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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 , |ω| < ωn priate sequence of control actions. Maximum overshoot is an H (ω) = (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  flaw corresponds to a poor transient performance index in 1 , 0 6 |ω| 6 (1 − ζ) ωn terms of the maximum overshoot. Consequently, as noted,  H (ω) = 0 , |ω| > (1 + ζ) ωn (2) this constitutes a fundamental limitation in the transient- h  i  1 π(|ω|−ωn) performance of higher-order transfer-functions designed us-  1 − sin , otherwise. 2 2ζωn ing the butterworth polynomial. This fundamental fact that ( 1 h  πω i 1 + cos , |ω| 2ωn the transient performance and sensitivity properties are not 2 2ωn 6 H (ω) = (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  πω  cos , |ω| 2ωn 2ωn 6 inator polynomial with a balanced (good) transient response Hn (ω) = (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, based on the maximally flat magnitude design the transient response for any order. criterion in both the pass band and the stop band, among all Consider the butterworth denominator polynomials (two- transfer functions of a given order [3], [6], [23]. They can be decimal place approximated) from (5–10). It can be observed represented as cascade of first and second-order polynomials that the binomial coefficients (excluding boundary coefficients) with relative damping constants [24]. More recently, in [25], are non-uniformly damped for all except the first three cases, the butterworth filter has been classified under a unified theory where the damping is constant. From this, it is clear that of critical monotonic amplitude characteristic (CMAC) all- the choice of damping constants for coefficients (excluding pole filters. The integer-order butterworth filter was gener- boundary coefficients) of the binomial polynomial influences alized to fractional-order in [26], [27]. There are other all- the transient-response characteristics of the filter. Therefore, pole transfer-functions detailed in [10] that outperform the proper choice of damping constants can avoid the flawed frequency response performance of the butterworth. However, transient-response of higher-order cases. none of them offer a better transient to frequency response compromise like the butterworth. Also, the butterworth filter D1(s) = s + 1 (5) phase-delay is linear when close to the origin, and is shorter D (s) = s2 + 2 (0.71) s + 1 (6) than either the bessel or binomial filter. 2 3 2 Interestingly, the butterworth-filter which shows no ripple D3(s) = s + 3 (0.67) s + 3 (0.67) s + 1 (7) 4 3 2 in its frequency response bands starts to show considerable D4(s) = s + 4 (0.65) s + 6 (0.57) s 3

+ 4 (0.65) s + 1 (8) A. Damped Binomial Coefficients 5 4 3 D5(s) = s + 5 (0.65) s + 10 (0.52) s Definition 1. Following, the standard definition of binomial + 10 (0.52) s2 + 5 (0.65) s + 1 (9) (combinatorial) coefficients, for any natural number n, the 6 5 4 3 damped binomial coefficient can be written in the form: D6(s) = s + 6 (0.64) s + 15 (0.49) s + 20 (0.46) s + 15 (0.49) s2 + 6 (0.64) s + 1 (10)  n  c a = C¯n ≡ ≡ (11) i i i ζ i!(n − i)! Further, to simplify, we impose two constraints. One, the where, ( damping constant should be uniform in the denominator ζ · n! , 0 < i < n polynomial. Two, the percentage maximum-overshoot should c = n! , . be at most five-percent, as realized from (6). This then reduces otherwise the design problem to one question. How exactly should the ¯n ¯n and n, i ∈ N, n > 0, n ≥ i ≥ 0. It follows that, C0 = Cn = 1, damping constant be defined for any nth-order denominator and also C¯n = C¯n . polynomial? i n−i

