numerical reconstruction of signals - ima · definition 1.1 ([9, 10]) let denote the unique complex...

September 14, 2011 NUMERICAL RECONSTRUCTION OF SIGNALS

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NUMERICAL RECONSRUCTION OF SIGNALS AND ORTHOGONAL POLYNOMIALS

Nicholas J. Daras

Department of Mathematics,

University of Military Education, Hellenic Military Academy,

166 73, Vari Attikis, Greece

E-mail: [email protected]

Abstract. Given only a few of initial Fourier coefficients for a continuous-time periodic signal, we construct efficient rational

approximants to the whole signal everywhere on his domain of definition. The convergence of these approximants depends

on the orthonormality of the chosen generating polynomial system into . The form of

each is characterized by recurrence relations dues to the connection between Schur and Szegö theories.

MSC2010: 41A20, 41A21, 42A16, 42C05, 65B99, 65D05, 65T40

Key words: approximation by rational functions, Padé-type and/or Padé approximants, Fourier coefficients, orthogonal

polynomials, acceleration of convergence, interpolation, trigonometric approximation and interpolation

INTRODUCTION Given a signal Fourier analysis easily calculates the frequencies and the amplitudes of those frequencies

which make up the signal. However, Fourier methods are not always a good tool to recapture the signal,

particularly if it is highly non-smooth. Especially, there is no analytical way to reconstruct with exactitude a

signal if only a few of its initial Fourier coefficients are known.

In this paper, we will consider a numerical version of the Carathéodory-Fejér interpolation problem ([4,

5], [13, 14]) in the trigonometric context. In particular, we will investigate a numerical method for

constructing efficient approximants to any continuous-time periodic signal by using only a few of its

initial Fourier coefficients. These approximants are real parts of rational functions with numerators

determined by the condition that the Fourier series expansion of the approximants matches the Fourier

series of the signal as far as possible. Motivated by this crucial property, the obtained approximants will be

called Padé-type approximants. The convergence of these approximants depends strongly on the

orthonormality of the chosen generating polynomial system into . The

form of is characterized by recurrence relations dues to the connection between Schur and Szegö

theories.

The detailed definition of a Padé-type approximant to a continuous-time periodic signal and their

principal properties are presented in Section 1. Section 2 is devoted to a study of the convergence of a

sequence of such approximants, with emphasis on the assumptions under which, for every sequence of

functions converging to a periodic continuous signal there is a Padé-type approximation sequence

converging point wise to that signal faster than the first sequence. Finally, in Section 3 numerical examples

are given making use of Padé-type approximants to signals.

1. CONSTRUCTION OF RATIONAL APPROXIMANTS TO A PERIODIC CONTINUOUS TIME SIGNAL Consider a periodic continuous time signal defined over an interval .

Suppose and has a finite number of extrema and discontinuities in any given

interval. Then, has a Fourier series expansion defined by

. The

problem we will investigate is the following: if only a few Fourier coefficients of the signal are given,

construct efficient rational approximants to the whole signal (almost) everywhere on .

Putting , the signal converts to a periodic continuous time signal defined over

the standard interval , with Fourier series expansion . It is clear that can be

NUMERICAL RECONSTRUCTION OF SIGNALS September 14, 2011

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identified with a real-valued function in of the unit circle by setting

. From the solution of the Dirichlet problem in the unit disk , it follows that the extended

real-valued function is harmonic in the open unit disk and such that

and the Fourier series expansion of the restriction of

to any circle of radius is given by .

We can consider Padé-type approximants to the harmonic function . Their fundamental property is that the Fourier series expansion of their restriction to any circle of radius coincides with the Fourier series expansion of as far as possible. Indeed, is the real part of an analytic function in . So, where denotes complex conjugate of . If

is the Taylor power series

expansion of around , then and

( ). Define the linear functional , where is the

vector space of all complex analytic polynomials. An application of Cauchy’s integral formula shows that

whenever . By density, there is a

continuous extension of into the space of all functions which are analytic in an open neighborhood

of . In particular, for every fixed point , the number is well defined and

equals . Hence, it holds for any . If the function

is replaced by a polynomial , then is approximated by . This is

an approximate quadrature formula and leads to a Padé-type approximant to the harmonic real-valued function .

DEFINITION 1.1 ([9, 10]) Let denote the unique complex polynomial of degree at most in , which interpolates the Cauchy kernel at points , i.e., for any

and . The real-valued function

is said to be a Padé-type approximant to , with

generating polynomial ( ). ■

The approximant in Definition 1.1 can be interpreted as real part of a rational function. Indeed, we put

and

.

