criteria for the existence of impulse responses and kernel representations for linear maps

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 18, 307-318 (1990)

CRITERIA FOR THE EXISTENCE OF IMPULSE RESPONSES AND KERNEL REPRESENTATIONS FOR LINEAR MAPS

DAVID BALL Applied Research Laborutories, The University of Te.Yas al Ausrin, Austin, TX 78712, U.S .A.

AND

I R W I N W . SANDBERG Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, U.S.A.

SUMMARY

Results are given that establish, for the first time, necessary and sufficient conditions for the existence of impulse responses and kernel representations for linear not-necessarily-time-invariant systems described by input-output operator equations.

These results concern systems whose inputs and outputs are real-valued functions on the real line R , the half-line [ 0 , ~ ) or R"'. They deal with causal as well as non-causal maps and considerably extend related previous results which concern causal maps defined on functions on the half-line.

1 . INTRODUCTION

The theory of input-output representations for linear systems, which centres around integral representations, is the basis for most modern engineering studies. For systems governed by differential equations the theory has been well developed for many years, but this is not so for more general linear systems described by operator equations of the form y = Hu. In fact, most readers will recall that the typical treatment found in books involves just approximating u by a piecewise-constant function and formally (i.e. without justification) passing to an integral limit. In this formal procedure one assumes also the existence of another limit: the system impulse response.

In a recent study' concerned in part with the representation of causal linear operators H mapping locally integrable functions on [ O , 00) t o functions on ( 0 , m), sufficient conditions are given under which H has the representation

for each input u, where the kernel h, the impulse response of the system, is an ordinary (as opposed to generalized) function and the integral is a Lebesgue integral. If H is time-invariant, (1) becomes

( H u ) ( t ) = 1' g(t - 7 ) u ( 7 ) d7, t > 0, 0

where, given 1 > 0, g(7) = h ( f , t - 7 ) for almost all in [0, / I . Necessary and sufficient conditions are given in Reference 1 under which g in (2) is continuous.

The relation (1) was obtained as a consequence of a representation result in Reference 1 for causal non- linear maps defined on functions o n the half-line. In this paper we take a different tack and obtain (1) and related results using linear operator theory and integration theory. By slightly generalizing the conditions in Reference 1 under which (1) holds, we obtain necessary and sufficient conditions for the existence of representations of the form (1). Our approach leads also to criteria for the existence of impulse

0098-9886/90/030307-12$06.00 0 1990 by John Wiley & Sons, Ltd.

Received 26 Ocfober 1988 Revised 23 May 1989

308 D. BALL AND I . W. SANDBERG

responses and kernel representations for systems that need not be causal, and for systems whose inputs and outputs are functions on the whole real line R or R"'.

Our results are given in Section 2, which begins with some preliminaries. Section 2.2 contains Theorems 1-3, which provide necessary and sufficient conditions for the existence of impulse responses and kernel representations for maps of functions defined on R , [0, 00) or R"' respectively. For example, Theorem 1 provides conditions, which are essentially continuity conditions, under which a map H defined on a space U of locally integrable functions u : R -+ R has the representation

(Hu)(t) = lim s n h( t , T ) U ( T ) d7, t E R , u E U n - m - n

and conditions under which (3) reduces to m

(Hu) ( t ) = i h(t, 7 ) u ( 7 ) d7, t E R - m

(3)

(4)

for certain inputs uE U. Section 2.2 also contains one of the highlights of the paper, a corollary to Theorem 1 which provides necessary and sufficient conditions under which a linear map H defined on a space of bounded inputs u has the representation (4). Theorems 1-3 are obtained using the same general approach adapted to specific cases, so we present a complete proof only for Theorem 1. Comments on changes in details of the proof are given for Theorems 2 and 3.

Notes on Theorems 1-3 are presented in Section 2.3. These include an example of a case in which (3) does not reduce to (4).

Results on time-invariant (or, in the R"' case, space-invariant) maps appear in Section 2.4. Theorem 4 gives necessary and sufficient conditions on a linear map H under which H has the representation

(Hu)(r)= lim i n g(t-7)u(7)d7, l € R n - m - n

for u E U , and additional conditions under which ( 5 ) reduces to

g(t - T ) U ( T ) d7, m

( H u ) ( t ) = i t E R - m

Section 2.4 also contains Theorem 5, which provides a complete solution to the problem of obtaining conditions under which g in ( 5 ) or (6) is continuous.

2. REPRESENTATION RESULTS

2.1. Preliminaries

We first consider some basically standard definitions and notation, in which R and R"' denote the set of real numbers and the set of real m-vectors respectively.

