A SAMPLING FORMULA FOR DISTRIBUTIONS

W. E. Leithead and E. Ragnoli

Hamilton Institute, NUI Maynooth

Keywords:

Sampled-data systems, Frequency response, Multirate systems, Hybrid systems, Discrete-time systems.

Abstract:

A key sampling formula for discretising a continuos-time system is proved when the signals space is a subclass

of the space of Distributions. The result is applied to the analysis of an open-loop hybrid system.

1 INTRODUCTION

Consider the hybrid system of Figure 1, where x(t)

and y(t) are input and output, (A/D)

is an A/D con-

verter with sampling period T, (D/A)

is a zero-order

hold (ZOH) and P andC are the plants of a continuous

time system and a discrete time system, respectively.

In order to perform the transform domain analysis of

the hybrid system of Figure 1, the transform domain

response of a sampled signal must be related to the

transform response of its correspondent continuous

time signal. This is done by building the transform

response of the sampled signal upon the superposi-

tion of inﬁnitely many copies of its continuous time

transform response, using the formula

) =

∞

∑

k=−∞

G(s+ jkω

) (1)

where G is the Laplace transform of a continuous time

signal g, G

is the z transform of the sequence of its

samples {g(kT)}

∞

k=0

and T and ω

= 2π/T are the

sampling period and the sampling frequency, respec-

tively.

Till 1997, with the publication of (Braslavsky

et al., 1997), 1 was mathematical folklore. In fact,

it was very often used in the digital control literature

((M.Araki and T.Hagiwara, 1996), (J.S.Freudenberg

and J.H.Braslavsky, 1995), (T.Hagiwara and M.Araki,

1995)), (Leung et al., 1991), (Y.N.Rosenvasser,

1995a), (Y.N.Rosenvasser, 1995b) and (Yamamoto

and Araki, 1994)) and it appeared in many control

textbooks ((K.J.Astrom and B.Wittenmark, 1990),

(T.Chen and B.A.Francis, 1995), (G.F.Franklin

and M.L.Workman, 1990)), (B.C.Kuo, 1992) and

(K.Ogata, 1987)), but it was not established by a rig-

orous proof that indicated the relevant classes of sig-

nals considered.

Attempts to provide 1 with a proof are in (E.I.Jury,

1958), (K.J.Astrom and B.Wittenmark, 1990) and

(T.Chen and B.A.Francis, 1995). Those proofs are

based on the use of impulse trains of impulse trains,

those deﬁned as the function

∞

∑

k=−∞

δ(x− n

where δ(x) is the impulse function or Dirac function

or Dirac impulse such that

δ(x) =



+∞ x = 0

0 otherwise

and

∞

−∞

δ(x)dx = 1

However, the proofs lack rigour, since the impulse

function, and hence the impulse trains, cannot be de-

ﬁned as functions.

In (J.R.Ragazzini and G.F.Franklin, 1958) it is

shown the similarity between 1 and the Poisson Sum-

mation Formula

∞

∑

n=−∞

f(n)

∞

∑

k=−∞

∞

−∞

f(s)e

−2πiks

Consequently, 1 is often indicated as the Poisson

Sampling Formula. In (G.Doetsch, 1971) a rigorous

117

E. Leithead W. and Ragnoli E. (2007).

A SAMPLING FORMULA FOR DISTRIBUTIONS.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 117-123

DOI: 10.5220/0001652201170123

 SciTePress

x(t)

(A/D)

(D/A)

y(t)

Figure 1: Open Loop Hybrid System.

proof,that avoids the use of the impulse trains, for

) =

g(0

)

∞

∑

k=−∞

G(s+ jkω

)

is derived under the assumption that the series

∑

G(s + jkω

) is uniformly convergent. However,

since this condition is a transform domain condition,

it is not obvious when a time domain function satisﬁes

it.

