There is a potential problem for frequency sampling realization of the FIR linear phase filter. The frequency sampling realization of the FIR filter introduces poles and zeros at equally spaced points on the unit circle.

The desired output of an FIR filter with an input h(n) and using a window of length M is given as

Since {h(n)} is real, we can easily show that the frequency samples {H(k+α)} satisfy the symmetry condition

We know that

Although the frequency sampling method provides us with another means for designing linear phase FIR filters, its major advantage lies in the efficient frequency sampling structure, which is obtained when most of the frequency samples are zero.

In the frequency sampling method, we specify the frequency response Hd(ω) at a set of equally spaced frequencies, namely ωk=\\(\\frac{2π}{M}(k+\\alpha)\\)

We know that

To reduce the side lobes, it is desirable to optimize the frequency specification in the transition band of the filter. This optimization can be accomplished numerically on a digital computer by means of linear programming techniques.

The symmetry condition, along with the symmetry conditions for {h(n)}, can be used to reduce the frequency specifications from M points to (M+1)/2 points for M odd and M/2 for M even. Thus the linear equations for determining {h(n)} from {H(k+α)} are considerably simplified.

In the ideal situation, the zeros cancel the poles and, consequently, the actual zeros of the H(z) are determined by the selection of the frequency samples H(k+α). In a practical implementation of the frequency sampling realization, however, quantization effects preclude a perfect cancellation of the poles and zeros.