B. Uniformly-Damped Pascal’s Rule B. Main Contributions Damped binomial coefficients form the famous universal symmetric pattern, known as the Pascal’s triangle. In the Therefore, the main goal of this paper is to present a closed- uniformly-damped case, this pattern can be defined. form solution to this above question. The uniformly-damped binomial polynomial is optimised on a transient-response Definition 2. The uniformly-damped Pascal’s rule is expressed criterion. This criterion is the maximum negligible overshoot as: value observed in the second-order butterworth response. The  ζ · C¯n + C¯n , i = 1 direct formula for the exact damping constant that makes this  i−1 i ¯n+1 ¯n ¯n possible, without the use of explicit numerical optimization is Ci = Ci−1 + Ci · ζ , i = n (12)  ¯n ¯n contributed in this paper. Ci−1 + Ci , otherwise. In all, the contributions of this paper are three-fold. Next, we will present the uniformly-damped binomial ex- One, extension of the binomial expansion theorem to the pansion theorem, where the damping constant ζ is spread uniformly-damped binomial theorem. Two, definition of the uniformly across all of the binomial coefficients, except for the exact damping-constant for a transient response with five- boundary (first and last) coefficients which are not damped. percent maximum overshoot. Three, design and analysis of the five-percent uniformly-damped binomial filter (FP-UDBMF) transfer-function. C. Uniformly-Damped Binomial Polynomial The rest of this paper is structured as follows: First, in The denominator polynomial of the all-pole transfer func- section II we start with the introduction of the damped tion, can now be defined, by stating the uniformly-damped binomial coefficient, and then extend the binomial theorem binomial theorem. to the uniformly-damped binomial theorem which leads to the uniformly-damped binomial polynomial. In section II-D, Theorem 1. For any natural number n ∈ N, n > 0, where n the solution to the exact uniform damping constant that is the order of the polynomial, the uniformly-damped binomial achieves the desired criterion is defined. Table I illustrates polynomial expansion can be written as: the coefficients (up to the tenth order) of the normal- n ¯n n ¯n n−1 ¯n n−2 2 ized uniformly-damped binomial polynomial that satisfies the Dn = (s + ωn) = C0 s + C1 s ωn + C2 s ωn + ··· n desired transient-response criterion. Definition and analysis X + C¯n sωn−1 + C¯nωn = C¯nsn−iωi (13) of the five-percent uniformly-damped binomial filter (FP- n−1 n n n i n i=0 UDBMF) transfer-function is presented in section III. A simu- lated comparison of the three filters is illustrated in section IV. where variables: s is the complex laplace variable, and ωn Finally, section V concludes the discussion in this paper. The represents the cut-off frequency. The proof of Theorem 1 is dynamical system which functions as the frequency-selective shown in Appendix A. filter is represented as a proper transfer function Hn, and Axiom 1. The sum of the coefficients in a uniformly-damped its denominator polynomial which is an improper transfer binomial polynomial is: 2 + (2n − 2) ζ, ∀n > 0. function is represented as D. Recall, we desire to achieve a maximum overshoot less or equal to that of the second-order butterworth filter. At this II.UNIFORMLY-DAMPED BINOMIAL THEOREM point, the problem is to determine, for each order, what the value of the exact damping constant applied uniformly to the In this section, we will introduce the first main result– binomial coefficients will be. This can be solved by numerical the uniformly-damped binomial theorem. First, we start by optimization, however a simpler and direct closed-formula that defining the damped binomial coefficients. solves this problem is defined next. 4

TABLE I 1 FIVE-PERCENT UNIFORMLY-DAMPED BINOMIAL FILTER TRANSFER FUNCTION Hn(s) = Dn(s)

n ζn Polynomial Dn(s) 1 1 s + ζ √ n 2 2/2 s2 + 2ζ s + 1 √ n 3 5/3 s3 + 3ζ s2 + 3ζ s + 1 √ n n 4 10/4 s4 + 4ζ s3 + 6ζ s2 + 4ζ s + 1 √ n n n 5 17/5 s5 + 5ζ s4 + 10ζ s3 + 10ζ s2 + 5ζ s + 1 √ n n n n 6 26/6 s6 + 6ζ s5 + 15ζ s4 + 20ζ s3 + 15ζ s2 + 6ζ s + 1 √ n n n n n 7 37/7 s7 + 7ζ s6 + 21ζ s5 + 35ζ s4 + 35ζ s3 + 21ζ s2 + 7ζ s + 1 √ n n n n n n 8 50/8 s8 + 8ζ s7 + 28ζ s6 + 56ζ s5 + 70ζ s4 + 56ζ s3 + 28ζ s2 + 8ζ s + 1 √ n n n n n n n 9 65/9 s9 + 9ζ s8 + 36ζ s7 + 84ζ s6 + 126ζ s5 + 126ζ s4 + 84ζ s3 + 36ζ s2 + 9ζ s + 1 √ n n n n n n n n 10 9 8 7 6 5 4 3 2 10 82/10 s + 10ζns + 45ζns + 120ζns + 210ζns + 252ζns + 210ζns + 120ζns + 45ζns + 10ζns + 1

