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| ARTICLE |
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| Year : 2012 | Volume
: 58
| Issue : 2 | Page : 166-170 |
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Wideband Digital Integrator and Differentiator
Maneesha Gupta1, Madhu Jain2, B Kumar3
1 Department of Electronics and Communication Engineering, Advanced Electronics Lab, Netaji Subhas Institute of Technology, Sector-3, Dwarka, India
2 Department of Electronics and Communication Engineering, Jaypee Institute of Information Technology, A-10, Sector 62, Noida, UP, India
3 Department of Electronics and Communication Engineering, Bhagwan Parshuram Institute of Technology, Rohini, New Delhi, India
| Date of Web Publication | 16-May-2012 |
Correspondence Address:
Maneesha Gupta
Department of Electronics and Communication Engineering, Advanced Electronics Lab, Netaji Subhas Institute of Technology, Sector-3, Dwarka
India
 DOI: 10.4103/0377-2063.96183
 Abstract | | |
Novel designs of third-order recursive wideband digital integrator and differentiator are presented. The integrator is obtained by interpolating two digital integration techniques, the Schneider-Kaneshige-Groutage (SKG) rule and the Trapezoidal rule. The differentiator is obtained by inverting the transfer function of the designed integrator after necessary modifications. The proposed integrator and differentiator approximate their ideal counterparts with absolute magnitude errors less than 0.02 and 0.14, respectively, over the entire frequency spectrum. Their performance compares favorably with existing integrators and differentiators. Keywords: Digital differentiator, Digital integrator, Schneider-Kaneshige-Groutage integration rule, Trapezoidal integration rule
How to cite this article:
Gupta M, Jain M, Kumar B. Wideband Digital Integrator and Differentiator. IETE J Res 2012;58:166-70 |
| 1. Introduction | |  |
Linear interpolation is a popular technique used in designing various types of digital integrators [1],[2],[3],[4] . A linear-programming optimization approach has also been proposed for designs of recursive digital integrators [5] . A digital integrator has been suggested by using Newton-Cotes integration formula [6] . Integrators proposed in [4],[6] are wideband while the integrators suggested in [1],[2],[3],[5] are not wideband, and their magnitude response can only approximate that of the ideal integrator for a fraction of the full band Nyquist frequency range.
Low-frequency recursive digital integrators have been proposed in [1],[2],[3] and [5] , the one suggested in [1] can also be used for midband frequency. Wideband recursive digital integrator has been suggested in [6] . In this paper, an interpolation approach has been proposed for the design of a third-order digital integrator, suitable for wideband frequency range.
Various linear phase nonrecursive differentiators suitable for low, mid, high, and wideband frequency ranges have also been suggested in [7],[8],[9],[10],[11],[12],[13] and [14],[15],[16] . These FIR wideband differentiators are of high filter order. We, in our design method, suggest a new stable wideband recursive digital differentiator with low filter order.
In this paper, a novel wideband digital integrator has been arrived at by performing a simple linear interpolation between the two popular digital integration techniques, viz. the Schneider-Kaneshige-Groutage (SKG) and the trapezoidal rule [1-3,17]. We observe that the response of the ideal integrator lies between the responses of the SKG and that of the trapezoidal rules type integrators [Figure 1]. Thus, it seems reasonable that interpolating the responses of above two integrators should yield a new integrator that could better approximate the ideal integrator. We shall show that the resulting integrator indeed outperforms these two integrators and accurately approximates the ideal integrator over the entire Nyquist frequency range. | Figure 1: Magnitude response of the ideal, the SKG, and the Trapezoidal rule integrators. Note that the ideal magnitude response lies in-between the magnitude responses of the other two.
Click here to view |
The design procedure for digital differentiator involves first obtaining an integrator and then modifying its transfer function (TF) appropriately to obtain the wideband stable differentiator. As will be shown, the proposed integrator and differentiator outperform the existing ones. The low-order and low-approximate errors of the suggested wideband integrator and differentiator makes them attractive for real-time applications. The paper is organized as follows: Section 2 presents design of the proposed integrator and its comparison with the existing integrators. Section 3 gives the design of the proposed differentiator and its performance. The conclusions are given in Section 4. Note that the sampling period "T" of the filter is normalized to unity (i.e., T = 1) for the frequency plots and the Nyquist frequency is taken as radians.