In (Braslavsky et al., 1997) it is pointed that for

1 to hold, it is not enough to require that the Laplace

transform G of g and its sampled version, G

, are well

deﬁned. It is shown that, for n

= 2

and the con-

tinuous function

g(t) = sin((2n

+ 1)t), t ∈ [pπ,(p+ 1)p], p ∈ N

1 does not hold, despite the fact that G

) and its

sampled version with period T = π, are both well de-

ﬁned in the open right-half plane. In fact, it is proved

that

lim

n=∞

∑

k=−n

G(s+ jkω

)

does not converges for any s ≥ 0. Because of the

rapid oscillations of g as t → ∞ the class of signals

is restricted to functions with bounded and uniform

bounded variation.

Deﬁnition 1 ((Braslavsky et al., 1997)). A function g

deﬁned on the closed real interval [a,b] is of bounded

variation (BV) when the total variation of g on [a,b],

(a,b) = sup

a=t

<...<t

n−1

∑

k=1

g(t

) − g(t

k−1

)

is ﬁnite. The supremum is taken over every n ∈ N and

every partition of the interval [a, b] into subintervals

k+1

] where k = 0,1,...,n − 1 and a = t

< t

... < t

n−1

< t

= b.

A function g deﬁned on the positive real axis is of

uniform bounded variation (UBV) if for some ∆ > 0

the total variationV

(x,x+∆) on intervals [x,x+∆] of

length ∆ is uniformly bounded, that is, if

sup

x∈R

−

(x,x+ ∆) < ∞

With the class of signals restricted to UBV func-

tions, a proof for

)

g(0

)

∞

∑

k=1

g(kT

) − g(kT

−

)

−skT

∞

∑

k=−∞

G(s+ jlω

)

a more general formulation of 1, is provided.

Note that the well posedness of 1 is proved for

an open loop context, when the system considered is

stable. Despite the fact that it is rather common to

analyse a hybrid feedback system with the help of

1, even if the class of signals is restricted to UBV

functions, there is no proof of the well posedness of

the feedback when applying 1.

The discussion about the consistency of Mathe-

matical Frameworks in Systems Theory that started

with the exposure of the Georgiou Smith paradox in

(Georgiou and Smith, 1995) made Leithead and al., in

(Leithhead and J.O’Reilly, 2003) and (W.E.Leithead

et al., 2005), to attempt a Mathematical Framework

that expands the class of signals to the class of Dis-

tributions (an advantage of a Framework using Distri-

butions is that signals like steps, train pulses and delta

functions can be rigorously deﬁned as distributions).

Consequently, when dealing with hybrid systems, as

the one of Figure 1, in a Distributions Framework, the

well posedeness of 1 must be proved again.

However, despite 1 being quoted in Theorem 16.8

of (D.C.Champeney, 1987), no proof could be found

in the literature. In this paper a rigorous proof of The-

orem 16.8 of (D.C.Champeney, 1987), establishing

1 in a Distributions context, is provided in the Ap-

pendix. Furthermore, an application of this result to a

open loop hybrid system is provided. In particular, a

correct formulation for the D/A and A/D converters

in a Distributions context is established.

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

118

2 SAMPLING THE

TRANSFORMS OF A

DISTRIBUTION

The following notations and conventions are adopted.

The value assigned to each φ(t) ∈ D, the class of

good functions with ﬁnite support, by the functional

x ∈

D , the class of distributions , is denoted by x[φ(t)].

The symbols for, respectively a regular functional in

D and the ordinary function by which it is deﬁned,

e.g. x and x(t), are distinguished by the explicit pres-

ence in the latter of the variable. The following sub-

classes of

D are required.

= {x ∈

D : x regular with x(t) BV on each

ﬁnite interval and

x(t)

≤ c(1+

)

for some c > 0};N ≥ 0

= {x ∈

D : x regular with x(t) BV on each

ﬁnite interval and

x(t)

≤ c(1+

)

for some N ≥ 0 and c > 0}

= {x ∈

D : x regular with

Var

[a+t,b+t]

{x(t)} ≤ c(1+

)

for each

ﬁnite interval [a,b]

for some N ≥ 0 and c > 0}

= {x ∈

D : x regular with

Var

[a+t,b+t]

{x(t)} ≤ c(1+

)

for each

ﬁnite interval [a,b] for some c > 0};N ≥ 0

= {x ∈

D : x =

∑

∞

−∞

};T > 0

= {x ∈

D : x =

∑

∞

−∞

with

≤ (1+

)

for some

c > 0 and N ≥ 0};T > 0

= {x ∈

D : x =

∑

∞

−∞

with

≤ (1+

)

for some c > 0};

