Signal Processing and Sampling Method for Obtaining Time Series Corresponding to Higher Order Derivatives

Abstract

For modeling and controlling dynamic phenomena it is important to establish with higher accuracy some significant quantities corresponding to the dynamic system. For fast phenomena, such significant quantities are represented by the derivatives of the received signals. In case of advanced computer modeling, the received signal should be filtered and converted into a time series corresponding to the estimated values for the dynamic system through a sampling procedure. This paper will show that present-day methods for computing in a robust manner the first derivative of a received signal (using an oscillating system working on a limited time interval and a supplementary differentiation method) can be extended to the robust computation of higher order derivatives of the received signal by using a specific set of second-order oscillating systems (working also on limited time intervals) so as estimative values for higher-order derivatives are to be directly generated (avoiding the necessity of additional differentiation or amplifying procedures, which represent a source of supplementary errors in present-day methods).

1. Introduction

For modeling and controlling dynamic phenomena it is important to establish with higher accuracy some physical quantities corresponding to the dynamic system. Usually this procedure is based on signal processing method applied upon the signal received from the dynamic system, implying some filtering methods (for noise rejection). In case of advanced computer modeling, the filtered signal should be converted further into a time series corresponding to the estimated values for the dynamic system through a sampling procedure. Many times these filtering and sampling devices consist of lowpass filters represented by asymptotically stable systems, the sampling moment of time being set after the transient regime of the filtering device has passed.

However, for fast phenomena, significant quantities are represented by the derivatives of the received signals. Usually the derivatives of a received signal 𝑦(𝑡)=𝑧(𝑡)+𝑛(𝑡) (where 𝑧(𝑡) represents the useful part of the signal, and 𝑛(𝑡) represents the noise) are computed by filtering the received signal and dividing the difference between the filtered values 𝑢(𝑡_𝑘) of the signal at two consecutive sampling moments of time by the time difference between these time moments (for first order derivative), by dividing the difference between the values of first-order derivative by the same time difference (for the second-order derivative), and so on. Yet this method requires very good filtering properties, while any difference in sampled values can drastically affect the estimation for the derivative.

The average value of the first derivative can be approximated by 𝑧(𝑡_𝑘)−𝑧(𝑡_𝑘−1)𝑡_𝑘−𝑡_𝑘−1=𝑧(𝑡_𝑘)−𝑧(𝑡_𝑘−1)Δ𝑡 (1.1) and can be estimated by the mathematical operation 𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1)𝑡_𝑘−𝑡_𝑘−1=𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1),Δ𝑡 (1.2) where 𝑢(𝑡_𝑘) represents the filtered values of the received signal 𝑦(𝑡) (as has been shown). In the ideal case, 𝑢(𝑡_𝑘) should be equal to 𝑧(𝑡_𝑘). This cannot be achieved. For avoiding significant errors the difference 𝑧(𝑡_𝑘)−𝑧(𝑡_𝑘−1) should be estimated with higher accuracy. This implies that the filtered values 𝑢(𝑡_𝑘) should be close to the values of the useful part of the received signal 𝑧(𝑡_𝑘).

As a consequence, the filtering device should reject the noise (supposed to present fast variation as compared to the variations of the useful part 𝑧(𝑡)) in a significant manner. For this purpose, the filtering and sampling devices based on asymptotically stable systems can be improved. They have the transfer function 1𝐻(𝑠)=𝑇₀𝑠+1 (1.3) (for a first-order system) and 1𝐻(𝑠)=𝑇²₀𝑠²+2𝑏𝑇₀𝑠+1 (1.4) (for a second-order system). They attenuate an alternating signal of angular frequency 𝜔≫𝜔₀=1/𝑇₀ (corresponding to noise) about 𝜔/𝜔₀ times (for a first-order system) or about (𝜔/𝜔₀)² times (for a second-order system). The response time of such systems at a 𝑐 useful signal is about 5𝑇₀ (5𝑇₀ for the first-order system and 4𝑇₀/𝑏 for the second-order system). If the signal generated by the first- or second-order system is integrated over such a period, a supplementary attenuation for the alternating signal of about 5𝜔/𝜔₀ can be obtained.

