Al-Alaoui Operator and the New Transformation Polynomials for Discretization of Analogue Systems

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. (9)or(10), of R = 0.5. Since Al-Alaoui operator is an interpolation counterclockwise, from ω = π to ω = 0 and back to ω = π. of the bilinear and the backward difference transformations, we expect the corresponding mapping of the left half of the 2.2.3 The mapping of the left-half of the s-plane s-plane to the z-plan to fill the gap between the bilinear and the backward-difference transforms circles in the z-plane The left half of the s-plane is mapped in the interior of the which is indeed the case as shown in Fig. 3. circle defined by Eq. (9)or(10). This is easy to see if the following equation is used: 2.2.1 The mapping equations a 1 Ts(1 + a) + 2 z − =− (11) (1 + a) 1 + a Ts(1 + a) − 2 The mapping equations consist of Eq. (3), repeated below Equation (11) is obtained by subtracting a/(1+a) from both and its solution for z shown as Eq. (5). Thus the resulting sides of Eq. (5). Set sT = b + jd in Eq. (11) where b and equation defining the new s-to-z transforms is given by d are real. Notice that for the left half s-plane, b < 0, the 2(z − 1) magnitude of the numerator of Eq. (11)issmallerthanthe s = (3) magnitude of the denominator in the last ratio. T [(1 − a) + (1 + a)z]

Equation (3) has a zero at z =1 and a pole at z =(a−1)/(a+1). 2.2.4 The mapping between the analog and discrete-time Solving Eq. (3)forz yields the corresponding inverse z-to-s frequency variables transform and is given by From the ratio of Eqs. (8) and (7) we obtain [T (a − 1)s − 2] z = [z] 4T [ ( + ) − ] (5) ω = arctan = arctan (12) T a 1 s 2 [z] 4 + (T )2(a2 − 1) 123 458 Electr Eng (2008) 90:455–467

Equation (12) can be used in a pre-warping design procedure 10 similar to that used in a pre-warping bilinear transformation 9 Analog Sekara design procedure. 8 Al-Alaoui Note that for a = 0, we obtain the familiar bilinear map- 7 ping after using the half angle trigonometric identities. The phase angle of [z − a/(1 + a)]is 6 5 () = 4 T arctan (13) Magnitude 4 [4 − (1 + a)(T )2] 3

3 Al-Alaoui operator and the α-approximation 1

0 The approximation of the α transformation starts from the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 well-known mapping of the s-to-z domains shown in Eq. (14) ω/π z = esT, (14) Fig. 4 Magnitude of ﬁrst order integrators where T is the sampling period. Thus Al-Alaoui operator and the “new transformation poly- Starting from Eq. (14), the following equivalent relation nomials for discretization of analogue systems“ are one and can be formulated: the same. (1−α) Ts (( −α) +α ) e z = esT = es 1 T T = ,α∈[0, 1] (15) e−α Ts 4 Al-Alaoui operator and the new transformation After the numerator and denominator on the right-hand polynomials side of Eq. (15) have been expanded in series, and all mem- bers of the second and higher orders neglected, expression It is shown above that the alpha-transform presented in [10, (15) becomes 11] is the same as the parameterized Al-Alaoui operator presented in [8]. The only novelty in [11] resides in stating ∞ [(1−α) Ts]n = ! 1 + (1 − α) Ts / n z = n 0 n ≈ (16) that the discretization of 1 s should be done independently ∞ (− )k (α Ts)k 1 − α Ts rather than by raising the discrete representation of 1/s to k=0 1 k! the power n. Several examples are given in [11] to back this Solving Eq. (16) for the complex variable s yields idea. In the following sections, we will show that the same results could be obtained with the parameterized Al-Alaoui 1 z − 1 s = f (z,α)= operator, and the examples given in [11] will be discussed, + α( − ) (17) T 1 z 1 one by one. Equation (17) deﬁnes the new transformation. 4.1 Integrators

