that bits that are adjacent in the code are sent far apart in time (more than T_c).
• A deep fade then produces scattered single-bit errors after de-interleaving, which the decoder corrects. Used in GSM, IS-95 and LTE.

 data->[encoder]->[interleaver]->channel
     ->[de-interleaver]->[decoder]->data

2. RAKE receiver (multipath time diversity)

• In spread-spectrum CDMA, multipath copies arriving more than one chip apart are separate in time.
• A RAKE receiver uses several correlators ("fingers"), each locked to one delayed path, and combines their outputs with maximal-ratio weights.
• Thus time-separated echoes of the same symbol provide diversity instead of interference.

Limitations of time diversity: it adds delay (interleaver depth must exceed T_c), and it gives little benefit when the mobile is slow or stationary because the coherence time becomes very long. Repetition coding and ARQ retransmission are simpler forms of the same idea.