The parameters {ak} are selected to minimize the sum of squares of the error sequence.

In this method, we consider the cascade connection of the desired filter Hd(z) with the reciprocal, all zero filter 1/H(z).

We know that we can perform pole-zero approximation for Hd(z) by using the least squares method.

By differentiating the square of the error sequence with respect to the parameters {ak}, it is easily established that we obtain the set of linear equations.

Let us assume that hd(n) is specified for n > 0, and the digital filter is an all-pole filter.

Nevertheless, we can use the desired response hd(n) for n < M to construct an estimate of hd(n).

For n > 0, y(n) represents the error between the desired output yd(n)=0 and the actual output.

The condition that yd(0)=y(0)=1 is satisfied by selecting b0=hd(0).

In a practical design problem, the desired impulse response hd(n) is specified for a finite set of points, say 0 < n <L where L ≫ N. In such a case, the correlation sequence can be computed from the finite sequence hd(n).

We excite the cascade configuration by the unit sample sequence δ(n). Thus the input to the inverse system 1/H(z) is hd(n) and the output is y(n). Ideally, yd(n)= δ(n).