3.1 Al-Alaoui operator and the new transformation In this section, the integrators presented in [11] are compared to the Al-Alaoui integrator of the same order; i.e. the In this section it will be shown that Al-Alaoui operator, representation of 1/sn in [11] is compared to Eq. (3), and the new transformation, Eq. (17) are one and −n the same. 8(z − 1) (19) Equation (3) may be reformulated as follows by letting + 1 7T z 7 α = (1 + a)/2, or equivilantly a = 2α − 1, yields 1 − α = (1 − a)/2 and thus we obtain by appropriate substitutions The comparison ﬁgures are shown in Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 . Al-Alaoui integrator clearly outperforms the 2(z − 1) (z − 1) integrators presented in [11] for the ﬁrst and fourth orders, in s = = [( − ) + ( + ) ] ( − ) ( + ) T 1 a 1 a z T 1 a + 1 a z addition to being superior at high frequencies for the second 2 2 and third orders. Besides, in the next paragraph, we show that (z − 1) 1 z − 1 = = (18) the same results obtained with the alpha transformation can T [(1 − α) + α z} T 1 + α(z − 1) be obtained with the parameterized Al-Alaoui operator. 123 Electr Eng (2008) 90:455–467 459

0.35 1000

Sekara 900 Analog 0.3 Al-Alaoui Sekara 800 Al-Alaoui

0.25 700

600 0.2 500 0.15 400 Magnitude Absolute Error 0.1 300 200 0.05 100

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω π ω/π /

Fig. 5 Absolute magnitude error with respect to analog integrator of Fig. 8 Magnitude of third order integrators ﬁrst order integrators 0.4

120 Sekara 0.35 Al-Alaoui Analog 100 Sekara 0.3 Al-Alaoui 0.25 80 0.2 60 0.15 Absolute Error Magnitude 40 0.1

0.05 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 ω/π 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω/π Fig. 9 Absolute magnitude error with respect to analog integrator of Fig. 6 Magnitude of second order integrators third order integrators

10 0.07 Sekara 9 Sekara 0.06 Al-Alaoui Al-Alaoui 8

0.05 7

6 0.04 5

0.03 4 Absolute Error Absolute Error 0.02 3 2 0.01 1

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω/π ω/π

Fig. 7 Absolute magnitude error with respect to analog integrator of Fig. 10 Absolute magnitude error with respect to analog integrator of second order integrators fourth order integrators (zoom in view)

123 460 Electr Eng (2008) 90:455–467

10000 4.2 Obtaining the same results with the parameterized 9000 Sekara Al-Alaoui transform Al-Alaoui 8000 The alpha transformation is deﬁned by 7000

6000 1 z − 1 5000 s = 0.5 ≤ α ≤ 1(20) T [1 + α(z − 1)] 4000 Absolute Error 3000 The parameterized Al-Alaoui transformation is deﬁned by 2000

1000 2(z − 1) s = 0 ≤ a ≤ 1 (21) 0 T [(1 − a) + (1 + a)z] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω/π In [2], it was proven that the two transformations are one and + Fig. 11 α = 1 a Absolute magnitude error with respect to analog integrator of the same. In fact, using 2 , the alpha transform yields fourth order integrators the Al-Alaoui transform. = 8(z−1) The conventional Al-Alaoui differentiator, s ( + 1 ) , 10 7T z 7 is obtained by using a = 0.75 in the Al-Alaoui transform or 9 Analog α = . Sekara 0 875 in the alpha transform. 8 Al-Alaoui The novelty in [11] is by stating that: (discretization 7 of 1/sn) = (discretization of 1/s)n, and by trying to approxi- / n 6 mate 1 s directly. The following steps describe the approach: 5

Magnitude 4 1 z − 1 1 s = = m (22) 3 T [1 + α(z − 1)] T 2

1 where

0 z − 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m = (23) ω/π [1 + α(z − 1)]

