THERE are two major approaches to designing infinite impulse

Transcription

1 338 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 Novel Approach to Analog-to-Digital Transforms Mohamad Adnan Al-Alaoui, Senior Member, IEEE Abstract A novel approach to analog-to-digital transforms, -to- transforms, is presented. The approach applies the Boxer Thaler expansion to a leaky differentiator or to a leaky integrator instead of an ideal differentiator or integrator. The bilinear (Tustin) and the matched pole zero (MPZ) transformation are special realizations of the new transforms, with an additional built-in prewarping and additional built-in zero placement, respectively. Examples are presented that demonstrate the viability of the approach where the proposed method is compared with the least-squares, bilinear (Tustin), and MPZ transformations. Index Terms Bilinear (Tustin) transformation, Boxer Thaler transformation, digital control, digital filters, digital signal processing, integrators, matched pole zero (MPZ), -to- transforms. I. INTRODUCTION THERE are two major approaches to designing infinite impulse response (IIR) digital filters [1] [7], [15]. One approach is the Direct Digital Design approach, which includes least-squares and pole zero placement methods. The second approach is the emulation approach that utilizes the wealth of the analog filter design techniques and appropriate transformations, or mappings, from the -plane to the -plane. These -to- transformations obtain a discrete-time equivalent transfer function from a continuous-time transfer function. In the following presentation, the subscripts will be omitted for simplicity. It is desirable that the mapping procedures have the following two properties: 1) they should map the left half of the -plane to the interior of the unit circle in the -plane to assure that real, causal, stable, and rational analog transfer functions result in real, causal, stable, and rational discrete-time transfer functions and 2) the imaginary axis of the -plane should be mapped onto the unit circle circumference in the -plane. The emulation approach may be further subdivided into three groups. A. Time-Invariant Response Methods (Hold Equivalence Methods) [3] [6] This approach preserves the time response of the original analog signal at the sampling instants only. The approach suffers from inherent aliasing which rers it unsuitable for designing high-pass and bandstop filters. Without aliasing, the magnitude of the frequency response will be exact. The aliasing stems from the fact that each section of width of the imaginary axis of the -plane, where is the sampling Manuscript received April 19, This work was supported in part by the University Research Board of the American University of Beirut. This paper was recommed by Associate Editor R. W. Newcomb. The author is with the Department of Electrical and Computer Engineering, the American University of Beirut, Beirut , Lebanon ( adnan@aub.edu.lb). Digital Object Identifier /TCSI period, maps onto the circumference of the unit circle in the -plane, the corresponding left half of the -plane strip maps to inside the unit circle in the -plane, and the corresponding right half of the -plane strip maps to the region that lies outside the unit circle in the -plane. The most famous member of these methods is the impulse-invariant (no-hold) method. Other important members of this approach are the step-invariant (zeroth-order-hold) and the ramp-invariant (first-order-hold) methods. The impulse-invariant method may be summarized as follows. 1) Determine the transfer function of the analog filter that corresponds to the specifications of the desired digital filter. 2) Expand, using partial fraction expansion, into firstand second-order sections. 3) Find the corresponding -transform of each term of the expansion obtained in 2). Obtain, the inverse Laplace transform of., the -transform of, is obtained by sampling with a sampling period to obtain and applying the -transform to. If, where and are constants, then, and the corresponding, which sug- -transform is gests obtaining from directly. 4) Obtain as. B. Numerical Approximation Methods [1] [8] Numerical integration or differentiation techniques are often employed to obtain -to- transformations. The most famous of the numerical integration-based mappings is the bilinear (Tustin) transformation, which is obtained from the trapezoidal integration rule, the backward difference rule, which is obtained from the backward rectangular integration rule, and the forward difference rule, which is obtained from the forward rectangular integration rule [1] [9]. Recently, approaches that interpolate the bilinear and the backward difference rules have been suggested [9] [11]. Also, approaches that interpolate the bilinear and the Simpson rules were proposed [12], [13]. The bilinear (Tustin) transform (BZT) is a conformal mapping that avoids aliasing of frequency components. It maps the imaginary axis in the -plane onto the circumference of the unit circle in the -plane only once, the left half of the -plane to the interior of the unit circle in the -plane, and the right half of the -plane to the region that lies outside the unit circle in the -plane. The method is suitable for designing frequency-selective filters. It preserves specific features of the magnitude response characteristics especially if the characteristics were piece-wise linear. However, it does not necessarily preserve the time-domain properties. The bilinear transformation is simple, time-tested, and a proven method of conversion. However, in many applications, such as professional audio, the distortion /$ IEEE

