New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties

In this paper we introduce new discrete-time derivative concepts based on the bilinear (Tustin) transformation. From the new formulation, we obtain derivatives that exhibit a high degree of similarity with the continuous-time Grünwald-Letnikov derivatives. Their properties are described highlighting one important feature, namely that such derivatives have always long memory.

Journal of Advanced Research 25 (2020) 1–10 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties Manuel D Ortigueira a,⇑, J.A Tenreiro Machado b a b CTS–UNINOVA and NOVA Faculty of Sciences and Technology of Nova University of Lisbon, Campus da FCT da UNL, Quinta da Torre, 2829 – 516 Caparica, Portugal Institute of Engineering, Polytechnic of Porto, Dept of Electrical Engineering, Porto, Portugal h i g h l i g h t s g r a p h i c a l a b s t r a c t  The paper introduces new discrete- time derivative concepts based on the bilinear transformation  Forward and backward derivatives having a high degree of similarity with the usual continuous-time Grunwald-Letnikov derivatives are introduced  Corresponding linear discrete-time systems are defined a r t i c l e i n f o Article history: Received 18 December 2019 Revised 12 February 2020 Accepted 16 February 2020 Available online 25 February 2020 Keywords: Discrete-time Fractional derivative Time scale bilinear transformation a b s t r a c t In this paper we introduce new discrete-time derivative concepts based on the bilinear (Tustin) transformation From the new formulation, we obtain derivatives that exhibit a high degree of similarity with the continuous-time Grünwald-Letnikov derivatives Their properties are described highlighting one important feature, namely that such derivatives have always long memory Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction Peer review under responsibility of Cairo University ⇑ Corresponding author E-mail addresses: mdo@fct.unl.pt (M.D Ortigueira), jtm@isep.ipp.pt (J.A Tenreiro Machado) The continuous/discrete unification introduced by Hilger [3] led to the definition of two discrete-time fractional derivatives, nabla and delta, that are essentially the usual incremental ratia In [11] the fractional versions of such derivatives were proposed together with the corresponding differential equations for discrete-time linear systems These versions have stability domains that are defined https://doi.org/10.1016/j.jare.2020.02.011 2090-1232/Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 2 M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 by the Hilger circles [11] Such domains not coincide with the stability domain of traditional causal discrete-time systems that is defined relatively to the unit circle As it is well known, the stability domain of causal continuous-time system is the right half complex plane (HCP) Therefore, there is a relation between the left (right) HCP and the interior (exterior) of the unit disk that can be expressed by a particular case the bilinear (or, Möbius) transformation Such map was proposed by Tustin [17] and used since then for the discrete-time approximation of continuous-time linear systems without considering the definition of any discrete-time derivative [18,13,14] Hereafter we formulate a discrete-time fractional calculus that mimics the corresponding continuous-time version, but that is fully autonomous The motivation for this study is to have in the discrete-time domain the tools and results available for the continuous-time fractional signals and systems [10] Another important characteristic of the proposed derivatives is that they are suitable to be implemented through the FFT with