Differentiators (Digital Filter Design Toolkit)

The ideal frequency response of a differentiator is as follows:

H(f) = jf

where f is the normalized frequency with the range [0, 0.5]. The frequency response has a constant 90° phase shift at all frequencies.

If you compare this equation with Equation B in the Linear Phase Filters topic, you can see that an ideal differentiator has a linear phase and is a Type III or Type IV linear phase FIR filter.

You can use the DFD Remez Design VI to design a differentiator by setting the filter type input to Differentiator. Because the passband magnitude response of a differentiator is sloped, a smaller approximation error at frequencies where the frequency response has a smaller absolute gain is useful. The DFD Remez Design VI uses an in-band weighting function inversely proportional to the frequency to achieve a constant amplitude percentage ripple size. The following sections describe how to create Type IV and Type III differentiators.

Designing Type IV Differentiators

If you need the frequency range from DC to the Nyquist frequency to have a differentiator response, you can design a Type IV differentiator. A Type III differentiator cannot maintain a strict differentiator response near the Nyquist frequency because the magnitude response of the Type III differentiator is constrained to zero at the Nyquist frequency.

When you design a Type IV differentiator using the DFD Remez Design VI, you can specify a single band that constrains two points with equal weights. For example, consider a frequency range of [0, 0.5] with an amplitude range of [0, 1]. To design a Type IV differentiator using this band, enter the specifications shown in the following figure into the DFD Remez Design VI.

In the previous figure, the odd order number 19 determines that the resulting differentiator is a Type IV differentiator. The following figure shows the magnitude response of the resulting Type VI differentiator.

In the previous figure, you can see that the magnitude response of the designed filter is sloped and the ripple size is small.

Designing Type III Differentiators

If you need to use a differentiator in combination with a lowpass filter, use a Type III differentiator. Type III differentiators have a differentiator response in a lowpass passband, and they have a stopband that extends to the Nyquist frequency. For example, you can create a Type III differentiator that covers a frequency range of [0, 0.45] with an amplitude range of [0, 0.9]. The following figure shows a single-band specification without the target response at 0.5 Hz specified.

The following figure shows the magnitude response of the resulting Type III differentiator.

In the previous figure, the even order number 20 determines that the resulting differentiator is a Type III differentiator. You can see that the magnitude response of the designed filter becomes zero at 0.5 Hz and the ripple size is larger than that of the Type IV differentiator. To achieve the same ripple size, you need to specify a larger value for the order input.