2 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 339 that results from applying the Tustin transformation is deemed to be unacceptable. Excellent discussions and considerable insights on this is provided by Clark et al. [14] and by Ifeachor and Jervis [15]. The bilinear (Tustin) method may be summarized as follows. 1) Find the transfer function of an analog filter that corresponds to the desired digital filter specifications. 2) Determine the corresponding band-edge or critical frequencies of the desired digital filter and prewarp the corresponding analog frequencies as follows:, where designates the analog frequency, designates the digital frequency, and is a design parameter that is usually assigned the value 1 or 2 and that will be cancelled by the next step, 3) Obtain by replacing the variable in by in some fashion. They up substituting approximations of, which is in effect a scaled value of, in the factors while keeping the exact value of, thus introducing additional distortions. However, the new transform approximates the factors in the transfer function. The approach presented in this paper is general and incorporates appropriately the poles or zeros in the factored form of. In Section II, the derivations of the new transforms are presented. In Section III, the new approach is presented in an algorithmic fashion. In Section IV, examples are presented that demonstrate the viability of the new transforms. A Matlab program is provided which makes the new method as easy to use as the Tustin transform, while avoiding its shortcomings. Additionally, it is shown that the Tustin transform and the MPZ transforms are special cases of the new proposed transforms. C. Heuristic Methods [1], [4], [6], [14], [15] These methods consist of sets of heuristic rules that establish the resulting -to- transforms. The most celebrated of these methods is the matched pole zero (MPZ) transform, also known as the matched -transform (MZT) method. Another proposed method is averaging of BZT and MPZ [14], [15]. The MPZ maps poles and zeros of on the -plane to poles and zeros on the -plane using. Note that the impulse-invariant method provides this mapping only to the poles of. Additionally, zeros at are mapped to the point, ostensibly because the point corresponds to the highest frequency in the -plane. The MPZ (MZT) method may be summarized as follows. 1) Determine a suitable analog transfer function that meets the specifications of the desired filter. 2) Find the locations of the poles and zeros of. 3) Map the finite poles and zeros from the -plane to the -plane as follows:, where is a constant, real or complex. 4) The zeros at at are mapped in to the point. A variant of this approach allows one zero at to be mapped to. The variant will result in, which is the inverse -transform of,tohave one unit delay. This will allow one sample period for the computation that corresponds to. 5) Combine the -plane equations appropriately to obtain up to a constant multiplier. The constant multiplier is determined by having the gain of equal to the gain of at a critical frequency. If the critical frequency were, then select the gain such that:. In general, the use of impulse-invariant or bilinear (Tustin) transformation is preferred over the MPZ. If the design is to preserve the temporal characteristics of the filter, then use the impulse-invariant transformation with anti-aliasing guard filters; otherwise, the Tustin transformation is preferred. In this study, new -to- transforms will be developed to overcome the deficiencies in the traditional transforms. The traditional transforms approximate of the analog transfer function II. DERIVATION OF THE NEW -TO- TRANSFORMATIONS Initially, the derivation applied the Boxer Thaler expansion [16] [19] to a leaky integrator instead of an ideal integrator. The new transform is obtained by taking the first term of the expansion. For the leaky differentiator,, the inverse of the leaky integrator transform was used. Then, the expansion was applied directly to the leaky differentiator and the first term approximation yielded the same result as that obtained from the inverse of the leaky integrator expansion. Since the expansion of the leaky differentiator is less cumbersome, the following derivation applies the Boxer Thaler expansion to a leaky differentiator and then takes the inverse of the resulting transform to obtain the transform corresponding to a leaky integrator. This yields a first-order -to- transform, which was obtained by retaining only the first term in the series expansion, similar to the bilinear transform, which is obtained in the case of an ideal differentiator. However, the new transform has a built-in prewarping as delineated below. The bilinear (Tustin) -to- transformation may be derived from the Taylor series expansion of or equivalently from, where denotes the natural logarithm of and denotes the sampling period in seconds. Without loss of generality, assume that we have the following transfer function, which could be the transfer function of a leaky differentiator [16] [20] where is a constant that could be a complex number. To obtain the -transform for the corresponding discrete equivalent of (1), substitute to obtain, with or, the following equation: Next, expand as the ratio (1) (2) as a Laurent series, by first expressing (3)