the corresponding advantages, from the numerical and calculation time perspectives The new derivatives On the Z and Discrete-Time Fourier Transforms In the following we consider that our domain is the time scale Th ẳ hZị ẳ f ; nh ; À2h; Àh; 0; h; 2h; ; nh; g with h Rỵ , that is called graininess [1,11] In the following the symbol n will represent any generic point in T In engineering applications where discrete signals are result of sampling continuoustime signals, h is the sampling interval Let xðnÞdenote any function defined on T, leaving implicit the graininess, unless it is convenient to display it The Z transform (ZT) is defined by X Xzị ẳ Z ẵxnị ẳ xnị zn ; z C: 1ị nẳ1 In some scientific domains, as Geophysics, z instead of zÀ1 is used In some domains, the ZT is often called ‘‘generating function” or ‘‘characteristic function” Definition (1) is the bilateral ZT that leads to the particular case of the unilateral ZT, defined by X X u zị ẳ xnịzn ; nẳ0 often adopted in the study of systems The existence conditions of the ZT are similar to those of the bilateral Laplace transform (LT) [7,15,16] Z Ysị ẳ Lẵytị ẳ ytị e À1 Àst dt; s C:  If the signal is right (i.e., xnị ẳ 0; n < n0 Z), then the ROC is the exterior of a circle centered at the origin (r ỵ ẳ 1): jzj > r À  If the signal is left (i.e., xnị ẳ 0; n > n0 Z), then the ROC is the interior of a circle centered at the origin (r ẳ 0): jzj < rỵ  If the signal is a pulse (i.e., non null only on a finite set), then the ROC is the whole complex plane, possibly with the exception of the origin In the ROC, the ZT defines an analytical function It should be noted that the ROC is included in the definition of a given ZT This means that we may have different signals with the same function as ZT, but different ROC If the ROC contains the unit circle, then by making pffiffiffiffiffiffiffi z ¼ eix ; jxj < p; i ¼ À1, we obtain the discrete-time Fourier transform, which we will shortly call Fourier transform (FT) This means that not all signals with ZT have FT The signals with ZT and FT are those for which the ROC is non-degenerate and contains the unit circle (r À < 1; r þ > 1) For some signals, such as sinusoids, the ROC degenerates in the unit circumference (r ẳ rỵ ¼ 1), and there is no ZT Definition The inverse ZT can be obtained by the integral defined by xnị ẳ 2pi I Xzịzn1 dz; 4ị c where c is a circle centred at the origin, located in the ROC of the transform, and taken in a counterclockwise direction In such situation the integral in (4) converges uniformly The calculation uses the Cauchy’s theorem of complex variable functions [16] Definition For functions that have a ROC including the unit circle or for functions having a degenerate ROC, as it is the case of the periodic signals, it is preferable to work with the discrete-time Fourier transform that can be obtained from the Z transform through the transformation z ẳ eix ; jxj < p Xeix ị ẳ X xnhịeixn 5ị nẳ1 2ị with the inversion integral Therefore, the existence conditions can be stated as follows If function xðnÞ is such that there are finite positive real numbers, r and rỵ , for which X jxnịjr n < xnị ẳ 2p Z p p Xðeix Þeixn dx ð6Þ meaning that a discrete-time signal can be considered as a synthesis of elementary sinusoids Xðeix Þeixn dx n¼0 and À1 X Definition A discrete-time signal xðnÞ is called an exponential order signal if there exist integers n1 and n2 , and positive real n numbers a; b; A, and B, such that A an1 < jxðnÞj < B b for n1 < n < n2 For these signals the ZT exists and the ROC is an annulus centred at the origin, generally delimited by two circles of radius r and rỵ , such that r < jzj < rỵ However, there are some cases where the