Decision Feedback Equalizer (DFE)

The DFE consists of a feedforward filter and a feedback filter. The feedforward filter shapes the incoming signal, while the feedback filter takes previously decided symbols and subtracts their estimated interference contribution from the current symbol. This closed-loop structure allows the DFE to cancel post-cursor ISI without amplifying noise in spectral nulls, a key advantage over linear equalizers like the Zero-Forcing Equalizer.

Unlike the Zero-Forcing Equalizer, which applies the inverse of the channel frequency response and amplifies noise in deep fades, the DFE's feedback path operates on quantized symbol decisions. Since decisions are noise-free (assuming correct detection), the ISI cancellation does not suffer from noise enhancement. This makes the DFE particularly effective in channels with deep spectral nulls where linear equalizers fail.

The primary vulnerability of a DFE is error propagation. If the slicer makes an incorrect symbol decision, that error is fed back through the feedback filter, corrupting subsequent symbol estimates. This can create burst errors in the output stream. Mitigation strategies include:

• Interleaving and channel coding to reduce raw error rates
• Soft-decision feedback using probabilistic estimates instead of hard decisions
• Periodic training sequences to reset the feedback state

DFE coefficients are typically adapted using the Least Mean Squares (LMS) or Recursive Least Squares (RLS) algorithms. LMS offers low computational complexity with slower convergence, while RLS provides faster convergence at the cost of higher complexity. The adaptation minimizes the error between the equalizer output and the slicer decision, driving the combined response toward an open eye pattern.

The MMSE-DFE is the optimal linear processing structure for channels with ISI. It jointly optimizes the feedforward and feedback filters to minimize the mean squared error at the decision point. The solution requires knowledge of the channel impulse response and noise power spectral density. The MMSE-DFE achieves the theoretical bound for symbol-by-symbol detection in ISI channels.

The DFE architecture is inherently causal: the feedback filter can only cancel interference from previously detected symbols (postcursor ISI). The feedforward filter must handle precursor ISI from future symbols. This asymmetry means the DFE performs best when the channel's energy is concentrated in the postcursor taps, which can be ensured through proper timing recovery and frame synchronization.