THEOREM 1.2 ([8, 9]) is the real part of a rational complex function of type : + = + .The error of the respective approximation equals

.

The fundamental property of these approximants is the following. The Fourier series

expansion

of the restriction of to the circle of

radius fulfills

for any . ■

Since the limit is uniform on – , the function is the

Poisson integral of a continuous function on the unit circle.

DEFINITION 1.3 ([8, 9])The radial limit

is said to be a Padé-type approximant to , with generating polynomial

( ). ■

By Theorem 1.2, we get the following properties of these approximants.

THEOREM 1.4 The Padé-type approximant is the real part of a rational complex

function of type : .

The error of the respective approximation is given by the following theoretical formula

where the limit is taken in the –norm. The Fourier series expansion

of

fulfills

for any .


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Proof We will use the standard identification of the periodic signal ( ) with the function ( , ). Let be any sequence such

that . Since, by Theorem 1.2, ,

the uniform convergence of the sequence

to the radial limit function

implies that , and the first assertion

is proved. To prove the second assertion, recall that

. Further, by the

uniform convergence of the sequence

to the radial limit function

we also get

. Letting ,

from Theorem 1.2, it follows that there exists a such that

for any . Now, set

. An application of the continuity property for the linear

functional shows that

.

Hence

,

for any , which proves the second assertion. It remains to prove the third assertion. Since every harmonic function in the unit disk, with continuous boundary values, is the Poisson integral of its continuous restriction to the unit circle, we have

.

Henceforth, the Fourier series expansion of is

. From Theorem 1.2, it

follows that

for any , and the proof is complete. ■

From Theorem 1.4, it follows immediately that if only a few Fourier coefficients of the signal are given, then one can approximate by an approximant in the Padé-type:

.

2. CONVERGENCE ACCELERATION AND ORTHOGONAL POLYNOMIALS In this Section, we shall study the convergence of a sequence of Padé-type approximants to a signal.

Let

be an infinite triangular interpolation matrix with complex entries

. For any fixed , let denote the unique

polynomial of degree at most in , which interpolates the Cauchy kernel in the nodes

of the th row of , i.e.,

for any .

THEOREM 2.1 Suppose the family of generating polynomials is an

orthonormal bounded system in and

, for a . Then for

any real-valued periodic continuous-time signal , the associated

sequence

of Padé-type approximants to , with

generating polynomials , converges to almost everywhere on .

Especially, if , the sequence

converges to everywhere on . Proof Let be a real-valued periodic continuous-time signal. Let also and be a strictly increasing sequence of positive numbers such that . Fix any . By

Theorem 1.4, there is a subsequence

of such that

for almost all . Denote by the set of all points with this property. For every , one can find a such that

( ) and


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the function is in , so, by Mercer’s Theorem, the

Fourier coefficients of this function with respect to the orthonormal family

, tend to zero. This means that there exists a = , such that

(for any and ).

Combination of these inequalities shows that for any . This implies that

for almost all . If , the convergence holds for

any . ■

COROLLARY 2.2 Let the generating polynomials of a Padé-type approximation

( ) be such that

( ) and

( ).

If there are two constants and fulfilling

( ) and ( and ).

Then for any real-valued periodic continuous-time signal , the associated

sequence

of Padé-type approximants

to , with generating polynomials , converges to almost everywhere

on . Especially, if , the sequence

= , converges to everywhere on , . ■

In [10], we gave a stronger sufficient condition in terms of the entries only. If the

interpolation points are chosen so that and

, then,

for any real-valued periodic continuous-time signal , the associated sequence

of Padé-type approximants to , with generating polynomials

, converges to everywhere on .

One can also obtain interesting results about the form of , dues to the connection between

Schur and Szegö theories. This connection is often attributed to Akhiezer ([1]), but it appears earlier and in

greater detail in the papers of Geronimo ([11, 12]). It based on important recurrence relations which were

first given in previous Szegö’s work ([15]). Denoting by the polynomial , the

recurrence relations written in terms of the monic polynomials are of

the general form for certain parameters ( ).

In current terminology the numbers are called Szegö parameters. Let us see what the

Schur parameters are. To do so, we may remind the following Schur’s construction.

THEOREM 2.3 ([13, 14])There is a one-to-one correspondence between the class of analytic

functions which are bounded by one on the unit disk and the set of all sequences of

complex numbers which are bounded by and such that if some term has unit modulus, then all

subsequent terms are zero.

Proof Given any , define a sequence

.