annulus can become infinite: 3ị jxnịjrnỵ < 1; nẳ1 then the ZT exists and the range of values for which those series converge defines a region of convergence (ROC) that is an annulus We must have in mind that this condition is sufficient, but not necessary The signals that verify (3) are the exponential order signals [16] Remark In fractional applications, we have branchcut points at z ẳ ặ1 Therefore, we have to avoid them by using an integration circle in (4) having a radius, r, greater (smaller) than for the causal (anti-causal) cases We have then f nị ẳ rn 2p Z p Àp Xðreix Þeixn dx ð7Þ M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 Forward and backward derivatives based on the bilinear transformation Df zn ¼ À zÀ1 ; h þ zÀ1 ð8Þ where h is the sampling interval, s is the derivative operator associated with the (continuous-time) Laplace transform and zÀ1 the delay operator tied with the Z transform Definition Let xðnhÞ be a discrete-time function, we define the order forward bilinear derivative DxðnhÞ of xðnhÞ as the solution of Dxnhị ỵ Dxnh hịị ẳ ½xðnhÞ À xðnh À hފ h ð9Þ Definition Similarly, we define the order backward bilinear derivative DxðnhÞ of xnhị as the solution of Dxnh ỵ hị ỵ Dxnhị ẳ ẵxnh ỵ hị xnhị h 10ị Definition The bilinear exponential es ðnhÞ is the eigenfunction of Eq (9) or (10) If we set xnhị ẳ es nhị; ynhị ẳ ses nhị; s C, with es 0ị ẳ 1, then es nhị ẳ  n þ hs ; n Z; s C: À hs ð12Þ The operator Hf ðzÞ defined by The Tustin transformation is usually expressed by s¼ À z1 n z : h ỵ z1 11ị Properties of the bilinear exponential es ðnhÞ  When n ! 1, this exponential is – Increasing, if ReðsÞ > 0, – Decreasing, if ReðsÞ < 0, – Sinusoidal, if ReðsÞ ¼ 0, with s – 0, – Constant equal to 1, if s ¼ 0,  It is real for real s,  It is positive for s ¼ jxj < 2h ; x R,  It oscillates for s ¼ jxj > 2h ; x R Following the procedure in [11] we could use this exponential to construct a bilinear discrete-time Laplace transform However, formula (8) suggests having z ẳ 2ỵhs that leads to the Z transform, 2Àhs since such transformation sets the unit circle jzj ¼ as the image of the imaginary axis in s, independently of which value of h is used Therefore, the exponential has the usual properties  When n ! 1, this exponential is – Increasing, if jzj > 1, – Decreasing, if jzj < 1, – Sinusoidal, if jzj ¼ 1, with z – 1, – Constant equal to 1, if z ¼ 1,  It is real for real z,  It is positive for z ¼ x > 0; x R,  It oscillates for z – Rỵ In what concerns to the derivative definitions, instead of considering (9) or (11), as in [1,11] where the nabla (causal) and delta (anti-causal) derivatives were introduced, we start from the ZT formulations Definition Let z C and h Rỵ Consider the discrete-time exponential function, zn ; n Z We define the forward bilinear derivative (Df ) as a discrete-time linear operator such that Hf zị ẳ z1 ; h þ zÀ1 jzj > 1; ð13Þ will be called foward transfer function (TF) of the derivative, borrowing the nomenclature used in signal processing [7,16] Definition The backward bilinear derivative (Db ) is defined as a discrete-time linear operator verifying Db zn ẳ z1 n z : h zỵ1 14ị where Hb zị is the operator Hb zị ẳ z1 ; h zỵ1 jzj < 1; 15ị called backward transfer function of the derivative By the repeated application of the above operators we obtain the forward and backward derivatives for any positive integer order However, we introduce the corresponding fractional derivatives, valid for any real order Definition Let a R The a-order forward bilinear fractional derivative is a discrete-time operator with TF Hf zị ẳ  a z1 ; h ỵ z1 jzj > 16ị such that Daf zn ẳ  a z1 zn ; h 1ỵz jzj > 1: ð17Þ Definition 10 The backward bilinear