However, such structures are very sensitive at the random variations of the integration period (for unity-step input, the signal, which is integrated, is equal to unity at the sampling moment of time). Even if we use oscillators with a very high accuracy, such random variations will appear due to the fact that the integration is performed by an electric current charging a capacitor. This capacitor must be charged at a certain electric charge 𝑄 necessary for further conversions; this electric charge cannot be smaller than a certain value 𝑄_lim, while it has to supply a minimum value 𝐼_min for the electric current necessary for conversions on the time period 𝑡_conv required by these conversions, the relation 𝑄_lim=𝐼_min𝑡_conv (1.5) being valid. So the minimum value 𝐼_int(min) for the electric current charging the capacitor in the integrator system is determined by the relation 𝐼_int𝑄(min)=^lim𝑡_int, (1.6) where 𝑡_int is the integration period required by the application (knowing the sampling frequency 𝑓_𝑠, we can approximately establish 𝑡_int using the relation 𝑡_int=1/𝑓_𝑠). So the current charging the capacitor cannot be less than a certain value; thus random variations of the integration period will appear due to the fact that the random phenomena are generated when a nonzero electric current is switched off.

2. Specific Aspects of Using Oscillating Systems for Filtering the Received Signal

The disadvantage of using asymptotically stable systems (previously mentioned) can be avoided by using an oscillating second-order system having the transfer function 𝐻_osc=1𝑇²₀𝑠²+1 (2.1) working on the time interval [0,2𝜋𝑇₀] (see [1] for more details). For initial conditions equal to zero, the response of the oscillating system at a step input with amplitude 𝐴 will have the form 𝑡𝑤(𝑡)=𝐴(1−cos(𝑇₀)). (2.2) By integrating this result on the time interval [0,2𝜋𝑇₀], we obtain the result 𝑢=2𝜋𝐴𝑇₀, and we can also notice that the quantity which is integrated and its slope are equal to zero at the end of the integration period. Thus the influence of the random variations of the integration period (generated by the switching phenomena) is practically rejected. Analyzing the influence of the oscillating system upon an alternating input, we can observe that the oscillating system attenuates about (𝜔/𝜔₀)² times such an input. The use of the integrator leads to a supplementary attenuation of about (2𝜋)(𝜔/𝜔₀) times. The oscillations having the form 𝑤_osc=𝑎sin(𝜔₀𝑡)+𝑏cos(𝜔₀𝑡) (2.3) generated by the input alternating component have a lower amplitude and generate a null result after an integration over the time interval [0,2𝜋𝑇₀]. As a conclusion, such a structure provides practically the same performances as a structure consisting of an asymptotically stable second-order system and an integrator (response time of about 6𝑇₀, an attenuation of about 6(𝜔/𝜔₀)³ times for an alternating component having frequency 𝜔) moreover being less sensitive at the random variations of the integration period. It is the most suitable for the operation 𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1),Δ𝑡 (2.4) where Δ𝑡=𝑡_𝑘−𝑡_𝑘−1. The difference 𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1) can be further divided by a constant value (corresponding to Δ𝑡) so as to estimate the first-order derivative 𝑧(𝑡_𝑘)−𝑧(𝑡_𝑘−1). For restoring the initial null conditions after the sampling procedure (at the end of the working period) some electronic devices must be added (see [2] for more details).

However, this method presents a major disadvantage: the filtering devices can generate an electronic voltage corresponding to 𝑢(𝑡_𝑘) within a certain range (less than 10 Volts, usually). This means that the difference 𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1) would correspond to a small voltage, implying the necessity of amplifying this voltage so as to achieve a result in a certain range (suitable for modeling, controlling, and data acquisition). This represents a supplementary source of errors (the resolution being limited by the resolution of the operation 𝑢(𝑡_𝑘)−𝑢(𝑡_𝑘−1)). It implies the necessity of using an oscillating second-order system so as the result of the integration on a working interval corresponds to the derivative of the useful part 𝑧(𝑡) of the received signal 𝑦(𝑡) (if possible).

