Turbo Equalization

The core mechanism involves a soft-input soft-output (SISO) equalizer and a SISO channel decoder exchanging extrinsic log-likelihood ratios (LLRs). The equalizer computes soft estimates of the transmitted bits based on the received signal and a priori information from the decoder. The decoder refines these estimates using the code's structure and feeds updated a priori information back to the equalizer. This loop continues until convergence, stripping away ISI and noise iteratively.

Optimal turbo equalization often employs a MAP symbol detector based on the BCJR algorithm. Unlike linear filters, the MAP equalizer computes the exact a posteriori probability of each transmitted symbol given the entire received sequence. It operates on a trellis representing the channel's memory, producing soft outputs that quantify the reliability of each bit decision. This probabilistic approach is critical for feeding meaningful extrinsic information to the decoder.

To avoid the exponential complexity of the MAP equalizer in channels with long delay spreads, Linear Minimum Mean Square Error (MMSE) equalizers with a priori information are used. These SISO equalizers compute soft symbol estimates by minimizing the MSE, incorporating feedback from the decoder to cancel known interference. The Extrinsic Information Transfer (EXIT) chart is a key tool for analyzing and predicting the convergence behavior of these sub-optimal but highly efficient iterative schemes.

A random interleaver between the encoder and the channel is fundamental. It decorrelates the bits entering the channel, ensuring that errors caused by a deep fade are spread out. Crucially, only extrinsic information—the novel information generated by one block independent of its input a priori information—is passed to the other block. This prevents self-reinforcement of errors and ensures the iterative process converges toward the true transmitted sequence.

Extrinsic Information Transfer (EXIT) charts visualize the iterative decoding trajectory by plotting the mutual information between the transmitted bits and the extrinsic LLRs at the output of the equalizer and decoder. The chart predicts the turbo cliff—the minimum SNR at which the iterative process converges to a low error floor. This tool is essential for designing the constituent codes and equalizers to ensure a narrow gap to the channel capacity.

Turbo equalization is the definitive technique for reliable communication over channels with long and severe intersymbol interference (ISI), such as underwater acoustic channels, magnetic recording channels, and high-frequency troposcatter links. In these environments, where a conventional decision feedback equalizer fails due to error propagation, the iterative exchange of soft information allows the system to approach the performance of an ISI-free channel, effectively turning a hostile multipath channel into a benefit through diversity.

The core component that accepts a priori log-likelihood ratios (LLRs) from the decoder and outputs extrinsic LLRs for each coded bit. Unlike hard-decision equalizers, a SISO equalizer—typically implemented via the BCJR algorithm or linear MMSE filtering—provides reliability metrics for its symbol estimates. This soft information prevents the premature discarding of ambiguous signal states, allowing the decoder to resolve uncertainty in subsequent iterations.

The optimal detection strategy for turbo equalization that computes the probability of each transmitted symbol given the entire received sequence. The BCJR algorithm efficiently implements MAP symbol detection on a trellis representing the channel's intersymbol interference (ISI) structure. While computationally intensive—scaling exponentially with channel memory length—MAP detection provides the most informative extrinsic outputs, maximizing the iterative gain between the equalizer and decoder.

The fundamental mechanism preventing positive feedback in the iterative loop. At each iteration, the equalizer strips its input a priori information from its output, passing only novel, independent evidence (extrinsic information) to the decoder. This strict separation ensures that information is not double-counted across iterations. The exchange follows the turbo principle: each component processor uses the other's extrinsic output as its a priori input, converging toward the joint optimal solution.

A permutation function that scrambles the bit ordering between the encoder and the channel. The interleaver serves two critical roles in turbo equalization: it decorrelates burst errors introduced by the ISI channel, spreading them into isolated errors that the decoder can correct, and it statistically decouples the extrinsic information exchanged between the equalizer and decoder. Random or S-random interleavers maximize this decorrelation, directly improving convergence speed and error floor performance.

Turbo equalization exhibits a characteristic waterfall region in bit error rate (BER) performance, where each iteration yields diminishing but significant gains. The process is visualized using EXIT (Extrinsic Information Transfer) charts, which plot the mutual information exchanged between components. A narrow tunnel between the equalizer and decoder EXIT curves indicates successful convergence; a closed tunnel predicts a decoding failure. This analysis guides the design of codes and equalizers to minimize the signal-to-noise ratio (SNR) convergence threshold.

A computationally efficient variant that performs equalization in the frequency domain using Fast Fourier Transforms (FFTs). By processing entire blocks of received symbols, this approach reduces the complexity of handling long channel impulse responses from quadratic to log-linear scaling. The MMSE-based frequency domain equalizer generates soft symbol estimates by incorporating a priori information from the decoder into the filter coefficients, making it practical for broadband systems like SC-FDE (Single-Carrier Frequency Domain Equalization).