fractional derivative has TF Hb zị ẳ  a z1 ; h zỵ1 jzj < 18ị such that Dab zn ẳ  a z1 zn ; h zỵ1 jzj < 1: ð19Þ Having defined the derivative of an exponential we are in conditions of defining the derivative of any signal having ZT Definition 11 From (4) and (17) we conclude that, if xðnÞ is a function with Z transform XðzÞ, analytic in the ROC defined by z C : jzj > a; a < 1; then Daf xnị ẳ 2pi I  c z1 h ỵ z1 a Xzịzn1 dz; 20ị with the integration path outside the unit disk This implies that h i 2 À zÀ1 a Z Daf xðnÞ ẳ Xzị; h ỵ z1 jzj > 1: 21ị Definition 12 Let xðnÞ be a function with Z transform XðzÞ, analytic in the ROC defined by z C : jzj < a; a > 1: We define Dab xnị ẳ 2pi I  c z1 h zỵ1 a Xzịzn1 dz; 22ị M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 with the integration path inside the unit disk and the branchcut line is a segment joinning the points z ẳ ặ1 This implies that  a  à zÀ1 Z Dab xnị ẳ Xzị; h zỵ1 jzj < 1: 23ị Remark We must note that: Df ;b eixn ¼ tan h eixn ; jxj < p; 250 200 150 100 50 -400 -200 200 400 Derivative of order 0.5 1z1 1ỵz1 10 Example In Figs and we represent the bilinear causal derivatives of orders a ¼ 0:5 and a ¼ 0:8 of a triangle function According to what we just wrote we can extend the above definitions to include sinusoids We define the derivative of xnị ẳ eixn ; n Z, through x!a Triangle function 300 Magnitude the unit disk, in z, and the left half-plane, in s ¼ 2h Fig Integration path modification for causal derivative Magnitude In (16) and (17) we have two branchcut points at z ẳ ặ1 The corresponding branchcut line is any line connecting these values and being located in the unit disk The simplest is a straight line segment (see Fig 1) In (18) and (19) we have the same branchcut points, but with branchcut line(s) lying outside the unit disk For simplifying, we can use two half-straight lines starting at z ẳ ặ1 on the real negative and positive half lines, respectively (see Fig 1) In both previous cases, we can extend the domain of validity to include the unit circumference, z ẳ eixn ; jxj 0; pị, with exception of the points z ẳ ặ1 In these cases the integration path in (4) must be deformed around such points, as it can be seen at Fig for the causal case This deformation is very important in applications where we use the fast Fourier transform (FFT) In such cases a small numerical trick can be used: push the branchcut points slightly inside (outside) the unit circle, that is, to z ẳ ỵ e and z ẳ e1 e; ỵ eị, with e being a small positive real number The ROC is independent on the scale graininess, h, and consequently we can establish a one to one correspondence between -5 -10 -1000 -500 500 1000 t Fig Derivative of order a ¼ 0:5 of a triangle function with h ẳ 24ị Triangle function 300 independently of considering the forward or backward derivatives Magnitude Definition 13 For a function having discrete-time Fourier transform (6), the bilinear derivative is expressed as: 250 200 150 100 50 -400 -200 200 400 Derivative of order 0.8 Magnitude -2 -4 -6 -1000 -500 500 1000 t Fig Derivative of order a ¼ 0:8 of a triangle function with h ẳ Df ;b xnị ¼ Fig ROC for causal and anti-causal derivatives and branchcut points and lines 2p Z p Àp Xðeix Þ x!a tan eixn dx h ð25Þ that is suitable for implementations with the FFT According to the existence conditions of the FT, we can say that, if xðnÞ is absolutely sommable, then the derivative (25) exists 5 M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 Properties of the derivatives we conclude that the TF in (16) and (18) can be expressed as power series, We present the main properties of the above derivatives The proofs are easily obtained from the corresponding FT  