Analysis and Design of Digital IIR Integrators and Differentiators Using Minimax and Pole , Zero , and Constant Optimization Methods

1 Department of Electronics and Communication Engineering, Jaypee Institute of Information Technology, A-10, Sector 62, Noida, Uttar Pradesh 201307, India 2 Advanced Electronics Lab, Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, Sector 3, Dwarka, New Delhi 110075, India 3 Instrument Design and Development Centre, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India


Introduction
Digital integrators and differentiators are integral parts of many systems like digital signal processing, control, audio, and video processing, communication, and medical applications.Frequency response of an ideal digital integrator is 1/ and of ideal digital differentiator is , where  = √ − 1 and  is the angular frequency in radians/second.Due to increasing number of applications involving digital signal processing, variety of requirements that have to be met by digital integrators and differentiators have increased as well.Digital integrators and differentiators can be classified as finite impulse response (FIR) and infinite impulse response (IIR), respectively.For a large number of applications, where high selectivity and efficient processing of digital signal are required, IIR digital integrators and differentiators are preferred.
Optimization [14][15][16][17][18][19][20][21][22][23] is a very popular design method.It is widely used to improve the performance of a system by reducing its runtime, bandwidth, memory requirement, or any other property.Optimization methods such as linear, simulated annealing, genetic algorithm, and pole-zero optimization have been used earlier to design IIR digital ISRN Electronics integrators and differentiators [14][15][16][17][18][19].Papamarkos and Chamzas [14] have used linear programming optimization method to design digital integrators.Al-Alaoui [15] has also proposed a family of digital integrators by using interpolation and simulated annealing optimization method.Upadhyay and Singh [16] have used pole-zero optimization method in design of digital integrators.Jiang and Kwan [17] have used semidefinite programming (SDP) relaxation algorithm to design IIR digital filters in the minimax sense.Lai and Lin [18] have used a sequential constrained least-squares method to design IIR digital filters in the minimax sense.Hsu et al. [19] have used genetic algorithm optimization method in the design of digital integrators.
Al-Alaoui [15,24] has proposed a method to design IIR digital differentiators, based on the inversion and magnitude stabilization of the transfer function of IIR digital integrators.This technique also became very popular and was used by various researchers like Bihan [1], Ngo [6], Gupta et al. [8][9][10], and Upadhyay and Singh [16] in the design of digital differentiators.The accuracy and range in terms of percentage absolute relative error (PARE) of the magnitude response of the designed differentiators are the same as those of integrators.
In this proposed work, a family of stable IIR digital integrators is designed via the use of minimax [17,18] and pole, zero and constant (PZC) optimization methods.The combination of two optimization techniques in a design is previously used by Renders and Flasse and Wu et al. [20,21].Renders and Flasse [20] have used genetic algorithm with hill-climbing optimization in order to generate a global optimization algorithm and Wu et al. [21] have used particle swarm with cultural algorithm to enhance the convergence characteristics of the original particle swarm optimization.
Here, minimax optimization method is used to design a family of second-, third-, and fourth-order digital integrators by optimizing the magnitude response in a min-max sense under the satisfactory condition of constant group delay.Then the PZC optimization is applied on minimax-optimized integrators to further improve frequency (magnitude and phase) response.This design strategy is motivated by the fact that there is a probability that minimax optimization does not provide global optimum solution so the PZC optimization is used over the result of minimax optimization to greatly increase the probability of finding the true global optimum solution.Thereafter, by modifying the transfer function of designed integrators appropriately, new second-, third-, and fourth-order digital differentiators are obtained which have the same accuracy as the designed integrators.These designed wideband operators (integrators and differentiators) outperform all the existing wideband operators in both magnitude and phase response.
All the figures in this paper have been obtained by using MATLAB 7.This paper is organized as follows.Section 2 presents the problem formulation.Section 3 presents the solution methodology consisting of minimax and pole, zero, and constant (PZC) optimization.Section 4 contains performance results by minimax and PZC optimization and comparison of the designed integrators with existing integrators.Section 5 presents the proposed differentiators and their comparison with the existing differentiators.The conclusions are given in Section 6.

Problem Formulation
It is known that IIR integrators have much better magnitude response than FIR integrators of the same order but their phase characteristics are not linear which can cause problems in some of the signal processing applications.In direct design method for IIR integrators, the desired digital transfer function is obtained directly from the given magnitude, with or without a specified phase specification.Let () be the transfer function of an IIR digital integrator with real coefficients (  ,   ); then where  is the order of the integrator.() can also be defined as where superscript  is used to define transpose.The frequency response (  ) of the integrator can be obtained from its transfer function () by simply evaluating it on the unit circle; that is, Here, |(  )| is the amplitude and () is the phase of the digital integrator, respectively.
Group delay is a measure of phase distortion; it is calculated by differentiating phase with respect to frequency.The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant.
The group delay of integrator (  ) is defined as The main limitation in the design of IIR digital integrators is to meet the specified magnitude and phase characteristics.An integrator should have linear phase response over the wideband frequency range; otherwise it would introduce phase distortion.Various researchers have been trying to improve the performance of digital integrators but a common difficulty faced by them is that when the magnitude response is improved, the phase response gets distorted and vice versa.