| 2. Design Of Integrator and Its Performance | | |
Consider the TF of the integrator as
where,, H SKG (z), H T (z), and H N (z) are the TFs of the SKG integrator [17] , the trapezoidal integrator [1],[2],[3] , and the proposed integrator, respectively, with
The proposed integrator was simulated for various values of taking the increment of 0.01. Minimum absolute magnitude error (0.02) is achievable for =0.95. Putting =0.95 in (1) and using (2)-(3), we obtain
Nam Quoc Ngo [6] has suggested a third-order digital integrator by applying the z-transform technique to the closed-form Newton-Cotes integration formula, and its TF is
Al-Alaoui [4] has recently proposed a class of integrators by interpolating the Newton-Cotes integration rules and then optimizing some of these operators. Some of these integrators are as follows:
Al-Alaoui's 2-Segment digital integrator has the TF
Al-Alaoui's optimized 3-Segment digital integrator has the TF
Al-Alaoui's optimized 4-Segment digital integrator has the TF
[Figure 2] shows the magnitude response and [Figure 3] gives the absolute magnitude error. It can be seen from [Figure 3] that H N (z) approximates the ideal integrator reasonably well (with absolute magnitude error ≤ 0.02) over the entire Nyquist frequency range of 0 ≤≤ radian and may thus be regarded as a wideband integrator. It can be observed that the proposed integrator H N (z) performs better than the Al-Alaoui's 2-Segment and Al-Alaoui's optimized 3-Segment integrators over the entire Nyquist frequency range. Also, H N (z) performs better than the Ngo's integrator over 0≤≤1.2 radian and 2.3≤≤ radian and it performs better than Al-Alaoui's optimized 4-Segment integrator for 0≤≤0.09 radian and 2.3≤≤ radian. The maximum deviation of the phase response of the proposed H N (z) from the ideal linear phase response is 35.3 o (which occurs at =1.87 radian). Note that the low order and high accuracy of the proposed wideband Integrator H N (z) makes it attractive for real-time applications. | Figure 2: Magnitude responses of the ideal integrator, the proposed third-order integrator HN(z), Ngo's integrator H1(z), Al-Alaoui's 2-Segment, optimized 3-Segment and optimized 4-Segment integrators, H2(z), H3(z), and H4(z), respectively. The ideal and the proposed integrator's response curves are indistinguishably close in the entire frequency range.
Click here to view |
 | Figure 3: Absolute magnitude errors of the proposed thirdorder integrator HN(z), Ngo's integrator H1(z), Al-Alaoui's 2-Segment, optimized 3-Segment and optimized 4-Segment integrators, H2(z), H3(z), and H4(z), respectively.
Click here to view |
| 3. Design of Proposed Differentiator and Its Performance | | |
A new differentiator has been obtained by inverting the TF H N (z), of the proposed wideband digital integrator and using the approach described in [18] for stability. If we take the inverse of H N (z), a pole appears outside the unit circle at z=−2.285. Replacing this undesired pole by inverting it to give a stable pole at z=−1/2.285 and multiplying the denominator by a factor of 2.285 to compensate for the change in amplitude, the resulting TF of the proposed recursive, stable digital differentiator is
Now, we give the TFs of some of the popular differentiators available in the literature.
The TF of the Nam Quoc Ngo's differentiator, given
in [6] is
The TF of the Al-Alaoui's 2-Segment digital differentiator [4] is
The TF of the Al-Alaoui's optimized 3-Segment digital differentiator [4] is
The TF of the Al-Alaoui's 4-Segment digital differentiator [4] is
The TF of the Pei-Hsu's digital differentiator [19] is
[Figure 4] shows the magnitude response and [Figure 5] gives the absolute magnitude error for the aforemen-tioned differentiators. It can be readily seen that the proposed H Ndiff (z) approximates the ideal differentiator reasonably well (with absolute magnitude error ≤ 0.14) over the entire Nyquist frequency range and may thus be regarded as a wideband stable differentiator. It is observed that the proposed H Ndiff (z) performs better than the Al-Alaoui's 2-Segment differentiator and the Al-Alaoui's 4-Segment differentiator over the entire Nyquist frequency range. It also performs better than the Ngo's differentiator for most of the Nyquist frequency range and it performs better than the Al-Alaoui's optimized 3-Segment digital differentiator for 0≤≤0.75 radian and 3.09≤ ≤ radian. Also, it performs better than the Pei-Hsu's differentiator for 0 ≤ ≤ 1.19 radian and 3.04 ≤≤ radian. | Figure 5: Absolute magnitude errors of the proposed third-order differentiator HNdiff (z), Ngo's differentiator
H1diff(z), Al-Alaoui's 2-Segment differentiator H2diff(z), Al- Alaoui's optimized 3-Segment differentiator H3diff(z), Al- Alaoui's 4-Segment differentiator H4diff(z), and Pei-Hsu's differentiator H5diff (z).