a X 1 À zÀ1 ¼ wak zÀk ; jzj > 1; ỵ z1 kẳ0 Linearity The linearity property of the fractional derivative is straightforward from the above formulae Time shift The derivative operators are shift invariant:  Df ;b xn n0 ị ẳ Df ;b xmịmẳnn where wak ; k ẳ 0; 1; , is the inverse ZT of 1z1 1ỵz1 a and represents the impulse response (IR) corresponding to the TF Let the discrete convolution be defined by xðnÞ Ã ynị ẳ  X xkịyn kị; n Z: k¼À1 This property is immedately obtained from (20) or (22) as a consequence of the shift property of the Z transform Z ẵxn n0 ị ẳ Xzịzn0 ; n0 Z Additivity and Commutativity of the orders Let a and b be two real values Then h i Da Db xnị ẳ Db Da xnị ẳ Daỵb xnị To prove this relation it is enough to observe that, in the forward  a  b  aỵb 1 1z1 case, we have 2h 1z ẳ 2h 1z and that the h 1ỵz1 1ỵz1 1ỵz1 product is commutative For the backward derivative, the situation is identical Neutral element This comes from the additivity property by putting b ¼ Àa, h i Daf ;b Df ;ba f nị ẳ D0 f nị ẳ f ðnÞ: h i h i Dj Da Db xðnÞ ẳ Djỵaỵb xnị ẳ Daỵbỵj f nị ẳ Da Dbỵj xðnÞ as a consequence of the additivity Derivative of the convolution P Let xnị ynị ẳ kẳ1 xðkÞyðn À kÞ be the discrete-time convolution Its ZT is XzịYzị Since we can write 1z1 h 1ỵz1 a o Xzị Yzị a n o Yzị ; ẳ Xzị 2h 1z 1ỵz1 ẵXzịYzị ẳ n 1z1 h 1ỵz1 wak ẳ k aịk 1ịk aịk X aịm 1ịkm aịkm ẳ ; k Zỵ0 : k! k! m! k mị! mẳ0 a we conclude that Df ẵxnị ynị ẳ Df xnị ynị ẳ xnị Df ynị : For the backward derivative of the convolution, we obtain an identical result Time formulations ð26Þ Performing this discrete convolution we obtain the following results The sequence wak ; k ¼ 0; 1; Á Á Á, that is obtained as the discrete convolution of two causal sequences, is causal and, therefore, is null for k < We will assume it below For any a R, we have wÀk a ẳ 1ịk wak k Zỵ0 This result is very important because it states the existence of inverse derivative Inverse element The existence of neutral necessarily implies that there is always an inverse element: for every a order derivative, there is always a Àa order derivative given by the same formula and so it does not need joining any primitivation constant We adopt the designation ‘‘derivative” for positive orders and ‘‘anti-derivative” for negative ones Associativity of the orders Let a; b, and j be three real values Therefore, we can write:  The IR, wak ; k ¼ 0; 1; Á Á Á, is obtained as the discrete convolution of the binomial coefficients sequence: ð27Þ The proof is immediate from (26) Initial value From the initial value theorem of the ZT, it is immediate that wa0 ¼ independently of the order Final value Let a From the final value of the ZT,  a z1 wa1 ẳ limz 1ị z!1 1ỵz that is 0, if a 0, and 2, if a ¼ À1 For a < À1 the sequence grows up to For a > we apply (27) If a R but a R Z , then wak ẳ 1ịk k aịk X aịm kịm 1ịm ; k Zỵ0 k! mẳ0 a k ỵ 1ịm m! 28ị Letting a ¼ N in (28), we get wNk ¼ ðÀ1Þk minðk;NÞ Nịk X Nịm kịm 1ịm ; k Zỵ0 k! mẳ0 N k ỵ 1ịm m! 29ị If a ZÀ , set a ¼ ÀN; N Zỵ We use wN k ẳ k X Nịm Nịkm 1ịm m! k mị! mẳ0 to obtain wN k ẳ mink;Nị Nịk X Nịm kịm 1ịm ; k Zỵ0 : k! mẳ0 N k ỵ 1ịm m! ð30Þ Comparing (30) with (29), we conclude that they differ only in In the previous sub-section, we introduced the derivatives using a formulation based on the ZT Here we obtain the corresponding time framework, getting formulae similar to the GrỹnwaldLetnikov derivatives From the binomial series [2] ặ wịa ẳ X ầ1ịk aị k kẳ0 k! wk ; jwj < 1; the factor 1ịk ; k Zỵ A recursion  à  À1 a P Let Wzị ẳ Z wak ẳ 1z As DWzị ẳ n n 1ịwan1 zn 1ỵz1  a     1 1ỵz1 D 1z , after some algebraic and DWzị ẳ a 1z 1ỵz1 1z1 1ỵz1 manipulation we obtain: M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10   2a a wak ẳ wak1 ỵ w ; k kÀ2 k k P 2; ð31Þ with wa0 ¼ and wa1 ¼ À2a This recursion shows that, if a < 0, then wak is a positive sequence As consequence, attending to (27), the sequence corresponding to positive orders is always oscillating: successive values have different sign Relation with the Hypergeometric function The first factor in (28), namely ðÀ1Þk ðak!