2 Signal Processing 107 (2015) 198–217
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Discrete-time differential systems Manuel D. Ortigueira a,1,n, Fernando J.V. Coito a, Juan J. Trujillo b a b
UNINOVA and DEE of Faculdade de Ciências e Tecnologia da UNL, Campus da FCT da UNL, Quinta da Torre, 2829 – 516 Caparica, Portugal Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain
a r t i c l e in f o
abstract
Article history: Received 17 November 2013 Received in revised form 1 February 2014 Accepted 9 March 2014 Available online 22 March 2014
In this paper we formulate a coherent discrete-time signals and systems theory taking derivative concepts as basis. Two derivatives – nabla (forward) and delta (backward) – are defined and generalized to fractional orders, obtaining two formulations that are discrete versions of the well-known Grünwald–Letnikov derivatives. The eigenfunctions of such derivatives are the so-called nabla and delta exponentials. With these exponentials two generalized discrete-time Laplace transforms are deduced and their properties studied. These transforms are back compatible with the current Laplace and Z transforms. They are used to study the discrete-time linear systems defined by differential equations. These equations although discrete mimic the usual continuous-time equations that are uniformly approximated when the sampling interval becomes small. Impulse response and transfer function notions are introduced and obtained. The Fourier transform and the frequency response are also considered. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or obtain the current discrete-time case based on difference equation. & 2014 Elsevier B.V. All rights reserved.
Keywords: Discrete-time Fractional Time scale Nabla Laplace transform Delta Laplace transform
1. Introduction 1.1. Integer order signal processing The discrete-time (integer order) signal processing [1–5] is a well established scientific area being responsible for important realizations in our daily life. It has a formulation based on difference equations and uses as important tools the Z transform, mainly suited for system study and the Fourier transform very useful in analyzing signals. With the Z transform we compute easily the transfer function and, from it, the impulse response. The discrete-time linear systems described by difference equations [6,8] have the discrete-time exponential, zn , n A Z, as eigenfunction and the eigenvalue is exactly the transfer function. For these systems the transfer function is a rational function. n
Corresponding author. E-mail addresses:
[email protected] (M.D. Ortigueira),
[email protected] (F.J.V. Coito),
[email protected] (J.J. Trujillo). 1 Also with INESC-ID, Lisbon. http://dx.doi.org/10.1016/j.sigpro.2014.03.004 0165-1684/& 2014 Elsevier B.V. All rights reserved.
All the tools developed in the context of the discretetime signal processing are based on stable coherent mathematical topics and simultaneously of simple utilization.
1.2. Difference vs. differential In the beginning discrete-time signal processing was merely a set of numerical techniques to solve differential equations. Of primordial importance we can refer the incremental ratio used to approximate the derivatives; it is the currently known Euler method [8]. However with small manipulation the substitution of this incremental ratio in the differential equations led to difference equations easier to treat, mainly when computers started their important role in signal processing. So the difference equations gained the protagonism that belonged to the differential equations. Any way the former procedure (that we continue calling differential, designation justified later) was not completely abandoned; it continued being important as an intermediate step to obtain difference equations
M.D. Ortigueira et al. / Signal Processing 107 (2015) 198–217
[4] and used in some applications under the delta system format. This consists in substituting the derivative by the incremental ratio ðf ðt þ TÞ f ðtÞÞ=T followed by a sampling with interval T. The delta systems were applied in approximating continuous-time systems for filter implementation and control [9–13] as well as modeling [14–17]. Although there is no formal introduction of these systems as we will do here they were studied and stability criteria formulated [18,19]. 1.3. Modern approach The modern approach to differential discrete equation dates back to Hilger's works on looking for a continuous/ discrete unification [20], nowadays called calculus on time scales. His methodology consists essentially on defining a general domain that can be continuous, discrete or mixed (time scales or more generally measure chains) [21–24]. He defined two derivatives, delta and nabla, that are the incremental ratio or their limit to zero when not at an isolated point. With these derivative definitions we can devise corresponding differential equations representations of linear systems. Using a current nomenclature we will call them nabla and delta systems in agreement with the used derivatives. The nabla derivative is causal while the delta is anti-causal. We will study them in parallel and derive general formulae valid for any real order (the generalization to complex is easy). 1.4. Fractional derivatives When we try to enlarge the application domain of difference equations into the fractional domain we have some problems hard to avoid, for two main reasons:
When fractionalizing the discrete-time systems using
the classic tools we obtain infinite dimension integerorder systems that are difficult to manipulate. While the continuous-time fractional systems have long memory, the integer order discrete-time systems currently used to approximate them have short memory, since their impulse responses go to zero exponentially.