N ≥ 0,T > 0

where Var

[a,b]

{x(t)} is the variation of x(t) on the in-

terval [a, b] and the functional δ

is the delta func-

tional in

D deﬁned by

[φ(t)] = φ(τ)

Each functional x ∈

D is related by a linear bijections

to a functional

U such that

x[φ(t)] = 2πX[Φ(ω]

for all φ(t) ∈ D with

Φ(ω) = F [φ(t)](ω)

The functionals x and X constitutes a Fourier trans-

form pair with

X =

F {x} and x = F

−1

{X}

The subclasses

and

are the Fourier transforms of the the corresponding

subclass of

D . The members of U

and its subclasses

are periodic with period 2π/T.

A multiplier in

D is an ordinary function f(x) that

is inﬁnitely differentiable at each real value of x. The

multipliers in D are denoted by M . The subclass M

is the class of periodic multipliers with period 2π/T.

The relations between the transform of a distribu-

tion and its sampled version is established in the fol-

lowing Theorem.

Theorem 2 (16.8 (D.C.Champeney, 1987)). Suppose

f ∈

U has a transform

F ∈ D that is regular and

equal to a function F that is of bounded variation

on each ﬁnite interval (though not necessarily on

(−∞,∞)): then

(i) F(y) will be equal a.e. to a function F

(y) such

that, at all y,

(Y) =

−

) + F

)]

(ii) also

∞

∑

−∞

f(x− nX) (2)

will converge in

U to deﬁne a periodic functional ˜g

whose Fourier coefﬁcients G

are given by

= F

(n/X), n = 0,±1, ±2,...

(iii) if in addition

f ∈

and F(y)/(1+

)

is of

bounded variation on (−∞,∞), then 2 will converge

A proof of 2 is given in the Appendix.

3 OPEN LOOP HYBRID

FEEDBACK SYSTEM

Reconsider the plants P and C of the open loop hybrid

system of Figure 1 as the stable systems on

and

, respectively.

C : x ∈

7→ y ∈

,y = Ψ ∗ x

P : x ∈

D 7→ y ∈ D ,y = Φ ∗ x

where Ψ and Φ are convolutes on D

and

D , respec-

tively. However, since it is required that the idealised

sampling of continuous time signal is well-deﬁned,

a more appropriate reformulation of continuous time

A SAMPLING FORMULA FOR DISTRIBUTIONS

119

signals is provided by the subclass of distributions

Consequently, the convolutes Ψ and Φ corre-

sponding to plants C and P must be restricted to

and

, respectively. In transform domain the Fourier

transforms of signals are represented by functionals

and the transfer functions of systems are func-

tionals in

, the class of multipliers on

mapping

into itself for all N ≥ 0. It remains to establish a

correct formulation of the D/A and A/D converters.

3.1 Frequency Domain Analysis - D/A

Converter

Consider an ideal D/A converter which acts, with a

time constant T, on a discrete time signal, {x[k]} to

produce a piecewise constant continuous time signal,

y(t); that is, it acts as an ideal zero-order-hold (ZOH).

The linear relationship between {x[k]} and y(t) in

the frequency domain is established by the following

Theorem.

Theorem 3. A discrete time signals {x[k]} is acted on

by a ZOH, with time constant T, to produce a piece-

wise constant time signal y(t) such that

y(t) =

∞

∑

k=−∞

x[k]h

(t − k)

where h

(t) = 1 when t ∈ [0, T), zero otherwise. Pro-

vided there exists a periodic functional X ∈

with

Fourier coefﬁcients {x[k]}, then y(t) deﬁnes a regular

functional, y ∈

∩

such that Y = H

X where

Y =

F {y} ∈ U

∩

and H

F {h

} with h

the functional in

D deﬁned by h

(t).