Fig. 12 Magnitude of fourth order integrators (zoom in view) Or, equivalently,

10 [1 + (1 − α)m] 9 Sekara z = (24) Al-Alaoui 1 − αm 8 7 Hence, 6 ∞ [1 + (1 − α)m] (−1)k−1(1 − α)k + αk 5 ln z = ln = mk 1 − αm k 4 k=1 Absolute Error 3 (25) ∞ ln z 1 (−1)k−1(1 − α)k + αk 2 s = = mk T T k 1 k=1 1 1 0 ⇒ = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n k−1 k k n s 1 ∞ (−1) (1−α) +α k ω/π T k=1 k m = n( n −n +···+ n −1 + n + ( )), ∈ Fig. 13 Absolute magnitude error with respect to analog integrator of T cn m c1 m c0 O m n N fourth order integrators (zoom in view) (26)

123 Electr Eng (2008) 90:455–467 461

The example below shows the computation details for ∞ −n z − 1 1 1 pk n = 2: m|α= = = p ⇒ = 1 z sn T k 1 1 k=1 = 2 2 = n( n −n +···+ n −1 + n + ( )), ∈ s 1 + −(1−α)2+α2 2 + (1−α)3+α3 3 + ( 4) T an p a1 p a0 O p n N T m 2 m 3 m O m (29) T 2 = (27) α2− α+ This leads to the same results. m2 + (2α − 1)m3 + 36 36 11 m4 +O(m5) 12 The above approach could be repeated exactly with the Applying the method of “long division”, we get parameterized Al-Alaoui operator as follows: 2 ( − ) 1 − − 12α −12α+1 2 z 1 2 = T 2 m 2 −(2α − 1)m 1 + + O(m) s = = m s2 12 T [(1 − a) + (1 + a)z] T 2 − − 12α − 12α + 1 (z − 1) z − 1 ≈ T 2 m 2 − (2α − 1)m 1 + m = = 12 [(1 − a) + (1 + a)z] z + 1 + a(z − 1) [1 + α(z − 1)]2 [1 + α(z − 1)] 1 + (1 − a)m = T 2 − (2α − 1) ⇒ z = (z − 1)2 z − 1 − ( + ) 1 1 a m α2 − α + +12 12 1 1 + (1 − a)m ⇒ ln z = ln 12 1 − (1 + a)m 2 T ∞ = 12[1+α(z−1)]2−12(2α−1)(z−1) (−1)k−1(1 − a)k + (1 + a)k 12(z−1)2 = mk (30) k ×[1 + α(z − 1)]+(12α2 − 12α + 1)(z − 1)2 k=1 ∞ (− )k−1( − )k + ( + )k T 2 z2 + 10z + 1 = ln z = 1 1 1 a 1 a k = (28) s m ( − )2 T T k 12 z 1 k=1 The result obtained was independent of α. Note that this is (31) ⇒ 1 = 1 the case for all the integrators presented in [11]. Since the n ∞ (− )k−1( − )k +( + )k n s 1 1 1 a 1 a mk results are independent of α, the formulas above could be T k=1 k α simpliﬁed by taking = 1 as follows: Below is a computational example for n = 3:

1 1 = 3 3 s 1 + (1+a)2−(1−a)2 2 + (1+a)3+(1−a)3 3 + (1+a)4−(1−a)4 4 T 2m 2 m 3 m 4 m

T 3 = 3 + 2 + 6a2+2 3 + ( 3 + ) 4 + ( 5) 2m 2am 3 m 2a 2a m O m

T 3 = (32) 8m3 + 24am4 + (48a2 + 8)m5 + (80a3 + 40a)m6 + O(m7)

3 1 T − − − = m 3 − 3am 2 + (3a2 − 1)m 1 − (a3 − a) + O(m) s3 8 3 T − − − ≈ m 3 − 3am 2 + (3a2 − 1)m 1 − (a3 − a) 8 T 3 [(z + 1) + a(z − 1)]3 3a[(z + 1) + a(z − 1)]2(z − 1) [(z + 1) + a(z − 1)](z − 1)2 = − + (3a2 − 1) − (a3 − a) 8 (z − 1)3 (z − 1)3 (z − 1)3

