The paper describes several improvements in a method reported recently by Antoniou for the design of digital differentiators satisfying prescribed specifications. The improvements include the use of a more efficient optimisation algorithm, and a simple and accurate technique for the prediction of the required differentiator order. The improved design method is illustrated by several examples, and is compared with the equiripple design method. List of symbols E(a, N, co) = error function £>ei> e 2,£3 = specific values of L(a,N) F(a),fi(a) = nonlinear functions of a H(z) = discrete-time transfer function h(nT) = impulse response 7 0 (x) = zeroth-order Bessel function of the first kind k = stage of minimisation L(pc,N) = maximum ofE(ot,N, co) over the passband m = number of functions //(a) M(a, N, co) = amplitude response of digital differentiator MJ(OJ) = amplitude response of ideal differentiator N = number of samples in impluse response ,-Wi, N 2 , N 3 = specific values of Af T = sampling period, s w(nT) = window function a = parameter of Kaiser window function a* = global minimum point a 0 , a fe , a, a = specific values of a 8 = prescribed maximum inband error e, €j = minimisation tolerances Co» ^10 = specific values of e and e x Tj= change in a for function //a) T max = maximum of 77 co = frequency variable, rad/s co p = passband edge, rad/s co fi = sampling frequency, rad/s