##  [Tomlinson-Harashima Precoding](/tomlinson-harashima-precoding-0) 

 Definition

A nonlinear transmitter pre‑equalization technique that cancels known intersymbol or inter‑stream interference before transmission by using a causal feedback filter combined with a modulo operation on finite constellations, thereby converting a problematic channel into one with reduced residual interference at the receivers.

 

 

 

 

 

 





## Principle

Principle

THP implements successive interference pre‑subtraction at the transmitter: a feedback filter removes the effect of previously generated transmitted symbols (given known channel or interference structure), and a modulo operation confines signal amplitude to the constellation region to control transmit power and maintain decodability via the receiver’s inverse modulo operation.

 

 

 

 

 





## Demonstration

Demonstration

Situation: A transmitter serving two spatial streams over a channel with known inter‑stream coupling. Recognition: The transmitter knows the channel coupling and the symbol sequence to be sent. Action: It applies a feedback filter that subtracts the predicted interference from the current symbol, then applies a modulo operation to wrap the result into the constellation region; the receiver applies the inverse modulo and decodes. Consequence: The receiver sees substantially reduced inter‑stream interference without relying on receiver‑side decision feedback; wrong or absent modulo inversion at the receiver produces symbol ambiguity.

 

 

 

 

## Misapplication

Misapplication

Treating THP as a linear precoder: the error is to ignore the feedback and modulo nonlinearity that distinguish THP; replacing THP by a linear matrix multiply removes the modulo mechanism and typically increases transmit power or residual interference for the same constellation constraints.

 

 

 

 

 





## Consequence

Consequence

When applicable, THP can achieve performance comparable to receiver decision‑feedback equalizers while shifting complexity to the transmitter and avoiding error propagation at receivers; misconfiguration (e.g., missing inverse modulo) creates decoding ambiguity and can increase symbol error rate.

 

 

 

 

## Reversal

Reversal

THP relies on accurate transmitter‑side knowledge of channel coupling and on finite constellations supporting modulo arithmetic; if channel knowledge is absent, rapidly varying, or if the system uses coding/constellations incompatible with modulo operations, THP’s benefits disappear and linear or receiver‑side equalization may be preferable.

 

 

 

 

 





## Boundary

Boundary

Clearly within: transmitter‑side preequalization using causal feedback filtering plus modulo mapping for finite constellations. Boundary case: precoding with feedback but no modulo (which raises transmit dynamic range). Clearly outside: purely linear precoding without feedback or receive‑side decision‑feedback equalizers acting after reception.

 

 

 

 

 





## Semantic Tension

Semantic Tension

THP ↔ Receiver Decision‑Feedback Equalization — both use successive interference cancellation concepts but place feedback at the transmitter (THP) versus at the receiver (DFE); the choice trades transmitter complexity and modulo operations against receiver complexity and error propagation risk.

 

 

 

 

 





## Synthesis

Synthesis

THP moves successive interference cancellation to the transmitter, using modulo arithmetic to bound signal excursions; this nonlinear strategy can reduce receiver complexity and error propagation but requires transmitter channel knowledge and compatible modulation structure.