Let ( X , .-M, p ) be a measure space. The set of functions f: X -+ R which are (Lebesgue) integrable with respect to p is denoted by 9 ' ( p ) . Given f € 9 ' ( p ) , we write the integral o f f over X as { f d p or, if f is a function of several variables and, say, f(a, * ) E Y ' ( p ) , we use 5 f (a , * ) dp to denote the integral of f (a , .) over X . The integral of fover a set A € dl is written as { ~ f d p or {f- X A dp, where X A denotes the characteristic function of A . The set of functions f: X - + R that are locally integrable, i.e. satisfy f - x A € 617'(p) for each set A E satisfying p ( A ) < m, is denoted by 9 ; o c ( p ) . The set of functions f: X + R which are p-measurable and bounded except on a set of p-measure zero is denoted by LZm(p).

We denote Lebesgue measure on R , [0, m) or R m by A, and write 9 ' ( R ) , 9 ' (0 , m) or Y"(R" ' ) instead of 9 ' ( A ) (and similarly for 9;oc(A) and &?"(A)). Given f in 617'(R), P ' ( 0 , m ) or Y ' ( R " ' ) , we write e.g. j R " l f ( 7 ) d r instead of SfdA and { , b f ( T ) d7 instead of {(o,b,fdA. Given f in 9 ' ( R f f f ) , we often write e.g.

[ ? m [ S m m f ( ~ ~ , 72) d n d n instead of S R 2 f ( 7 ) d7, where 7 = ( 7 1 ~ 7 2 ) (of course this is permissible by Fubini's Theorem). The standard 9 ' ( R ) (617'(0, a), 9 ' ( R f " ) ) and LZm(R) (LZm(O, m), Y P ( R " ' ) ) norms are denoted

IMPULSE RESPONSES AND KERNEL REPRESENTATIONS 309

by ( 1 * ( 1 and ( 1 - I(m respectively. Whenever we say that a property holds almost everywhere, or that a function or set is measurable, we are referring to Lebesgue measure unless otherwise indicated.

Given a set X , we define the unit constant function 1 : X + R by l ( t ) = 1, t E X . Finally, whenever we say that a limit exists, we mean that it exists as a real number.

2.2. Representations for time-varying maps

Theorems 1-3 of this section provide necessary and sufficient conditions for the existence of impulse responses and kernel representations for linear systems whose inputs and outputs are real-valued functions defined on R , [0, 00) or R”’ respectively. As mentioned in Section 1, each of these theorems can be proved using the same general approach. We include in the proof of Theorem 1 an outline of the steps in obtaining a proof of any of these theorems, and we state Theorems 2 and 3 with only a few comments concerning changes in the hypotheses and conclusions of Theorem 1.

2.2.1. Theorems 1 and 2. We use the following notation in connection with maps of functions defined on the whole line R or the half-line [0, 00).

Let M (62) denote the set of functions f : R + R ( f : [0 , 00) -+ R ) . Given r in R ([0, a)), we define the delay or shift operator T,: M + M by

The delay operator T, : a -+ a, r 2 0, is defined by

We use W ( W ) to denote the set of functions of bounded support in M (fi) that are piecewise-constant and have a finite number of discontinuities. Let U (0) denote any linear space over R ([0, m)) satisfying

(b) U ( 0 ) is closed under T, for each r in R ( [O,m)) and is closed under multiplication with the characteristic functions of intervals in R ( [ O , 00)) (e.g. u E U implies u * x [ ~ , ~ I E U for all a and b in R with a < 6).

(a) w s U S P:,,(R) ( W G 0 s PL(o, m)>,

For example, U can be taken to be W or, at the other extreme, we can take U = P!,,(R). Given t E R , let L ( t ) be the set of h e a r maps H : U-+ Msuch that, given uE U, limn-m ( c X [ - n , n ] u ) ( t )

exists and equals (Hu)( t ) . Let L( t ) , t 2 0, denote the set of linear maps H : o+ M such that limn+- (H~[o ,n lu ) ( t ) exists and equals (Hu) ( t ) for each u E 0. The (very reasonable) hypothesis that an input-output map belongs to L ( t ) or L ( t ) (or the set L,,,(t) defined in Section 2.2.2) plays a central role in our results.

We say that a map H : U-+ M is causal if ( H u ) ( t ) = ( H ~ ( - ~ , ~ l u ) ( t ) , u E U, t < a , for each a E R. A map H : o+ a is causal if for each u E 0 and a 2 0, ( H u ) ( t ) = ( H ~ [ o , ~ ] u ) ( t ) , t < a. We say that a map H : U + M ( H : o+ a) is time-invariant if T,H = HT, for each a in R ([0, m)).

