1 Uniformly-Damped Binomial Filters: Five-percent Maximum Overshoot Optimal Response Design Oluwasegun A. Somefun1 Kayode F. Akingbade2 and Folasade M. Dahunsi3 Abstract—In this paper, the five-percent maximum overshoot therefore, often restricted to the second-order butterworth design of uniformly-damped binomial filters (transfer-functions) filter, because of the characteristics of five-percent maximum is introduced. First, the butterworth filter response is represented overshoot, satisfactory noise-reduction, and smaller phase- as a damped-binomial filter response. To extend the maximum- overshoot response of the second-order butterworth to higher delay. Phase-delays represent another problem. However in orders, the binomial theorem is extended to the uniformly- the case of step-tracking of signals, this delay consideration damped binomial theorem. It is shown that the five-percent is sometimes secondary and could be reduced with the extra uniformly-damped binomial filter is a compromise between the complexity overhead of lead-lag filters [2], [5]. butterworth filter and the standard binomial filter, with respect to the filter-approximation problem in the time and frequency All-Pole Transfer Functions. Interestingly, frequency- domain. Finally, this paper concludes that in applications of selective low-pass filters for noise attenuation are simply all- interest, such as step-tracking, where both strong filtering and pole transfer-functions [6]. It is well known in the literature a fast, smooth transient-response, with negligible overshoot are that the maximum overshoot is related to the presence of desired, the response of the normalized five-percent uniformly- damped binomial form is a candidate replacement for both the a damping constant in the transfer-function (input–output) butterworth and standard binomial filter forms. response of such filters. However, except for the all-pole second-order standard response, exact values for this damping Index Terms—Filter Design, Damping, Binomial Polynomial, Feedback Control, Maximum Overshoot constant that obtains a defined negligible amount of maxi- mum overshoot from higher-order all-pole transfer-functions, remains unknown. I. INTRODUCTION Linear filtering, a quintessential operation in signal pro- A. Background cessing and control, can be viewed abstractly as a unity- Transient Response. The requirement of a good transient gain transfer-function mapping [7], [8]. The unity-gain all- response is important in the selection of filters that condition pole filter transfer-function design problem can be simplified the input signals of a feedback controller like the proportional- to specifying only the filter order and a cut-off frequency integral-derivative (PID) controller [1], [2]. This need for [9]. Many interesting properties of these transfer-functions are filtering is mostly motivated by the need to attenuate mea- strongly related to polynomial theory. The filtering problem surement or set-point noise fed in as input to a controller. This then reduces to using a standard all-pole filter-design form to indirectly connotes the design objectives of smooth tracking specify the denominator polynomial of the transfer-function. (or regulation) and strong filtering [3]. Often, the controller The denominator polynomial gives the filter’s characteristic and the filter are implemented as digital signal processing equation. This then simplifies the filter-design problem to systems but conveniently designed as analog signal processing specifying the poles of the filter in terms of the positive real systems. coefficients of the denominator polynomial. arXiv:2007.00890v2 [eess.SY] 9 Dec 2020 In this context, the oscillatory transient behaviour of most analog filter design forms are not acceptable, as it leads to misinterpretation of the input sequence and therefore inappro- ( 1 ; j!j < !n priate sequence of control actions. Maximum overshoot is an H (!) = (1) 0 ; otherwise: important and visible index of transient response performance [4]. The transient response of signals, in this context, require negligible maximum overshoot. Selection of these filters is The behaviour of these standard all-pole filter transfer-function *This work was not supported by any funding. forms is then fully described by the denominator polynomial 1O. A. Somefun is with the Department of Computer Engineer- selected to minimize a certain performance criteria [4], [6], ing, Federal University of Technology Akure, PMB 704 Ondo, Nigeria [email protected] [10]. These standard forms are used to approximate the 2K. F. Akingbade is with the Department of Electrical and Electronics ideal transfer-function (frequency) response given by (1). As Engineering, Federal University of Technology Akure, PMB 704 Ondo, the transfer-function