3. Analog Signal Processing Methods Suitable for Derivative Procedures

A general mathematical method for obtaining the derivatives of the useful part 𝑧(𝑡) of a received signal 𝑦(𝑡) in a robust manner (with good filtering properties and also with a good resolution, avoiding a supplementary amplification of the difference between two previously sampled values) consists in using a signal processing device with the transfer function 𝐻(𝑠)𝑠𝐻(𝑠)=^𝑁(𝑇²₁𝑠²+1)(𝑇²₂𝑠²+1)⋯(𝑇²_𝑘𝑠²+1)⋯(𝑇²_𝑛+1𝑠²,+1) (3.1) which can be also written as 𝐻(𝑠)=𝑠^𝑁𝑛+1_𝑘=11(𝑇²_𝑘𝑠².+1) (3.2) The degree of the denominator polynomial must be greater or equal to the degree of numerator polynomial so as the transfer function is to be implemented using electronic devices; this means that 𝑁≤2(𝑛+1) (as can be easily noticed).

The output of this signal processing system for an input corresponding to 𝑦(𝑠)=1/𝑠^𝑁+1 (the Laplace transformation of the time function 𝑦(𝑡)=(1/𝑁!)𝑡^𝑁) is represented by 1𝑤(𝑠)=𝑠^𝑁+1𝑠^𝑁1(𝑇²₁𝑠²1+1)(𝑇²₂𝑠²⋯1+1)(𝑇²_𝑛𝑠²,+1) (3.3) which can be also written as 1𝑤(𝑠)=𝑠1(𝑇²₁𝑠²1+1)(𝑇²₂𝑠²⋯1+1)(𝑇²_𝑛𝑠²+1) (3.4) or 1𝑤(𝑠)=𝑠^𝑛+1_𝑘=11(𝑇²_𝑘𝑠²,+1) (3.5) which represents a set of multiplication of a unity step input 1/𝑠 by transfer functions 1/(𝑇²_𝑘𝑠²+1).

This means that it can be written as 𝑤₁1(𝑠)=(𝑇²₁𝑠²1+1)𝑠,𝑤₂1(𝑠)=(𝑇²₂𝑠²𝑤+1)¹𝑤(𝑠),₃1(𝑠)=(𝑇²₃𝑠²𝑤+1)²(𝑠) (3.6) or (in a general form) 𝑤_𝑘+11(𝑠)=(𝑇²_𝑘+1𝑠²𝑤+1)^𝑘(𝑠) (3.7) until 𝑤_𝑛1(𝑠)=(𝑇²_𝑛−1𝑠²𝑤+1)^𝑛−1(𝑠). (3.8) As was shown in previous paragraph, the function 𝑤₁(𝑡) will be represented by a unity step function and by an alternating function −cos(𝑡/𝑇₁) (which can be also written as sin(𝑡/𝑇₁+𝜙₁)=sin(𝜔₁𝑡+𝜙₁), where the angular frequency 𝜔₁=1/𝑇₁). This represents the input for the transfer function 1/(𝑇²₂𝑠²+1). Its output 𝑤₂ will be represented by the sum of the output of this transfer function for a unity step input 1/𝑠 and the output of this transfer function for the input sin(𝑡/𝑇₁+𝜙₁)=sin(𝜔₁𝑡+𝜙₁). The output generated by the unity step input will be represented once again by an unity step function and by an alternating function with angular frequency 𝜔₁=1/𝑇₁; the output generated by this transfer function for the alternating input with angular frequency 𝜔₁ will be represented by a sum of two alternating functions with angular frequencies 𝜔₁ and 𝜔₂, respectively. It results that 𝑤₂ can be written as 𝑤₂(𝑡)=1+𝑎₁⁽²⁾sin(𝜔₁𝑡+𝜙₁⁽²⁾)+𝑎₂⁽²⁾sin(𝜔₂𝑡+𝜙₂⁽²⁾) (3.9) (there are just three terms because alternating functions with the same angular frequency 𝜔₁ were grouped together in 𝑎₁⁽²⁾sin(𝜔₁𝑡+𝜙₁)).

