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

A New Approach for the Design of WidebandDigital Integrator and Differentiator

Nam Quoc Ngo, Member, IEEE

AbstractThis brief presents a general theory of theNewtonCotes digital integrators which is derived by applyingthe -transform technique to the closed-form NewtonCotes 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 orders

and high accuracies of the novel wideband trapezoidal integratorand the new wideband differentiator make them attractive forreal-time applications.

Index TermsDigital differentiator, digital filter, digitalintegrator.

I. INTRODUCTION

DIGITAL integrators and differentiators form an integral

part 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 is

given 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 based

on the definitions given in (1) and (2), respectively.

Recursive digital integrators have been designed by per-

forming a simple linear interpolation between the magnitude

responses of the classical rectangular, trapezoidal and Simpson

digital integrators [1][3]. A linear-programming optimizationapproach has also been proposed to design recursive digital

integrators [4]. However, these integrators are not wideband

because their magnitude responses only approximate the mag-

nitude response of the ideal integrator up to a fraction of the

fullband Nyquist frequency range [1][4]. In [1], a first-order

integrator 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:[email protected]).

Digital Object Identifier 10.1109/TCSII.2006.881806

low and midband frequency ranges. References [3] and [4] have

reported on the design of low-frequency second-order digital

integrators. In this brief, an alternative approach is proposed

for the design of a novel wideband digital integrator.

Nonrecursive or finite-impulse response (FIR) digital differ-

entiators with linear phase characteristics have been designed

using various techniques for low-frequency [5][9], midband

frequency [10], high-frequency [11] and wideband [12][14]

operations. The FIR wideband differentiators, however, have a

long filter length or high filter order [12][14]. In this brief, a

new wideband recursive digital differentiator with a low filter

order is proposed.In this brief, we present a general transfer function of the

NewtonCotes digital integrators, which is obtained by per-

forming the -transform on the closed-form NewtonCotes

integration formula. From this theory, we find a class of trape-

zoidal digital integrators, in which a third-order trapezoidal

digital integrator is found to have its magnitude response

closely match the magnitude response of the ideal integrator

over the entire Nyquist frequency range. Using the designed

wideband third-order trapezoidal integrator, a new wideband

digital differentiator is designed, which approximates the ideal

differentiator reasonably well over the entire Nyquist frequency

range. The proposed integrator and differentiator comparefavourably with the existing integrators and differentiators, re-

spectively. The brief is organized as follows. Section II presents

a general theory of the NewtonCotes digital integrators, the

design of the proposed integrator and performance compar-

isons of the proposed integrator with the existing integrators.

Section III describes the design of the proposed differentiator

and performance comparisons of the proposed differentiator

with the existing differentiators. The conclusion is given in

Section IV. Note that (the sampling period of the filter)

is used in the frequency plots and the Nyquist frequency is

radians.

II. PROPOSED INTEGRATOR AND COMPARISON

WITH EXISTING INTEGRATORS

A general transfer function of the th-order NewtonCotes

digital 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)

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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, Simpsons 1/3 (or Simpson), Simpsons 3/8

and Booles 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 givenby (7), shown at the bottom

of the page, which can be shown to be in the form of(6). By per-

forming a linear interpolation between the magnitude response

of the rectangular integrator and the magnitude response of the

trapezoidal 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 good

magnitude response as described below. Using a linear-pro-

gramming optimization technique, Papamarkos and Chamzas

have designed a class of second-order digital integrators which

is described by [4]

(10)

Fig. 1 shows the magnitude responses of the ideal in-

tegrator, the proposed third-order trapezoidal integrator,

, the Al-Alaouis first-order integrator, , the

Al-Alaouis second-order integrator, , and the Papa-

markosChamzass integrator, . It can be seen that the

Fig. 1. Magnitude responses of the ideal integrator, the proposed third-ordertrapezoidal integrator

H ( z )

, Al-Alaouis first-order integrator H ( z ) ,Al-Alaouis second-order integrator H ( z ) , and the PapamarkosChamzassintegrator

H ( z )

.

