For an FIR filter, when M is odd, the real valued frequency response of the FIR filter Hr(ω) has the characteristic that Hr(0)=0. A zero response at zero frequency is just the condition that the differentiator should satisfy.

In the frequency range 2πfp ≤ ω ≤ π, the desired response may be either left unconstrained or constrained to be zero.

We know that the unit sample response of an ideal differentiator is given as

An ideal differentiator is defined as one that has the frequency response

Since we know that the unit sample response of an ideal differentiator is anti-symmetric,

In view of the fact that the ideal differentiator has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

We know that, for an ideal differentiator, the frequency response is given as

An ideal differentiator has a frequency response that is linearly proportional to the frequency.

In most cases of practical interest, the desired frequency response characteristic need only be linear over the limited frequency range 0 ≤ ω ≤ 2πfp, where fp is the bandwidth of the differentiator.

Full band differentiators cannot be achieved with an FIR filters having odd number of coefficients, since Hr(π)=0 for M odd.