Click here to view |
 | Figure 4: Magnitude responses of the ideal differentiator, the proposed third-order differentiator HNdiff (z),
Ngo's differentiator H1diff(z), Al-Alaoui's 2-Segment differentiator H2diff(z), Al-Alaoui's optimized 3-Segment
differentiator H3diff(z), Al-Alaoui's 4-Segment differentiator H4diff (z), and Pei-Hsu's differentiator H5diff
(z). The ideal and the proposed differentiator's response curves are indistinguishably close in the entire frequency range.
Click here to view |
The maximum deviation of the phase response of the proposed H Ndiff (z) from the ideal linear phase response is 12° (which occurs at =2.23 radian). The proposed digital integrator and differentiator can be exploited to obtain new s-to-z transformation for integer orders (i.e., Non-fractional z). Note that the low order and high accuracy of the proposed wideband stable differentiator H Ndiff (z) makes it useful for real-time applications.
| 4. Conclusion | | |
A novel digital integrator has been proposed by interpolating the two popular digital integration techniques, the SKG rule and the trapezoidal rule. The proposed wideband third-order integrator H N (z) accurately approximates the ideal integrator over the entire Nyquist frequency range (with absolute magnitude error ≤ 0.02) and compares favorably with the existing integrators. By modifying its TF appropriately, a stable differentiator H Ndiff (z) has been obtained. The suggested wideband third-order differentiator also accurately approximates the ideal differentiator over the entire Nyquist frequency range (absolute magnitude error≤0.14) and compares favorably with the existing differentiators. The low orders and high accuracies of the novel wideband integrator and differentiator make them particularly useful for real-time applications.
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| Authors | | |
 Maneesha Gupta received B.E. and M.E. in Electronics & Communication Engineering from Government Engineering College, Jabalpur in 1981 and 1983 respectively and Ph.D. in Electronics Engineering from Indian Institute of Technology, Delhi in 1990. She is currently working as Professor in Electronics & Communication Engineering Department of Netaji Subhas Institute of Technology, New Delhi since 2000. Her teaching and research interests are Switched Capacitors Circuits and Analog Signal processing. She has co-authored over 20 research papers in the above areas in various international/ national journals and conferences.
 Madhu Jain received B.E. degree in Electronics and Communication Engineering with Honours, topping from University of Rajasthan in 2003. She obtained P.G diploma in Embedded System Design from University of Pune in 2004 and M. Tech. in Signal Processing from Netaji Subhas Institute of Technology, New Delhi in 2009. She is pursuing Ph. D. in IIT Delhi and currently working as Sr. Lecturer in Electronics & Communication Engineering Department of Jaypee Institute of Information Technology, Noida since January 2012. Her teaching and research interests are Digital Signal Processing, Signal System, Circuit System and Embedded System. She has co-authored 6 research papers in the above areas in various international journals and conferences.
 B. Kumar received his B.A. (Hons.) from Punjab University in 1959. He obtained his B.Sc. Engg. (Hons.) from Punjab Engineering College, Chandigarh in 1963. He obtained M.Tech. (Radar Engg.) and Ph.D in 1972 and 1988, respectively from IIT Delhi. He worked as the director of Netaji Subhas Institute of Technology, Guru Premsukh Memorial College of Engineering, and HMR. He was with Maharaja Surajmal Institute of Technology from 2006-2009. He is currently working as Addl. Director in Bhagwan Parshuram Institute of Technology, Delhi since 2009. His area of interest is Microelectronics, Signals & Systems, Digital Signal Processing, Communication, and Radar Engineering. Prof. Kumar has published more than seventy technical papers in national and international journals of repute. Two students have been awarded Ph.D. degree under his guidance, and two more are pursuing Ph.D. under his guidance.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
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