3 340 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 Solving (3) for yields dc scaling is desired, then the constant value in (10) would have the From (3) and the properties of the logarithm, the following equation is obtained: Applying the Taylor expansion to each of the two right-hand terms of (5) yields the following two equations: (4) (5) (11) If instead of (1), we had a factor of in the denominator of, then the resulting equivalent discrete-time transfer function for this factor would be the inverse of (10). In particular, the transfer function (12) (6) (7) would map into the following transfer function in the -plane: From (5) (7), the following expansion is obtained: In practice, approximations of (8) are obtained by taking appropriately a few of the right-hand-side terms. In the following, a first-order approximation is obtained by retaining only the first term in the expansion in (8). A. Case of Simple Real Zeros or Real Poles A first-order -to- transform is obtained by retaining only the first term in the expansion in (8) to obtain Thus, from (2), (9), and (1), following : (8) (9) maps to the (10) The new transformation maps a zero at to a zero at and a pole at, which are symmetrical with respect to the origin in the -plane. The pole lies inside the unit circle only if is positive, i.e., only if the zero lies in the left half of the -plane. Note that, when, then, and (10) reduces to the bilinear (Tustin) transformation. The above approximation, however, destroys the scale of the magnitude, as compared with (1). One notion of preserving the scale is to consider a scaling frequency such that, when the frequency, in radians per second, is set to in and in, which results in and in, both the analog and discrete transfer functions have the same magnitude. The constant in (10) is chosen to assure the above-mentioned scaling. If there are many factors involved, then the constant resulting from the product of all such factors is evaluated at the to assure scaling. For example, if contains only one factor, as in (1), and when the scaling frequency is set to zero, i.e., when (13) Thus, a pole at maps to a pole at and a zero at, which are symmetrical with respect to the origin in the -plane. Thus, if (12) corresponds to a stable system, i.e., the real part of the constant is positive, then the absolute magnitude of the pole at is less than one, and the resulting pole lies indeed inside the unit circle. is being used as a general constant factor chosen to achieve the desired scaling. If there are many factors involved, then the constant resulting from the product of all such factors is evaluated at the to assure scaling. For example, if contains only one factor, as in (11), and when the scaling frequency is set to zero, i.e., when dc scaling is desired, then the constant in (12) would have the value (14) Thus, from the above, it is concluded that only minimumphase systems whose poles and zeros lie in the left half of the -plane would map to stable systems in the -plane. In fact the resulting stable systems would also be minimum-phase systems, whose poles and zeros lie inside the unit circle. B. Case of Simple Complex Conjugate Pairs of Poles or Zeros Note that the above apply if the simple pole, or zero, had a complex value. Since poles and zeros occur in complex conjugate pairs, suppose that we have the following analog transfer function, where, and are real constants: Applying (10) to each of the factors of (15) yields (15) (16)

4 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 341 TABLE I PROPOSED FIRST-ORDER s-to-z TRANSFORMS As in the case of (13), (16) asserts that, if the original analog system of (15) is stable, then the resulting discrete-time system would also be stable. An alternative derivation would be to expand (15) using partial fraction expansion and (13). The transfer function of the resulting discrete approximation yields (17) If the complex-conjugate pairs appear in the numerator of the analog transfer function, then the transfer function of the corresponding discrete-time system will have the inverse form of (16). Again, the resulting system will be stable only if the zeros of the analog system are located in the left half of the -plane. For the sake of completeness, assume the following transfer function of an analog system, where, and, are real constants: (18) The transfer function of the corresponding discrete-time system will be (19) An alternative derivation would be to use the corresponding inverse of (17). The transfer function of the resulting discrete approximation yields (20) Table I tabulates the above proposed first-order -totransforms. The case of stable but nonminimum-phase analog systems can be handled in at least two ways. One method is to map the analog zeros that are in the right-hand side of the -plane using the bilinear transformation and use the new transforms for the rest of the analog transfer function. Another method is to apply the stabilizing approach outlined in [21] [24] to the resulting poles that are located outside the unit circle in the -plane. The stabilization method consists of the following two steps: 1) reflect the poles that lie outside the unit circle at a radius to inside the unit circle at a radius of in the -plane and 2) compensate for the magnitude of the resulting transfer function by multiplying it by. The credit for the above stabilizing approach, as far the author could determine, goes to Steiglitz [24]. In practice, the offing poles would be close to the unit circle, thus the phase will not be severely affected by the stabilization. C. Case of Multiple Poles or Zeros Higher degree factors appear in many transfer functions due to multiple poles and multiple zeros. Without loss of generality, consider the following transfer function, which could be a factor in a more general transfer function, where is an integer: (21) The method would be to apply the