Solution Methodologies
In this paper, a family of second-, third-, and fourth-order stable wideband digital integrators are designed via use of minimax and pole, zero, and constant optimization.The flow chart of complete design method is shown in Figure 1.Integrators (2nd, 3rd, and 4th order) Very efficient IIR integrators (2nd, 3rd, and 4th order) Figure 1: Flowchart of the complete design method for digital integrator.

Design of IIR Digital Integrators Using Minimax Optimization.
A minimax is one of the optimization techniques in which the task is to find the solution with the minimum of the worst (maximum) value.It has been popularly used to find the unknown system parameters by minimizing the maximum difference between system output and design specifications [17,18].The flow chart of the applied minimax optimization for second-order digital integrator is shown in Figure 2.
The transfer function of an integrator with unknown coefficients is defined as (  ).A vector containing numerator and denominator coefficients is defined as  = [  ,   ]  .
In order to explicitly show the dependence of the frequency response of integrator (  ) on the numerator and denominator coefficients, the integrator is defined as ( The applied minimax optimization method can be defined as minimize In order to obtain exact approximation of the magnitude response, the PARE values should be as small as possible.For the case of second-order integrator ( = 2), the minimum (  ), maximum (  ), and initial value ( 0 ) of coefficients are defined as follows The lengths of the coefficients are taken as 1 integer and 4 decimal points.The same algorithm is also applied for 1 integer 3 decimal and 1 integer 5 decimal length but the difference in the efficiency is negligible.
The constraint of linear phase is applied by restricting the maximum value of absolute group delay (  (, )) of integrator (  , ) to be less than 5 samples.The frequency range is defined as 0 ≤  ≤  radians/second.
The internal function of minimax optimization in MAT-LAB 7 (fminimax) is used to obtain the coefficients for minimum of maximum PARE(, ) over the defined frequency range.
The following command is used for this purpose: where  is the vector of coefficients having minimum of maximum PARE for the defined frequency range.PAREval and maxPARE are the vectors of PARE values over the defined frequency range and maximum among these corresponding to coefficients .
Four blank square brackets are used to define that there are no linear equalities and inequalities.fminimax function works in following manner.
In first iteration, a set of coefficients are selected; then the values of objective function (PARE) are calculated by using (5), corresponding to these coefficients.The selection criteria of the coefficients consist of two points: first the value of coefficient should be within defined minimum and maximum bound, and the absolute value of group delay of the corresponding integrator should be less than 5 samples.In next iteration, another set of coefficients are selected and PARE values are calculated.This process is repeated till suitable coefficients are obtained.After last iteration, maximum PARE of each iteration is compared and coefficients corresponding  to minimum among maximum PARE are the desired coefficients.
After this, frequency range is defined as 0 ≤  ≤ 0.99  radians/second, by reducing maximum frequency by 0.01  radians/second and again coefficients are obtained for minimum of maximum PAREs using the command defined in (7).This process is repeated eleven times (0 ≤  ≤  radians/second to 0 ≤  ≤ 0.90  radians/second), so that designed integrators can be considered as wideband.In the last, all eleven sets of coefficients are listed in a table along with mean and maximum PARE of the corresponding integrator.By comparison, the optimum coefficients are the one having minimum among maximum PAREs.By substituting these coefficients in (2), a second-order digital integrator is obtained having magnitude response close to the ideal one and almost linear phase response.
The same algorithm is used to design third-and fourthorder digital integrators.In order to improve the optimization result, PZC optimization method is applied on the result of minimax optimization.

Pole, Zero, and Constant (PZC) Optimization Method.
The location of poles and zeros of the transfer function is very important for digital system analyses and synthesis.According to the location of poles, it is possible to test stability of the system.Variation of poles and zeros has a significant effect on the response of a design.The available literature [16,22,23] shows that pole-zero optimization is popular among researchers to improve their responses in various fields.The basic general strategy of designing frequency selective filters is also based on pole-zero placements.As placing a pole near the frequency  0 on the unit circle will increase the gain of the frequency response near  0 , similarly, a zero near  0 will diminish the gain near  0 .Here, as all the three parameters, namely, poles, zeros, and constant, are optimized, the results show tremendous improvement as compared to the previously proposed operators, where only poles and zeros have been optimized.
To obtain the poles, zeros, and constant of an integrator (), the numerator and denominator polynomials of (1) are rewritten as or The numerator and denominator can be factorized and expressed as where  =  0 is a scaling constant,   are the zeros, and   are the poles of integrator (), respectively.For PZC optimization, the modified transfer function   () can be written as The amplitude of the modified digital integrator   (  ) is PARE of integrator   (  ) is defined as Phase response of the modified digital integrator is defined as Equations ( 12)- (14) show that the optimization parameters Δ  , Δ  , and Δ  affect not only magnitude but phase response also.
The group delay of the modified integrator   (  ) is defined as Various steps used in the design of PZC optimized second-order IIR digital integrator are as follows.
(i) First of all, minimax optimized second-order integrator is defined in terms of poles, zeros, and constant.
(ii) Percentage absolute relative error PARE() as defined in ( 13) is specified as an objective function.
(iv) The step size is taken as +0.0001.
(v) All the optimization parameters are varied for 21 values, and those which gives minimum PARE are defined as optimum parameters.
(vi) The above mentioned process is repeated for thirdand fourth-order minimax optimized integrators.
In this way PZC optimized second-, third-, and fourth-order wideband digital integrators are obtained.By substituting the above mentioned optimum coefficients in (2), the resulting transfer functions of second-, third-, and fourth-order digital integrators ( 2MO (),  3MO (), and  4MO ()) are obtained as

