some aspects of dynamic reduction of transient duration in delay-equalized chebyshev filters

1718 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 8, AUGUST 2008

Some Aspects of Dynamic Reduction of TransientDuration in Delay-Equalized Chebyshev Filters

Jacek Piskorowski

Abstract—One important problem of analog signal process-ing in measurement and data acquisition systems is designing acontinuous-time filter that provides both a constant group delayover the filter passband and a selective magnitude response and,at the same time, ensures the transient to be as short as possible.In this paper, we propose a new theoretical method for dynamicshortening of transients in phase-compensated continuous-timefilters. Using the compensated Chebyshev filter, we introducetime-varying parameters to its structure for the purpose of mini-mization of the transient that was lengthened due to compensation.This paper describes the methodology of the choice of optimalfunctions that vary the filter parameters, discusses the stabilitypreservation of the designed filter, and presents a detailed blockdiagram of such a filter. Results verifying the effectiveness of theproposed method are presented and compared with traditionalfilters.

Index Terms—Chebyshev filters, continuous-time filtering, dataacquisition, group delay compensation, time-varying systems,transient state.

I. INTRODUCTION

IN MANY measurement processes, there is a need for the ap-plication of analog continuous-time filters whose magnitude

response is possibly selective, and the group delay is as constantas possible. If the group delay response is also an importantconsideration and the group delay response associated with thenetwork function that gives the required magnitude responseis not satisfactory, the usual solution is to introduce additionaldelays so that the total delay is nearly flat over the desiredfrequency band in which a constant group delay is important.These additional delays are furnished by the all-pass filter.

However, the process of group delay compensation is alwayscarried out at the cost of an extension of the filter transient. Thelonger the transient of the filter, the longer the time after whichthe filter works out the useful signal component. This extendedtransient may cause significant delays in measurement and data-acquisition processes.

An acceleration of the filter run is related to the improvementof the filter properties in the time domain. The problem ofimproving the transient performance has been considered inmany areas of measurement, such as sensor response correction

Manuscript received July 3, 2007; revised March 31, 2008. This work wassupported in part by the Polish State Committee for Scientific Research underGrant 3 T10C 032 28 and in part by Szczecin University of Technologyunder Grant 041-0205/18-06. An earlier version of this paper was presented atthe IEEE Instrumentation and Measurement Technology Conference, Warsaw,Poland, May 1–3, 2007.

The author is with the Institute of Control Engineering, Szczecin Universityof Technology, 70-313 Szczecin, Poland (e-mail: [email protected]).

Digital Object Identifier 10.1109/TIM.2008.923781

in weighing processes [1] and fast processing of the brainstemauditory-evoked potential signal [2]. The first technique isbased on the adaptive algorithm, and the second technique isbased on the linear parameter-varying (LPV) approach. In ad-dition, very good results were achieved in adaptive and controlsystems [3], [4] and inductor motor drives [5].

The simultaneous improvement of the filter properties in thetime and frequency domains is not possible, taking into accounttraditional time-invariant filters [6]–[8]. For time-invariant fil-ters, there are only small possibilities for transient reductionsince the filter parameters are calculated based on the assumedmethod of the frequency response approximation. This factguarantees that the frequency requirements are satisfied, with-out taking into consideration the character of the transient state.If the frequency response requirements are imposed, we canslightly influence the transient reduction of the nth-order filterby choosing different approximation methods. The uncertaintyprinciple says that it is not possible to achieve a shorter rise timeof the low-pass filter output signal when the filter passband isconstant. However, it is possible to obtain significant changesof the transient duration by variation of the filter passband [2],[9]–[12]. This procedure is related to the change of the valueof the filter coefficients. The theory of linear time-varyingcontinuous-time systems is well established and was widelydescribed [13]–[17].

In this paper, the question of how to design a Chebyshevlikefilter that ensures a low transition-band ratio of the magnituderesponse and a constant delay over the desired frequency bandand, at the same time, possesses fast response in the timedomain will be addressed. The outline of this paper is given asfollows: In Section II, the problems of the group delay compen-sation are elucidated. The main assumptions of the time-varyingfiltering with detailed analysis of the stability, stationarity, andselection of optimal parameters of the functions that vary thefilter parameters are discussed in Sections III and IV. Section Vthen presents the results of simulations. The conclusions arepresented in Section VI. This paper is an updated and extendedversion of [18] and [19].

