IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008 881 An Alternative Derivation of the Al-Alaoui Operator Verica Radisavljevic Abstract—It is shown in this letter that the well-known class of first-order digital differentiators and integrators can be simply de- rived from a classical continuous-time approximate differentiator by using the bilinear transformation. The result obtained repre- sents a simple and alternative method of deriving the Al-Alaoui op- erator (also known as Al-Alaoui’s differentiator, integrator, trans- form). Index Terms—Al-Alaoui operator, digital differentiators, digital integrators, discretization. I. INTRODUCTION A class of first-order digital integrators and differentiators was derived in [1] by generalizing the results of [2]. In [2], the first-order digital filter is constructed by interpolating the rectangular and trapezoidal integrators. In [1], first the non- minimum phase digital integrator is obtained as a weighted sum (1) where and are, respectively, the rectangular and trapezoidal integrators [2], leading to (2) with representing the sampling period. This integrator is then converted using a rule from [2] into the corresponding minimum phase digital integrator (3) By inverting the transfer function of the integrator in (3), using the technique of [3], the minimum phase differentiator is obtained in [1] as (4) see also [4]–[6]. It is interesting to point out that in Chen and Moore’s paper [6], the mixed scheme [1] composed of the rect- angular (Euler) and trapezoidal (Tustin) operators was termed Manuscript received June 01, 2008; revised September 01, 2008. The asso- ciate editor coordinating the review of this manuscript and approving it for pub- lication was Dr. Yuan-Pei Lin. The author is with the Mathematics Department and the Electrical and Com- puter Engineering Department, Rutgers University, Piscataway, NJ 08854 USA (e-mail: vericag@ece.rutgers.edu). Digital Object Identifier 10.1109/LSP.2008.2008211 as the Al-Alaoui operator. This terminology was also used in [5]. II. MAIN RESULT It is the purpose of this letter to show that the minimum phase digital differentiator (4) can be directly obtained from a classic continuous-time approximate differentiator [7] simply by map- ping it into the discrete-time domain via the bilinear transfor- mation [8]. The approximate continuous-time differentiator has been used for a long time in proportional-integral-derivative (PID) control [7] to approximate the pure derivative (5) where is a small positive parameter. Usually, the parameter is taken as . Applying the bilinear transformation, defined by [7] (6) to the approximate continuous-time differentiator in (5), we readily obtain (7) which with gives the digital differentiator identical to (4). Since in (4), is restricted to , the design parameter should be restricted to , which is always the case in control engineering practice. The corresponding digital integrator can be obtained by in- verting the transfer function in (7), which gives the digital in- tegrator introduced in [2] and given in formula (3) with . Since this letter presents a very simple method to derive the Al-Alaoui operator, it should be pointed out that there are other alternative ways of deriving the Al-Alaoui operator, see for ex- ample [9]–[14]. III. CONCLUSIONS We have shown that well-known digital differentiator can be obtained by simply mapping the classical analog approx- imate differentiator into the discrete-time domain via the bi- linear transformation. This provides a simple way to derive the Al-Alaoui operator. As suggested by a reviewer of this letter, it 1070-9908/$20.00 © 2008 IEEE