D. Uniform-Damping Constant

The core result of this paper lies in the explicit definition of 1 the uniform-damping constant. The definition of this constant, provides a solution to the five-percent maximum overshoot 0.8 design. Consider the nth order uniformly-damped binomial polyno- 0.6 mial in Theorem 1. The exact uniform-damping constant that satisfies the maximum overshoot optimization criterion Mp of 0.4 the second-order butterworth filter can be defined as follows: 0.2 Definition 3. The uniform-damping constant that satisfies a M ≤ 5%, for the nth order uniformly-damped binomial p 0 polynomial in Theorem 1 is: 0 10 20 30 pn (n − 1) − (n − 2) ζ = ζn = (14) n (a) The expression in (14) represents the solution to the ringing problem present in the specification of the damped binomial polynomial. This result, further leads to the synthesis of 1 the five-percent uniformly-damped binomial transfer-function, whose transient step and impusle response is illustrated in 0.8 (Fig. 1), and discussed in the next section. 0.6

III.UNIFORMLY-DAMPED BINOMIAL FILTER 0.4 (TRANSFER-FUNCTION) Applying Theorem 1 and Axiom 3 to the synthesis of the 0.2 unity-gain continuous-time transfer-function, where k0 = 1. The uniformly-damped binomial filter Hn is defined as: 0 n k0 ωn k0 0 10 20 30 Hn (s) = = (15) D (s) Pn ¯n n−i n i=0 Ci (s/ωn) The expressions in (16) and (17) respectively give the (b) magnitude, and squared-magnitude of the FP-UDBMF. Fig. 1. Step (1a) and Impulse Response (1b) plot of the uniformly-damped binomial low-pass filter transfer-function with normalized cut-off frequency 1 for values of n = 1 (blue) to n = 10 (brown). |Hn (ω)| = (16) q 2n (ω/ωn) + κ + 1

2 1 |Hn (ω)| = Hn (s) Hn (−s) = (17) r¯ 2n 2 X r (ω/ωn) + κ + 1 ¯n
¯n ¯n αt = Ct + 2 (−1) Cj Ck (19) 1 r=1 X 2i κ = αt (ω/ωn) (18) i=n−1 and 5

 ( n  i > 2 and even n n − i , (n−1) 1 1 r¯ = i > 2 and odd n  i , otherwise. 0.5 0.5 where, t = n − i, j = t − r, k = t + r and αt = αn−t. Also, the minimum attenuation at a given frequency, and 0 0 the bandwidth given a minimum attenuation can be obtained 10-2 100 10-2 100 respectively by: (20), and reducing (21) to a quadratic equa- (a) (b) tion. From inspection of the slope of the magnitude (in dB) plot in Fig. 2a, as ω >> ωn, it is clear that high-frequency 1 1 roll-off is −20 n dB/decade.

 2n  0.5 0.5 AdB = 10 log (ω/ωn) + κ + 1 (20)   2n AdB/10 0 0 (ω/ωn) + κ + 1 − 10 = 0 (21) 10-2 100 10-2 100

The UDBMF becomes a digital infinite impulse response filter (c) (d) by s → z bilinear transformation. Also, the denominator Fig. 3. Magnitude Response Comparison of the FP-UDBMF (solid blue), polynomial can be directly applied as a digital finite impulse BWF (dashed red), and BMF (dotted brown) with normalized cut-off fre- response filter (moving average filter) by directly replacing quency for orders n = 1 (3a) to n = 4 (3d). s = z and ωn = 1 as shown in (22).

