 The coefficient arrays formula_296 and formula_297 are determined such that the output response formula_61 of a filter matches the desired response formula_299.Let us denote the error signal as

formula_300

And let formula_301 denote it's the Fourier transform

formula_302

By multiplying both sides by formula_303, we get the modified error spectrum, converted in discrete domain as

formula_304

The total mean-squared error is obtained as

formula_305

Let the input signal be formula_306.Now, the numerator coefficient formula_296 is zero outside region formula_308 because of the ROS of input signal.Then equation becomes

formula_309 for formula_310 or formula_311 or ... formula_312

Substituting the result formula_313 into formula_314 and differentiating formula_315 with respect to denominator coefficients formula_316, the linear set of equations is obtained as

formula_317
for formula_318

Now, taking the double summation for the region "R" i.e.