II. COMPENSATION PROBLEM

A delay equalizer usually consists of one or more all-passnetworks, each of which has the same gain for all frequencies.They do not disturb the magnitude response. The procedures ofthe group delay compensation are well known from filter theoryand have been widely described [7], [8], [19].

As the number of all-pass sections increases, the group delayof the equalized filter approaches the constant group delay

0018-9456/$25.00 © 2008 IEEE

PISKOROWSKI: DYNAMIC REDUCTION OF TRANSIENT DURATION IN DELAY-EQUALIZED CHEBYSHEV FILTERS 1719

Fig. 1. Comparison of the step responses of the original and phase-compensated Chebyshev filters.

response. However, the process of group delay compensationis always carried out at the cost of an extension of the filtertransient state [19]. Fig. 1 presents the step responses of theoriginal and phase-compensated third-order Chebyshev filters.This figure shows that the response of the phase-compensatedfilter was occupied not only by longer transient state but alsoby undesirable undershoot. These undesirable compensationeffects can be eliminated by the variation of the filter parametersduring the transient state of the original filter.

III. LPV APPROACH

A. Model

The dynamic properties of a low-pass filter are entirelydescribed by damping factor β, natural frequency ω0, and timeconstant T (only for odd filter orders). Similarly, the all-passfilter is described by natural frequency ω0p and damping factorβp. A time variation of these parameters can be useful for thereduction of the transient of the phase-compensated filter.

A time-varying filter design is the result of the modelingof the system of ordinary differential equations with vary-ing coefficients. For the nth-order filter, the model can bewritten as

T (t)y′1(t) + y1(t) = x(t) (1a)

ω−20 (t)y′′

2(t) + 2β(t)ω−10 (t)y′

2(t) + y2(t) = y1(t) (1b)

...

ω−20p (t)y′′

n(t) + 2βp(t)ω−10p (t)y′

n(t) + yn(t)

= ω−20p (t)y′′

n−1(t) − 2βp(t)ω−10p (t)y′

n−1(t) + yn−1(t) (1c)

where x(t) and yn(t) are the input and output of the filter,respectively. Equation (1a) describes the dynamics of the first-order element. This equation is omitted if the modeled filteris of an even order. Moreover, the model of the filter con-tains equations describing the dynamics of the second-order

Fig. 2. Example of function F (t) and its parameters.

elements (1b) and the dynamics of the all-pass filtersection (1c), which, in our case, is the second-order element.

B. Function of Filter Parameters

It is well known that the higher the value of natural frequencyω0, the shorter the transient of the filter. On the other hand, thesmaller the value of damping factor β, the smaller the rise timeof the filter, and the larger the overshoot. Based on the computersimulations and the aforementioned rules, the function of thefilter parameters was formulated in the following form:

F (t) = d · F ·[1 − d − 1

d· h(t)

], where d =

F (0)F

(2)

where F is the value of the filter parameters, following theChebyshev approximation and all-pass filter calculus, and d isthe variation range of function F (t). For d > 1, function F (t)decreases in the variation interval; for d ∈ (0, 1), it increases;and for d = 1, the function is constant, and F (t) = F .

Function (2) can easily be generated in the analog technique,and this time dependency works well [9], [10], [19]. Functionh(t) in (2) describes the step response of the second-ordersupportive system HS(s) given by

HS(s) =1

ω−20f s2 + 2βfω−1

0f s + 1(3)

and has the following form: h(t) = L−1[s−1 · HS(s)]. L−1 isthe inverse Laplace transform, ω0f determines the variationrate of function F (t), and βf determines the oscillations offunction F (t).

An example of function F (t) and its more important pa-rameters are presented in Fig. 2, and a block diagram of theproposed filter structure is depicted in Fig. 3. This diagrampresents, in a general way, how the supportive system influencesthe dynamics of the Chebyshev and all-pass filters.


Fig. 3. Block diagram of the proposed filter.