Fig. 2. Absolute values of the percent relative errors of the magnitude re-sponses of the proposed third-order trapezoidal integrator

H ( z )

, Al-Alaouisfirst-order integrator H ( z ) , Al-Alaouis second-order integrator H ( z ) ,and the PapamarkosChamzass integrator H ( z ) .

proposed approximates the ideal integrator reasonably

well (to within 6.5% error, see Fig. 2) over the whole Nyquist

frequency range and may thus be regarded as a wideband

integrator.

(7)

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The absolute values of the percent relative errors of the mag-

nitude responses of the integrators shown in Fig. 1 are shown in

Fig. 2. From Fig. 2, the proposed performs better than

over the whole Nyquist frequency range of

radian. 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 of

and are exactly over the whole Nyquist frequency

range. The maximum deviations of the phase responses of the

proposed and from the ideal linear phase re-

sponse are (which occurs at radian) and

(which occurs at radian), respectively. Note that the

low order and high accuracy of the proposed makes it

attractive 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 inverting

the transfer function (see (7)) of the wideband third-order trape-

zoidal digital integrator. In taking the inverse of (7), an unstable

pole 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 for

the amplitude, the resulting transfer function of the new design

of 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 given

by [1]

(12)

Fig. 3 shows the magnitude responses of the ideal differen-

tiator, the proposed differentiator, , and the Al-Alaouis

differentiator, , 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 entire

Nyquist frequency range and may thus be regarded as a wide-

band differentiator. The proposed differentiator almost outper-

forms the Al-Alaouis differentiator in the high-frequency range

of radian, while the Al-Alaouis differentiator outper-

forms the proposed differentiator for radian. It is noted

that the proposed differentiator also outperforms the following

Al-Alaouis 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-Alaouis 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-Alaouis differentiatorH ( z )

.

Simpson-trapezoidal differentiator [3] because these differen-

tiators are low-pass differentiators which can only approximate

the 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 compares

favourably with the 10-point low-pass differentiator reported by

Oppenheim and Schafer which has anerror of 12% for a range

of 0.8 of the Nyquist frequency range [6]. The maximum de-

viations of the phase responses of the proposed differentiator

and the Al-Alaouis differentiator from the ideal linear phase

response are (which occurs at radian) and

(which occurs at radian), respectively. Note that the

(11)

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

low order and high accuracy of the new wideband differentiator

makes it attractive for real-time applications.

IV. CONCLUSION

We have presented a general theory of the NewtonCotes

digital integrators which has been derived by performing the

-transform on the closed-form NewtonCotes integration for-

mula. Using the developed theory, a new third-order trapezoidal

digital integrator has been found to be a class of the trapezoidal

digital integrators. The novel wideband third-order trapezoidal

integrator accurately approximates the ideal integrator over the

entire Nyquist frequency range and compares favourably with

the existing integrators. Using the new design of the wideband

third-order trapezoidal integrator, a new wideband digital differ-

entiator has been designed. The novel wideband digital differen-

tiator approximates the ideal differentiator reasonably well over

the whole Nyquist frequency range and compares favourably

with the existing differentiators.

APPENDIX

DERIVATION OF A GENERAL THEORY OF THE

NEWTONCOTES 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 number

of equal segments with each segment having a step size of .

The underlying principle of the numerical integration algorithm

is shown in Fig. 5.

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

(A3)

where represents the area of the hatched region ofFig. 5. The

-transform of (A3) is given by

(A4)

where is the -transform of and

is 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) Newtons Interpolating Polynomial: The discrete-time

variables in Fig. 5 are re-defined as and ,

where and . Using these definitions,

in (A3) becomes

(A5)

For the integration interval in (A5), the curve

can be approximated by the th-order Newtons interpolating

polynomial, 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 NewtonCotes Closed Integration

Formula: Substituting (A11) into (A5) results in

(A12)

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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 the

NewtonCotes closed integration formula.

4) General Theory of the NewtonCotes 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 NewtonCotes digital integrator

can be generally described by (3).

ACKNOWLEDGMENT

The author is thankful to the reviewers for their constructive

criticisms of the work and useful suggestions for improving the

manuscript.

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