Given u > 0, we define wo in W or W by

In either space, wu approaches the unit impulse at the origin (in the usual intuitive sense). In Theorem 1 we are concerned with subsets B(w, t ) of L ( t ) defined as follows: for each t E R and

measurable W E M , B(w, t ) = ( H E L ( t ) : I(Hu)(t)l < JIwu(I for all u E U satisfying wu E &?‘(R)). We now present Theorem 1.


Theorem 1

Let H : U -+ A4 be linear and let t E R . Then

(1) the condition that HE B(w, t ) for some w E Y m ( R ) is necessary and sufficient for the following to hold: (a) h( t , 7) 5! limo-.o (HT,w,)(r) exists for almost all 7 in R (b) the function h(t;) belongs to Y " ( R ) (c) limn-,j"l ,,h(t, 7 ) u ( 7 ) d7 exists for each u E U and we have

(Hu) ( t )= lim [" h(t , T ) U ( ~ ) d7, u E U n - m - n

(7)

(2) (a)-(c) above hold with h( t ; ) belonging to Y ' ( R ) if an only if H c B ( w , t ) for some wE@(R)nK'" (R) .

Notes

1 . If ( H u ) ( t ) can be expressed as in (7) with h( t , .) E Y m ( R ) and if u E -Y"(R), then m

( H u ) ( t ) = h( t , 7)u(7) d7 -m

Similarly, if ( H u ) ( t ) is given by (7) with h(t , - ) E LF"(R) and u E Y'"(R), then (8) holds. 2. The following important corollary to Theorem 1 provides representations for a class of linear maps

on Y'"(R) (similar corollaries to Theorems 2 and 3 are easily obtained? and these are given in the Appendix).

Corollary I

that I(Hu)(t)l < IIwu(I for all u E Y'"(R) is necessary and sufficient for the following to hold: Let H : Y m ( R ) --* M b e linear and let I E R . Then the condition that there exists w E Y ' ( R ) f l Y m ( R ) such

(a) h(r, 7) 4 (b) the function h(r , .) belongs to LP1(R) fl Y / ' " ( R ) (c) we have

(HT,w,)(r) exists for almost all 7 in R

m

( H u ) ( t ) = h( t , 7)u(7) d7, u E Y " ( R ) --m

By modifying the proof of Theorem 1, one may obtain a similar result in which w is integrable but not necessarily bounded (in this case one gets h( t , .) E LP1(R) but not h(t , .) E Y " ( R ) ) . Note that, adopting the viewpoint that (H( . ) ) ( t ) is a linear functional, Corollary 1 provides Riesz-type representations for a class of bounded linear functionals on P m ( R ) .

3. Theorem 1 provides a representation for (Hu) ( r ) for arbitrary t and u ; it therefore yields a representation for H. Specifically, it is clear from the theorem that if there is a function w : R x R -+ R such that w ( t , .) E P m ( R ) and H E B(w(t, -), t ) for each t E R , then

(Hu) ( t ) = lim i n h ( t , 7)u(7) d7, tE R , u E U n - m - n

where h( t ; ) is defined as in (a) for each t E R . 4. The condition of Theorem 1 that HE B(w, t ) for some w E Y m ( R ) is essentially a continuity condition

TConcerning the proof of Corollary 1 , observe that we have H E L(r) when U = P(f?), W E Y " ( R ) and l (Hu) ( t ) l < ~ ~ M ~ u ~ ~ for U E u.


on H. It requires that H be in L ( t ) , i.e. that the response ( H u ) ( t ) be the limit of a sequence of responses to truncations of u, and that (H( . ) ) ( t ) be a bounded (and therefore continuous) linear functional on the space of functions u E U satisfying wu E Y, ' ( R ) (see (0) in the proof of Theorem 1 below).

5. It will become clear that similar comments apply to Theorems 2 and 3 and to Theorem 4 in Section 2.4. Further notes on Theorems 1-3 appear in Section 2.3.

Proof of Theorem 1. The steps in the proof are as follows.

(1) Necessity. Observe that H E B ( h ( t , s), t ) . Sufficiency. ( a ) Define a positive measure p on the Lebesgue-measurable sets A in R by

Show that U'(p) is the set of functions f satisfying wfE LP'(R). (0) Notice that, since H € B ( w , t ) , ( H ( . ) ) ( t ) is a bounded linear functional on the subspace U n U'(cp) of Y'(cp). Use the Hahn-Banach Extension Theorem to extend (H( . ) ) ( t ) to a bounded linear functional on P ' ( c p ) . (7) Use the Riesz Representation Theorem and a fundamental result on the differentiation of an integral to obtain (a) and (b) and the representation

m

( H u ) ( t ) = 1 h( t , 7 ) u ( 7 ) d7 - m

for each u E A4 satisfying wu E Y ' ( R ) . (6) Notice that (c) follows from (y) and the fact that H E L(1) .