T 3 = (z + 1)[(z + 1)2 − (z − 1)2] 8(z − 1)3

T 3 z(z + 1) = (33) 2 (z − 1)3 123 462 Electr Eng (2008) 90:455–467

Step Response As expected, the result was independent of “a”. In addition, 2.5 this is the same result obtained in [11] with the alpha trans- analog form. Hence, all the results obtained with the alpha transform Tustin are obtainable with the parameterized Al-Alaoui operator. 2 Similarly to the alpha transform, since the results are independent of “a”, the formulas above could be simpliﬁed by 1.5 taking a = 1 as follows: z − 1 1 1

Amplitude 1 m|a=1 = ⇒ = n k n 2z s 1 ∞ 2k 1 z−1 T k=1 k 2k z 0.5 −n ∞ k = 1 p T k k=1 0 0102030405060 = n( n −n + ··· + n −1 + n + ( )), ∈ T an p a1 p a0 O p n N Time (sec) z − 1 withp = (34) z Fig. 15 Step response—analog vs. Tustin Again, this leads exactly to the same results obtained with Step Response the alpha transform. In the following sections, we will discuss 2.5 the examples presented in [11]. analog Euler 4.3 Examples 2

Example 1 1.5

Example 1 in [11] consists of approximating the analog transfer function describing the suppressed oscillations: Amplitude 1 1 s−2 G(s) = = (35) 0.5 s2 + 0.4s + 0.68 1 + 0.4s−1 + 0.68s−2 Using the approximations for 1/sn from Table 2 in [11], 0 Fig. 14 is obtained. Using the bilinear (Tustin) approxima- 0 10 20 30 40 50 60 tion, we get Fig. 15. Euler Approximation yields Fig. 16. Time (sec) The conventional (a = 0.75) Al-Alaoui yields Fig. 17.Itis Fig. 16 Step response—analog vs. Euler

Step Response 2.5 Step Response 2.5 analog analog s^(-n) Approximation 2 Al-Alaoui 2

1.5 1.5

1 Amplitude 1 Amplitude

0.5 0.5

0 0 10 20 30 40 50 60 0 0102030405060 Time (sec) Time (sec) Fig. 14 Step response—analog vs. approximations from Table 2 in Fig. 17 Step response—analog vs. Al-Alaoui [11]

123 Electr Eng (2008) 90:455–467 463 obtained by Example 2 8(z − 1) s = The transfer function describing stationary oscillations below T (z + 1 ) 7 7 is used in [11]: 49z2 + 14z + 1 ⇒ G(z) = (36) 0.64 119.72z2 − 137.68z + 61.48 G(s) = (42) s2 + 0.64 The system obtained from the Al-Alaoui transform has a lower overshoot and a faster convergence time than all the with zero-order hold: other systems. −3 − 0.64s − To get the best approximation, the author of [11]usesthe G (s 1) = (1 − e sT ) (43) HO + . −2 following transfer function: 1 0 64s Using the approximations for 1/sn from Table 2 in [11], we Gα ,α (z) 1 2 get = 1 z−1 1 z−1 1 z−1 1 +α ( − ) +α ( − ) + 0.4 +α ( − ) + 0.68 T 3 z(z+1) 1 1 z 1 T 1 2 z 1 T 1 1 z 1 T 0.64 2 (z−1)3 − (37) G R (z) = ( − z 1) HO 2 ( 2+ + ) 1 1 + 0.64 T z 10z 1 12 (z−1)2 The he selects the pair (α1; α2) so that the poles of the discrete transfer function match those of the analog transfer function. 24(z + 1) = (44) This method gives the best results for Example 1 in [11]. 79z2 − 110z + 79 After matching the poles, the results obtained in [11]are: The result of the alpha transform is

16 α3(z − 1)3 + 3α2(z − 1)2 + 3α(z − 1) + 1 Gα = HO z (16α2 + 25)z2 − 2(16α2 − 16α + 25)z + (16α2 − 32α + 41)

16 α3z3 + (3α2 − 3α3)z2 + (3α3 − 6α2 + 3α)z + (3α2 − α3 − 3α + 1) = (45) z (16α2 + 25)z2 − 2(16α2 − 16α + 25)z + (16α2 − 32α + 41)