Proof. y(t) is of bounded variation on any ﬁnite inter-

val, and, since X ∈

implies

x[k]

≤ c(1 +

)

for some c,

y(t)

< c

∗

(1+

)

for some c

∗

. Hence

y =

∑

∞

k=−∞

x[k]h

∈

. Furthermore for all b

∈

{−1,1} and {τ

,τ

,...,τ

n+1

} satisfying a ≤ τ

< τ

... < τ

n+1

≤ b

∑

i=1

(y(t + τ

i+1

) − y(t + τ

))

¯n

∑

i=1

(y(t + τ

i+1

) − y(t + τ

))

≤

¯n

∑

i=1

(

y(t + τ

i+1

)

y(t + τ

)

≤

¯n

∑

i=1

∗

(1+

t + τ

i+1

) + c

∗

(

t + τ

)

≤ 2c

∗

¯n(1+

t + b

)

where ¯n = int(t/(kT)). Hence, Var

[a+t,b+t]

{y(t)} ≤

¯c(1+

)

, for some ¯c > 0, and y ∈

. In addition,

since h

is a convolute on

D ,

y = lim

n→∞

∗

∑

k=−n

x[n]h

= lim

n→∞

∗

∑

k=−n

x[k]δ

= h

∗ lim

n→∞

∑

k=−n

x[k]δ

= h

∗ x

with x = F

−1

{X} and Y = H

X as required. 

Therefore, a D/A converter is represented in the

frequency domain by the multiplier H

mapping

into

∩

. Moreover, as a consequence, a dis-

crete time subsystem positioned before a D/A con-

verter is equivalent to a continuous time subsystem

positioned after the D/A converter, provided their fre-

quency response functions are the same.

3.2 Frequency Domain Analysis - A/D

Converter

Consider an ideal A/D converter which samples, with

a sampling interval T, a continuous time signal, x(t),

to produce a discrete time signal {y[k]} = {x[k]}. The

linear relationship between x(t) and {y[k]} in the fre-

quency domain is established by the following Theo-

rem.

Theorem 4. A continuous time signal, x(t), is acted

by a sampler with sampling interval T to produce a

discrete time signal {y[k]}. Provided there exists a

regular functional x ∈

deﬁned by x(t) then

(i) x(t) is equal almost everywhere to a function

(t) such that, at all t,

(t) =

−

(t) + x

(t))

and so sampling is well deﬁned with y[k] = x

(kT).

(ii) the summation

∑

∞

k=−∞

2πk/T

converges in

U , where X = F {x} ∈ U

, and {y[k]} are the

Fourier coefﬁcients for a periodic functional Y ∈

with period 2π/T such that Y =

[X] =

∑

∞

k=−∞

2π/T

Proof. Since X ∈

, x(t) is of bounded variation

on each ﬁnite interval and part (i) follows from The-

orem 2. In addition, there exists a periodic func-

tional Y ∈

U , with period 2π/T and Fourier co-

efﬁcients y

[k] = x

(kT) such that the summation

∑

∞

k=−∞

2πk/T

converges in

U and Y = O

[X] =

∑

∞

k=−∞

2πk/T

. Furthermore, since x ∈

, y =

−1

{Y} ∈

as required by part (ii). 

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

120

Therefore, an A/D converter is represented in the

frequency domain by the linear operator

mapping

into

for all N ≥ 0. Further proper-

ties of the operator

are established in the following

Theorem.

Theorem5. If X is a functional in

with n

deriva-

tive X

(n)

, Y is a functional in

and M

is a periodic

multiplier in

with period 2π/T then

(i)

[X] is a periodic multiplier in

with pe-

riod 2π/T provided j

(n)

∈

for all n ≥ 0;

(ii)

X] = M

[X];

(iii)

[X]] =

[Y]

[X] provided

(n)

∈

for all n ≥ 0.

Proof. (i)The regular functional x =

−1

{X} ∈

deﬁned by a function x(t), which by Theorem 4 part

(i) is equal almost everywhere to a function x

(t) such

that, at all t,

(t) =

−

(t) + x

(t))

For all n ≥ 0, since j

(n)

∈ U

, y ∈

, where

y is the functional deﬁned by t

x(t), and the se-

ries

∑

∞

k=−∞

(kT)

(kT)e

− jkωT

converges for all ω.

