936 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006

A New Approach for the Design of WidebandDigital Integrator and Differentiator

Nam Quoc Ngo, Member, IEEE

Abstract—This brief presents a general theory of theNewton–Cotes digital integrators which is derived by applyingthe -transform technique to the closed-form Newton–Cotes inte-gration formula. Based on this developed theory, a new widebandthird-order trapezoidal digital integrator is found to be a class oftrapezoidal digital integrators. The novel wideband third-ordertrapezoidal integrator accurately approximates the ideal inte-grator over the whole Nyquist frequency range and comparesfavourably with existing integrators. Based on the designed wide-band third-order trapezoidal integrator, a new wideband digitaldifferentiator is designed, which approximates the ideal differen-tiator reasonably well over the whole Nyquist frequency range andcompares favourably with existing differentiators. The low ordersand high accuracies of the novel wideband trapezoidal integratorand the new wideband differentiator make them attractive forreal-time applications.

Index Terms—Digital differentiator, digital filter, digitalintegrator.

I. INTRODUCTION

DIGITAL integrators and differentiators form an integralpart of many physical systems. Therefore, the design of

sufficient wideband integrators and differentiators is of consid-erable interest. The frequency response of an ideal integrator isgiven by

(1)

and the frequency response of an ideal differentiator is given by

(2)

where and is the angular frequency in radians. Dig-ital integrators and differentiators are normally designed basedon the definitions given in (1) and (2), respectively.

Recursive digital integrators have been designed by per-forming a simple linear interpolation between the magnituderesponses of the classical rectangular, trapezoidal and Simpsondigital integrators [1]–[3]. A linear-programming optimizationapproach has also been proposed to design recursive digitalintegrators [4]. However, these integrators are not widebandbecause their magnitude responses only approximate the mag-nitude response of the ideal integrator up to a fraction of thefullband Nyquist frequency range [1]–[4]. In [1], a first-orderintegrator can only approximate the ideal integrator well in the

Manuscript received August 16, 2005; revised February 9, 2006. This paperwas recommended by Associate Editor C.-T. Lin.

The author is with the School of Electrical and Electronic Engi-neering, Nanyang Technological University, Singapore 639798 (e-mail:eqnngo@ntu.edu.sg).

Digital Object Identifier 10.1109/TCSII.2006.881806

low and midband frequency ranges. References [3] and [4] havereported on the design of low-frequency second-order digitalintegrators. In this brief, an alternative approach is proposedfor the design of a novel wideband digital integrator.

Nonrecursive or finite-impulse response (FIR) digital differ-entiators with linear phase characteristics have been designedusing various techniques for low-frequency [5]–[9], midbandfrequency [10], high-frequency [11] and wideband [12]–[14]operations. The FIR wideband differentiators, however, have along filter length or high filter order [12]–[14]. In this brief, anew wideband recursive digital differentiator with a low filterorder is proposed.

In this brief, we present a general transfer function of theNewton–Cotes digital integrators, which is obtained by per-forming the -transform on the closed-form Newton–Cotesintegration formula. From this theory, we find a class of trape-zoidal digital integrators, in which a third-order trapezoidaldigital integrator is found to have its magnitude responseclosely match the magnitude response of the ideal integratorover the entire Nyquist frequency range. Using the designedwideband third-order trapezoidal integrator, a new widebanddigital differentiator is designed, which approximates the idealdifferentiator reasonably well over the entire Nyquist frequencyrange. The proposed integrator and differentiator comparefavourably with the existing integrators and differentiators, re-spectively. The brief is organized as follows. Section II presentsa general theory of the Newton–Cotes digital integrators, thedesign of the proposed integrator and performance compar-isons of the proposed integrator with the existing integrators.Section III describes the design of the proposed differentiatorand performance comparisons of the proposed differentiatorwith the existing differentiators. The conclusion is given inSection IV. Note that (the sampling period of the filter)is used in the frequency plots and the Nyquist frequency isradians.