α1 = 0.71320; α2 = 0.32120 0.22908(z + 2.1133)(z + 0.40213) Al-Alaoui approximation yields ⇒ Gα ,α (z) = 1 2 1.2843z2 − 1.4651z + 0.86086 3 2 2 α= . 219.52z + 94.08z + 13.44z + 0.64 0.22908z + 0.576234z + 0.194677 G 0 875 = (46) = (38) HO . 3 − . 2 + . 1.2843z2 − 1.4651z + 0.86086 762 88z 952 32z 517 12z Note that the same results are obtained by using the parame- The Euler approximation yields terized Al-Alaoui operator with the correct values for (a1,a2): 1 G , (z) = (39) a1 a2 − − − z 1 2 z 1 2 + 0.4 z 1 2 + 0.68 [(1−a1)+(1+a1)z] T [(1−a2)+(1+a2)z] T [(1−a1)+(1+a1)z] T

α1 = 0.71320 ⇒ a1 = 0.4264; (40) α2 = 0.32120 ⇒ a2 =−0.3576

0.91632z2 + 2.30496z + 0.7787 Ga ,a (z) = 1 2 5.137014z2 − 5.86046z + 3.443446 4 0.22908z2 + 0.576234z + 0.194677 = 4 1.2843z2 − 1.4651z + 0.86086

= Gα1,α2(z) (41)

2 Figure 18 shows the result of the alpha transform (and equiv- α= z G 1 = (47) alently the parameterized Al-Alaoui) with pole matching. HO 41z2 − 50z + 25 123 464 Electr Eng (2008) 90:455–467

Step Response Step Response 2.5 2 analog analog 1.8 alpha transform s^(-n) Approximation 2 1.6

1.4

1.5 1.2

1 Amplitude Amplitude 1 0.8

0.6

0.5 0.4

0.2

0 0 0102030405060 0 102030405060 Time (sec) Time (sec)

Fig. 18 Step response—analog vs. alpha transform Fig. 19 Step response—analog vs. approximations from Table 2 in [11]

Step Response The step responses of the above transfer functions are shown 2 in Figs. 19, 20, 21 and 22. analog 1.8 The Al-Alaoui: a = 0.75, (α = 0.875) and Euler: a = Tustin 1(α = 1) yield stable systems not oscillating ones. In fact, 1.6 any value of α>0.5 will yield a stable system. The approx- 1.4 imations from Table 2 in [11] yield good results in approxi- 1.2 mating the analog system. 1

0.8 Amplitude

Example 3 0.6

0.4 This example considers the transfer function of a sixth order butterworth ﬁlter: 0.2 0 0102030405060 G (s) BF Time (sec) 1 = √ 2 + 5π + 2 + π + 2 + + Fig. 20 Step response—analog vs. Tustin s 2 cos 12 s 1 s 2 cos 12 s 1 s 2s 1 (48) Step Response 2 analog Applying the Al-Alaoui transform, we get 1.8 Euler 1.6 8(z−1) s = ⇒ G(z) 1.4 7T z+ 1 7 1.2 ( 2 + + )3 = ⎡ 49z 14z 1 ⎤ (49) 1 (362.9754z2 −547.6932z+248.7178)x ⎢ ⎥ Amplitude 0.8 ⎢ ( . 2 − . + . ) ⎥ ⎣ 521 3672z 683 4576z 226 0904 x⎦ 0.6 ( . 2 − . + . ) 463 3918z 633 7644z 234 3726 0.4

0.2 The ﬁgures comparing the transfer function above to those 0 obtained in [11] are shown in Figs. 23 and 24. 0 10 20 30 40 50 60 Again, the system obtained from the Al-Alaoui transform Time (sec) has a lower overshoot and a faster convergence time than the other systems. Fig. 21 Step response—analog vs. Euler 123 Electr Eng (2008) 90:455–467 465

Step Response 4.4 Prewarping issues 2

1.8 analog Al-Alaoui In Example 3, the transfer function of an analog Butterworth 1.6 filter is given. Usually, in a digital filter design problem, the 1.4 specifications of the digital filter are given (passband and

1.2 stopband frequencies, stopband attenuation, passband ripple, etc.), then an analog prototype is determined after prewar- 1 ping the critical frequencies of the discrete-time ﬁlter to the

