This Design Brief introduces a computationally efficient network that computes
the derivative of a discrete time-domain (digital) sequence. While recursive
differentiating networks exist, this article describes a simple tapped-delay
line (finite impulse response or FIR) differentiator that has guaranteed linear
phase.
Although the idea of differentiation is well-defined in the world of continuous signals, that's not the case for discrete signals. Fortunately, however, we can approximate the calculus of a derivative operation in the domain of discrete signals. (While DSP purists prefer to use the terminology "digital differencer," we'll use the term " differentiator.")
To review the notion of differentiation, think about a continuous sine wave,
whose frequency is rad./s, represented by:
The derivative of that sine wave is:
Therefore, the derivative of a sine wave is a cosine wave whose amplitude is
proportional to the original x(t) sine wave's frequency. Equation 2 says that
an ideal differentiator's frequency magnitude response is a straight line increasing
with frequencyω. Keeping that in mind, consider two common discrete-time FIR
(non-recursive) differentiators: a first-difference and a central-difference
differentiator. They are computationally simple schemes for estimating the derivative
of a digital x(n) time-domain signal sequence with respect to time.
The first-difference differentiator, which involves the simple process of computing
the difference between successive x(n) signal samples, is defined in the time
domain by:
The frequency magnitude response of that differentiator is the dashed |Hfd(ω)|
curve in Figure 1. For comparison, Figure 1
also shows an ideal differentiator's straight-line |Hidea(ω)|
= magnitude response. The frequency axis in the figure covers the positive frequency
range 0≤ ω ≤ π samples/radian, corresponding to a cyclic frequency
range of 0 to fs/2, where fs is the x(n) sample rate in
hertz.
Equation 3 is sweet in its simplicity, but unfortunately its |Hfd(ω)|
tends to amplify high-frequency noise that often contaminates real-world signals.
For that reason, the central-difference differentiator is often used in practice.
The time-domain expression of the central-difference differentiator is:
