LTI representations of adaptive systems with tap delay-line regressors under sinusoidal excitation

It is shown that an adaptive system whose regressor is formed by tap delay-line (TDL) filtering of a multitone sinusoidal signal is representable as a parallel connection of an linear time-invariant (LTI) block and a linear time-varying (LTV) block. The induced 2-norm of the LTV subsystem decreases as N-1 where N is the number of taps. Hence the adaptive system becomes LTI in the limit as the number of taps goes to infinity.