approximation of the firstdegree factor times. The same approach would apply for the case of multiple zeros as in (21) (22) D. Frequency Selectivity and Pole Zero Aliasing Throughout the previous presentation, we have discussed the stability of the transform. We have seen that all left-hand-side analog poles map to discrete poles inside the unit circle, preserving stability of all-pole systems. For the case of zeros, if the analog system is a nonminimum-phase system, has zeros in the right half of the -plane, then discrete-time systems will have poles outside the unit circle, but these cases can be handled by all-pass stabilization [21] [24]. This discussion supports the claim that the new transform satisfies the second desirable behavior that we put forth in our introduction, namely that of stable mapping. Here, we address the first desirable behavior, that of preserving the frequency selectivity of the analog system. For this, we start by observing that all different variants of our method, which we will also see in more algorithmic form in Section III, map poles and zeros directly to the positions in which they would have been with the ideal frequency map, in addition to companion zeros and poles, respectively. It is in this sense that the presented transform has built-in prewarping, in contrast to the bilinear (Tustin) transformation, which does direct frequency mapping and avoids pole zero misplacement through explicit prewarping. By virtue of this property, we obtain excellent matching of frequency selectivity in the examples that we present in Section IV. This pole zero mapping property, however, suffers from an aliasing-like phenomenon. This phenomenon is unlike the aliasing of the impulse response method, which consists of the overlap of the periodically exted frequency responses. Here, the placement of the poles and zeros is aliased, hence the pole zero aliasing expression. To see this, reconsider the leaky integrator transfer function (12). We have seen how this maps to (13). We observed that the pole at maps to a pole at and a zero at. Now, consider a pole at, where is an arbitrary integer. Clearly, this

5 342 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 pole maps exactly to the same discrete-time pole and zero as, due to the periodicity of the complex exponential. This result generalizes to the case of a zero, and to poles and zeros of variants of the method. As a consequence, with being the imaginary part of, the presented transform is best applied to systems with poles and zeros that lie within the strip, which is the discretized baseband strip This is a reasonable restriction, since systems that do have poles and zeros, and consequently considerable response, beyond this range will usually require a higher sampling rate for convenient discretization, which will in turn satisfy the said restriction. III. NEW TRANSFORM ALGORITHMS Two algorithms are presented below. The first is a cascade form algorithm, and the second is a parallel form of the algorithms. While the bilinear transformation preserves the order of the poles in the analog transfer function, only the parallel form of the new algorithms preserves the order, if the analog system is strictly proper, i.e., the degree of the numerator of the analog transfer function is less than that of the denominator. The cascade form would double it for an exactly proper system, where the degree of the numerator and denominator of the analog transfer function are equal and all of the zeros are real. However, if the poles and zeros are pure conjugates, which is often the case, the order is preserved. Hence, the order obtained by applying the new cascade transform will increase by the number of real zeros of the analog system, and thus the order will be preserved for the case of all-pole filters. A. Cascade-Form Algorithm Without loss of generality, assume in the following that and that the poles and the zeros are distinct. The following recommations and observations are relevant to the implementation of the algorithm. 1) Express the transfer function of the desired analog prototype filter in the factored form given by (23) Construct that corresponds to (23) in factored form, by introducing the appropriate factors as outlined in the following steps. 2) Each simple factor in the denominator of (23) contributes a pole and a zero as in (13). Equation (13) can be entered with the appropriate value of ; for, it would be the same as the bilinear transform. 3) Each simple factor in the numerator of (23) contributes a pole and a zero as in (10). Equation (10) can be entered with the appropriate value of ; for, it would be the same as the bilinear. 4) Complex poles and zeros may be treated as in Steps 2 and 3 above, or they may be grouped in complex-conjugate pairs using (16) or (17) and (19) or (20), respectively. 5) Higher degree factors, say of degree, are obtained by applying the first degree factors times. 6) The constant factor in the transfer function of the resulting discrete-time system of (24) is evaluated to obtain the same magnitude as in (23) at the scaling frequency. B. Parallel-Form Algorithm Without loss of generality, assume that and that the poles are distinct. The following recommations and observations are relevant to the implementation of the algorithm. 1) Express the transfer function of the desired analog prototype filter in the partial fraction expansion form given by (25) Construct that corresponds to (25) in parallel form by introducing the appropriate factors as outlined in the following steps. 