(2) Use an argument similar to that in (y) above to show that h( t , - ) E P ' ( R ) when W E Y ' ( R ) n Y m ( R ) . For the converse, notice that since HE B(h( t , -), t ) when (a)-(c) hold, we have that h(r, ' ) E Y"(R) implies H E B ( ~ , ~ ) for some ~ E Y ' ( R ) ~ . Y " ( R ) .

What follows is a detailed proof of Theorem 1 . (1) Necessity. Part (c) implies that H E L ( t ) . Given u~ U such that h(r,-)u(.)E Y ' ( R ) , by (c)

I(Hu)(t)l = 1 ~-~ lim S",! h(f , 7 ) u ( 7 ) d7 1 m

h( t , ~ ) U ( T ) d7 = I I-, m

< lh(fy7)u(7)l d 7 = Ilh(t,.>u<.>Il -co

so HE B(h( t , * ), t ) .

measure, and given a Lebesgue-measurable function f on R , Suficiency. ( a ) By Theorem 1.29, p. 23, of Reference 2, the set function cp defined by (9) is a positive

where we stretch the notation to allow SI f I d p = 00. Thus f satisfies wf E P ( R ) if and only i f f E 9 ' ( ~ ) , and in this case 1 ) f \ I c = 1 ) wf 1 1 (here 1 1 - [Ip denotes the Y'(p) norm), so .Y'(cp) consists precisely of those functions f satisfying wfc P'(R).

( p ) By ( a ) we have

l(Hu)(t)l < l I w 4 l = l l4 lv , u€,Ung( , ' (P)

The set I/ n 9 ' ( c p ) is a linear space, so using the Hahn-Banach Extension Theorem (Reference 2, p. 104),

312 D. BALL AND I . W . SANDBERG

there is a linear extension of (If(.))(?), which we also denote by (H( . ) ) ( t ) , to P'(cp) satisfying

I(Ifu)(t)l < IIuIIw uE P"(cp) (12) (y) By (12), ( H ( - ) ) ( t ) is a bounded linear functional on S'(cp), so using the Riesz Representation

Theorem (Reference 2, p. 127), there is a unique (up to a set of measure zero) h(t, .) E L?'"(cp) satisfying

(Hu)( t )= 1 h ( t , . ) u ( . ) dcp, uEP"(cp)

Then defining h(t , .) = h(t, -)w( .), it easily follows from ( 1 1 ) that

h(?, 7)u (7 ) d7 m

(Hu) ( t ) = { -m

for each u E A4 satisfying wu E 9 ' (R) . For each n, w - X I - n , n l E 9 ' ( R ) , so by (a), X I - n , n l E 9' ( c p ) , implying (since h(t, .) E 9m(cp)) that

h(t, -)x[ - n , n ~ ( * ) E P ' ((0) and thus h(t, *)x[ - n , n ~ ( . )w( . ) = h( t , *)x[ - n , n ~ ( * ) E 9 ' (R) . This implies that h ( f , - ) € P l l o c ( R ) .

Since h(t , * ) E PlOc(R) and since T,wu E U satisfies wTTwo E P' ( R ) for each u > 0, we have

(HTTwu)(t) = h ( t , 7)(TTwu)(7) d7 m

-m

T + U =I h ( t , 7 ) dT+h(t ,7) u r

as u -+ 0 for almost all 7 in R , and (a) holds. Let A be a Lebesgue-measurable set in R and define

I T E A : - h ( t , 7) > ess sup lw(7)I TEA

T E A :h( t , 7) > ess sup lw(7)I TEA

Since h(t , . ) is Lebesgue-measurable, BA+ and B,i are Lebesgue-measurable. Define BA+ (n) = BA+ n [ - n, n ] for each n. Then using (10) and (12), unless X(BA+ (n)) = 0 ,

OD

ess sup 1 w(7)I - X(B2 (n)) < 1 h(t , T ) X B ; ( ~ ) ( ~ ) d7= (If~~i(n))(t)

< 1 1 w ' X B i ( n ) l l

< ess sup I w ( T ) ( - X(BA (n))

TEA -m

TEA

This is an impossibility, so X(B2 (n)) = 0 for each n and X(B2) = limn-+mX ( B i (n)) = 0. Similarly,

ess sup Ih(t, 7)1 < ess sup I w(7)I A ( & ) = 0 , so

TEA TEA

We therefore have h(t , .) E P"(R), because

ess SUP (h( t , 7)1 < ess sup l ~ ( 7 ) 1 = 1 1 w l l m < 00 r € R

(6) Given u E U, we have un = u x[ - n , n ~ E 9 ' ( R ) for each n, so wun E P ' ( R ) and from (y)