n A. Flatness and Selectivity of Filter n X ¯n n−i D (z) = (z + 1) = Ci z (22) Given the magnitude of the filter, the magnitude flatness i=0 and selectivity of the filter can be investigated by respectively finding the derivative of the magnitude with respect to the frequency and the negative derivative of the magnitude with respect to the frequency at the origin. 0  2n−1 ! d |Hn (ω)| 3 n ω dκ = − |Hn (ω)| + (23) dω ωn ωn dω -50 1 2i−1 dκ X i  ω  where, = α (24) dω t ω ω i=n−1 n n -100 From (23), it is easy to see that, since (16) is positive, the derivative monotonically decreases with no ripple. The kth, k = 1, ...∞ -150 ( ) derivatives of the gain are zero at the origin 10-2 100 (ω = 0) of the closed left-half complex plane for all n, except for even indexed kth derivatives, k = 2i up to 2n, where i = 1, 2, ..., n, provided n > 2. As illustrated in Fig. 3, this (a) results in a maximal flatness for n <= 2 and a flatness in between that of the BWF and BMF for n > 2. 1 n P1 i + αt d |Hn (ω)| ωn i=n−1 ωn − = 3 (25) dω h i 2 ω=ωn P1 2 + i=n−1 αt It is obvious from (25), that the frequency selectivity is the 0.5 same as the BWF for n <= 2 and in between the BWF and BMF for n > 2.

B. Phase-Delay and Group-Delay 0 -2 0 The phase response of the FP-UDBMF is illustrated in 10 10 (Fig. 4). The linear response can be investigated through the expressions (26) for the phase-delay and (27) for the group- (b) delay.

Fig. 2. Magnitude Response Plots of the uniformly-damped binomial low-pass (i−1) i  n    filter with normalized cut-off frequency for orders n = 1 (blue) to n = 10 P (−1) 2 C¯n ω ω i=odd i ωn (brown): shown in decibels in (2a) compared to (2b). n   τp (ω) = arctan i (26) ω  n i    P (−1) 2 C¯n ω i=even i ωn 6

0 10

8 -5 6

4 -10 2

-15 0 10-2 100 10-2 100

Fig. 4. Phase Response Plot (in radians) of the uniformly-damped binomial Fig. 5. Phase Delay of the uniformly-damped binomial low-pass filter with low-pass filter with normalized cut-off frequency for orders n = 1 (blue) to normalized cut-off frequency for orders n = 1 (blue) to n = 10 (brown). n = 10 (brown).

  τ (ω) = |H (ω)|2 (ω/ω )2n−2 + δ + n (27) g n n 12 1 X 2i δ = λt (ω/ωn) (28) 10 i=n−2 8 r¯ X r−1 ¯n ¯n λt = (−1) Cj Ck (29) 6 r=1 and, 4  ( n  i > 2 and even n 2 n − i , (n−1) r¯ = i > 2 and odd n  0 i + 1 , otherwise. 10-2 100 where t = n−1−i, j = t+1−r, k = t+r and λt = λn−t−1. At ω = ωn, the total phase in radians is nπ/4. At the origin ω = 0, the phase-delay (Fig. 5) and group-delay (Fig. 6) seen Fig. 6. Group Delay (in seconds) of the uniformly-damped binomial low-pass at the output of the filter is ζ n and n respectively, which is filter with normalized cut-off frequency for orders n = 1 (blue) to n = 10 ωn ωn (brown). exactly proportional to the order of the filter. Therefore both delays increase nonlinearly as the order is increased. The transient and frequency response analysis of the FP- filter is a uniformly damped binomial filter for lower orders, UDBMF poses it as a balanced compromise between the and then non-uniformly damped binomial filter for higher butterworth and binomial standard forms. It provides a balance orders. In contrast to this, the five-percent uniformly-damped of: excellent transient performance in its time-response (Fig. 1) binomial filter (FP-UDBMF) with a uniform-damping constant with a good sensitivity and selectivity performance across the optimized on the 5% maximum-overshoot criterion of the passband and stopband in its frequency response (Fig. 2 and second-order butterworth filter was introduced. This class of Fig. 4). filter or standard form represent a compromise of the strong merits of both the butterworth and the standard binomial IV. SIMULATION filter. The filter meets all imposed constraints (time-invariant, causal, linear, proper rational transfer function of finite order, In application to higher-order digital infinite impulse re- with positive real coefficients given by the uniformly-damped sponse filtering of a noisy signal, with negligible overshoot, the binomial polynomial) that assure the realization of a practical five-percent uniformly-damped binomial filter is contrasted to analog or digital filter. both the butterworth and standard binomial filter as illustrated Other applications of this filter include: the design of in (Fig. 7). It can be observed that the FP-UDBMF is a com- linear output-state (derivatives) estimators, and the design promise response with respect to the other two filters, useful of characteristic equations for higher-order control systems where fast, smooth transient response and strong filtering are with balanced transient response (no ringing) and frequency both required. filtering objectives. One important limitation, of this work, is that unlike the V. CONCLUSIONS butterworth filter, there is currently no exact closed-form A new class of filter (or transfer-function) design was formula for determining the pole positions of the uniformly- discussed in this paper. It was first shown that the butterworth damped binomial filter. This problem poses as a interesting 7