2) Each fraction of (25) contributes a pole and a zero as in (13). Equation (13) can be entered with the appropriate value of ; for, it would be the same as the bilinear transformation. 3) Complex poles may be treated as in Step 2, or they may grouped in complex conjugate pairs using (16) or (17). 4) Higher degree factors, say of degree, are obtained by applying the first degree factors times. 5) The constant factor in the transfer function of the resulting discrete-time system of (26) is evaluated to obtain the same magnitude as in (25) at the scaling frequency. C. Modified Cascade-Form Algorithm In all the examples, the following variant of the cascade form algorithm is used. 1) Real poles and zeros are treated using the original cascade transformation approach, i.e., the following apply: 2) Higher degree factors are treated by applying successive first-degree factors. 3) Complex poles and zeros are treated using the recombined partial fraction method, since that reduces the overall order. The following apply:

6 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 343 4) The constant factor in the transfer function of the resulting discrete-time system is evaluated to obtain the same magnitude as the analog transfer function at the scaling frequency,. Unless otherwise specified, dc scaling was used, i.e., was set to zero. A corresponding Matlab code of the above procedure is included in the Appix. (a) IV. EXAMPLES Here, we present several examples that demonstrate the viability of the new method in comparison with the bilinear, MPZ, and the least-squares methods. 1 Six examples are presented below. The first example compares the sum of squared errors obtained by applying the Boxer Thaler approach, The second is a dc controller, the third is a notch filter, the fourth is a sixth-order elliptic low-pass filter, the fifth is a fourth-order filter, and the sixth is a fourth-order Bessel filter. All of them demonstrate the viability of the new approach in comparison with the bilinear (Tustin), MPZ, and least-squares methods [25], [26]. (b) (c) A. Example 1 This example is Example 2 in Wang et al. [19] and they used the sum of squared errors (SSE) of the time-domain responses as the measure of goodness. The example considers the fifth-order system (27) The sampling time is s. The example was implemented using 500 samples as in [19]. Simulations of the impulse and step responses were carried out using the bilinear (Tustin), the Boxer Thaler, and the proposed new transforms. Wang et al. showed that the Boxer Thaler method yielded smaller SSE from the analog response than the Tustin transformation for the step response. The proposed new transforms yielded the smallest SSE from the analog, for both the impulse and step responses. For the impulse responses, the SSE was for the Tustin, for the Boxer Thaler, and for the new approach. For the step responses, the SSE was for the Tustin, for the Boxer Thaler, and for the new approach. B. Example 2 This example consists of designing a series controller, connected to the plant in the forward path of a unity feedback system as shown in Fig. 1(a). This example was developed in the award-winning textbook by Franklin et al. [1]. Fig. 1(b) and (c) shows the corresponding mixed control system and the pure discrete equivalent system, respectively. The corresponding analog closed-loop transfer function and the plant transfer function are 1 The Matlab function inv freqz was used. (28) (d) Fig. 1. (a) Analog control system. (b) Mixed control system. (c) Pure discrete equivalent of Fig. 1(b). (d) Step responses of the unity feedback system using the analog compensation (ideal) and the discrete-time approximations using the bilinear transform, the MPZ transform, the direct digital design, and the new approach. Equations (28) (35). The design should obtain a closed-loop natural frequency rad and a damping ratio. In the analog case, the specifications can be met by the lead compensation [1] (29) The discrete model representation of the plant preceded by a zeroth-order hold filter is for s (30) In the discrete design, the plant is preceded by a zeroth-order hold. To digitize as shown in Fig. 1(b) and (c), first an appropriate sampling rate is chosen. Since the bandwidth is approximately equals to 0.3 rad/s, a safe sampling rate would be more than 20 times the bandwidth, which is equal to 6 rad/s or about 1 Hz. Thus, the selected sampling period is s. Note that, since the pole has no imaginary part, it lies at the center of the baseband strip and is thus immune to pole zero aliasing.

7 344 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 is Using a digital compensator with a transfer function, the corresponding overall closed-loop transfer function An th digitization scheme will obtain a corresponding and. The bilinear digitization of (30) yields The MPZ digitization of (30) yields (31) (32) (33) The direct digital design adds a derivative term to the proportional term to obtain [1] (34) The new design is obtained by using, (10) for the denominator of (29), the inverse of (10) for the numerator of (29), and evaluating the constant multiplier to obtain the same dc gain as the dc gain of (29). The transfer function resulting from the new approach is given by (35) Fig. 1(d) exhibits the step response of the overall compensated system, corresponding to (31), using all of the above discrete approximations to the analog controller in addition to the continuous control design. The response corresponding to the new approach yields a better transient response than those of the other approximations in terms of both magnitude overshoot and settling times. C. Example 3 Considering the sixth-order inverse Chebyshev band-rejection