II. PROPOSED INTEGRATOR AND COMPARISON

WITH EXISTING INTEGRATORS

A general transfer function of the th-order Newton–Cotesdigital integrator is shown in the Appendix to be given by

(3)

where is the feedback delay and is the-transform parameter. The th coefficient in (3) is given by

(4a)

1057-7130/$20.00 © 2006 IEEE

NGO: A NEW APPROACH FOR THE DESIGN OF WIDEBAND DIGITAL INTEGRATOR AND DIFFERENTIATOR 937

where the binomial coefficient is defined as

(4b)

The th difference equation in (3) is given by

(5)

When (3) (4)(5) are solved for certain values of and , (3)will take the form of

(6)

where is a real-valued coefficient of the feedforward section.For example, (where ), (where

), (where ) and (where) are, respectively, the well-known transfer functions

of the trapezoidal, Simpson’s 1/3 (or Simpson), Simpson’s 3/8and Boole’s integrators, and these integrators have poor approx-imation of the ideal integrator [1]–[4], [16]. By solving (3)(4)(5)for and , the transfer function of a new third-ordertrapezoidal digital integrator is given by (7), shown at the bottomof the page, which can be shown to be in the form of (6). By per-forming a linear interpolation between the magnitude responseof the rectangular integrator and the magnitude response of thetrapezoidal integrator, Al-Alaoui has proposed a first-order dig-ital integrator which is described by [1], [2]

(8)

Similarly, by performing a linear interpolation between the mag-nitude response of the Simpson integrator and the magnitude re-sponse of the trapezoidal integrator, Al-Alaoui has proposed aclass of second-order integrators which is described by [3]

(9)

where . Here we use because it gives a goodmagnitude response as described below. Using a linear-pro-gramming optimization technique, Papamarkos and Chamzashave designed a class of second-order digital integrators whichis described by [4]

(10)

Fig. 1 shows the magnitude responses of the ideal in-tegrator, the proposed third-order trapezoidal integrator,

, the Al-Alaoui’s first-order integrator, , theAl-Alaoui’s second-order integrator, , and the Papa-markos–Chamzas’s integrator, . It can be seen that the

Fig. 1. Magnitude responses of the ideal integrator, the proposed third-ordertrapezoidal integrator H (z), Al-Alaoui’s first-order integrator H (z),Al-Alaoui’s second-order integrator H (z), and the Papamarkos–Chamzas’sintegrator H (z).

Fig. 2. Absolute values of the percent relative errors of the magnitude re-sponses of the proposed third-order trapezoidal integratorH (z), Al-Alaoui’sfirst-order integrator H (z), Al-Alaoui’s second-order integrator H (z),and the Papamarkos–Chamzas’s integrator H (z).

proposed approximates the ideal integrator reasonablywell (to within 6.5% error, see Fig. 2) over the whole Nyquistfrequency range and may thus be regarded as a widebandintegrator.

(7)

938 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006

The absolute values of the percent relative errors of the mag-nitude responses of the integrators shown in Fig. 1 are shown inFig. 2. From Fig. 2, the proposed performs better than

over the whole Nyquist frequency range ofradian. Also, the proposed only performs better than

for radian and radian. In addi-tion, the proposed only performs better thanfor radian and radian. The phases ofand are exactly over the whole Nyquist frequencyrange. The maximum deviations of the phase responses of theproposed and from the ideal linear phase re-sponse are (which occurs at radian) and(which occurs at radian), respectively. Note that thelow order and high accuracy of the proposed makes itattractive for real-time applications.

III. PROPOSED DIFFERENTIATOR AND COMPARISON

WITH EXISTING DIFFERENTIATORS

Using the approach described in [17], we here describe the de-sign of a new type of recursive digital differentiator by invertingthe transfer function (see (7)) of the wideband third-order trape-zoidal digital integrator. In taking the inverse of (7), an unstablepole that lies outside the unit circle at is ob-tained. Replacing this unstable pole at by in-verting it to give a stable pole at and multi-plying the denominator by a factor of to compensate forthe amplitude, the resulting transfer function of the new designof a third-order recursive digital differentiator is given by (11),shown at the bottom of the page.

The transfer function of a “rectangular-trapezoidal”first-order differentiator proposed by Al-Alaoui is givenby [1]

(12)

Fig. 3 shows the magnitude responses of the ideal differen-tiator, the proposed differentiator, , and the Al-Alaoui’sdifferentiator, , and the absolute values of the rel-ative percent errors of the magnitude responses of these dif-ferentiators are shown in Fig. 4. It can be seen that the pro-posed differentiator approximates the ideal differentiator rea-sonably well (to within 5% error, see Fig. 4) over the entireNyquist frequency range and may thus be regarded as a wide-band differentiator. The proposed differentiator almost outper-forms the Al-Alaoui’s differentiator in the high-frequency rangeof radian, while the Al-Alaoui’s differentiator outper-forms the proposed differentiator for radian. It is notedthat the proposed differentiator also outperforms the followingAl-Alaoui’s second-order differentiators which are the “Tick”differentiator [17], the “Simpson” differentiator [18] and the

Fig. 3. Magnitude responses of the ideal differentiator, the proposed differen-tiator H (z), and Al-Alaoui’s differentiator H (z).

