  This paper introduces a novel constraint adaptive filtering algorithm based
on a relative logarithmic cost function which is termed as Constrained Least
Mean Logarithmic Square (CLMLS). The proposed CLMLS algorithm elegantly adjusts
the cost function based on the amount of error thereby achieves better
performance compared to the conventional Constrained LMS (CLMS) algorithm. With
no assumption on input, the mean square stability analysis of the proposed
CLMLS algorithm is presented using the energy conservation approach. The
analytical expressions for the transient and steady state MSD are derived and
these analytical results are validated through extensive simulations. The
proposed CLMLS algorithm is extended to the sparse case by incorporating the
$\ell_1$-norm penalty into the CLMLS cost function. detailed Simulations
confirms the superiority of the sparse CLMLS over the state-of-the-art.