filter, the transfer function of the normalized continuous model with stopband attenuation db is given by (36), shown at the bottom of the page. We discretize this transfer function with This choice places the poles and zeros within the baseband strip and avoids the risk of pole zero aliasing. Using the bilinear transformation with prewarping yields (37), shown at the bottom of the page. Using (17) and (20) to transform, respectively, the denominator and numerator of (36), the new transforms, after scaling the dc gain and using the cascade form of the new approach, yield (38), shown at the bottom of the page. Additionally, applying the MPZ transform yields a result similar to the new method and will be omitted Finally, applying the least-squares method for a filter of the same order as the proposed method yields the transfer function (39), shown at the bottom of the page. Fig. 2 shows the magnitude and phase responses of the three designed filters together with the analog response. The magnitude response of the new method nearly coincides with the analog one and shows a better response than the bilinear (Tustin) method and the least-squares method. Additionally, the phase response of the proposed method shows the best approximation to the phase of the analog filter. (36) (37) (38) (39) (40)

8 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 345 (a) (a) (b) Fig. 2. (a) Magnitude responses of the filter in example 3. Equations (36) (39). (b) Phase responses of the filter in example 3. Equations (36) (39). D. Example 4 In this fourth example, we will consider a sixth-order lowpass elliptic filter with the transfer function (40), shown at the bottom of the previous page. The new method will give the transfer function (41), shown at the bottom of the page. Also, in this example we apply the MPZ transform which yields a result similar to the new method and will be omitted for brevity. To make a fair comparison, we will design a sixth-order leastsquares approximation of the desired filter, given by (42), shown at the bottom of the next page. (b) Fig. 3. (a) Magnitude responses of the filter in example 4. Equations (40) (42). (b) Phase responses of the filter in example 4. Equations (40) (42). Fig. 3 shows that the magnitude response of the new method is better than that of the same order least-squares method. Similarly, the phase response of the proposed method shows a better approximation. E. Example 5 In this fifth example, we will consider the following transfer function: (43) The new transformation yields (44), shown at the bottom of the next page. The least-squares method yields the fifth-order approximation (45), shown at the bottom of the next page. (41)

9 346 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 (a) (b) Fig. 4. (a) Magnitude response of the filter in example 5. Equations (43) (46). (b) Phase response of the filter in example 5. Equations (43) (46). Applying the MPZ transform yields the following transfer function: (46) Fig. 5. Magnitude response for the same number of multipliers in example 5. Equations (43) (45) and (47). It is evident here that the proposed method gives better performance in terms of magnitude response whereas the MPZ method yields a poor approximation, as shown in Fig. 4(a). Fig. 4(a) also shows that both the new approach and the leastsquares method approximate well the desired response with a better result for the least-squares method. However, it is evident by examining and that the latter has a higher number of numerator coefficients and thus has a higher number of multipliers ( ). On the other hand, has only seven multipliers, which is another advantage of the proposed method. Repeating the least-squares design by choosing only two multiplier coefficients for the numerator and five coefficients for the denominator reduces the overall number of multipliers from 11 to seven, yielding (47), shown at the bottom of the next page. Fig. 5 shows that the new approach has a better magnitude response than the least-squares method with the same number of multipliers. F. Example 6 In this example, we will consider the fourth-order Bessel filter (48) (42) (44) (45)

10 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 347 (a) Fig. 7. Magnitude response for the same number of multipliers in example 6. Equations (48), (49), and (51). The least-squares method yields (50), shown at the bottom of the page. In Fig. 6(a) and (b), it is clear that the least-squares method has a better magnitude response and phase response, respectively, than the proposed method for the same order filter, but it is clear that has a higher number of multipliers. We will repeat the design method for the same number of multiplier coefficients. The least-squares method yields (b) Fig. 6. (a) Magnitude response of the least-squares method and the new method in example 6. Equations (48) (50). (b) Phase response of the least-squares method and the new method in example 6. Equations (48) (50). We will compare the proposed method with the same order least-squares method. The proposed method yields (49) (51) In Fig. 7, the least-squares magnitude response reduces to a poor one when it is forced to have the same order of multipliers. The phase response also deteriorates and is omitted for brevity. This tr was noticed in the last two examples. V. CONCLUSION This paper presents a novel approach to discretization of analog systems, using a leaky integrator/differentiator concept. The new method subsumes the Tustin (bilinear) and the MPZ transformations. Significantly, the resulting new -totransforms are viable alternatives to the Tustin transform, the MPZ transforms, and the least-squares design methods, as was evidenced by the supporting examples. We can arrive at the following conclusions. (47) (50)

