Electr Eng (2008) 90:455–467 DOI 10.1007/s00202-007-0092-0 ORIGINAL PAPER Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems Mohamad Adnan Al-Alaoui Received: 22 August 2007 / Accepted: 1 December 2007 / Published online: 8 January 2008 © Springer-Verlag 2007 Abstract The “new transformation polynomials for discret- The bilinear transform meets the above requirements. ization of analogue systems” was recently introduced. The However, it introduces a warping effect due to its nonlinear- work proposes that the discretization of 1/sn should be done ity, albeit it can be ameliorated somewhat by a pre-warping independently rather than by raising the discrete representa- technique. tion of 1/s to the power n. Several examples are given in to The backward difference transform satisfies the first con- back this idea. In this paper it is shown that the “new transfor- dition, but the second condition is not completely satisfied, mation polynomials for discretization of analogue systems” since the imaginary axis of the s-plane maps onto the cir- is exactly the same as the parameterized Al-Alaoui opera- cumference in the z-plane centered at z = 1/2 and having a tor. In the following sections, we will show that the same radius of 1/2. The mapping meets condition 2 rather closely results could be obtained with the parameterized Al-Alaoui for low frequencies [1–6]. operator. Other transforms were introduced in attempts to obtain better approximations [7–11]. In particular, in [7,8]the Keywords Al-Alaoui operator · Digital filters · approach interpolates the rectangular integration rules and Discretization · s-to-z transforms the trapezoidal integration rule. In [7] a fixed weighting of 0.75 was assigned to the rectangular rule and 0.25 for the trapezoidal rule, while in [8] the interpolation was param- 1 Introduction eterized with an a-parameter. The resulting operator in [7] was designated Al-Alaoui operator and applied in fractional A popular method for designing IIR digital filters is to map order discretization schemes by Chen and Moore in [13]. It the transfer function of a corresponding analog filter using an is also called Al-Alaoui differentiator, Al-Alaoui rule, and s-to-z transformation [1–6]. It is desirable that the mapping Al-Alaoui transform [20,24,32,33]. The operator developed procedures have the following two properties: (1) they should in [8] may be designated as the parameterized Al-Alaoui map the left half of the s-plane to the interior of the unit circle operator. However often, for brevity, Al-Alaoui operator is in the z-plane which would insure that stable analog filters used to refer to either of them. In [10]the“α-approximation” map into stable digital filters, and (2) the imaginary axis of the for discretization of analog systems is proposed, while in [11] s-plane should be mapped onto the unit circle circumference a further elaboration of the “α-approximation” is carried out in the z-plane. as the “new transformation polynomials for discretization of analogue systems”. It was proved in [12] that the “α-approx- This research was supported, in part, by the University Research imation” presented in [10] is the same as the parameterized Board of the American University of Beirut. Al-Alaoui operator presented in [8]. In [11], the same “α-approximation” of [10] is renamed as “fractional approx- M. A. Al-Alaoui (B) imation”. A rather unfortunate choice because it might cause Department of Electrical and Computer Engineering, confusion with fractional order differentiation and integra- American University of Beirut, P.O. Box 11-0236, 179 Bliss Street, Beirut 1107 2020, Lebanon tion. The only novelty in [11] resides in stating that the dis- e-mail: [email protected] cretization of 1/sn should be done independently rather than 123 456 Electr Eng (2008) 90:455–467 3 1 new 0.9 ideal 2.5 trapezoidal 0.8 rectangular Simpson 0.7 2 0.6 1.5 0.5 0.4 Magnitude 1 Magnitude 0.3 0.2 0.5 new differentiator 0.1 ideal differentiator 0 0 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency Frequency Fig. 1 Magnitude response of new integrator, and ideal, trapezoidal, Fig. 2 Magnitude response of the new differentiator for a = 0.75 and and rectangular integrators, and Simpson Integrators ideal differentiator following class of integrators [7,8]: by raising the discrete representation of 1/s to the power n. Several examples are given in [11] to back this idea. In the H(z) = aHRe ct (z) + (1 − a)HTrap(z) (1) following sections, we will show that the same results could Using the backward rectangular rule for the rectangular inte- be obtained with the parameterized Al-Alaoui operator, and gration