An attempt to overcome such problems of the fractional delay difference systems were proposed in [25]. However, a bridge between these systems and the continuous-time fractional systems was not found. An alternative is in using the derivative based systems that mimic the continuous-time formulation that will emerge from the discrete here presented as a limit when the sample rate increases without bound (the inter-sample interval goes to zero) according to what we wrote above. The motivation for this study is the desire of bringing to the discrete-time domain the tools and results of the continuous-time fractional signals and systems [26]. However, these were got by using the integral formulations that were proposed by Liouville. We must refer that these derivatives must not be confused with the mostly used Riemann–Liouville and Caputo derivatives [27–31]. These have several inconvenients as shown in [33,34]. The
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formulation based on the incremental ratio does not present such difficulties, although may have some analytical aspects hard to deal with [38–40]. It is this formulation that will be presented in this paper using the time scales setup. We start from the above referred derivatives. Each one has an exponential as eigenfunction [35]. With such exponentials we define two discrete-time Laplace transforms that we will use for the study of the general fractional discrete-time linear systems. These transforms are backward compatible with the classic two-sided Laplace transform and with the Z-transform. From these transforms and with suitable variable changes we will arrive at the discrete-time Fourier transform [36]. 1.5. Outline of the paper The paper outlines as follows. In Section 2, we revise the current Laplace and Z transforms. The nabla and delta derivatives are introduced and generalized in Section 3. Their properties are studied and exemplified. We show that there are two different exponentials that are eigenfunctions of these derivatives. Those exponentials are also studied and used in Section 4 to define discrete-time Laplace transforms suitable for dealing with the systems obtained with the above referred derivatives. For these transforms we study the main properties and compute some examples. The discrete-time linear systems are considered in Section 5. Their impulse responses and transfer functions are presented and particularized. In Section 6, we introduce the Fourier transform obtained from the discrete Laplace transforms by suitable variable change. At last we will present some conclusions. 2. On the Laplace and Z transforms 2.1. Relations between these transforms and the linear time invariant systems The importance of the Laplace and Z transforms in the study of the linear time invariant (LTI) systems is unquestionable. Many books enhance such fact. However, most of them do not show clearly why they are important and this led to misinterpretations that conditioned their generalizations to time scales. Let us consider the continuous time case first. In this case, the input–output relation for LTI systems is given by the convolution [1,5,31]: Z þ1 yðtÞ ¼ hðτÞxðt τÞ dτ ð1Þ 1
where xðtÞ is the input signal and h(t) is the impulse response. Let xðtÞ ¼ est for t A R. As it is easy to see, the output is given by yðtÞ ¼ HðsÞest with HðsÞ given by Z þ1 hðtÞe st dt ð2Þ HðsÞ ¼ 1
with a suitable region of convergence. It is called transfer function. As we see HðsÞ is the bilateral (two-sided) Laplace transform of the impulse response of the system. We showed that the exponential, defined on R is the eigenfunction of the continuous-time LTI systems. This shows why we must use the two-sided Laplace transform, although the one-sided has greater popularity. However
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the causal exponential used in this transform is not an eigenfunction of the LTI systems. It is currently used because of the initial conditions, but it fails in the fractional case. Moreover the two-sided has as a special case the Fourier transform: we only have to make s ¼ iω. In the following we will consider always the two-sided Laplace transform (LT) – we obtain easily the one-sided LT by multiplying the function to be transformed by the Heaviside unit step. Now we will consider the LT of a signal resulting from the uniform sampling of a given function, f ðtÞ defined in R. The ideal sampler is the comb defined by a sequence of Dirac deltas: þ1
pðtÞ ¼ ∑ δðt nhÞ
ð3Þ
1
where h 40 is the sampling interval. The ideal sampled function is þ1
f p ðtÞ ¼ ∑ f ðnhÞδðt nhÞ
ð4Þ
1
h; 0; h; 2h; 3h; …g with hA R þ . The results we will obtain are readily generalized to the situation where this chain is slided by a given a o h [21,23]. We start by introducing two well-known derivatives. In the following t will be any generic point in T ¼ hZ ¼ fkh: k A Zg. We define the nabla derivative by 0
f ∇ ðt Þ ¼
f ðtÞ f ðt hÞ h
and the delta derivative by 0
f Δ ðt Þ ¼
f ðt þ hÞ f ðtÞ h
þ1
and
ð5Þ
1
that can be considered as the discrete Laplace transform of the function f n ¼ f ðnhÞ. With a variable change z ¼ esh we obtain the Z transform þ1
FðzÞ ¼ ∑ f n z n
ð6Þ
1
As it is easy to verify, the exponential zn is also the eigenfunction of the discrete-time linear systems defined by linear difference equations. It is important to remark that z 1 represents a delay in time. Frequently we convert continuous-time systems to discrete-time systems by doing a conversion from s to z through the Euler transformation s¼
1z1 h
ðNÞ
f Δ ðt Þ ¼ ð 1ÞN
εðnhÞ ¼
ð11Þ
1
n Z0
0
n o0
ð12Þ
The anti-causal unit step is given by εð nhÞ. Using (9) we verify immediately that 8 < 1 n ¼ 0 DΔ εð nhÞ ¼ ð14Þ h : 0 na0 This leads us to introduce the discrete delta (impulse) function by δðnhÞ ¼ D∇ εðnhÞ
The two-sided transforms appear naturally as conse-
In the following we will consider that our domain is the measure chain (time scale) T ¼ ðhZÞ ¼ f…; 3h; 2h;
N
It is a simple task to show that the nabla derivative of the unit step is 8