C. Function Constraints

The main assumption that must be imposed on function F (t)is the necessity of settling during the transient of the originaltime-invariant filter. This condition can be written as

∀t>tsα, F (t) = F ± α (4)

where tsα is the settling time (with assumed accuracy α) of theoriginal time-invariant filter.

D. Efficiency Factors

The optimal parameters of all functions of the filter co-efficients were selected based on the computer simulations,because analytical solutions of differential equations with vary-ing coefficients are impossible to obtain in our case. Optimalvalues of the filter parameters were selected based on the totalefficiency factor η, which is defined as the product of timeefficiency factor ηt and frequency efficiency factor ηω , i.e.,η = ηt · ηω.

Time efficiency factor ηt is defined as the ratio of the settlingtimes t̃scα and tsα of the phase-compensated time-varyingfilter and the original time-invariant filter (not compensated);frequency efficiency factor ηω is defined as the ratio of cutofffrequencies ω̃c0 and ωc of the time-varying filter (for t = 0) andthe original time-invariant filter (not compensated), i.e.,

ηt =t̃scα

tsαηω =

ω̃c0

ωc. (5)

The smaller the value of efficiency factor η, the better theproperties of the designed time-varying filter. If the value ofthe total efficiency factor satisfies condition η < 1, then thedesigned filter has better properties than the original time-invariant filter.

E. Stationarity

To analyze the spectral properties of time-varying filters,one can use the methods that are applicable to time-invariantsystems under the conditions that the filter parameters stabilize(with α accuracy) after the transient expiration. To show this,the theorem on the spectral density of the output signal afterthe transient expiration was used. Kaszynski proved [20] thatthis result also held for systems with time-varying parametersif their values stabilized when t → ∞. The proof described

in [20] allows one to apply the spectral relations that hold inthe steady state for linear time-invariant systems to systemswith time-varying parameters if these parameters stabilize theirvalues with time.

F. Stability

To check the stability of the time-varying system, one canuse the second Lyapunov method. For the second-order filter

y′′(t) + 2β(t)ω0(t)y′(t) + ω20(t) = 0 (6)

the Lyapunov function can be assumed to be in the followingform [21]:

V [t, y(t), y1(t), ω0(t)] =12y2(t) +

12ω2

0(t)y21(t) (7)

where y1(t) = y′(t). Function (7) is a positive-definitequadratic form, and its time derivative is given by

V ′ = y(t)y1(t) − ω′0(t)

y21(t)

ω30(t)

+ y′1(t)

y1(t)ω2

0(t). (8)

Therefore

y′1(t) = −2β(t)ω0(t)y1(t) − ω2

0(t)y(t). (9)

Combining now (9) and (8), the stability condition takesthe form

V ′ = − 1ω3

0(t)ω′

0(t)y21(t) − 2β(t)

ω0(t)y21(t) < 0. (10)

Since y21(t) > 0 for y1(t) �= 0, this gives

− 1ω3

0(t)ω′

0(t) −2β(t)ω0(t)

< 0. (11)

For the considered system to satisfy the requirements for thelow-pass filters, the following conditions have to be fulfilled,due to the following properties of the spectrum:

ω0(t)>0 and β(t)>0, or ω0(t)<0 and β(t)<0.(12)

Taking (12) into account in inequality (11), one has toconsider two cases: ω′

0(t) > 0 and ω′0(t) < 0. The analysis of

these conditions leads to the stability condition of the second-order parametric system:

|ω′0(t)| <

∣∣2β(t)ω20(t)

∣∣ . (13)

It follows from (13) that, for the second-order system to bestable, it suffices that the rate of changes of the characteristicfrequency is bounded by the product of functions β(t) andω0(t). According to (12), the functions β(t) and ω0(t) have tobe of the same sign. If the following condition is satisfied:

limt→∞

ω′0(t) → 0 (14)


Fig. 4. Placement of the poles and zeros in the proposed filter.

then the sufficient condition for the homogeneous system (6)to be asymptotically stable and, consequently, for the nonho-mogeneous system to be stable, is that both functions β(t) andω0(t) at every t must have the same sign. If (14) does not hold,then (13) must hold, which bounds the rate of changes of theparameters.

