IN SIGNAL processing theory, a linear, time-invariant (LTI),

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1 106 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 1, JANUARY 2006 A Note on Impulse Response for Continuous, Linear, Time-Invariant, Continuous-Time Systems Maurizio Ciampa, Marco Franciosi, and Mario Poletti Abstract In the paper by Sandberg in 2003, he proved that, even if a continuous, linear, time-invariant, continuous-time system admits an impulse response, such a response does not always give a complete description of the system. In this paper, a theorem by Schwartz is used to define an impulse response under almost general assumptions, and to understand what we really know about two systems with the same impulse response. These results are applied to a survey of systems (significant by themselves and as leading examples), showing that, apart from three classes of exceptions, all of them are completely described by their impulse response. Concerning the first two classes of exceptions, counterexamples were given by Sandberg; concerning the remaining third class, a counterexample is deduced here from Sandberg s results. Index Terms Continuous linear time-invariant (LTI) systems, continuous-time signals, distributional signals, impulse response. I. INTRODUCTION IN SIGNAL processing theory, a linear, time-invariant (LTI), continuous-time system is a map where (input space) and (output space) are linear spaces of signals defined on, both closed under translation, 1 and is a linear map which commutes with translation. 2 In recent papers [1] [3], Sandberg considered the Banach space of bounded uniformly continuous complex-valued functions defined on, equipped with the usual sup-norm and the class of all continuous (with respect to the sup-norm) LTI maps admitting an impulse response, in the sense that there exists a function such that for every sequence of progressively taller and narrower unit-area functions of, centered at, the sequence is pointwise convergent on to. In this class of maps, Sandberg proved the existence of a non null map (in particular, of a non null causal map) with impulse response. Manuscript received January 19, This paper was recommended by Associate Editor K. Thulasiraman. The authors are with the Dipartimento di Matematica Applicata U. Dini, Università di Pisa, Pisa, Italy ( mciampa@dma.unipi.it; franciosi@dma.unipi.it). Digital Object Identifier /TCSI A space X is closed under translation when for every f (t) 2Xand 2 IR, also f (t 0 ) 2X: 2 That is: for every f 2Iand 2 IR, defined y(t) =L(f (t)) it is L(f (t 0 )) = y(t 0 ). This unexpected result of Sandberg shows in particular that: even if a continuous LTI map admits an impulse response, such a response does not always give a complete description of, i.e., there may exist continuous LTI maps different from with the same impulse response. In this paper, we consider continuous LTI maps and face the two problems arising from the results of Sandberg, namely, how to define an impulse response for a continuous LTI map and what we really know about two continuous LTI maps, with the same impulse response. In order to exclude spaces of too generic functions (e.g., the Banach space of all bounded, measurable or not, complex valued functions defined on ) but to allow handling more general signals (e.g., the derivatives, of any order, of bounded measurable complex valued functions defined on ), and to guarantee that the mildest available signals are allowed inputs, we assume the following. Assumption 1: and are subspaces of the set of complex valued distributions on, both equipped with a notion of convergence and limit (denoted -lim and -lim, respectively) for sequences, such that if a sequence is convergent in (respectively, in ) with limit, then is also convergent in with the same limit. Assumption 2: The space of complex valued functions defined on with compact support, is a subspace of, and the notion of convergence in is such that if a sequence is convergent in with limit, then is also convergent in with the same limit. Concerning the meaning of continuity, we assume the following. Definition 1: A map is said to be continuous if it is sequentially continuous, i.e., if for every and every sequence such that - it is -. The preliminary problem of defining an impulse response for every continuous LTI map (obviously under Assumptions 1 and 2) is treated in Section II. A Theorem of Schwartz allows us to solve this problem in a way that agrees with the usual impulse response for every admitting one, and with whenever the Dirac impulse is a member of and, as usual, there is a sequence convergent, both in the space of the distributions with compact support and in,to. The crucial problem of understanding what we really know about two continuous LTI maps,, with the same impulse response, is treated in Section III. By (transfinite) induction we construct the widest set of members of related to through limits of sequences, and we prove that for at least all /$ IEEE

2 CIAMPA et al.: IMPULSE RESPONSE FOR CONTINUOUS-TIME SYSTEMS 107 As a consequence of this result, whenever,any continuous LTI map is completely described by its impulse response. This result agrees with a recent result of Sandberg concerning a wide and significant class of LTI maps continuous with respect to suitable notions of convergence and limit [4, Th. 1]. To better understand the extent of the above result and to give methods to determine, in Section IV, we analyze some (in our opinion particularly significant by themselves and as leading examples) spaces of functions or distributions, showing in each case whether it is or not. To be more precise, we prove that the following input spaces : with ; (weak convergence) with ; (strong convergence) with ;,, ; (where is the natural extension of into and is the space of tempered distributions) all verify,so proving that for such every continuous LTI map is completely described by its impulse response. On the other hand, we prove that the following input spaces : all verify. As a consequence, when is one of these last spaces, there may exist different continuous LTI maps, with the same impulse response. Concerning, the result of Sandberg proves that there really exist different causal continuous LTI maps, with the same impulse response. Concerning an analogous result has been proved by Sandberg in [4]. In Section V, as corollaries of the result of Sandberg on, we give a new proof that there exist different causal continuous LTI maps, with the same impulse response, and we prove that, whenever is considered with the strong convergence, there exist different causal continuous LTI maps, with the same impulse response. II. IMPULSE RESPONSE Here, we recall some basic notions on distributions then, applying a theorem of Schwartz, we give a definition of distributional impulse response and we show its first relation with convolution. Finally, we analyze the relation of this impulse response with the one adopted by Sandberg for continuous LTI maps and with the response to the Dirac impulse. Let be the linear space of complex-valued functions defined on with compact support. Let be a sequence of members of, and let ; if there is a compact subset of such that supp for every, and moreover for every the sequence converges to uniformly on, then we write -. Let be the linear space of distributions on, i.e., the space of the linear and continuous functionals from into. As usual, for every and every, the complex number is denoted by, and, whenever is a locally integrable function, it is Let be a sequence of members of, and let ; if for every it is then we write -. In this section, we need also the space of distributions with compact support. It is well known that can be introduced via a duality pairing as a space of functionals as follows [5, Ch. 3, Th. XXV]. Let be the linear space of complex-valued functions defined on. Let be a sequence of members of, and let ; if for every the sequence converges to uniformly on every compact subset of, then we write -. is the set of the linear and continuous functionals from into. As usual, for every and every, the complex number is denoted by, and, whenever is an integrable function with compact support, it is Let be a sequence of members of, and let ; if for every it is then we write - ; moreover it is well known that this condition is verified if and only if there exists a compact subset of such that supp for every, and -. In order to handle linear changes of variables for distributions, we agree to denote an element by a function-like symbol, so that the name of the current variable is pointed out. In this way, for every, such that, we denote by the distribution defined by for every. In particular, for,, we obtain defined by and, for,, we obtain defined by For every,, the convolution is the function defined, for every, by

3 108 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 1, JANUARY 2006 [5, Ch. 6, Th. XI ]. Observe that, whenever is a locally integrable function, this definition agrees with the usual definition that Proof: For, let be the set of all such for every Concerning continuous LTI maps, the remark following [5, Ch. 6, Th. XXIII] can be rewritten as follows. Theorem 2.1 (Schwartz): Let be a continuous LTI map. Then the following statements hold. a) For every sequence with -, the sequence is convergent in, and - is independent of the particular sequence. b) For every it is. This theorem allows us to define an impulse response for every continuous LTI map (obviously under Assumptions 1 and 2). Indeed: let be a continuous LTI map. Then the map defined by for every, is a continuous LTI map. By Theorem 2.1 applied to we obtain the following. a) There exists a unique such that, for every sequence with -, it is -. b) For every it is. The distribution will be called the impulse response of. The following theorems relate the above defined impulse response to the one adopted by Sandberg for continuous LTI maps and to the response to the Dirac impulse. Let. Since Assumptions 1 and 2 are verified, every continuous LTI map has an impulse response. As proven by the identity map, it may be that does not have an impulse response in the usual sense. The following theorem proves that, whenever has an impulse response in the usual sense, here denoted by, then it is. Theorem 2.2: Let be a continuous LTI map, admitting an impulse response in the usual sense, and let be a sequence such that for every it is for every Obviously, it is. For every, and every, let be the diameter of Obviously, it is. Observe that for every it is. Otherwise, there would exist and positive integers such that every. Hence, there would exist and such that, for every, it would be,. This is absurd, since it is,. As a consequence of the time invariance of, it is easily proven that, for every, every such that, and every,itis and hence it is. Let ; the previous result proves that, for every, and every, itis as a consequence Let be a compact subset of, and let. For every, there exists such that ; hence, for every it is. An usual trick, depending on the compactness of, proves that there exists such that. Let be as in the text of the theorem. Let be a compact subset of, and let. Since every, then for every, and every it is The following statements hold. a) The sequence converges to uniformly on every compact subset of. b) is a continuous (not necessarily bounded and uniformly continuous) function. c). This result proves that the sequence is uniformly convergent on, and hence proves statements a) and b). Concerning c), let be as above, but with the ulterior condition that every ; observe that -. By the definition of impulse response, it is -. Since is continuous, it is. By a) it is also -. Hence,. Let the Dirac impulse be an allowed input (i.e., ), let be a continuous LTI map, and let be its impulse response. Then we may expect the impulse response of

4 CIAMPA et al.: IMPULSE RESPONSE FOR CONTINUOUS-TIME SYSTEMS 109 the system being the response of the system to the Dirac impulse, i.e., we may expect that. The following unusual example proves that it may be. Let be the subspace of spanned by and by the family, with, of the translated of. It is easily seen that every may be uniquely written, apart from zero summands, in the form with,, every and every. A sequence will be called convergent in if there exist such that every with the sequence convergent in, and the sequences convergent in ; in that case the -limit of the sequence is defined by Let, and let be the continuous LTI causal map defined by It is easily seen that: a) the impulse response of is ; b) the response of to is ; so, choosing it is but, choosing for instance it is. The following theorem gives a usually verified sufficient condition on in order that. Theorem 2.3: Let be a continuous LTI map, and let be the impulse response of.if, and there exists a sequence such that - and -, then it is. Proof: Let be a sequence such that - and -.Bydefinition it is -. By assumption it is - ; hence by continuity it is -. Since, by Assumption 1,, then it is. III. CONTINUOUS LTI MAPS WITH THE SAME IMPULSE RESPONSE. SEQUENTIAL CLOSURE Let be an input space. In this section, we find a sufficient condition on in order that every continuous LTI map is completely described by its impulse response. To this aim, for every finite or transfinite ordinal, with, let be the subset of inductively defined by the following. a) is the set of the such that there exists a sequence with -. b) For, is the set of the such that there exists a sequence with - It is immediately seen that: i) every is a subspace of ; ii) for every it is ; iii) if, for it is, then, for every such that it is. As a consequence, since the cardinality of every is bounded by the cardinality of, there exists a unique finite or transfinite ordinal such that: iv) for every,itis ; v) for every,itis. The space will be denoted by and will be called the sequential closure of in. The significance of rests on the following two results. Theorem 3.1: Let, be two continuous LTI maps with the same impulse response. Then, for every,itis. Proof: By transfinite induction on, we prove that for every it is. Let, and let. Since there exists a sequence such that -, then by Theorem 2.1 it is Hence,. Let, and assume that the inductive statement holds for every. Let. Bydefinition, there exists a sequence such that every, with a suitable, and that - ; then it is Since and, by the inductive assumption it is. Hence. Theorem 3.2: Let. Let, be two continuous LTI maps with the same impulse response. Then, it is. Proof: The statement is a straightforward consequence of Theorem 3.1. The above theorem proves that, whenever, every continuous LTI map is completely described by its impulse response. This result agrees with a recent one obtained by Sandberg [4, Th. 1] concerning a wide and significant class of continuous LTI maps, where: is equipped with the following notion of convergence and limit for sequences: - if either, and -,or

5 110 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 1, JANUARY 2006 and, where is the characteristic function of the interval ; is again but equipped with the following notion of convergence and limit for sequences: - if -. Indeed we have: a) and verify Assumptions 1 and 2, hence, every continuous LTI map admits an impulse response (in ); b) whenever an is in the class considered by Sandberg and moreover admits an impulse response in the sense specified in [4, Th. 1, (a)], then ; c) since for all it is - and for all there exists a sequence such that - then it is It is well known that every is the limit of a sequence of members of, which implies as a direct consequence that, hence Let, and let be the usual Banach space with the norm defined by The argument adopted for proves that Concerning the spaces of distributions,,, by the proof of [6, proof of Th. 1.20, Cor 1.5; proof of Th. 1.31], it is, and ; hence hence, every continuous LTI map is completely described by its impulse response; in particular, given any sequence with -, since -, then the behavior of the sequence uniquely determines the behavior of ; d) whenever an is in the class considered by Sandberg, the representation of for every given in [4, Th. 1, eq. (8)] agrees with the above remark c) and makes apparent how the behavior of on particular sequences converging to in, uniquely determines the behavior of on the whole input space. IV. SURVEY OF SEQUENTIAL CLOSURES AND CONTINUOUS LTI MAPS COMPLETELY DESCRIBED BY THEIR IMPULSE RESPONSE In this section, in order to apply Theorem 3.2, we test the condition where is one of the following spaces:,,,,,. In our opinion these spaces are particularly significant both as spaces of signals and as leading examples to determine for other possible choices of. Concerning, since it is a Banach space, we have Obviously,, where is the space of the such that ; moreover, it is well known that every is the limit, with respect to the sup norm, of a sequence of members of. Hence, it is and then Concerning the spaces we distinguish between the case and the case. Let, and let be the usual Banach space with the norm defined by Last, we consider the spaces, which we are going to illustrate. For each the space is the natural extension of in the space of distributions, namely the subspace of spanned by itself and by the distributional derivatives (of any order) of its elements. Despite their easy definition and the obvious reasons of their introduction e.g. an voltage across a capacitor results in a current, in these spaces plays a fundamental role a phenomenon which is not perceived working with,,. As, and (for this last, see [5, Ch. 7, Sec. 4]), the spaces can be introduced via a duality pairing, as spaces of functionals, as follows [5, Ch. 6, Th. XXV]. Let be the space of all complex-valued functions defined on, such that for any. Given a sequence of members of, and we will write - if for every, the sequence converges to in.for, denotes the subspace of, whose elements are the such that for every, equipped with a similar notion of convergence and - for sequences. For, is the space of linear and continuous functional from into, where is defined by. As usual, for every,, the complex number is denoted by, and whenever it is For, is the space of linear and continuous functionals from into. Working with spaces of distributions introduced via a duality pairing, two notions of convergence need to be considered: a strong convergence and a weak one. As far as we are concerned with, and and only sequences are taken into account, there is no distinction: a sequence is strongly convergent if and only if it is weakly convergent [5, Ch. 3, Th. XIII; Ch. 3 Sec. 7; Ch. 7, Sec. 4]. But this is no longer true for spaces [5, Ch. 6, Sec. 8]. Thus, we first introduce the notion of bounded set in

6 CIAMPA et al.: IMPULSE RESPONSE FOR CONTINUOUS-TIME SYSTEMS 111 with and in, and then we explain strong and weak convergence for sequences in. Let, and let ; is called a bounded subset of if there exist positive real numbers such that, for every, itis Let. Every ; for all it is ; moreover and, for every and every it is.as a consequence of Lebesgue Theorem it is Bounded subsets of have a similar definition. Let. Let be a sequence of members of, and let. If, for every bounded subset of,itis uniformly with respect to strongly converges to If, for every, it is, then we say that the sequence, and we write then we say that the sequence weakly converges to, and write Strong and weak convergence for sequences in have similar definitions by using bounded subsets in and. The sequential closure of in with respect to the strong and the weak convergence will be denoted, respectively, by In order to analyze the above sets, we need some lemmas. Lemma 4.1: Let. For every there exists a sequence such that, and hence such that. Proof: Let. Let be such that for every, supp and, and consider the sequence It is easily seen that. Hence, since, then is a sequence of members of which strongly converges to in (see the results on regularization in [5, Ch. 6, Sec. 8]). Lemma 4.2: Let. For every there exists a sequence such that, and hence such that. Proof: Let. Since, by[5, Ch. 6, Sec. 8] there exists a sequence of members of such that -, and it is easily seen that