order increases, the presence of exces- Nigeria [email protected] sive ringing (oscillations or ripples) becomes visible in the 3F. M. Dahunsi is with the Department of Computer Engineer- ing, Federal University of Technology Akure, PMB 704 Ondo, Nigeria transient-response of these frequency-selective filters. This [email protected] phenomenon is an important fundamental limitation in many 2 control and filtering applications [3], [11]. ringings in its transient response as the order increases. This 8 flaw corresponds to a poor transient performance index in >1 ; 0 6 j!j 6 (1 − ζ) !n terms of the maximum overshoot. Consequently, as noted, <> H (!) = 0 ; j!j > (1 + ζ) !n (2) this constitutes a fundamental limitation in the transient- h i > 1 π(j!|−!n) performance of higher-order transfer-functions designed us- :> 1 − sin ; otherwise: 2 2ζ!n ing the butterworth polynomial. This fundamental fact that ( 1 h π! i 1 + cos ; j!j 2!n the transient performance and sensitivity properties are not 2 2!n 6 H (!) = (3) consistent with each other, indicates trade-offs inherent in the 0 ; otherwise: design of the denominator polynomial. It turns out that the Raised cosine functions. It is known that the study of polynomial coefficients of these frequency-selective filters are raised-cosine functions (2) illustrate how much this transient optimised for frequency response performance at the expense ringings in the time-domain can be damped, while still retain- of the transient performance or vice-versa in the case of the ing an approximation to the ideal-filter (ζ = 0) [11]. bessel filter [11]. Damped Binomial Polynomials. The synthesis of a denom- ( n π! cos ; j!j 2!n 2!n 6 inator polynomial with a balanced (good) transient response Hn (!) = (4) 0 ; otherwise: and frequency response for all positive integer orders is therefore very useful. For higher orders, in connection to the The raised-cosine function in (3) with ζ = 1, is a compact design of the characteristic equation of transfer functions, representation of the standard binomial polynomial with no this problem has been attacked by the use of binomial fil- transient overshoot. The raised cosine function (4) was dis- ters (BMF) [28]–[31] which directly correspond to (3). This cussed in [12], as an alternative form of binomial expansion. denominator polynomial is specified by the use binomial coef- The binomial polynomial is a widely used finite impulse ficients with uniform damping-constant ζ = 1. The binomial response filter in computer vision and image processing for filter can be viewed as the upper-limit of the ideal all-pole approximating the gaussian filter function [13], [14]. The transfer-function. With its smoother (no overshoot) transient- literature on the use of this polynomial is quite rich. For performance, it poses as a superior design choice for desired instance, the binomial window was introduced for interpo- transfer functions compared to the butterworth filter. Although lating narrow-band signals in [15], [16]. [17] approximated of higher sensitivity, the inherent requirement of no overshoot the ideal fractional-delay operator using generalized binomial present in this standard binomial polynomial (real poles only) coefficients. may not always be a practical choice. It leads to a much slower Notwithstanding, in [9], [18], [19], it is claimed that in rise-time in the transient response and a poorer filter-selectivity practise, most filtering require only a unity dc-gain and but- [3]. A faster transient-response with some form of negligible terworth response. The butterworth filter (BWF) proposed in overshoot is usually preferred. The design of the denominator [20], therefore remains arguably the most widely used among polynomial in the standard binomial form is therefore limiting. the class of available frequency-selective filters. Although, of In applications where both fast, smooth transient-response slower roll-off (attenuation), the characteristic equation of the and strong frequency filtering characteristics are design objec- butterworth lead to evenly distributed poles in the unit-circle tives, we would desire a compromise between the butterworth of the normalized complex s-plane, with an added single real and binomial standard response, side-stepping the main flaws, pole in the case of odd transfer-function orders [21], [22]. while keeping a balanced set of merits, namely: a maximally Further, it was shown in [12], that the butterworth filter is a flat monotonic amplitude in the second-order sense; a quicker flat-top raised cosine filter. This butterworth transfer-function roll-off around the cut-off frequency with increasing order; is regarded as the best achievable transfer-function approxima- and a faster rise-time with negligible maximum overshoot in tion to the ideal,