IV. FUNCTIONS OF FILTER PARAMETERS: SOME DETAILS

The best results of the transient reduction (in the sense ofthe settling time) in phase-compensated Chebyshev filters wereobtained when ω0i, ω0p, and T−1 were varied according to thesame function (with the same variation range denoted by dω).On the other hand, βi and βp should be varied according tothe different functions (different variation ranges denoted bydβ and dβp

, respectively) [19].Functions ω0i(t), ω0p(t), and T−1(t) start from a larger value

than ω0i, ω0p, and T−1

, respectively, which means that thesefunctions decrease (d > 1) in variation interval t ∈ 〈0, tsα〉.Such a run of functions ω0i(t) and T−1(t) shifts the cutofffrequency to a larger value in the initial phase of the filter run.

Function βi(t) also starts from a larger value than βi, whichmeans that this function decreases (d > 1) in the variation in-terval t ∈ 〈0, tsα〉. Such a run of function βi(t) causes strongerdamping of the filtered signal in the initial phase of the filter runand suppression of undesirable overshoot in the step response.

Function βp(t) starts from a smaller value than βp, whichmeans that this function increases [d ∈ (0, 1)] in variation inter-val t ∈ 〈0, tsα〉. This function is responsible for the undershootelimination from the step response.

Fig. 4 presents the poles and zeros of the third-order phase-compensated time-varying Chebyshev filter. The poles andzeros at the center of the figure belong to the filter when itsparameters are settled, i.e., for t > tsα, and the others describethe filter when it starts, i.e., for t = 0.

Fig. 5 presents the step responses of the original, phase-compensated time-invariant, and phase-compensated time-varying Chebyshev filters.

Fig. 5. Comparison of the step responses of the original, compensated, andcompensated time-varying Chebyshev filters.

It is easy to notice that the time-varying approach enabledconsiderable shortening of the transient, which was lengtheneddue to the group delay compensation. Moreover, the proposedfilter is even faster than the original filter but not the com-pensated filter. Last, but not least, undesirable overshoots andundershoots were eliminated from the filter response.

For the original filter, the 2% settling time is ts = 12.20 s; forthe compensated filter, tsC = 14.30 s; and for the compensatedtime-varying filter, t̃sC = 2.50 s. Based on these values, it iseasy to calculate that the settling time of the phase-compensatedtime-varying filter is more than 5.7 times shorter than that ofthe compensated time-invariant filter and more than 4.8 timesshorter than that of the original filter.

Fig. 6 presents a detailed model of the third-order time-varying Chebyshev filter, which has been compensated with theaid of the second-order all-pass filter. A classical implementa-tion of the time-varying filter described in this paper requiresthe use of multipliers, adders, and two additional integrators,which form the supportive system. As one can notice, the over-all complexity of the system underwent a significant increase.However, in situations in which the transient should be as shortas possible, this complexity increase may be profitable.

Fig. 7 presents the step responses of the ideal and realtime-varying filters that were considered in this paper. Thecharacteristic labeled by the simulated (ideal) filter presents theresponse of the filter in which all product systems (see Fig. 6)are ideal, and the characteristic labeled by the simulated (real)filter presents the response of the filter in which all productsystems are simulated as real (analog simulator MEDA 43TC)with nonlinearities, which are typical for these kinds of circuits.As one can see, the influence of the nonlinearities is noticeable;however, it is not significant.

V. CONTINUOUS RUNNING MODE

In order for the time-varying filters to be useful in thecontinuous running mode, it is necessary to design a specialsystem that will be able to detect step changes in the useful


Fig. 6. Detailed model of the third-order phase-compensated time-varying Chebyshev filter.

Fig. 7. Step responses of the ideal and real time-varying filters.

input signal [11], [19], which is particularly possible when arectangular input signal distorted by a small-level additive noiseis filtered, as in Fig. 8. For this purpose, one can make useof a system that delays the input signal by a given period oftime denoted by τ . The time should be chosen in this way tocapture all significant (assumed) changes of the level of inputsignal x(t).

A block diagram of the continuous running mode of time-varying filters is shown in Fig. 9.

The absolute value of the difference between the input signaland the delayed input signal ξ(t) = |x(t) − x(t − τ)| is fedto the comparator input. Then, this comparator compares theactual value of signal ξ(t) with activation threshold c. The

Fig. 8. Rectangular input signal with additive noise.