convergence in implies strong convergence in, then it is. Lemma 4.3: For every there exists a sequence such that. Proof: Let. For, let be such that for every, and that for every. Let. i.e.,. As a consequence of Lemma 4.1, 4.2, and 4.3, we obtain that for it is for it is The analysis of - is slightly more difficult. Let denote the space of distributions convergent to 0 at infinity, i.e., of the distributions such that -. Lemma 4.4: It is -. Proof: The statement follows from the second and the last subsections of [5, Ch. 6, Sec. 8]. Lemma 4.5: For every there exists a sequence such that. Proof: Let. Let and be as in the Proof of Lemma 4.1, so that. Since -, then for every and it is ; hence every. Lemma 4.6: For every there exists a sequence such that. Proof: Let, and let, be as in the Proof of Lemma 4.3. Let be a bounded subset of, and let be a positive real such that. For every it is Since, then it is. By Lemma 4.6 it is then, by Lemma 4.5 it is so that Hence, by Lemma 4.4 it is

7 112 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 1, JANUARY 2006 The above results can be summarized in the following. Theorem 4.1: a) Let be one of the following spaces: then. b) Let be one of the following spaces: Obviously, and the maps defined by for every, and for every, are causal continuous LTI maps. Now let be a causal continuous LTI map such that: for every, and existing by the result of Sandberg, and consider the causal continuous LTI map then. As a consequence of Theorem 3.2 and 4.1, we obtain the following. Corollary 4.1: Let be one of the spaces listed in part (a) of Theorem 4.1, and let be any output space (verifying Assumption 1). Then any continuous LTI map is completely described by its impulse response. The Section V proves that Corollary 4.1 cannot be extended to any of the three spaces listed in part (b) of Theorem 4.1. V. THREE PATHOLOGICAL INPUT SPACES In [2], Sandberg showed that there exist non null causal continuous LTI maps For every, itis ; hence. Since,itis ; hence, which concludes the proof. Proposition 5.2: Consider with the strong convergence. There exist non null causal continuous LTI maps with impulse response. Proof: The proof is identical to the proof of Proposition 5.1, merely substituting the space with the space (strong convergence). To understand why the argument of the proof of Proposition 5.2 cannot be applied to LTI maps continuous with respect to the weak convergence, observe that the map with impulse response. We recall that all the in his proof verify the conditions: for every, and is a non zero constant function, where 1 is the function with constant value 1. The following Propositions 5.1 and 5.2 are corollaries of this result. Propositions 5.1, with a different proof, is due to Sandberg [4, Th. 2 and related remarks ]. Proposition 5.1: There exist non null causal continuous LTI maps with impulse response. Proof: Let be such that which must be used in the proof of Proposition 5.2 is continuous with respect to the strong convergence in but is not continuous with respect to the weak convergence in. Indeed, the sequence is weakly convergent to 0 in, but the sequence is not convergent in. REFERENCES [1] I. Sandberg, Continuous-time linear systems: Folklore and fact, in Proc. ISCAS 2002, vol. 1, 2002, pp [2], Causality and the impulse response scandal, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, pp , Jun [3], Countinuous multidimensional systems and the impulse response scandal, Multidim. Syst. Signal Process., vol. 15, pp , [4], Bounded inputs and the representation of linear system maps, Circuits, Syst., Signal Process., vol. 24, pp , [5] L. Schwartz, Théorie des distributions. Paris: Hermann, [6] E. Beltrami and M. Wohlers, Distributions and the Boundary Value of Analytic Functions. New York: Academic, By [5, Ch. 6, Th. XXV], for every it is, and, by the results on regularization in [5, Ch. 6, Sec. 8], the linear map defined by is continuous. Since supp, is causal. Hence, is a causal continuous LTI map. Maurizio Ciampa received the degree in electronic engineering in 1990 and the Ph.D. degree in 1993, both from the University of Pisa, Pisa, Italy. He is currently a senior researcher with the Department of Applied Mathematics, University of Pisa. His research interests include circuit theory, systems theory and numerical computations.

8 CIAMPA et al.: IMPULSE RESPONSE FOR CONTINUOUS-TIME SYSTEMS 113 Marco Franciosi received the degree in mathematics from the University of Pisa, Pisa, Italy, in From 1993 to 1996, he was Perfezionando of Scuola Normale Superiore, Pisa. He is currently a senior researcher with the Department of Applied Mathematics, University of Pisa. His research interests is in pure and applied mathematics and include mainly algebraic geometry and systems theory. Mario Poletti received the degree in mathematics from the University of Pisa, Pisa, Italy, in He is currently a Full Professor with the Department of Applied Mathematics, University of Pisa. His research interests include algebraic geometry, distributions, and circuit and systems theory.