(Hun) ( t )= i n h ( t , 7 ) ~ ( 7 ) d7 - n

Part (c) clearly follows from the fact that H E L ( t ) . (2) By (a), since w = w - 1 E ~ ' ( R ) , 1 E 9 ' ( c p ) and &(t;)l(-)=h(t ,*)E9'(cp), where h(t, a ) is the

Riesz kernel in (7). Then by (a) we have h( t , a ) = h(t, .)w( .) E B ' (R ) . Conversely, if h ( t , .) in (a)-(c) belongs to g ' ( R ) , we have H E B(w, t ) with w = h ( t , .) E P ' ( R ) n P"(R). This completes the proof.


0 a

Now consider ‘f the proof o f ’ measurable w

systems whose inputs and outputs are defined on the half-line [0, 00). An examination Theorem 1 reveals that similar results hold for maps H : 0 -+ A? in i ( t ) for t 2 0. Given E M and t 2 0, define B ( w , t ) = { H E i ( r ) : J(Hu)(t)l < 1) wuJJ for each u E 0 satisfying

wu E Y * (0, a) 1.

Theorem 2

Let H : 0-+ &i be linear and let t 2 0. Then the condition that HE B ( w , t ) for some w E Y “ ( 0 , w ) is necessary and sufficient for the following to hold:

(a) h(t , 7) & limo40 (HT,w,)(t) exists for almost all T in [0, w ) (b) the function h ( t , a ) belongs to Y”(0, w ) (c) limn-m j,“h(t, T ) U ( T ) d7 exists for each u E 0 and we have

( H u ) ( t ) = lim 5‘ h( t , T ) U ( T ) d7, u E . 0 n - m 0

Furthermore, (a)-(c) above hold with h( t , a ) belonging to Y ‘ ( 0 , ~ ) if and only if HE B ( w , 1 ) for some

2.2.2. Theorem 3. We now address the matter of extending Theorem 1 to cover the case of maps of

Let M,n denote the set of functions f: R”’ --t R . Given T E R”‘, we define the shift operator T, : M,,, + M,,,

w E ~ l ( o , ~ ) n ~ m ( o , w).

functions defined on R”. We will use the following notation.

by

(TTf ) ( t ) = f ( r - T ) , t E R”’, f E M,,,

Define W,,, as the set of functions of bounded support in M,,, that are piecewise-constant and have a finite number of discontinuities. Let Urn denote any linear space over R”’ satisfying

(a) Wn C U,,, C 9POc(R”‘) (b) U,,, is closed under T, for each 7 E R”’ and is closed under multiplication with the characteristic

Given t ER”’, let L,, , ( t ) be the set of linear maps H : U,,,+ M,,, such that, given M E U,,,, limn-m (Hx[ - n , n l ” ’ u ) ( t ) exists and equals (Hu) ( t ) .

Given u > 0, we define a function wu E W,,, by

functions of intervals in R”’ (an interval in R”’ is the direct product of intervals in R ) .

l / ~ ’ ” , t E [0, u)”’ otherwise w o w =

As u + 0, wo approaches the unit m-dimensional impulse at the origin (in the usual intuitive sense). As is the case with Theorem 2, conclusions similar to those of Theorem 1 hold for H € L,, ,( t) , t E R”’.

Given a measurable w E M,,, and t E R”’, define B,,,(w, t ) = (HE L,,,(t) : I(Hu)(t)l < 1 ) wull for each u E U,,, satisfying wu E 9 I ( R “ ) ) .

Theorem 3

Let H : U,,, + M,n be linear and let r E R”’. Then the condition that HE B,,,(w, t ) for some w E Y m ( R r ” )

(a) h ( t , 7) 4 lim,-o (HT,w,)(t) exists for almost all 7 in R”’ (b) the function h( t , . ) belongs to Ym(R‘”) (c) limn-m j [ - n , n ] ” ’ h ( t , T ) U ( T ) dT exists for each u E U,n and we have

is necessary and sufficient for the following to hold:

( H u ) ( t ) = lim 5 h(t , T ) U ( T ) d7, u 6 U,,, n - m [ - n , n ] ” ’


Furthermore, (a)-(c) above hold with h(t , .) belonging to 9 (R'") if and only if HE B,,,(w, t ) for some w E Y (R'") f l Ym (R'").