The whole procedure can continue by analyzing the output 𝑤₃ of the transfer function 1/(𝑇²₃𝑠²+1) for the input represented by 𝑤₂(𝑡). It results that a unity step output 1/𝑠 will appear once again, together with three alternating components with angular frequencies 𝜔₁, 𝜔₂, and 𝜔₃, respectively. This means that 𝑤₃(𝑡) can be written as 𝑤₃(𝑡)=1+𝑎₁⁽³⁾sin(𝜔₁𝑡+𝜙₁⁽³⁾)+𝑎₂⁽³⁾sin(𝜔₂𝑡+𝜙₂⁽³⁾)+𝑎₃⁽³⁾sin(𝜔₃𝑡+𝜙₃⁽³⁾), (3.10) where 𝜔₃=1/𝑇₃. In the general form, the output 𝑤_𝑘(𝑡) can be written as 𝑤_𝑘(𝑡)=1+𝑎₁^(𝑘)sin(𝜔₁𝑡+𝜙₁^(𝑘))+⋯+𝑎_𝑘^(𝑘)sin(𝜔_𝑘𝑡+𝜙_𝑘^(𝑘)) (3.11) or 𝑤_𝑘(𝑡)=1+^𝑘_𝑖=1𝑎_𝑖^(𝑘)sin(𝜔_𝑖𝑡+𝜙_𝑖^(𝑘)) (3.12) (it can be noticed that the coefficient and phase corresponding to a certain angular frequency 𝜔_𝑖 are changed from 𝑎_𝑖^(𝑘−1),𝜙_𝑖^(𝑘−1) to 𝑎_𝑖^(𝑘),𝜙_𝑖^(𝑘) at each step due to the mixture of alternating functions with the same angular frequency 𝜔_𝑖 generated by the transfer function 1/(𝑇²_𝑘+1) for input represented by a sum of the unity step function and by alternating functions with angular frequencies 𝜔_𝑗, 𝑗<𝑖 (as was shown for 𝑤₂(𝑡), where certain functions with the same angular frequency were grouped together)). Finally, the output of the signal processing system 𝑤(𝑡) (at step 𝑛+1) will be represented by function 𝑤_𝑛 which can be written as 𝑤_𝑛+1(𝑡)=1+𝑎₁^(𝑛+1)sin(𝜔₁𝑡+𝜙₁^(𝑛+1))+⋯+𝑎^(𝑛+1)_𝑛+1sin(𝜔_𝑛+1𝑡+𝜙_𝑘^(𝑛+1)) (3.13) or 𝑤(𝑡)=𝑤_𝑛+1=1+^𝑛+1_𝑖=1𝑎_𝑖^(𝑛+1)sin(𝜔_𝑖𝑡+𝜙_𝑖^(𝑛+1)). (3.14) By integrating this function 𝑤(𝑡) on a time interval 𝑇 represented by a multiple of all time periods 2𝜋𝑇₁,2𝜋𝑇₂,…,2𝜋𝑇_𝑛 (this means that any ratio 𝑇_𝑖/𝑇_𝑗 should be expressed by a rational number) the influence of all alternating components vanishes. As a consequence, the result 𝑢(𝑇) of the integration will be 𝑢(𝑇)=2𝜋𝑇 (3.15) for a received signal corresponding to 𝑦(𝑡)=(1/𝑁!)𝑡^𝑁 which has been processed by 𝐻(𝑠). This means that the result 𝑢(𝑇) is proportional to the derivative of order 𝑁 of the received signal (supposed to have the form 𝑦(𝑡)=(1/𝑁!)𝑡^𝑁).