If for some , then is constant and we taken for any . This occurs if and

only if is finite Blaschke product of factors:

,

where and …, are points in The numbers

( ) are

defined to be the Schur parameters for . ■

Since

has positive real part in the open unit disk and

value at the origin, the function belongs to . In 1943,

Geronimo showed that the Shur parameters

for coincide with the numbers :

whenever ([15]).


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We shall now study assumptions under which, for every sequence of signals converging to a continuous periodic signal, there is always a Padé-type approximation sequence converging pointwise to the signal faster than the sequence. This property, due to the free choice of the interpolation points permits us to construct better and better approximations to continuous functions.

Using techniques similar to those proposed by Bromwich and Clark ([3, 6]), we can prove the

PROPOSITION 2.4 Let be the operator of differences. Let also and be two

convergent sequences of real numbers. Suppose is strictly monotone and .

If , the sequence converges faster than the

sequence . ■

Combination of Proposition 2.4 with Theorem 2.1 gives the following result. THEOREM 2.5 Let be any strictly monotone converging sequence. Suppose the family is an orthonormal bounded system in and

, for a . Then, for any real-valued periodic continuous-time

signal , the associated sequence

of

Padé-type approximants converges to faster than .everywhere on

.■

3. NUMERICAL EXAMPLES In this section, we give examples making use of Padé-type approximants to periodic continuous-

time signals . EXAMPLE 4.1 Let be the signal ( ). If and are the roots of the fifth Legendre

polynomial in , then and therefore

( ). Indicatively, we have and

. If and , then and therefore

( ). Indicatively, we have ,

, , and

.

EXAMPLE 4.2 Let be the signal ( ). If and

, then and therefore

( ).

In particular, it holds

,

,

,

,

and

.

If and and , then and therefore


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( ). Observe that is well defined everywhere on , with the exception of

. This is a consequence of our choice , which in particular implies . However,

,

,

and

.

EXAMPLE 4.3 Let finally be the signal

If and , , then and

( ).

In particular, we have , ,

, and

.

REFERENCES [1] N. I. Akhiezer: The Classical Moment Problem, Hafner, 1965.

[2] C. Brezinski: The Asymptotic Behavior of Sequences and New Series Transformations Based on the Cauchy

Product, Rocky Mountain I. Math. 21(1) (1991), 71-84.

[3] T. J. Bromwich: An Introduction to the Theory of Infinite Series, 2nd edition, Macmillan, London, 1949.

[4] C. Carathéodory: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte

nicht annehmen, Math.Ann. 64(1907), 95-115.

[5] C. Carathéodory: Über den Variabilitätsbereich der Fourierschen Konstanten von positive harmonischen

Functionen, Rend. di Palermo 32(1911), 193-217.

[6] W. D. Clark: Infinite Series Transformations and their Applications, Thesis, University of Texas, 1967.

[7] T. Constantinescu: Schur parameters, Factorization and Dilation Problems, Operator Theory: Advances

and Applications, 2, Birkhäuser, 1996.

[8] N. J. Daras: Rational approximation to harmonic functions, Numer. Algorithms 20 (1999), 285-301.

[9] N. J. Daras: Padé and Padé-type approximation to periodic functions, Acta Applicandae

Mathematicae 65 (2000), 245-343.

[10] N. J. Daras: Interpolation methods for the evaluation of a periodic finite Baire measure,

Approximation Theory and its Applications 17(2) (2001), 1-27.

[11] J. Geronimo: On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied

Carathéodory and Schur functions, C.R. (Doklady) Acad.Sci.USSR (N.S.) 39 (1943), 291-295.

[12] J. Geronimo: On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied

Carathéodory and Schur functions, Mat. Sb. 105 (147) (1978), English transl. in Math. USSR Sb. 34(2)

(1978), 131-145.

[13] I. Schur: On power series which are bounded in the interior of the unit circle (I. Gohberg, editor),

Methods in Operator Theory and Signal Processing, Operator Theory: Advances and Applications, 18

(1986), pp. 31-59 (Original (in German): J. Reine Angew. Math. 147 (1917), 205-232).

[14] I. Schur: On power series which are bounded in the interior of the unit circle II (I. Gohberg, editor),

Methods in Operator Theory and Signal Processing, Operator Theory: Advances and Applications, 18

(1986), pp. 68-88 (Original (in German): J. Reine Angew. Math. 184 (1918), 122-145).

[15] G. Szegö: Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23 (4th edition with revisions, 1975;

1st edition 1939), Amer. Math. Soc., Providence, RI.

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