2.3. Notes on Theorems 1-3

Theorems 1-3 are related to Theorem 2 of Reference 1, which is obtained as an application of a representation result in Reference 1 for non-linear causal maps of functions defined on the half-line [0, m). Our Theorems 1 and 3 address the doubly infinite interval and R"' cases, which are not considered in Reference 1. In addition, we provide necessary and sufficient conditions for the existence of representations, while only sufficient conditions are given in Reference 1. Our Theorem 2 is more general than Theorem 2 of Reference 1. The latter is in fact a consequence of the following corollary to Theorem 2, in which @(k, t ) for t >, 0 and k 2 0 denotes { H : 0 - f i : H is causal and linear, and I(Hu)(t)l < kll~ro,~lull for each u E 0) (notice that B ' (k , t ) E i ( t ) ) .

Corollary 2

to hold: The condition that HE p ( k , t ) for some k 2 0 for each t 2 0 is necessary and sufficient for the following

(a) h(t , 7) (b) the function h(t , * ) belongs to g'"(0, I ) for each t 2 0 (c) (Hu) ( t ) = Skh(t, 7 ) ~ ( 7 ) d7, t >, 0, u E 0. Proof of Corollary 2. The corollary follows from Theorem 2, the causality of H and the observation

that, given uE 0, if (c) of Theorem 2 holds then

limo+,, (HT,w,)(t) exists for almost all 7 in [0, t ] for each t 2 0

(Hu) ( t ) = ( ~ x r o , l l u ) ( ~ )

= lim !'I h(t , T ) X I O , ~ I ( ~ ) U ( ~ ) d7 n - m 0

Much of Theorems 1-3 may be obtained as consequences of the representation results for non-linear maps given in Reference 3. The approach used here yields stronger results and has the advantage of directness.

In the proof of Theorem 1, in order to show that H E B(w, t ) for some w E Y m ( R ) implies that h( t , * ) defined in (a) belongs to Pm(R) , we obtained a bound on h( t , .) in terms of w. This bound, which might be useful in applications, is given in the following corollary (similar corollaries hold for Theorems 2 and 3, and for Theorem 4 in Section 2.4).

Corollary 3

1 satisfies the inequality If Hbelongs to B(w, t ) for some w E Y " ( R ) and t E R , then the function h(t , .) defined in (a) of Theorem

ess sup (h ( t , 7)1 < ess sup l w ( ~ ) I T E A rEA

for each Lebesgue-measurable set A in R. In particular, h( t , - ) E Y m ( R ) and

llh(t, .) l lm < IIwIIm.

As we mentioned in Section 2.2.1, the requirement that H E B(w, t ) for some W E Y'/'"(R) is essentially a continuity condition on the map H : U --+ M . The following result makes this point more explicitly.


Lemma 1

Let t E R and let H E L(1). Then in order for there to exist some w 6 Y " ( R ) such that HE B(w, t ) , it is necessary and sufficient that there is a function w' E LP"(R) such that (Hun)(t) + 0 as n + w for any sequence (u , ) in U satisfying w f un E Y 1 ( R ) for each n and 11 w'un(l + 0 as n -+ 00.

Proof of Lemma 1. Setting w' = w, the necessity clearly holds. As t o sufficiency, we need only show that there is a K 2 0 such that, with w = K w ' , I(Hu)(t)l < IIwull = Kllw'ull for each uE U satisfying w ' u ~ 9 ' ' ( R ) . Thus suppose that no such K exists; then there is a sequence in U such that w ' u n E K " ( R ) and I(Hun)(t)l > n( lw'unI( for each n. We may assume without loss of generality that I Iw'unl( > 0 for each n , so we may define a sequence in U by sn = (l/nllwfunll)un for each n . Then, since H is linear, I(Hsn)(t)I > 1 for each n , but I / w'snl( = l / n -+ 0 as n + co. This is a contradiction, so the desired K > 0 must exist. This completes the proof. (Similar results are easily obtained for maps of functions on [0, 00) or R'".)

The function h( t , - ) defined in Theorems 1-3 may be defined in several equivalent ways (see the sections on 'nicely shrinking sets' in Reference 2, pp. 140-141). For example, we may define h ( t , * ) in (a) of Theorem 1 by

for almost all r in R , because the set ( r - a, r ] shrinks 'nicely' t o r as a -+ 0 (recall that, in this context, wo = ( l /a)x,7,7+ u ) ) . Similarly, in R 2 the set [ T I , 71 + a) x [n, 7 2 + a ) shrinks nicely to r = ( T I , 7 2 ) as a - 0, so we may define h( t , .) in (a) of Theorem 3 by

for almost all r in R2. However, the set [ T I , 71 + a) x [ 7 2 , ~ ~ + a') does not shrink nicely to r as a + 0, so we may not define h( t , .) in terms of ( H T T ( ( 1 / ~ 3 ) x [ 7 , , 7 , + u ) [ r 1 . r 2 + u z ) ) ) ( t ) .