Þk , represents the binomial coefficient, while the second is a sequence from the Gauss Hypergeometric function fn ẳ m k X aịm kịm 1ị ẳ2 F ðÀa; Àk; À k À a; À1Þ; n Zỵ0 : a k ỵ 1ị m! m mẳ0 32ị N ẳ k 11 The coefficient of ak decreases with increasing k P For simplifying the proof, let wak ¼ km¼1 pm am and P kÀ1 wakÀ1 ¼ m¼1 qm am From (31) we conclude that pk ¼ À 2ka qkÀ1 , because the second term in the right hand side of (31) only affects the lower order coefficients of the polynomial As this happens for k ¼ 2; 3; Á Á , we can write pk ẳ 1ịk ẳ 43 a3 þ 23 a ð35Þ that is a polynomial of degree n À in a Inserting (35) into (34) and the resulting expression in (28), we obtain wak ẳ 1ịk g k ; k ¼ 2a2 wÀ3 a wÀ4 a wÀ5 a wÀ6 a wÀ7 a wÀ8 a Similarly, for positive orders a ¼ 0:2k; k ¼ 1; 2; Á Á Á ; 6, the bilinear sequences are plotted in Fig obtain [18] gn ¼ wÀ2 a 2k k! that decreases with k In fact, after simplifying the common fack tors between and k!, the denominator is the largest odd divisor of n! The numerator is always a power of corresponding to the factors that were not used when removing the common factors (see below 37) [6] Example We are going to present wỈN for some values of N Zỵ k and for any real order obtained by recursive computation Definition 14 In agreement with the meaning attributed to the sequence wak ; k ¼ 0; 1; Á Á Á, we define the a-order forward and backward derivatives as aị Df xnị ẳ  a X wa xn kị h kẳ0 k 38ị and aị Db xnị ẳ eiap  a X wa xn ỵ kị: h kẳ0 k 39ị The use of the terms forward and backward is due to the ‘‘time flow”, from past to future or the reverse [10] This terminology is the reverse of the one used in some mathematical literature We can remove the exponential factor, eiap , in (39) to obtain a right derivative In the following we will consider the causal derivative (38) represented by the simplified notation Da and with ZT given by (21) Other properties  The first is causal while the second is anti-causal aị In fact, if xnị ẳ 0; n < n0 Z, then Df xnị ẳ 0; n < n0 and we obtain ðaÞ Df xðnÞ ¼  a nÀn X0 a w xðn À kị h kẳ0 k 40ị that is null for n < n0 For the backward the proof is similar using xnị ẳ 0; n > n0 Z, leading to M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 Magnitude 0.8 0.6 0.4 0.2 Magnitude 1.8 1.6 1.4 Magnitude Alpha= -0.5 50 40 30 20 10 0 50 100 150 200 250 200 250 200 250 200 250 200 250 Alpha= -1 50 100 150 Alpha= -1.5 50 100 150 Magnitude Alpha= -2 1000 800 600 400 200 0 50 100 150 Magnitude Alpha= -2.5 20000 15000 10000 5000 0 50 100 150 k Fig Bilinear sequences corresponding to orders a ¼ À0:5k; k ¼ 1; 2; Á Á Á ; 5: Magnitude Alpha= 0.25 0.5 - 0.5 -1 50 100 150 200 250 150 200 250 150 200 250 150 200 250 Magnitude Alpha= 0.5 0.5 - 0.5 -1 50 100 Magnitude Alpha= 0.75 1.5 0.5 -0.5 -1 -1.5 50 100 Magnitude Alpha= -1 -2 -3 50 100 k Fig Bilinear sequences corresponding to orders a ¼ 0:25k; k ¼ 1; 2; Á Á Á ; 4; 8 M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 110 aị Db xnị ẳe iap  a nỵn X0 a wk xn ỵ kị h kẳ0 41ị nẳ0 dnị ẳ n0 ytị ẳ : & nP0 n 1, as expected According to the n above properties, we can obtain the fractional derivative of the unit step function We have 2 eðnÞ ẳ w1 n ỵ dnị: Knowing that ẳ s, we can write meaning that YðsÞ is the LT of the (continuous-time) derivative of xðtÞ This relation states a compatibility