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Truncation error for the generalized Bessel type sampling series
Truncation Error for the Generalized Bessel Type Sampling Series by A. J JERRI Department Technology,
of Mathematics Potsdam,
NY
and Computer 13676, U.S.A.
Science,
Clarkson
College
of
and I.A.JOSLIN International
Business
Machines
Corp.,
Poughkeepsie,
NY
12601,
U.S.A.
Numerous upper bounds for the truncation error of the Shannon sampling expansion associated with the Fourier transforms are available in the literature. In this paper we derive an upper bound for the truncation error of the generalized sampling expansion associated with the Hankel (Bessel) transform. A clear example of the application of such series is in the analysis of optical systems with circular symmetry. A lower bound for the Bessel function J,(j& +iy), which agrees with the known asymptotic one and is necessary to this analysis, is derived. The method, which employs complex contour integration, can be applied to other generalized sampling expansions.
ABSTRACT:
I. Introduction
The Shannon sampling expansion
(1,2) for band-limited
signals states that
if e’“‘F(x) dx,
(1)
then f(t) = 2 f (gsi;a(pf!-;). “E--m
The generalized version (3) of this expression transforms with kernel K(x, t).
f(t) = I, ,&)K(x,
f(t) = lim N*Jn
0 TheFranklin Institute 001~32/82/110323~$03.00/0
F
W(x)
f(t.)s”(t)T
(2) involves
dx,
finite limit integral
(3)
(4)
323
A. J. Jerri and I. A. Joslin
and
s”(t) = s(t, t”) =
I’ dxW(x, t)K(x, t,) dx Idx)lWx, tn)l*dx ’ I
An example is the finite limit (bandlimited) Hankel transform, with a kernel K(x, t) = J,,,(xt), the Bessel function of the first kind of order m: xJ,(xt)F(x)
f(t) =
dx,
(6)
(7) I$f(t~,.~a(t~,“~~~~~t~af ); m+l
m,n
J,,,(at,,,,) = 0, n = 1,2,. . . ; tm,” = 5.
For our development, we may note here that the sampling functions {S(t, t,)} in (5) constitute an orthogonal set for integral transforms (3) with Fourier-type symmetric inverse (4):
F(x) =
1dt)Wt,
x)f(t) dt.
(8)
This is the case for the Hankel transform. f(t) =
IrnxJ,(xt)F(x)
dx,
(9)
0
F(x) = Ix tJ,,,(tx)f(t)
dt.
(10)
0
The truncation
error ET(t) is defined as
%-w=fW - fhw = ,“& f(tnxw9 t”). One of the earliest, but elementary, truncation for the Shannon sampling expansion (2)
lez-(t)l5 where complex 324
4M dN(
contour
1 - r)’
(11)
error bounds was derived (5)
M = Maxlf(t)l for all t; - UJ< t < 00,
integration
was employed
(12)
and a smaller finite band Journal of the
Franklin Institute Pergamon Press Ltd.
Truncation Error for the Generalized Bessel Type Sampling Series
limit ra ; 0 < r < 1, is used instead of the required a in (1). Numerous improved truncation and other relevant error bounds for (2) can be found in (2), Sec. VI. The only known attempt for a truncation error of the generalized sampling expansion (4) is due to Yao (6), whose expansion was shown to be a realization of the abstract reproducing kernel Hilbert space. His treatment and notation can be greatly simplified if we realize that the sampling functions {S(t, t,)} in (4-5) constitute an orthogonal set for transforms with Fourier type symmetric inverse (4), which is the case for the transform we are to consider; i.e. the Hankel transform (9-10). Let C;=I(S(
7 ti)iI:=/
S(t, ti)S(t, ti)dt;
thus (4) takes the form, f(t) = lim N+-
F
fCti)Ci4Ct9
(13)
ti).
