Also known as: DFE, decision-feedback equalizer
A decision-feedback equalizer (DFE) is a nonlinear equalizer that pairs a feedforward filter with a feedback filter fed by the receiver’s own past symbol decisions, using those already-detected symbols to subtract off the interference their tails leave on later symbols.1 Because it cancels ISI with known (decided) symbols rather than by inverting the channel, a DFE avoids the noise enhancement that plagues linear equalizers on channels with deep nulls.
How it works
The channel’s smeared impulse response has a cursor (the main symbol) with pre-cursor tails from future symbols and post-cursor tails from past symbols. A DFE splits the job:
- The feedforward filter operates on the received samples and, like a linear MMSE equalizer, cleans up pre-cursor ISI and shapes the response — but it is deliberately not asked to invert deep nulls, so it enhances little noise.
- The decision device slices the summed signal to the nearest constellation point, producing a hard symbol decision.
- The feedback filter takes those past decisions, reconstructs the post-cursor interference they contribute to the current symbol, and subtracts it at the summing junction. Since decisions are noise-free symbol values, this cancellation adds no noise.
Both filters are typically adapted together by an LMS or RLS rule under an MMSE criterion, trained on a known sequence and then run decision-directed.
Error propagation
The DFE’s strength — feeding decisions back — is also its weakness. If the decision device slices a symbol wrong (likely at low SNR), the feedback filter subtracts the wrong interference, which corrupts the next few symbols and can trigger a short burst of errors before the loop recovers. This error propagation is the DFE’s defining caveat; it bounds how aggressively feedback can be used and is why the gap to an MLSE receiver widens on the harshest channels. Mitigations include precoding (e.g. Tomlinson–Harashima, which moves the feedback to the transmitter) and reduced-state or soft-decision feedback.
Relevance to SDR
The DFE is the workhorse equalizer for severe-ISI channels where a purely linear equalizer would either leave too much interference or enhance too much noise — it appears in digital TV (ATSC), voiceband and cable modems, DSL, and many microwave and HF links. It tightens the constellation on multipath channels that would defeat a zero-forcing design. GopherTrunk’s narrowband land-mobile decoders (P25, DMR, NXDN) rely on matched filtering and synchronisation rather than a DFE, so it is described here as a standard equalization architecture from the broader RF world.
Sources
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Decision feedback equalizer — Wikipedia, on the feedforward/feedback structure, post-cursor cancellation, and error propagation. ↩