Fig. 9. Block diagram of the continuous running mode of time-varyingChebyshev filters.

value of this threshold is chosen based on the additive noiselevel (e.g., the variance of the noise). The higher the level ofthe input noise, the larger the value of the activation threshold


Fig. 10. Comparison of the time-varying and time-invariant filter responses inthe continuous running mode.

that must be adjusted. In many cases of signal processing(e.g., in measurement systems), the noise is small, and we areable to estimate its parameters (e.g., variance). Therefore, withknowledge of the noise signal, we can possibly fix the activationthreshold of the designed filter.

If condition ξ(t) ≥ c is met, the comparator generates signalζ(t) = 1 (detection of the edge in the useful input signal);otherwise, ζ(t) = 0 (no detection). Then, signal ζ(t) is fed tothe integrators (their reset inputs), which form the supportivefilter structure. Signal ζ(t) fed to the reset inputs of integratorsis responsible for the generation of all the functions of the filterparameters. Any rising edge of signal ζ(t) is synonymous withrestarting the integrators from the supportive system, whichcauses the cyclic generation of the functions.

Fig. 10 presents the results of filtering by using the traditionaltime-invariant Chebyshev filter and the proposed time-varyingfilter (with ideal and real product systems). It is easy to noticethat the application of time-varying filters to the processing ofthe rectangular signals distorted by additive noise (like that inFig. 8) gives much better results than that of time-invariantfilters. If the edge in the useful input signal is detected, thetime-varying filter is considerably faster than the traditionaltime-invariant filter. It is necessary to add that the group delayresponse is also equalized (of course, when the filter parametersare settled).

The proposed algorithm may be applied in many measure-ment and data acquisition systems. There are, of course, somelimitations. The noise level must be smaller than the jumpsin the useful input signal, and the changes in the input sig-nal level should be no faster than the settling time of thefilter. However, there are also data acquisition processes inwhich it is known when the level changes in the input signaloccur, and in this case, the edge-detection system is not re-quired. A good example of such a case is a weighing processwhere the proposed filters can improve the response of themass sensor and thereby considerably speed up the process ofmeasurement.

VI. CONCLUSION

As has been proven, the application of time-varying coeffi-cients in delay-equalized Chebyshev filters yields good results.By using the described time-varying parameter approach, it ispossible to obtain an efficient filter that ensures a low transition-band ratio of the magnitude response and, at the same time,provides a constant delay over the desired frequency band.In addition, the designed filter is considerably faster than thetraditional filter.

It seems that further examinations of time-varying filterswith application to the dynamic correction of sensor responseare needed. In the future, the proposed filter configurationwill be implemented with the aid of the dynamic translineartechnique [22]. By using the dynamic translinear principle,it is possible to implement linear and nonlinear differentialequations, using only transistors and capacitors. Dynamictranslinear circuits are excellently tunable across a wide rangeof several parameters, such as cutoff frequency, quality factor,and gain, which increases their designability and makes themattractive for use as standard cells or programmable buildingblocks. In fact, the dynamic translinear principle facilitatesa direct mapping of any function, which is described bydifferential equations, onto silicon.

It is worth adding that the proposed filter structures caneasily be transformed into digital filters. For that purpose, thecontinuous-time integrators from Fig. 6 should be transformedinto their digital equivalents with the aid of the well-knownbilinear transform.

ACKNOWLEDGMENT

The author would like to thank the anonymous reviewers fortheir constructive comments and useful suggestions.

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Jacek Piskorowski was born in Pila, Poland, in1977. He received the M.Sc. degree in electronicengineering and the Ph.D. degree from SzczecinUniversity of Technology, Szczecin, Poland, in 2002and 2006, respectively.

Since 2002, he has been with the Institute of Con-trol Engineering, Szczecin University of Technology,where he is currently an Assistant Professor. Hisresearch activity is mainly focused on signal process-ing and measurements, with particular interest inthe analysis, synthesis, and design of systems and

circuits with time-varying parameters.

some aspects of dynamic reduction of transient duration in delay-equalized chebyshev filters

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