Amplitude 0.8 corresponding frequencies of the continuous-time filter. In 0.6 [11], the purpose seems to approximate the response of the Analog Butterworth Filter, not performing a digital filtering 0.4 problem. Therefore, no information is given about the critical 0.2 frequencies of the discrete-time filters. 0 The analog transfer function of the Butterworth filter is 0102030405060 Time (sec) GBF(s) Fig. 22 Step response—analog vs. Al-Alaoui 1 = √ 2 + 5π + 2 + π + 2 + + s 2 cos 12 s 1 s 2 cos 12 s 1 s 2s 1 Step Response 1.4 (50) analog = 1.2 s^(-n) Approximation corresponds to a cutoff frequency C 1. The correspond- Al-Alaoui ing discrete-time frequency when applying the bilinear trans- 1 form with prewarping is given by 0.8 −1 C T ωc = 2tan = 0.49 (51) 2 0.6

Amplitude And the discrete-time frequency when applying the Al-Ala-

0.4 oui transform with prewarping is given by − 4C T 0.2 ω = 1 = . c tan 2 2 0 475 (52) 4 + (C T ) (0.75 − 1) 0 0 10 20 30 40 50 60 Applying the bilinear transform to the analog transfer func- Time (sec) tion gives

Fig. 23 Step response 2(z − 1) s = T (z + 1) 2 3 Bode Diagram (z + 2z + 1) 50 ⇒ G(z) = ⎡ ⎤ (19.0704z2 − 30z + 14.9296)x (53) 0 ⎢ ⎥ ⎢ 2 ⎥ -50 ⎣ (24.72736z − 30z + 9.27264)x⎦ -100 (22.6568z2 − 30z + 11.3432) -150 analog s^(-n) Approximation

Magnitude (dB) -200 Applying the Al-Alaoui transform, we get Al-Alaoui -250 8(z − 1) 180 s = ⇒ G(z) + 1 0 7T z 7 ( 2 + + )3 -180 = ⎡ 49z 14z 1 ⎤ (54) -360 (362.9754z2 −547.6932z + 248.7178)x ⎢ ⎥ Phase (deg) -540 ⎢ 2 ⎥ ⎣ (521.3672z −683.4576z + 226.0904)x⎦ -720 2 10-2 10-1 100 101 102 (463.3918z −633.7644z + 234.3726) Frequency (rad/sec) The ﬁgures comparing the bilinear transform to Al-Alaoui Fig. 24 Bode diagram are shown in Figs. 25 and 26. 123 466 Electr Eng (2008) 90:455–467

Step Response 1.4 Acknowledgments This research was initiated during my visit to the analog Adaptive Systems Laboratory at UCLA in 2007–2008. It is indeed a s^(-n) Approximation pleasure to acknowledge Professor A. H. Sayed for his invitation and 1.2 Bilinear for providing the atmosphere conducive to research. I am grateful to Al-Alaoui Cassio Lopes and Qiyue Zou for their help during my stay at UCLA. I 1 am grateful to the highly talented graduates of the ECE Department at the American University of Beirut. It is indeed a pleasure to acknowl- 0.8 edge Elias Yaacoub and Jimmy Azar for their invaluable help in the production of this work. 0.6 Amplitude

