  We propose two sparsity-aware normalized subband adaptive filter (NSAF)
algorithms by using the gradient descent method to minimize a combination of
the original NSAF cost function and the l1-norm penalty function on the filter
coefficients. This l1-norm penalty exploits the sparsity of a system in the
coefficients update formulation, thus improving the performance when
identifying sparse systems. Compared with prior work, the proposed algorithms
have lower computational complexity with comparable performance. We study and
devise statistical models for these sparsity-aware NSAF algorithms in the mean
square sense involving their transient and steady -state behaviors. This study
relies on the vectorization argument and the paraunitary assumption imposed on
the analysis filter banks, and thus does not restrict the input signal to being
Gaussian or having another distribution. In addition, we propose to adjust
adaptively the intensity parameter of the sparsity attraction term. Finally,
simulation results in sparse system identification demonstrate the
effectiveness of our theoretical results.