As was noted in Section 2.2.1, i f the conditions of Theorem 1 are met for a linear map H : U -+ M and some t c R , and if h ( t ; ) defined in (a) of Theorem 1 belongs to Y 1 ( R ) , then we have (8) for each u E U n (Y ' ( R ) U Pm(R) ) , i.e. for each (essentially) bounded or integrable u in U. In this case we have a representation for ( H u ) ( t ) for uE U of the form (7) but not of the form (8) only if U E U is neither bounded nor integrable. Such inputs are usually of limited interest.

However, if h( t , . ) is not in Y ' ( R ) , then we have (8) only for those inputs uE U which are integrable, and we must represent ( H u ) ( t ) as a limit of the form (7) for some interesting inputs, such as the unit constant function 1. As an example, let U denote the space of piecewise-constant functions on R (i.e. functions which take on finitely many values and have only a finite number of discontinuities) and define a map H : U + M by

sin(t - r ) ( H u ) ( t ) = lim u ( r ) d r , t c R , uE U

It is clear that the conditions of Theorem 1 are satisfied for each t E R and that h ( t , * ) , t E R , is defined by h(r, r ) = sin(t - r ) / ( t - r ) for almost all 7 in R. Thus, given t E R , h ( t , * ) is not in P ' ( R ) and, as is clear from (13), we have a representation of the form (8) for (Hu) ( r ) only for those inputs u E U which have bounded support. Given f E R , (Hl ) ( t ) must be expressed as a limit of the form (7); in fact,

sin(t - 7) ( H l ) ( t ) = lim h( t , r ) d r = lim d r , t E R

n - m S", so since h( t ; ) is not in P ' ( R ) , we cannot reduce the limit t o the form (8).


2.4. Represen tations for time-invariant maps

In this section we consider linear maps that are time-invariant. The results given concern maps of functions defined on the whole line R. Similar results for the half-line and R"' cases are easily obtained.

The theorems in this section are related to Theorems 3 and 4 of Reference 1, which concern causal linear maps of functions defined on the half-line [0, a). The notation used is that of Section 2.2.1.

Theorem 4 is our main representation result for linear time-invariant maps. We will refer to the following hypothesis in our statement of the theorem.

(Al) Given t and 7 in R and u > 0, one has (HT,u,)(t) = (Hv,)(t - T), where uu E W is defined by uu = xlo,O) (notice that u0 = uwu).

We now present Theorem 4.

Theorem 4

Let H : U - t M be linear and suppose that (Al) holds. Then

( 1 ) the condition that HE B(w, t ) for some w E g m ( R ) for each t E R is necessary and sufficient for the following to hold: (a) g(7) 4 lim,-o (Hw,)(7) exists for almost all 7 in R (b) the function g belongs to g m ( R ) (c) limn-m j l ,g ( t - T ) U ( T ) d7 exists for each u E U and t E R and we have

(Hu)( t )= l im 1' g(t-T)u(7)d7, tER, u E U n - m - n

(2) (a)-(c) above hold with g belonging to Y ' ( R ) if and only if HE B (w,O) for some W E 9 ' ( R ) n g'"(R) .

We note the following.

Corollary 4

Under the conditions of Theorem 4, a linear map H : U -+ M is time-invariant.

Proof of Theorem 4

( 1 ) Necessity. With w E g m ( R ) given by w(7) = g(t - 7) for r E R , an argument similar to that for Theorem 1 shows that H E B(w, t ) for each t E R. Suflciency. Using (a) of Theorem 1, the linearity of H and (Al),

h(t , t - 7) 4 lim (HT(,- T)w,,)( t ) = lim (HwU)(7)

exists for almost all 7 in R. Thus (a) holds and g(7) = h(t , t - 7) for almost all 7 in R for each t E R. Parts (b) and (c) follow from (b) and (c) of Theorem 1 .

(2) If HEB(w,O) for some wEP'(R)nLPm((R), then h(0;) defined by (a) of Theorem 1 is in Y ' ( R ) , so since g(7) = h(0, - 7) for almost all 7 in R , g E 9 ' ( R ) . Conversely, if g in (a)-(c) belongs to g l ( R ) , then defining w ~ . 9 ' ( R ) n 6 / 7 ~ ( R ) by w ( ~ ) = g ( - ~ ) , T E R , we have H E B (w,O). This completes the proof.

Theorem 5 provides necessary and sufficient conditions under which the function g defined in (a) of

0 - 0 0 - 0

Theorem 4 is continuous.