Hence, by Theorem 4 part (ii),

[X] is an inﬁnitely

differentiable regular functional. Furthermore, the n

derivative of O

[X] is continuous and periodic and so

bounded. Consequently,

[X] is a multiplier in

with period 2π/T.

(ii) For any X ∈

, M

X ∈

and by Theo-

rem 4 both

[X] ∈

and

X] ∈

exist.

Moreover, since M

is a multiplier in

with period

2π/T,

lim

n→∞

∑

k=−n

lim

n→∞

∑

k=−n

lim

n→∞

∑

k=−n

= M

lim

n→∞

∑

k=−n

and

X] = M

[X] as required.

(iii) It follows directly from part (i) and (ii). 

A consequence of Theorem 4 part (ii) is that, in

frequency domain, a continuous time sub systems po-

sitioned before an A/D converter is equivalent to a

discrete time subsystem positioned after the A/D pro-

vided their frequency response functions are the same.

3.3 The Response of the Open Loop

Hybrid Feedback System

In time domain the stable hybrid feedback system of

Figure 1 has solution

y = Φ ∗ [(D/A)

(Ψ∗ [(A/D)

x])] (3)

Deﬁne K

and K

the multipliers in

, the trans-

fer functions of the convolutes Ψ and Φ, respectively.

Therefore, by Theorems 3 4 and 5, in Frequency Do-

main, to 3 corresponds the solution

Y = K

[

X])]

where Y and X are functionals in

, the Fourier

transforms of y and x.

4 CONCLUSION

In this paper the proof of the well posedness of the

sampling of a the transform of a distribution is given,

establishing the correctness of the Sampling Theorem

16.8 quoted in (D.C.Champeney, 1987). Moreover,

the result is applied to the frequency domain response

of an open loop hybrid system, through the correct

formulation for the D/A and A/D converters.

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APPENDIX

Theorem 2 (D.C.Champeney, 1987)

Proof. (i) and (ii) Let

∈

D be the regular func-

tional deﬁned by f

(x) where

(x) =

∑

n=−N

jn(2π/X)x

sin(π(2N + 1)x/(2X))

sin(πx/(2X))

is a multiplier on

D and f

(x) is periodic with

period X such that

X/2

−X/2

(x)dx = X

For any regular ˜g ∈

D , with g(x) of bounded variation

on any ﬁnite interval, and any ψ(x) ∈ D,

(

˜g)[ψ(x)] = ˜g[ f

(x)ψ(x)] =

∞

−∞

g(x) f

(x)Ψ(x)dx

Since ψ(x) is of ﬁnite support, ∃K such that ψ(x) = 0

for

> (K +

)X. Hence,

˜g[ψ[x]] =

(K+1/2)X

−(K+1/2)X

g(x) f

(x)ψ(x)dx

X/2

−X/2

(

∑

k=−K

(x)g(x+ kX)ψ(x+ kX)

)

X/2

−X/2

(x)φ

(x)dx

X/2

−X/2





sin



π(2N+1)x







(

(x)x

sin



πx



)

where

(x) =

∑

k=−K

g(x+ kX)ψ(x+ kX)

For all k, g(x) is of ﬁnite variation on [(k −

1/2)X,(k + 1/2)X] and so φ

(x)x/(sin(πx/(2X))) is

of ﬁnite variation on [(k−1/2)X,(k+1/2)X]. Conse-

quently, by Theorem 5.10 of (D.C.Champeney, 1987),

x = 0 is a Dirichlet point and

lim

N→∞

X/2

−X/2

(sin(π(2N + 1)x/(2X))/x)

{φ

(x)x/sin(πx/(2X))}dx = X(φ

) + φ

−

))/2

It follows that

lim

N→∞

(

˜g)[ψ(x)]

= X

∑

k=−K

(g(kX

−

) + g(kX

))ψ(kX)

= X

∑

k=−K

(g(kX

−

) + g(kX

))

[ψ(x)]

Hence,

˜g converges to

h =

∑

k=−K

(g(kX

−

) + g(kX

))

D . Furthermore,

H =

F {

h} =

∞

∑

k=−∞

(g(kX

−

)+ g(kX

)) ˜e

k(2π/X)

∈

and by Theorem 16.3 of (D.C.Champeney, 1987),

is periodic with period 2π/X and Fourier coefﬁcients



(g(kX

−

) + g(kX

))



. However



˜g



F {

} ∗ F { ˜g}

∑

n=−N

n(2π/X)

∗

G =

∑

n=−N

n(2π/X)

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122

It immediately follows that

∑

∞

n=−∞

n(2π/X)

∈

and is equal to

H. Thus part (i) part and (ii) are

established.