11 348 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 The new transforms are capable of providing a good approximation of the desired responses. They gave a better approximation to the analog filters than the bilinear and MPZ methods. In some applications, the proposed transforms gave better performance than the same order least-squares methods. In all of the applications, the proposed method gave better performance than the same order least-squares method with the same number of multipliers. The examples demonstrate the efficacy of the new transforms and that they could outperform other transforms. The new transforms should be considered as serious alternatives to the traditional transforms and the least-squares methods. The advantages of the new transforms stem from the fact that the approximations include, in a natural fashion, the poles and zeros of the transfer functions, and thus better approximations of the analog systems are obtained, and no prewarping is needed, as is evident from the included examples. The derivations of the transforms start from basic principles and employ the poles and zeros of the analog transfer function. Indeed, this is more logical than carrying out approximations on then placing the resulting approximations in factors of the form, where is an exact constant. Subjecting the possibly complex constant, to the same approximations as the complex variable yielded -to- transforms that could perform better than the traditional transforms. In other terms, the property of mapping poles and zeros to the exact positions given by the ideal frequency map confers to the new method its excellent frequency response matching ability. This is in contrast to the bilinear (Tustin) transform, which handles misplacement of poles and zeros through prewarping. Unlike the MPZ method, which maps zeros at to the point, zeros are provided naturally with every pole. APPENDIX The following is the corresponding MATLAB code of the procedure outlined in Section IV. function [zd; pd; kd] = alalaoui(z; p; k; fs; we) % ALALAOUI Al-Alaoui Novel Approach to %Analog-to-Digital Transorms. % %[Zd;Pd;Kd] = ALALAOUI(Z; P; K; F s; W e) converts %the s-domain transfer %function specified by Z, P, and K to a %z-transform discrete %equivalent obtained using the following %transformation: % %H(z) =H(s)j % 1 (s + a) = C 1+exp 3 0 a 3 z^ 0 1 Fs 1 0 exp 0 a 3 z 0 1 Fs %K specifies the gain, and Fs is the sample %frequency in hertz. %To %obtain the constant C, the digitized transfer %function s amplitude %should be equalized to the original analog %one at the equalization %frequency We. By default, We = 0, i.e., dc values are set equal. If %there is a zero or a pole at dc, a value at %the proximity is tried %out, by 0.1 rad/s decrements, until an %acceptable value is found. % %[NUMd; DENd] = ALALAOUI(NUM; DEN;Fs;We), %where NUM and DEN are %row vectors containing numerator and %denominator transfer %function coefficients, NUM(s)=DEN(s), in %descing powers of %s, transforms to z-transform coefficients %NUMd(z)=DENd(z). %N.B.: The gain k encountered throughout is %defined as the constant %multiplying a rational function where the %highest order coefficients %of both the numerator and denominator are %normalized to 1. [mn; nn] =size(z); [md; nd] =size(p); if (nd == 1 & nn < 2 ) % In zero-pole-gain form if mn > md error( Numerator cannot be higher order than denominator. ) elseif (md == 1 & mn == 1 ) % Transfer function case if nn > nd error( Numerator cannot be higher order than denominator. ) num = z; den = p; fs = k; z = roots(num); p = roots(den); k = num(1)=den(1); else error( First two arguments must have the same orientation. ) % %where column vectors Z and P %zeros and poles,scalar specify the num = poly(z); den = poly(p); if( exist(`we`) ) we =0;

12 AL-ALAOUI: NOVEL APPROACH TO ANALOG-TO-DIGITAL TRANSFORMS 349 while(sum(find(z == we))jsum(find(p == we)) ) wn = wn 0 0:1; % Do Al-Alaoui s-to-z transformation z = cplxair(z); p = cplxair(p); zi =1; pi =1; zd =[]; pd =[]; if (nd == 1 & nn < 2 ) % In zero-pole-gain form kd = C; elseif (md == 1 & mn == 1 ) % Transfer function case zd = C 3 numd; pd = d; % Decode values while(zi <= (length(z) 0 1) ) ACKNOWLEDGMENT if(z(zi) ==conj(z(zi + 1)) ) The author would like to thank Dr. A. H. Sayed for providing pd =[pd;0]; the atmosphere conducive to research through his invitation to zd =[zd;exp(z(zi)=fs); exp(z(zi +1)=fs)]; zi = zi +2; else sp the summer of 1999 in his Adaptive Systems Laboratory at UCLA. The author would also like to thank the graduate students T. Al-Naffouri, R. Merched, R. Wang, N. Yousef, J. Zhao, and Z. Zhao for their help during his stay at UCLA. The author would also like to thank the highly talented graduates of the pd =[pd; 0exp(z(zi)=fs)]; zd =[zd; exp(z(zi)=fs)]; Electrical and Computer Engineering Department at the American zi = zi +1; University of Beirut. It is indeed a pleasure to acknowledge G. Deeb, M. El Choubassi, R. Ferzli, B. Ghanem, K. Joujou, M. Ohanessian, A. Slim, and E. Yaacoub for their invaluable help in the production of this work. Thanks are also due to the if(zi == length(z) ) outstanding reviewers and the Associate Editor whose constructive comments contributed to the improvement of the paper. pd =[pd; 0exp(z(zi)=fs)]; zd =[zd; exp(z(zi)=fs)]; REFERENCES while(pi <= (length(p) 0 1) ) [1] F. G. