Givens rotations are unitary transformations that selectively annihilate a single vector component while leaving one axis intact, by rotating a two-dimensional coordinate plane through an angle determined from the ratio of the components being combined; because they are unitary, they preserve vector norm. Applying a sequence of such rotations to a data matrix drives it to upper-triangular form, which is the QR decomposition underlying least-squares problems. In recursive least-squares (RLS) adaptive filtering, this QR/Givens machinery is applied to the exponentially-weighted (forgetting-factor) data and desired-response vectors so that the upper-triangular factor can be updated recursively from one time index to the next rather than recomputed from scratch, with the optimal filter coefficients obtained by back-substitution and the minimum residual given by the norm of the annihilated (zeroed) partition.
Givens rotations are unitary transformations that selectively annihilate a single vector component while leaving one axis intact, by rotating a two-dimensional coordinate plane through an angle deter…