On the design of efficient second and higher degree FIR digital differentiators at the frequency π/(any integer)

Abstract

In a number of signal processing applications, digital differentiators (DD) of degree greater than unity performing over a narrow band of frequencies are required. The minimax relative error DDs are especially suitable for broad band applications, but they become inefficient when adopted for narrow band situations. This paper proposes second and higher degree DDs which are maximally accurate at the spot frequency: π/(any integer). Mathematical relations have been established between the weighting coefficients of the first degree FIR digital differentiators which are maximally linear at the frequency π/(any integer) and those of the proposed (second and higher degree) differentiators. It has been shown that very high accuracies in the frequency response of the approximation are achievable with attractively low order of the structure for the suggested differentiators. As an example, with just 16 multiplications per input sample of the signal, it is possible to obtain a third degree differentiator over a frequency bandwidth of 0.20π centred around ω = π/3, with an accuracy no worse than 99.999%. The phase error is zero over the entire frequency band 0 ≤ ω ≤π of operation.