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control if(p(pi) ==conj(p(pi + 1)) ) of Dynamic Systems. Reading, MA: Addison-Wesley, pd =[pd; exp(p(pi)=fs); exp(p(pi +1)=fs)]; [2] S. K. Mitra, Digital Signal Processing, 3rd ed. New York: McGraw- Hill, zd =[zd;0]; [3] A. V. Oppenheim, R. W. Schafe, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pi = pi +2; else [4] C. L. Philips and H. T. Nagle, Digital Control System Analysis and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1995, ch. 11. [5] J. G. Proakis and D. G. Manolakis, Introduction to Digital Signal Processing, zd =[zd; 0exp(p(pi)=fs)]; 3rd ed. Englewood Cliffs, New Jersey: Prentice-Hall, pd =[pd; exp(p(pi)=fs)]; [6] L. R. Rabiner and B. Gold, Theory and Applications of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, pi = pi +1; [7] K. Steiglitz, A Digital Signal Processing Primer: With Applications to Computer Music. Reading, MA: Addison-Wesley, [8] H. M. Yassine, General analog to digital transformation, Proc. Inst. Elect. Eng., vol. 133, no. 2, pt. G, pp , [9] M. A. Al-Alaoui, Novel digital integrator and differentiator, Electron. if(pi == length(p) ) zd =[zd; 0exp(p(pi)=fs)]; Lett., vol. 29, no. 4, pp , [10], Filling the gap between the bilinear and the backward difference transforms: An interactive design approach, Int. J. Elect. Eng. pd =[pd; exp(p(pi)=fs)]; Education, vol. 34, no. 4, pp , Oct [11] E. Gurova and V. I. Georgiev, Transformation of P-Z based on digital integration methods, Izvestia VUZ Radioelektronika, vol. 39, no. 4, pp. 3 18, numd = poly(zd); d = poly(pd); [12] M. A. Al-Alaoui, A class of second-order integrators and lowpass differentiators, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 4, pp , Apr [13], A class of numerical integration rules with first-order derivatives, % Determine C by equalization analog = k 3 polyval(num;we)=polyval(den;we); digital = polyval(numd; exp(j 3 we))=polyval(d; exp(j 3 we)); ACM SIGNUM Newsletter, vol. 31, no. 2, pp , Apr [14] R. J. Clark, E. C. Ifeachor, G. M. Rogers, and P. W. J. Van Eetvelt, Techniques for generating digital equaliser coefficients, J. Audio Eng. Soc., vol. 48, no. 4, pp , [15] E. C. Ifeachor and B. W. Jervis, Digital Signal Processing A Practical Approach, 2nd ed. Harlow, U.K.: Prentice-Hall,, [16] R. C. Boxer and S. Thaler, A simplified method of solving linear and C = analog=digital; nonlinear systems, Proc. IRE, vol. 44, pp , Jan

13 350 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 54, NO. 2, FEBRUARY 2007 [17] R. Boxer, Frequency analysis of computer systems, Proc. IRE, vol. 43, pp , Feb [18], A note on numerical transform calculus, Proc. IRE, vol. 45, pp , Oct [19] C. H. Wang, M. Y. Lin, and C. C. Teng, On the nature of the Boxer- Thaler and Madwed integrators and their applications in digitizing a continuous-time system, IEEE Trans. Autom. Control, vol. 35, no. 10, pp , Oct [20] M. A. Al-Alaoui, A stable differentiator with a controllable signal-to-noise ratio, IEEE Trans. Instrum. Meas., vol. IM-37, no. 4, pp , Sep [21], Novel approach to designing digital differentiators, Electron. Lett., vol. 28, no. 15, pp , [22], Novel IIR differentiator from the Simpson integration rule, IEEE Trans. Circuits Systems I, Fundam. Theory Appl., vol. 41, no. 2, pp , Feb [23], Novel stable higher order s-to-z transforms, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 11, pp , Nov [24] K. Steiglitz, Computer aided design of recursive digital filters, IEEE Trans. Audio Electroacoust., vol. AU-18, no. 2, pp , Jun [25] E. C. Levi, Complex-curve fitting, IRE Trans. Autom. Control, vol. AC-4, pp , [26] J. E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, Mohamad Adnan Al-Alaoui (S 70 M 83 SM 91) received the B.S. degree in mathematics from Eastern Michigan University, Ypsilanti, in 1963, the B.S.E.E. degree from Wayne State University, Detroit, MI, in 1965, and the M.S.E.E. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology (Georgia Tech), Atlanta, in 1968 and 1974, respectively. He served as an Assistant Project Engineer with the Avionics Department, Bix Radio Division, Baltimore, MD, from 1966 to At Georgia Tech, he held teaching assistant positions with the School of Electrical Engineering and the School of Mathematics. After receiving the Ph.D. degree, he joined the Electrical Engineering Department, Royal Scientific Society, Amman, Jordan, where he was responsible for the communications area. He served as an Assistant Professor with the Electrical Engineering Department, American University of Beirut (AUB), Beirut, Lebanon, from 1977 to He was a visiting Assistant and Associate Professor with the Electrical Engineering and Computer Science Department, University of Connecticut, Storrs, from 1980 to 1982 and in the summer of He served as an Associate Professor of Electrical Engineering with the Hartford Graduate Center, Hartford, CT, from 1983 to He was the Chairman of the Automatic Control Department at the Higher Institute for Applied Science and Technology, Damascus, Syria, from 1985 to In 1988, he rejoined AUB, where he is currently a Professor. He served as the Chair of the Department of Electrical and Computer Engineering at AUB from September 1, 2001, to August 30, His research interests are in neural networks and their applications and in analog and digital signal and image processing and their applications in biomedical engineering, communications, controls and instrumentation. He was a Visiting Scholar with Stanford University, University of Southern California, the University of California, Santa Barbara, and the University of California, Los Angeles. Dr. Al-Alaoui was the recipient of the First Research Award in Engineering for by AUB. He is currently serving as the first Chair of the newly established IEEE Lebanon Section.