Fig. 4. Absolute values of the percent relative errors of the magnitude re-sponses of the proposed differentiator H (z) and Al-Alaoui’s differentiatorH (z).

“Simpson-trapezoidal” differentiator [3] because these differen-tiators are low-pass differentiators which can only approximatethe ideal differentiator up to radian. In addition,the new wideband differentiator outperforms the two-point dif-ference differentiator and the three-point central difference dif-ferentiator [5]. The new wideband differentiator also comparesfavourably with the 10-point low-pass differentiator reported byOppenheim and Schafer which has an error of 12% for a rangeof 0.8 of the Nyquist frequency range [6]. The maximum de-viations of the phase responses of the proposed differentiatorand the Al-Alaoui’s differentiator from the ideal linear phaseresponse are (which occurs at radian) and(which occurs at radian), respectively. Note that the

(11)

NGO: A NEW APPROACH FOR THE DESIGN OF WIDEBAND DIGITAL INTEGRATOR AND DIFFERENTIATOR 939

low order and high accuracy of the new wideband differentiatormakes it attractive for real-time applications.

IV. CONCLUSION

We have presented a general theory of the Newton–Cotesdigital integrators which has been derived by performing the

-transform on the closed-form Newton–Cotes integration for-mula. Using the developed theory, a new third-order trapezoidaldigital integrator has been found to be a class of the trapezoidaldigital integrators. The novel wideband third-order trapezoidalintegrator accurately approximates the ideal integrator over theentire Nyquist frequency range and compares favourably withthe existing integrators. Using the new design of the widebandthird-order trapezoidal integrator, a new wideband digital differ-entiator has been designed. The novel wideband digital differen-tiator approximates the ideal differentiator reasonably well overthe whole Nyquist frequency range and compares favourablywith the existing differentiators.

APPENDIX

DERIVATION OF A GENERAL THEORY OF THE

NEWTON–COTES DIGITAL INTEGRATORS

1) Definition of Numerical Integration: Assume that a con-tinuous-time signal is given and that its integral

(A1)

is to be determined from a sequence of samples of at thediscrete time where and

is the period between successive samples. Thus, (A1)can be written as

(A2)

in which the integration interval is divided into a numberof equal segments with each segment having a step size of .The underlying principle of the numerical integration algorithmis shown in Fig. 5.

From Fig. 5, (A2) can be divided into two integrals

(A3)

where represents the area of the hatched region of Fig. 5. The-transform of (A3) is given by

(A4)

where is the -transform of andis the -transform of . In (A4), is the

-transform parameter, where is the angular fre-quency, and is the sampling period of the integrator. in(A4) will be determined in the Appendix part 4.

Fig. 5. Graphical illustration of the numerical integration technique.

2) Newton’s Interpolating Polynomial: The discrete-timevariables in Fig. 5 are re-defined as and ,where and . Using these definitions,in (A3) becomes

(A5)

For the integration interval in (A5), the curvecan be approximated by the th-order Newton’s interpolatingpolynomial, which passes through data points, as [15]

(A6)

where

(A7)

and

(A8)

(A6) can be further simplified by defining a new quantity

(A9)

which is substituted into (A7) to give

(A10)

Substituting (A10) into (A6) results in

(A11)

3) General Form of the Newton–Cotes Closed IntegrationFormula: Substituting (A11) into (A5) results in

(A12)

940 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006

From (A7) and (A9), and the limits of integration in(A12) are changed from to and from to

. Substituting these parameters into (A12) results in

(A13)

where

(A14)

Using , (A8) can be further simplified to

(A15)

(A13)(A14)(A15) describe the general form of theNewton–Cotes closed integration formula.

4) General Theory of the Newton–Cotes Digital Integrators:Taking the -transform of (A13) results in

(A16)

where is the -transform of (A15), which is given by

(A17)

where is the -transform of . Equation (A17)can be recognized as

(A18)

where

(A19)

Substituting (A18) into (A16) gives

(A20)

Substituting (A20) into (A4), the th-order transfer function,, of the Newton–Cotes digital integrator

can be generally described by (3).

ACKNOWLEDGMENT

The author is thankful to the reviewers for their constructivecriticisms of the work and useful suggestions for improving themanuscript.

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