Theorem 5

Let H : U + A4 be linear and time-invariant. Then the following three statements are equivalent:


(i) there is a bounded continuous function k E M such that

( H u ) ( t ) = lim j " k ( r - 7)u(7) d7, f c R , u c U n - m - n

(ii) H E B(w, t ) for some W E Y m ( R ) for each t E R ,

exists for each t E R , and g is continuous on R

convergence to the limit uniform with respect to t in any bounded interval in R . (iii) H E B(w, t ) for some W E g m ( R ) for each t E R , and g ( t ) given by (15) exists for each C E R with

Proof of Theorem 5. If (iii) holds, then it is easy to see that (b) and (c) of Theorem 4 hold, so given to and t in R and (T > 0, by the linearity and time invariance of H ,

I(Hw,)(t) - (Hwu)(to)l = I(HT(2/,,- / ) W 0 ) ( 2 t O ) - (HT/,wu)(2tO)l = I(H(T(2l0- t ) - Tr0)wu)(2to)l

= 1 lim i n g(t- 7)((7'(2rf , - l ) - T d w u ) ( 7 ) d r l

= I j n - m - n

m

g(t - 7)((T(2r0- I ) - Td)wu(7) d7 - m

< Ilgllmll(T2ro-t)- Tdwul l -+ 0

as t -+ to. Thus, for each a > 0, (Hw,) ( t ) depends continuously on t. The uniform convergence of (Hw,) ( t ) with respect to f on bounded intervals implies the continuity of g, and (ii) holds.

If (ii) is satisfied, defining k = g and using Theorem 4 we have (i). Finally, suppose (i) holds. For each t E R ,

lim (Hw,)( t ) = lim k ( t - 7) d7 = k ( t ) 0 - 0

by the continuity of k, so (a) of Theorem 4 holds with g = k, and g ( t ) is defined by (15) for each f E R . Statements (b) and (c) of Theorem 4 are implied by the boundedness of k and (14). Thus H E B ( w , t ) for some w E P m ( R ) for each t E R .

Now let Z be a bounded interval in R with endpoints a and b ( a < 6). Given y > 0, ?:= [ a - y, b ] is compact, so g is uniformly continuous on f Then, given E > 0, we may choose 6 E (0, y) such that l g ( n ) - g(n)1 < E for all 71 and 7 2 in fsatisfying 171 - 721 < 6. Let t E ZC f a n d aE (0,6). Using (14), we have

< E

This establishes the uniform convergence to g = k. Thus (i) implies (iii), which completes the proof.

ACKNOWLEDGEMENT

We wish to thank Dr. Klaus Bichteler for pointing out the bound on h( t , - ) in Corollary 3.


APPENDIX: COROLLARY 1-LIKE RESULTS FOR MAPS O F BOUNDED FUNCTIONS ON [0, m) OR R"'

The following are direct corollaries of Theorems 2 and 3 respectively.

Corollary 5

Let H : Y m ( O , ..)-A? be linear and let t 2 0. Then the condition that there exists W E Y ' ( 0 , m)nY"(O, m) such that I(Hu)(r) l < 1 1 wull for all u c P m ( O , 00) is necessary and sufficient for the following to hold:

(a) h( t , 7) k lim,+o ( ~ ~ , w , ) ( t ) exists for almost all 7 in [O, 00) (b) the function h(t , .) belongs to Y'(0, m) n Ym(O, 00) (c) we have

m

( H u ) ( t ) = i h(t , T ) U ( T ) d7, u E P m ( O , m) 0

Corollary 6

Let H:g '" (R" ' ) - t MI, be linear and let t € R"'. Then the condition that there exists W E Y"(R"')nYm(R"') such that I(Hu)(t)l < )Iwull for all u € Y m ( R r " ) is necessary and sufficient for the following to hold:

(a) h(r, 7) & lim,+o (HT,w,)(t) exists for almost ail 7 in R"' (b) the function h( t , .) belongs to Y ' (R" ' ) n Y m ( R f " ) (c) we have

( H u ) ( t ) = h(t , T ) U ( T ) dr , u E Pm(R"') R

REFERENCES

1 . I . W . Sandberg, 'Linear maps and impulse responses', IEEE Trans. Circuits and Systems, CAS-35, 201-206 (1988). 2. W. Rudin, Real and Complex Analysis, 3rd Edn, McGraw-Hill, New York, 1981. 3 . D. Ball and 1. W. Sandberg, 'g- and h-representations for nonlinear maps', J . Math. Anal. Appl. in the press.

criteria for the existence of impulse responses and kernel representations for linear maps

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