rule in Eq. (1), yields the examples given in [11] with the discretization of analog systems will be discussed, one by one. Tz T z + 1 H(z) = a + (1 − a) The paper is divided into five sections including the z − 1 2 z − 1 introduction and conclusion. The second section presents T [(1 − a) + (1 + a)z] = ; 0 ≤ a ≤ 1(2) Al-Alaoui first order integrator and differentiator, with the 2(z − 1) latter often designated as Al-Alaoui operator, together with where T is the sampling period. Equating H(z), the transfer the corresponding s-to-z transformation in addition to pre- function of the resulting IIR digital integrator as expressed senting the intuitive derivations and the corresponding in (2) to the transfer function 1/s of an ideal analog integra- s-plane to z-plane mappings elaborated in [7,8]. The third tor, yields the parameterized s-to-z transformation shown in section presents the α-approximation introduced in [10] and Eq. (3). The right-hand side of Eq. (3) corresponds to the utilized in [10,11] for the dicretization of analog systems. digital differentiator obtained from taking the inverse of the Additionally this section presents the proof given in [12] digital integrator obtained in (2): that the “α-approximation” is one and the same as Al-Alaoui operator. The fourth section shows that the “new transforma- 2(z − 1) s = (3) tion polynomials for discretization of analogue systems“ of T [(1 − a) + (1 + a)z] [11] and Al-Alaoui operator operator obtain the same results. In reference [7] the fixed value of a = 0.75 that obtains The fifth section concludes the paper. almost the ideal response at the midfrequency range was used, and the s-to-z transformation of Eq. (4) is obtained. 8(z − 1) s = (4) 2 Al-Alaoui integrator and differentiator + 1 7T z 7 2.1 Al-Alaoui integrator Figure 2 shows the magnitude of the frequency response, obtained by substituting z = e jω, in the differentiator corre- Al-Alaoui integrator was obtained by observing that the mag- sponding to the right-hand side of Eq. (3). The phase respon- nitude of the Fourier transform of the analog (ideal) integra- ses of the resulting differentiator and integrator are almost tor lies between the magnitudes of the trapezoidal and the linear and are omitted for brevity. rectangular integration rules as shown in Fig. 1. The resulting differentiator yields a good approximation Al-Alaoui integrator is obtained by interpolating the trap- of the ideal differentiator up to 0.8 of full range. Note that ezoidal and the rectangular integration rules to obtain the the frequency had been normalized by dividing it by π. 123 Electr Eng (2008) 90:455–467 457 S-Plane Z-Plane 2.2.2 The mapping of the imaginary axis of the s-plane Im[s] Im[z] To obtain the mapping of the imaginary axis of the s-plane, substitution of s = j in Eq. (3) yields R = 1 [ jT (a − 1) − 2] ω a/(1+a) = = j Re[s] Re[z] z re (6) R = 1/2 [ jT (a + 1) − 2] R = 1/(1+a) It can be verified that = 0 maps into the point (r = 1,ω = 0) in the z-plane and the point =±∞maps into the point Unit Circle (r = (1 − a)/(1 + a), ω = π) which is the pole of Eq. (2). The real and imaginary parts of z are given by Fig. 3 Mapping of the s-plane on the z-plane, Al-Alaoui (solid circle), the backward difference transform (dash-dotted circle), and the bilinear [4 + (T )2(a2 − 1)] transform (lined circle) [z]= (7) [[(T )2(a + 1)2 + 4] [4T ] ( ) = T [z]= (8) The forward rectangular rule, HForward Re ct z z−1 ,is [(T )2(a + 1)2 + 4] used in [7] which results in a non-minimum phase transfer = function H(z) and the stabilizing approach of [14]isusedin Thus the mapping of the line s j in the z-plane is the circle described [7]. The same results are obtained, without the need for sta- bilization, by using the backward rectangular rule. The above a 2 1 [z]− +{[z]}2 = (9) procedure is equivalent to interpolating directly the bilinear (1 + a) (1 + a)2 operator (Tustin), and the backward difference operator [8]. Equation (7) represents a circle centered at z = a/(1 + a) and having a radius with a value of 1/(1 + a). Note that the 2.2 Mapping s-plane to z-plane circle crosses the negative real axis at the value of the pole of Eq. (3). Equation (9) is more compactly written as The Tustin (bilinear) transformation maps the left half of the a 1 s-plane onto the disk defined by the interior of the unit circle z − = ( + ) ( + ) (10) in the z-plane while the backward difference transform maps 1 a 1 a the left half of the s-plane onto the disk defined by the interior Thus, as goes from −∞ to 0 to +∞, its transform in of the circle in the z-plane with center at z = 0.5 and a radius the z-plane moves on the circle defined by Eq.