The analysis of the action of transfer function 𝐻(𝑠) upon an input represented by 1𝑦(𝑡)=𝑠^𝑀,𝑀<𝑁 (3.16) (corresponding to the time function 𝑦(𝑡)=(1/(𝑁−𝑀)!)𝑡^𝑁−𝑀) can be considered as 1𝑤(𝑠)=𝐻(𝑠)𝑠^𝑀=𝐻(𝑠)𝑠^𝑁−𝑀1𝑠^𝑁=𝑠^𝑁−𝑀1𝐻(𝑠)𝑠^𝑁. (3.17) According to Laplace transformation properties, the operator 𝑠^𝑁−𝑀 corresponds to a derivative procedure applied (𝑁−𝑀) times upon a certain function. Since 𝐻(𝑠)1/𝑠^𝑁 corresponds to the sum of a step function and a set of alternating components previously presented, it results that the output 𝑤(𝑡) corresponding to the input 1/𝑠^𝑁−𝑀 can be represented as 𝑑𝑤(𝑡)=^𝑁−𝑀𝑑𝑡^𝑁−𝑀(1+^𝑛+1_𝑖=1𝑎_𝑖^(𝑛+1)sin(𝜔_𝑖𝑡+𝜙_𝑖^(𝑛+1))). (3.18) But the derivatives of a constant function equal zero, and the derivatives of alternating functions of certain angular frequency are represented also by alternating functions with the same angular frequency. This means that the integration of this function on the time interval 𝑇 represented by a multiple of all time periods will generate a null result. As a consequence, if the received signal 𝑦(𝑡) can be written as a sum 𝑦(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑁𝑡^𝑁 (3.19) (according to properties of Taylor series), the action of a transfer function 𝐻(𝑠) is represented by 𝐻(𝑠)=𝑠^𝑁𝑛+1_𝑘=11(𝑇²_𝑘𝑠².+1) (3.20) The integration of the output 𝑤(𝑡) of this filtering device on a time interval 𝑇 represented by a multiple of all time periods 𝑇_𝑖 will generate a result proportional to 𝑁!𝑐_𝑁, being proportional to the derivative of order 𝑁 of the received signal. Thus this signal processing method based on a set of oscillating second-order systems and an integrator can generate a sequence of sampling values corresponding to the 𝑁th derivative of the received signal at the end of each working interval, the derivative of order 𝑁 being transformed into a time series suitable for modeling, control, and/or data acquisition. Filtering properties are still good, as long as any angular frequency corresponding to noise (𝑛(𝑡) is several times greater than any angular frequencies 𝜔_𝑖 from the set 𝜔₁,…,𝜔_𝑛+1.

We must point the fact that certain limitations appear as the order 𝑁 of the derivative to be estimated increases.

(i) For a great number of alternating functions, it is quite possible for the maximum value of the sum of alternating functions of angular frequencies 𝜔_𝑖 (part of 𝑤(𝑡)) to become several times greater than the constant part of 𝑤(𝑡) (corresponding to the derivative to be estimated); so the resolution of the method decreases (the voltage range of estimated derivative decreases since it represents a small part of the maximum voltage allowed by electronic devices).

(ii) Taylor series for 𝑦(𝑡) was restricted to (𝑁+1) terms. This means that the influence of derivatives of order 𝐾>𝑁 on the integration period 𝑇 was neglected. This approximation should be carefully checked in any signal analysis.

4. Conclusions

This paper has presented a possibility of obtaining the derivatives of the received electrical signal using a filtering device consisting of a sequence of certain oscillating second-order systems and an integrator. The oscillating systems are working on a time period for filtering a received electrical signal, with initial null conditions. The output of this system is integrated over a time period corresponding to a multiple of all time periods of the second-order systems which are part of the signal processing device (at the end of this period the integrated signal being sampled). The influence of all alternating components is rejected due to the integration performed on a multiple of all time periods, and thus the final result corresponds to the integration of a constant function which is proportional to the derivative having to be estimated. The proposed method has shown that present-day methods for computing in a robust manner the first derivative of a received signal (using an oscillating system working on a limited time interval and a supplementary differentiation method) can be extended to the robust computation of higher-order derivatives of the received signal by using a specific set of second-order oscillating systems (working also on limited time intervals) so as estimative values for higher order derivatives are to be directly generated (avoiding the necessity of additional differentiation or amplifying procedures, which represent a source of supplementary errors in present-day methods). It can be used for decreasing the phase delay for signal processing methods, but without using a weighted sum of real and filtered derivatives of the received signal, as in [3]. The proposed method is similar to other attempts for computing the derivative without using additional procedures (see also [4] where the need of sampled and digitised data is avoided).

In future studies, the analysis will continue by trying to use nonlinear dynamical equations able to generate practical test functions for estimating in robust manner and with greater accuracy the derivatives of the signal transmitted by dynamic systems (see [5] for general properties of nonlinear differential equations able to generate practical test functions). The results presented in this paper can be extended for modeling phenomena described by partial differential equations as traveling waves and wavelets inside certain materials [6, 7] or as general transformations of waves when the material reference system is changed (see [8] for classical field of interaction and [9] for quantum field of interaction) due to advantages presented by the accurate estimation of higher-order derivatives. It could be also used for integrating derivative procedures in procedures of analyzing time series (as presented in [10–12]) or directly into machine learning algorithms or emergent dynamic routing by establishing certain sampling moments (see [13, 14]).