(iii) Let f

as above. For any function g(x), with

g(x)/(1+

)

of bounded variation on (−∞,∞) for

some M > 0, and any ψ(x) ∈ S

g(x)ψ(x)

< c/(1+

)

for some c > 0. Hence,

∞

−∞

g(x) f

(x)ψ(x)dx

= lim

K→∞



(K+1/2)X

−(K+1/2)X

(x)g(x)ψ(x)dx



= lim

K→∞

X/2

−X/2

(x)

(

∑

k=−K

g(x+ kX)ψ(x+ kX)

)

In addition, for any x,

g(x+ kX)ψ(x+ kX)

< c/(1+

)

for some c > 0 and the series

(x) =

∑

k=−K

g(x+ kX)ψ(x+ kX)

is absolutely convergent. Hence, there exists a func-

tion, φ(x), such that φ

(x) converges pointwise to

φ(x) and there exists a constant, A, such that, for all

K > 0,

(x)

< A, ∀x ∈ [−X/2,X/2]. Consequently,

by Theorem 4.1 of (D.C.Champeney, 1987),

lim

K→∞

X/2

−X/2

(x)

(

∑

k=−K

g(x+ kX)ψ(x+ kX)

)

X/2

−X/2

(x)φ(x)dx

X/2

−X/2





sin



π(2N+1)x

















φ(x)



sin

(

πx

)













Furthermore, φ(x)x/(sin(πx/(2X)) is of bounded

variation on [−X/2,X/2]. By Theorem 5.10 of

(D.C.Champeney, 1987), x = 0 is a Dirichlet point and

lim

N→∞

X/2

−X/2





sin



π(2N+1)x







(

(x)x

sin



πx



)

dx = X(φ

) + φ

−

))/2

Since, for

< X/2,

g(kX + x)ψ(kX + x)

< c/(1+

)

for some c > 0

φ(0

) =

∞

∑

k=−∞

g(kX

)ψ(kX

)

and

φ(0

−

) =

∞

∑

k=−∞

g(kX

−

)ψ(kX

−

)

and it follows that

lim

N→∞

∞

−∞

(x)g(x)ψ(x)dx

∞

∑

k=−∞

((g(kX

)ψ(kX

))

+(g(kX

−

)ψ(kX

−

)))

∞

∑

k=−∞

(g(kX

) + g(kX

−

)ψ(kX

−

))ψ(kX)

Let

∈

be the regular functional deﬁned by

(x) then

is a multiplier on

. For the regular

functional ˜g ∈

deﬁned by g(x)

(

˜g)[ψ(x)] = ˜g[ f

(x)ψ(x)] =

∞

−∞

g(x) f

(x)ψ(x)dx

From the foregoing, it follows that

lim

N→∞

(

˜g)[ψ(x)]

= X

∞

∑

k=−∞

(g(kX

−

) + g(kX

))ψ(kX)

= X

∞

∑

k=−∞

(g(kX

−

) + g(kX

))

[ψ(x)]

Hence,

˜g converges to

h =

∞

∑

k=−∞

(g(kX

−

) + g(kX

))

. Furthermore,

H =

F {

h} =

∞

∑

k=−∞

(g(kX

−

) + g(kX

)) ˜e

k(2π/X)

∈

and by Theorem 16.3 of (D.C.Champeney, 1987),

is periodic with period 2π/X and Fourier coefﬁcients



(g(kX

−

) + g(kX

))



. However



˜g



F {

} ∗ F { ˜g}

∑

n=−N

n(2π/X)

∗

G =

∑

n=−N

n(2π/X)

It immediately follows that

∑

∞

n=−∞

n(2π/X)

∈

and is equal to

H. Thus part (iii) is established. 

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123