1.2

1

0.8 Noisy Signal 0.6 BWF BMF 0.4 FP-UDBMF

0.2

0

0 0.2 0.4 0.6 0.8 1

Fig. 7. Comparison of the 7th-order filtered output response of a unit-step noisy signal. and challenging future work, which readers may attempt to Thus, we have shown by inductive hypothesis that the damped discover. Please see the repository at [32] for the MATLAB binomial theorem is true for an arbitrary natural number a, scripts used for this paper. then it is true for a + 1 that is P (a) → P (a + 1). P (1) is true, it follows therefore from mathematical induction that APPENDIX A P (n) is true for all natural numbers and so the theorem PROOFOFTHE UNIFORMLY-DAMPED BINOMIAL THEOREM is established. Since the normalized expression is true, the First, consider the normalized form of the binomial expan- denormalised expression also follows as true. This is proved sion by noting that n n  n n X ¯n n−i i X ¯n n−i n n s P (n) = (s + 1) = Ci s 1 = Ci s (s + ωn) = ωn + 1 i=0 i=0 ωn From the principle of mathematical induction, we show that " n n−1 n−i 0# n = 1 is true.  s  X  s   s  = ωn C¯n + C¯n + C¯n P (1) = (s + 1) ≡ (s + 1) 0 ω i ω n ω n i=1 n n Assume n = a is true. n−1 n a X X a X ¯a a−i = C¯nsn + C¯nsn−iωi + C¯nωn = C¯nsn−iωi P (a) = (s + 1) = Ci s 0 i n n n i n i=0 i=1 i=0 Then, we show that n = a + 1 is true. a+1 REFERENCES a+1 X ¯a+1 a+1−i P (a + 1) = (s + 1) = Ci s [1] W. S. Levine, “Signal Processing for Control,” in Handbook of Signal i=0 Processing Systems, S. S. Bhattacharyya, E. F. Deprettere, R. Leupers, Expanding the left-hand side expression, we have: and J. Takala, Eds. New York, NY: Springer, 2013, pp. 261–279. a+1 a [2] T. Hagglund,¨ “A Unified Discussion on Signal Filtering in PID Control,” (s + 1) = (s + 1) (s + 1) = Control Engineering Practice, vol. 21, no. 8, pp. 994–1006, 2013. a ! a ! [3] Y. Bavafa-Toosi, “Fundamental limitations,” in Introduction to Linear X X C¯asa+1−i + C¯asa−i Control Systems, Y. Bavafa-Toosi, Ed. Academic Press, Jan. 2019, pp. i i 847–974. i=0 i=0 [4] M. M. Seron, J. H. Braslavsky, and G. C. Goodwin, Fundamental Further expanding and then applying the Damped Pascal rule, Limitations in Filtering and Control, ser. Communications and Control Engineering. London: Springer-Verlag, 1997. a−1 [5] H. Kennedy, “Recursive Digital Filters With Tunable Lag and Lead X = C¯asa+1 + C¯a + C¯a sa−i + C¯as0 Characteristics for Proportional-Differential Control,” IEEE Transac- 0 i+1 i a tions on Control Systems Technology, vol. 23, no. 6, pp. 2369–2374, i=0 Nov. 2015. a−1 [6] W. B. Ribbens, “The Systems Approach to Control and Instrumentation,” ¯a+1 a+1 X ¯a+1 a−i ¯a+1 0 in Understanding Automotive Electronics (Seventh Edition). Oxford: = C0 s + Ci+1 s + Ca+1 s Butterworth-Heinemann, Jan. 2013, pp. 1–49. i=0 [7] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Shifting the summation index, 3rd ed. Pearson Education, 2014. a a+1 [8] I. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles ¯a+1 a+1 X ¯a+1 a+1−i ¯a+1 0 X ¯a+1 a+1−i and Applications, ser. The Springer International Series in Engineering = C0 s + C s + Ca+1 s = C s i i and Computer Science. Springer US, 1990. i=1 i=0 8

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