Convergence analysis of a frequency-domain adaptive filter with exponential power averaging and generalized window function

One of the advantages of a Frequency-Domain Adaptive Filter (FDAF) is that one can achieve convergence at a constant rate over the whole frequency range by choosing the adaptation constant for each frequency bin
l
equal to the overall adaptation constant divided by an estimate of the input power at this frequency bin. A commonly used method, applied in this paper, to estimate the input power is to do an exponentially weighting with smoothing constant
\beta
on the magnitude squared of the input values at each frequency bin
l
. Furthermore, it is known that a correctly implemented FDAF, using the overlap-save method, contains five 2
N
-points Fast Fourier Transforms (FFT). Two of these are used to force the last
N
points of the time-domain augmented impulse response to zero by applying a particular window function. In this paper, an analysis is given of the FDAF where the window function is generalized. Using these results, the convergence behavior of FDAF's with various window functions is compared. Furthermore, the analysis describes the influence of
\beta
on the convergence behavior of the FDAF over the whole convergence range.