/n aN
We also have ((4) Eq. (17)), Ci = IlS( , ti)ll = l/IjK( , ti)ll. The truncation error (11) now can be handled with the help of the usual Schwartz inequality, Parseval’s equality, and the orthogonality of S(t, ti) = C;+(t, ti), to obtain
5 p,
f2(ti)c~]“z[~,
5 [fob p(t)f2(t) -f
dt
-2
4*(t, tj)]“2
lK( 3ti)II~2f2(ti)]“2 *[labP(X)K2(& t) dx
S2(t9ti)llK( 9ti)l12]“2*
(14)
i=l
We note that, in contrast I(K(- , ti)ll, we have this plicates the evaluation of K(x, t) = J,,,(xt) with a = lET(
to the Shannon sampling expansion and its constant norm to evaluate for each transform, which comthe indicated series in (14). For example, the case of 1 becomes
5 [I,’ tf’(t) dt - 2 2 J;;i;i)]“2.
[$J:(t)
+ J:+,(t)}-
2 $ $i~~(;;]“2. m.1 (15)
In comparison with existing bounds for the Shannon sampling expansion, which are given as more simple and direct functions of the truncation limit N, Vol. 314, No. 5, pp. 323-328, Printed in Great Britain
November
1982
325
A. J. Jerri and I. A. Joslin
this bound is very complicated. Thus, a more computationally-feasible is needed, which is the subject of the next section. II. Truncation
Error Bound for the &-Bessel
bound
Sampling Expansion
In the following we present a more computationally-feasible bound for the &,-Bessel sampling expansion. This can be improved and the method extended to other sampling expansions,
(16) where the sampling series (7) is truncated by ra, O< r< 1, and .Il(j&)
at N, with the band limit a replaced
= 0, s = 1,2,. . . ,
If(z)/ 5 K =
I0
I($$*,
(17)
’ x*F*(x) dx.
(18)
The method employs complex contour integration, which parallels that for deriving the error bound (12) for the Shannon sampling expansion. The present result (16), however, required the derivation of the lower bound for .I,,,(jA,,+ iy) at any j& (7), the zeros of Jo, (19) where I,,,(y) is the modified Bessel function of order m. This allows us to consider a truncation error for any N which is not necessarily large, the condition for the existing asymptotic lower bounds ((8) p. 584),
where z is on the line joining A, - im to AN + im and provided that N exceeds a value dependent on V. The choice of j;,, is to maximize J,,,(jA,,) and hence improve the bound (19). In deriving the truncation error (11) for the sampling expansion (7), we consider a rectangular contour C = C, + C2 + C, + C,, where C, and C, are the vertical left and right sides that extend from j6N+1 - im to C, and C, are the j,k,N+,+ im, and -jA,,N+1- im to -j&.,+1 + im, respectively. horizontal lines. For f(t) in (6), we consider the contour integral
dz I II(2 - OJAZ) f(z)
(21)
c
326
Journal
of the Franklin Institute Pergamon Press Ltd.
Truncation
Error for the Generalized
Bessel Type Sampling Series
which vanishes along C, and C, as y + 7 03, since it can be shown (9), using the Schwartz inequality, that for f(z) in (6), with band limit ra instead of a, we have
If(t)15
K[$$]‘;‘,0 < r < 1
(22)
and that jJm(z)l is bounded as in (19). Now it is easy to see that the truncation error (11) can be expressed in terms of the integrals along C, and C, as
I4t)l = If(t) - Mt)l = ~lJ_(‘)IIj-+c3 (x”‘t;;~z)~,
(23)
which, after evaluating the residues at z = t, z = t,,,, and using the bounds (17) and (19), yields the desired truncation error bound (16).