between the new formulation described above and the well known results from the continuous-time derivative formulation [12] If we used the backward formulation, we would obtain the same result, but with a ROC valid for ReðsÞ < Taking in account the above equations and (38), we conclude that, for t R, we can write:  a X aị Df xtị ẳ lim wak xt khị: h!0 h kẳ0 46ị Similarly, we can obtain from (39)  a X aị Db xtị ẳ eiap lim wak xt ỵ khị h!0 h kẳ0 47ị Relations (46) and (47) state two new ways of computing the continuous-time fractional derivative that are similar to the Grünwald-Letnikov derivatives However, it may be interesting to remark that we can compute derivatives with (44) instead of (22) The above derivatives lead us to consider systems defined by constant coefficient differential equations with the general form  aÀ1  a 2 D enị ẳ wna1 ỵ wan : h h a  Fractional derivative of the w function We are interested in computing the derivative of wan , for any a with n Z From (42) and the additivity property, we can write  a !  aỵb 2 wan ẳ wnaỵb enị h h N M X X ak Dak ynị ẳ bk Dbk unị kẳ0 43ị with aN ẳ The operator D is the forward (or backward) derivative above defined, assuming orders ak and bk ; k ¼ 0; 1; 2; Á Á Á The coefficients ak and bk ; k ¼ 0; 1; 2; Á Á Á are real numbers and N and M represent any given positive integers Let gðnÞ be the IR of the system defined by (48) that is, v nị ẳ dnị The output is the convolution of the input and the IR, ð49Þ If v ðnÞ ¼ z , then the output is given by: n " ynị ẳ zn X # gnịzn : nẳ1 Backward compatibility Often, discrete-time systems are viewed as mere approximations to the continuous-time counterpart However, and as seen above, the discrete-time systems exist by themselves and have properties that are independent from, although similar to, the continuous-time analogues Nonetheless, this observation does not prevent us from establishing a continuous path from each other In fact, we can go from the discrete into the continuous domain by reducing the graininess To see it, let us return to (20) and rewrite it as 48ị kẳ0 ynị ẳ gnị v nị: that leads to  b  à Db wan ¼ wnaỵb enị: h Xsị ẳ Lẵxtị The differential discrete-time linear systems Consequently Db Da dnị ẳ Db and 45ị Ysị ¼ sa XðsÞ; ReðsÞ > 0; ; jzj > 1: À zÀ1  a wan eðnÞ; h Ysị ẳ Lẵytị hs limh!0 1eh As we can see, the derivative of any order of the Kroneckker impulse is essentially given by the wan coefficients In fact, from (38) we get Da dnị ẳ 44ị  a X  a 2 À eÀhs wak eÀkhs XðsÞ ẳ Xsị; h kẳ0 h ỵ ehs where and its ZT is given by Z ẵenị ẳ  a X wa xt khị: h kẳ0 k The LT of (44) is The Heaviside discrete unit step is usually defined by enị ẳ  a X wa xnh khị: h kẳ0 k Assume that xðnhÞ resulted from a continuous-time function xðtÞ and define a new function, yðtÞ, by that is null for n > n0  Fractional derivative of the impulse Let introduce the Kroneckker impulse, dðnÞ; n Z, by & ðaÞ Df xnhị ẳ The summation expression will be called transfer function as usually and it is the ZT, GðzÞ, of the IR With the definition of forward derivative and mainly formula (21) we write M  X bk 2h GðzÞ ẳ 1z1 1ỵz1 kẳ0 N  X ak 2h kẳ0 1z1 1ỵz1 bk ak ; jzj > 1; 50ị M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 IR Alpha= 0.5 0.5 0.4 0.3 0.2 0.1 -0.1 50 100 150 200 250 300 200 250 300 200 250 300 200 250 300 IR Alpha= 0.5 0.4 0.3 0.2 0.1 -0.1 50 100 150 IR Alpha= 1.5 0.8 0.6 0.4 0.2 -0.2 50 100 150 IR Alpha= 0.8 0.6 0.4 0.2 -0.2 -0.4 - 0.6 -0.8 50 100 150 t Fig Impulse responses corresponding to (53) for orders a ¼ 0:5k; k ¼ 1; 2; Á Á Á ; for the causal case, and M  X bk 2h Gzị ẳ kẳ0 N X ak  z1 zỵ1 z1 h zỵ1 Conclusions bk ak ; jzj < 1; 51ị kẳ0 for the anti-causal case We