0.4 References

0.2 1. Enden VD (1989) Discrete-time signal processing. Prentice- Hall, Englewood Cliffs 0 0 10 20 30 40 50 60 2. Franklin FG, Powell JD (1980) Digital control of dynamic systems. Addison-Wesley, Reading Time (sec) 3. Lam HY-F (1979) Analog and digital filters. Prentice-Hall, Engle- Fig. 25 Comparison of step responses wood Cliffs 4. Mitra SK (2006) Digital signal processing, 3rd edn. McGraw-Hill, Boston 5. Phillips CL, Nagle HT (1984) Digital control system analysis and Bode Diagram 200 design. Prentice-Hall, Englewood Cliffs 6. Rabiner LR, Gold B (1975) Theory and applications of digital sig- 0 nal processing. Prentice-Hall, Englewood Cliffs -200 7. Al-Alaoui MA (1993) Novel digital integrator and differentia- analog tor. IEE Electr Lett 29(4):376-378, 18 February 1993. (See also -400 s^(-n) Approximation ERRATA, IEE Electr Lett 29(10):934, 1993.) -600 Bilinear 8. Al-Alaoui MA (1997) Filling the gap between the bilinear and the Magnitude (dB) Al-Alaoui backward difference transforms: an interactive design approach. -800 Int J Electr Eng Educat 34(4):331–337 180 9. Al-Alaoui MA (2001) Novel stable higher order s-to-z transforms. 0 IEEE Trans Circuits Syst I: Fundam Theory Appli 48(11):1326– 1329 -180 10. Šekara TB, Stojic MR (2005) Application of the α-approximation -360 for discretization of analogue systems. Facta Universitatis, -540 Ser: Elec. Energ (3):571–586 2005. http://factaee.elfak.ni.ac.yu/ Phase (deg) fu2k53/Sekara.pdf -720 11. Šekara T (2006) New transformation polynomials for discretiza- -2 -1 0 1 2 10 10 10 10 10 tion of analogue systems. Electr Eng 89(2):137–147 Frequency (rad/sec) 12. Al-Alaoui MA (2006) Al-Alaoui operator and the α-approximation for discretization of analog systems. Facta Universitatis (NIS), Fig. 26 Comparison of bode diagrams Ser: Elec Energ 19(1):143–146 13. Chen YQ, Moore KL (2002) Discretization for fractional order differentiators and integrators. IEEE Trans Circuits Syst I 49:363–367 And a Al-Alaoui provides a better approximation for the 14. Al-Alaoui MA (1992) Novel approach to designing digital differ- magnitude response than the bilinear transform at high fre- entiators. IEEE Electron lett 28(15):1376–1378 quencies, but the bilinear transform approximates better the 15. Al-Alaoui MA (1994) Novel IIR differentiator from the simpson integration rule. IEEE Trans Circuits Syst I 42(2):186–187 step response, although the Al-Alaoui led to a step response 16. Al-Alaoui MA (1996) A class of numerical integration rules with with faster convergence and lower overshoot. first order derivatives. ACM SIGNUM Newslett 31(2):25–44 17. Al-Alaoui MA, Ferzli R (2006) An Enhanced first-order sigma– delta modulator with a controllable signal-to-noise ratio. IEEE Trans Circuits Syst I 53(3):634–643 5 Conclusion 18. Al-Alaoui MA (2007) Linear phase lowpass IIR digital differentiators. IEEE Trans Signal Process 55(2):697–706 It was shown that Al-Alaoui operator and the “new transfor- 19. Al-Alaoui MA (2007) Novel Approach to Analog to Digital Transforms. IEEE Trans Circuits Syst I: Fundam Theory Appl mation polynomials for discretization of analogue systems” 54(2):338–350 are one and the same. Al-Alaoui operator is a credible alter- 20. Al-Alaoui MA (2007) Using fractional delay to control the mag- native to other discretization methods with wide range of nitudes and phases of integrators and differentiators. IET Signal applications in integer and fractional order discretization and Process 1(2):107–119 21. Aoun M (2004) Numerical simulations of fractional systems: an other applications that require accurate and real time differ- overview of existing methods and improvements. Nonlionear entiation or integration [15–44]. Dynam 38(1–2):117–131 123 Electr Eng (2008) 90:455–467 467