Results after Minimax and Pole, Zero, and Constant Optimization Methods
For comparison, various existing integrators have been considered.These are Ngo integrator ( N ()) [6] [15], and Upadhyay and Singh integrator ( US ()) [16].Their transfer functions are Here the sampling period of the integrator "  " is assumed as 1 second and the Nyquist frequency is  radians/sample.The PARE response of the designed integrators and above mentioned integrators over Nyquist frequency range is shown in Figure 3.The useful frequency range, mean, and maximum PAREs in that frequency range for above mentioned integrators are shown in Table 4.The group delay response of the above mentioned integrators is shown in Figure 4.
It is verified from Figures 3 and 4 and Table 4 that the minimax optimized designed second-, third-, and fourthorder integrators outperform all the existing integrators over entire Nyquist frequency range.
By substituting all these values in (19), respectively, the new integrators are obtained as All the designed integrators (( 18) and ( 20)) are stable as their respective poles are inside the unit circle.The PARE response of the designed integrators  2MO (),  3MO (),  4MO (),  2MO+PZC (),  3MO+PZC (), and  4MO+PZC () is shown in Figure 5.The comparison of designed and existing integrators on the basis of mean PARE, maximum PARE and useful frequency range is shown in Table 4.
The square wave response of designed and existing integrators is shown in Figures 7 and 8, respectively.It is seen that the response of designed integrators  2MO (),  3MO (),  4MO (),  2MO+PZC (),  3MO+PZC (), and  4MO+PZC () is exact triangular wave, while the response of existing integrators is distorted triangular wave.

Design of IIR Digital Differentiators and Their Comparison with the Existing Differentiators
A digital differentiator can also be designed by using transfer function of digital integrator in a similar way to that used in the design of analog differentiator, as suggested by Al-Alaoui [24].This method consists of four design steps.In the first step, an integrator is designed that has the same range  [10], Al-Alaoui  A () [15], and Upadhyay and Singh integrator  US () [16].and accuracy as the desired differentiator.Then, the transfer function of this integrator is inverted and stabilized by reflecting the poles that lie outside the unit circle to inside the unit circle, and in the last step, the magnitude is compensated appropriately.
The PARE response of the designed differentiators  2MO (),  3MO (),  4MO (), and above mentioned existing differentiators (for   = 1) is shown in Figure 9.The comparison of designed and existing differentiators on the basis of mean PARE, maximum PARE, and useful frequency range is shown in Table 5.The group delay response of the designed differentiators  2MO (),  3MO (),  4MO (), and above mentioned existing differentiators is shown in Figure 10.It is seen that the maximum deviation from constant group delay in case of  2MO (),  3MO (), and  4MO () are 0.5 samples, 0.8 samples and 0.4 samples, respectively.While, the maximum deviation from constant group delay in case of the existing differentiators Ngo [6], Gupta et al. 1 [8], Gupta et al. 2 [9], Gupta et al. 3 [10], Al-Alaoui [15], and Upadhyay and Singh [16] are 1.5 samples, 0.55 samples, 0.45 samples, 1.0 samples, 1.4 samples, and 3.3 samples, respectively.
It is verified from Figures 9 and 10 and Table 5 that the minimax optimized designed second-, third-, and fourthorder differentiators outperform all the existing differentiators over entire Nyquist frequency range.
The triangular wave response of designed and existing differentiators is shown in Figures 13 and 14, respectively.It is seen that the response of designed differentiators  2MO (),  3MO (),  4MO (),  2MO+PZC (),  3MO+PZC (), and  4MO+PZC () is exact square wave, while the response of existing differentiators is distorted square wave.

Conclusion
This paper is focused on the use of two optimization methods, namely, minimax and pole, zero, and constant (PZC).It has been proved that the efficiency of minimax optimized integrators is remarkably improved by PZC optimization.Subsequently, by modifying the transfer function of these designed integrators appropriately, new differentiators are obtained which have the same accuracy as the designed integrators.The family of designed recursive integrators and differentiators are important when excellent magnitude and linear phase response is required at the same time.

Figure 2 :
Figure 2: Flowchart of the applied minimax optimization in design of second-order digital integrator.

Table 1 :
Coefficients and maximum PARE for different frequency ranges of second-order integrator.

Table 2 :
Coefficients and maximum PARE for different frequency ranges of third-order integrator.

Table 3 :
Coefficients and maximum PARE for different frequency ranges of fourth-order integrator.

Table 4 :
Comparison between existing and designed integrators.

Table 5 :
Comparison between existing and designed differentiators.