can give to expressions (50) and (51) a form that states their similarity with the classic fractional linear   À1 systems [13,4] For example, for the first, let v ẳ 2h 1z We have 1ỵz1 Gv Þ ¼ M X b k v bk k¼0 N X 52ị ak v ak kẳ0 Compliance with Ethics Requirements This article does not contain any studies with human or animal subjects Declaration of Competing Interest Remark It is important to note that the factors À2Áak ; k ¼ 1; 2; Á Á Á, not have any important role in the compuh tations Therefore, they can be merged with ak and bk coefficients Example Consider the simple system with transfer function Gv ị ẳ a : v ỵ1 In this paper, we introduced new discrete-time fractional derivatives based on the bilinear transformation We obtained both time and frequency representations The corresponding impulse responses are always finite, contrarily to their continuous-time analogs We illustrate the behaviour of the forward derivative through the computation of the impulse response of a simple system ð53Þ In Fig we represent the impulse responses for several values of the order, a ¼ 0:25k; k ¼ 1; 2; Á Á Á ; It is interesting to verify that all the IR assume a finite value at the origin, contrarily to the continuous-time system analog to (53) described by Gv ị ẳ v a1ỵ1 ; Rev ị > The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper Aknowledgements This work was partially funded by National Funds through the Foundation for Science and Technology of Portugal, under the projects UIDB/00066/2020 References [1] Bohner M, Peterson A Dynamic Equations on Time Scales: an introduction with applications Boston: Birkäuser; 2001 [2] Henrici P Applied Computational Complex Analysis vol 2, Wiley-Interscince Publication; 1991 10 M.D Ortigueira, J.A Tenreiro Machado / Journal of Advanced Research 25 (2020) 1–10 [3] Hilger S Analysis on measure chains - a unified approach to continuous and discrete calculus Res Math 1990;18(1–2):18–56 [4] Magin R, Ortigueira MD, Podlubny I, Trujillo J On the fractional signals and systems Signal Process 2011;91(3):350–71 [5] Maione G, Lazarevic´ MP On the symmetric distribution of interlaced zero-pole pairs approximating the discrete fractional tustin operator In: 2019 IEEE international conference on Systems, Man and Cybernetics (SMC), Bari, Italy; 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Fractal Fractional 2017;1(1):3 [13] Petráš I Fractional-order nonlinear systems: modeling, analysis and simulation Springer; 2011 [14] Petráš I A new discrete approximation of the fractional-order operator In: Proceedings of the 13th International Carpathian Control Conference (ICCC), May p 547–51 [15] Proakis JG, Manolakis DG Digital signal processing: principles, algorithms, and applications, Prentice-Hall; 2006 [16] Roberts MJ Signals and systems: analysis using transform methods and matlab International ed McGraw-Hill; 2003 [17] Tustin A A method of analysing the behaviour of linear systems in terms of time series J Inst Electrical Eng – Part IIA: Automat Regul Servo Mech 1947;94 (1):130–42 [18] Strehl V Personal mail communication; 2003  ... proposed derivatives is that they are suitable to be implemented through the FFT with the corresponding advantages, from the numerical and calculation time perspectives The new derivatives On the Z and. .. fractional derivatives based on the bilinear transformation We obtained both time and frequency representations The corresponding impulse responses are always finite, contrarily to their continuous-time... (30) with (29), we conclude that they differ only in In the previous sub-section, we introduced the derivatives using a formulation based on the ZT Here we obtain the corresponding time framework,