22. Barbosa RS, Machado JAT, Ferreira IM (2005) Pole-zero approx- 34. Mrad F, Dandach SH, Azar S, Deeb G (2005) Operator friendly imations of digital fractional-order integrators and differentiators common sense controller with experimental verification using Lab- using signal modeling techniques. In: 16th IFAC world congress, VIEW. In: Proceedings of the 2005 IEEE international symposium Prague, Czech Republic, July 4–8, 2005 on, intelligent control Mediterrean conference on control and auto- 23. Barbosa RS, Machado JAT (2006) Implementation of discrete- mation time fractional order controllers based on LS approximations. Acta 35. Ngo NQ (2006) A new approach for the design of wideband digi- Polytech Hungarica 3(4):5–22 tal integrator and differentiator. IEEE Trans Circuits Syst II 53(9): 24. Barbosa RS, Machado JAT, Silva MF (2006) Time domain design 936–940 of fractional differintegrators using least-squares. Signal Process 36. Papamarkos N, Chamzas C (1996) A new approach for the design 86(10):2567–2581 of digital integrators. IEEE Trans Circuits Syst I 43(9):785–791 25. Barbosa RS, Machado JAT, Galhano AM (2007) Performance 37. Romero M, de Madrid AP, Manoso C, Hernandez R (2006) Dis- of fractional PID algorithms controlling nonlinear systems with cretization of the fractional-order differentiator/integrator by the saturation and backlash phenomena. J Vib Contr 13(9–10):1407– rational chebyshev approximation. In: CONTROLO 2006, 7th Por- 1418 tuguese conference on automatic control, Instituto Superior Tec- 26. Backmutsky V, Shenkman A, Vaisman G, Zmudikov V, Kottick D, nico, Lisboa, Portugal, September 11–13, 2006 Blau M (1999) New DSP method for investigating dynamic behav- 38. Tomov L (2006) Matrix method for generalized discretization of ior of power systems. Electric Mach Power Syst 27(4):399–427 transfer functions through formal replacement with first order oper- 27. Chen Y, Vinagre BM, Podlubny I (2003) A new discretization ator. J Fundam Sci Appl 13(4):61–68 method for fractional order differentiators via continued fraction 39. Tseng C-C (2006) Digital integrator design using simpson rule and expansion. In: Proceedings of the ASME design engineering tech- fractional delay filter. IEEE Proc Vis Image Signal Process 153(1) nical conference 5 A, pp 761–769 40. Tseng C-C (2007) Closed form design of digital IIR integrators 28. Chen YQ, Vinagre BM (2003) A new IIR-type digital fractional using numerical integration rules and fractional sample delays. order differentiator. Signal Process 83(11):2359–2365 IEEE Trans Circuits Syst I 54(3):643–655 29. Chen YQ, Vinagre BM, Podlubny I (2004) Continued fraction 41. Varshney P, Gupta M, Visweswaran GS (2007) New switched expansion approaches to discretizing fractional order derivatives— capacitor fractional order integrator. J Active Passive Electron Dev an expository review. Nonlinear Dynam 38(1–2):155–170 2:187–197 30. Dabroom AM, Khalil HK (2000) Output feedback sampled-data 42. Wong CX, Worden K (2007) Generalized NARX shunting neural control of nonlinear systems using high-gain observers. IEEE Trans network modelling of friction. Mech Syst Signal Process 21(1): Automatic Control 46(11, 1):1712–1725 553–572 31. Ferdi Y, Boucheham B (2004) Recursive filter approximation of 43. Worden K, Wong CX, Parlitz U, Hornstein A, Engster D, digital differentiator and integrator based on Prony’s method. In: Tjahjowidodo T, Al-Bender F, Rizos DD, Fassois SD (2006) Proceedings of the first IFAC workshop on fractional differentia- Identification of pre-sliding and sliding friction dynamics: grey box tion and its applications (FDA’04), Bordeaux, France, July 19–21, and black-box models. Mech Syst Signal Process 21(1):514–534, 2004, pp 428–433 2007; see also vol 13(4), 2006 32. Ferdi Y (2006) Computation of fractional order derivative and inte- 44. Zhang J, Zhang K, Grenfell R, Li Y, Deakin R (2005) Real-time gral via power series expansion and signal modelling. Nonlinear Doppler/Doppler rate derivation for dynamic applications. J Global Dynam 46(1):1–15 Position Syst 4(1–2):95–105 33. Li Y,Dempster A, Li B, Wang J, Rizos C (2005) A low-cost attitude heading reference system by combination of GPS and magnetom- eters and MEMS inertial sensors for mobile applications. In: The International Symposium on GPS/GNSS

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