where (, )gt x is the IR of the system [55]. The IR characterizes completely the system. In the time-invariant case (, )( ) gt gt xx =- and the input-output relation assumes a convolutional form [55]-[57] () () () . yt =-xxx dx 3 #+3 - Remark II.1. If (, )(/)/, ,, then (7) gt gt xx ! R0xx= + t becomes a multiplicative (Mellin) convolution and the corresponding systems are not time-invariant, but they are essentially scale invariant. Among these types of systems we can consider the LS based on the Hadamard [31] and quantum derivatives [58], [59]. Returning to (8), if () ,, , yt Gs et R st () = with () = gt6 ! xt es CRst () ,, = !! then (9) Gs L ()@ the LT of the IR that is the TF. Therefore, 1) The exponentials are the eigenfunctions of the LTIS and the eigenvalues, G(s) are the LT of their IR [60]. 2) If the system is causal, then its IR is a right function and () ht ,0= for t 01 implying that () c , has a ROC defined by ()Re sa R2 ! . 3) Similarly, if the system is anti-causal, then () ,, ht = a 00tfor 2 Hsa defined by ()Re sb R1 ! . 4) Consider a bilateral system that is the sum of a causal and an anti-causal, () Re () Gs Gs Gs=+ (). , ca () If its ROC is non void, as b11 the corresponding IR is two-sided, () () (). gt gt gt =+ ca 5) If the region of convergence (ROC) of G(s) contains the imaginary axis, then we can set sj ,,R!~~= so that the response of a LTIS to a sisoid is also a sisoid with the same frequency. In this case, the LT becomes the FT and originates the FR, denoted by ().Gj~ It is frequently represented by the Bode diagrams that are logarithm plots of the amplitude, () ()AG j;; z~ gGj~ ~~= phase, () = ar (). Therefore, we require that the IR, (),gt [23] ■ is continuous almost everywhere, ■ has bounded variation, ■ is of exponential order. Concerning the stability of a system, we prefer usually the BIBO (bounded input-bounded output) stability criterion. This implies that the IR is absolutely integrable (AI). Therefore, any stable LS with TF given by G(s) has frequency response, ().Gj~ For a given stable (unstable) causal there is always a unstable (stable) anti-causal with the same G(s) as TF, but different ROC. These considerations created a framework for the definition of LS through the IR and corresponding TF. 40 IEEE CIRCUITS AND SYSTEMS MAGAZINE and Gs = ax + ` x + a sa sa 6) Phase lead/lag compensator [24] () implying that () has a ROC Hsc gt (8) B. Some Examples of TF The above considerations do not suggest any particular form that the TF (or FR) can assume. The practical problem at hand can give insights into the one we must choose. In Engineering, it is frequent to use the information depicted by means of the Bode diagrams. Some interesting models are well-known and were suggested in different studies of electromagnetic media or electrical circuits (with suitable ROC): 1) Oscillator Gs = 2 () ,; sp 1 + 1 p ! R+ 2) Cole-Cole dielectric model () ,; Gs = a + sp Gs = sp s a + a Gs ,; sp 1 = () + a p ! R+ Gs = 6 +() @ 1 s 1 x p ! R+ 3) Causal highpass filter () ,; p R! + 4) Cole-Davidson dielectric model [61] () 5) Havriliak-Negami dielectric model [62] () a b ; j ,, ,. a ! R+ 21 These examples suggest we can introduce some non common TF. Let N and M be positive integers and the polynomial coefficients ak and ,, ,,bk 01= f be k real numbers. We introduce also the parameters ak and bk k 01 f= ,, ,, that, without loss of generality, we can assume to form positive real increasing sequences. 1) The most used model is the rational TF corresponding to a continuous-time autoregressivemoving average (CT-ARMA) model: M Gs () = N / / k=0 k=0 where the N and M are the orders of the model that correspond to the degrees of denominator and numerator polynomials, respectively. 2) The fractional continuous-time autoregressivemoving average (CT-FARMA) [63]. SECOND QUARTER 2022 as bs k , k k (10) k

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