Novel Differentiator Handles Discrete Time-Domain Signals

The central-difference differentiator's frequency magnitude response is the dotted |H_cd(ω)| curve in Figure 1. The tradeoff for |H_cd(ω)|'s desirable high-frequency (noise) attenuation is that its frequency range of linear operation is only from zero to roughly = 0.16 π samples/radian (0.08f_s Hz). Unfortunately, that's less than the frequency range of linear operation of the first-difference differentiator.

As noted, this Design Brief describes a third alternative, a computationally efficient differentiator that maintains the central-difference differentiator's beneficial high-frequency attenuation behavior, but extends its frequency range of linear operation. The differentiator is defined by:

This novel differentiator's normalized frequency magnitude response is the solid |H_dif(ω)| curve in Figure 1. Its frequency range of linear operation extends from to zero to approximately = 0.34π samples/radian (0.17f_s Hz). This is twice the usable frequency range of the central-difference differentiator.

The implementation of the differentiator is shown in Figure 2, where a delay block comprises two unit delays. The folded-FIR structure for this differentiator is presented in Figure 3, which shows that only a single multiply need be performed per y_dif(n) output sample. The really slick aspect of the y_dif(n) differentiator is that its non-unity coefficients (±1/16) are integer powers of two. This means that a multiplication can be implemented with an arithmetic right shift by four bits. Happily, such a binary right-shift implementation is a linearphase, multiplier-less differentiator.

Another valuable feature of the y_dif(n) differentiator is that its time delay (group delay) is exactly three sample periods (3/f_s), making it convenient for use with popular FM demodulation methods.¹

Reference:

1. Lyons, Richard, Understanding Digital Signal Processing, 2nd Edition, Prentice Hall, Upper Saddle River, N.J., 2004, p. 549-552