III. Truncation Error for Other Sampling Expansions
The method of the last section can be extended to other sampling series, provided, of course, that a lower bound for the kernel K(x, t) is available. Another possibility is to derive the truncation error bound when the samples f(t,) and their derivatives f’(tJ are involved (9). It is expected that such error bound may have the usual N-’ dependence. It should be stressed that the work here represents only a starting point in finding practical truncation error bounds, and efforts should be continued to derive more efficient and tighter bounds, as has been done for the Shannon sampling expansion ((2) Sec. VI). Our present efforts, aimed toward improving the convergence of the generalized sampling series (13), and introducing a tighter truncation error bound, consist of convolving the sampling function 4(t, ti) with higher degree weight of related form. In the frequency domain, this is equivalent to multiplying the gate function Pa(X) =
I1, 0,
Ixl
Id> a
by a high order general hill function (10)associated with the general kernel K(x, t), instead of the known hill functions (B-spline) associated with the Fourier transform. We are continuing to develop and study the properties of these very new but useful “generalized hill functions”. References (1) C.
E. Shannon, “Communications in the presence of noise”, Proc. IRE, Vol. 37, pp. lo-21,1949. (2) A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review”, Proc. IEEE, Vol. 65, pp. 1565-1596, 1977.
Vol. 314, No. 5, pp. 32M28, Printed in Great Britain
November
,982
327
A. J. Jerri and I. A. Joslin (3)H. P. Kramer, “A generalized sampling theorem”, J. math. Phys., Vol. 38, pp. 68-72, 1959. (4) A. J. Jerri, “Application of the sampling theorem to time varying systems”, J. Franklin Inst., Vol. 293, pp. 53-58, 1972. (5) H. D. Helms and J. B. Thomas, “On truncation error of sampling theorem expansion”, Proc. IRE, Vol. 50, pp. 179-184, 1962. (6) K. Yao, “Application of reproducing kernel Hilbert spaces-band-limited signal models”, Inf. Contr., Vol. 11, pp. 427-444, 1967. (7) A. J. Jerri, “A lower bound for Bessel functions”, to be submitted. (8) G. N. Watson, “A Treatise on Theory of Bessel Functions”, Cambridge University Press, London, 1964. (9) A. J. Jerri and D. W. Kreisler, “Sampling expansions with derivatives for finite Hankel and other transforms”, Siam J. math. Anal., Vol. 6, pp. 262-267, 1975. (10)A. J. Jerri, “On general transforms hill functions”, Appl. Analysis, 1982, in press.
328
Journal
of the Franklin Institute Pergamon Press Ltd.
of Mathematics Potsdam,
NY
and Computer 13676, U.S.A.
Science,
Clarkson
College
of
and I.A.JOSLIN International
Business
Machines
Corp.,
Poughkeepsie,
NY
12601,
U.S.A.
Numerous upper bounds for the truncation error of the Shannon sampling expansion associated with the Fourier transforms are available in the literature. In this paper we derive an upper bound for the truncation error of the generalized sampling expansion associated with the Hankel (Bessel) transform. A clear example of the application of such series is in the analysis of optical systems with circular symmetry. A lower bound for the Bessel function J,(j& +iy), which agrees with the known asymptotic one and is necessary to this analysis, is derived. The method, which employs complex contour integration, can be applied to other generalized sampling expansions.
ABSTRACT:
I. Introduction
The Shannon sampling expansion
(1,2) for band-limited
signals states that
if e’“‘F(x) dx,
(1)
then f(t) = 2 f (gsi;a(pf!-;). “E--m
The generalized version (3) of this expression transforms with kernel K(x, t).
f(t) = I, ,&)K(x,
f(t) = lim N*Jn
0 TheFranklin Institute 001~32/82/110323~$03.00/0
F
W(x)
f(t.)s”(t)T
(2) involves
dx,
finite limit integral
(3)
(4)
323
A. J. Jerri and I. A. Joslin
and
s”(t) = s(t, t”) =
I’ dxW(x, t)K(x, t,) dx Idx)lWx, tn)l*dx ’ I
An example is the finite limit (bandlimited) Hankel transform, with a kernel K(x, t) = J,,,(xt), the Bessel function of the first kind of order m: xJ,(xt)F(x)
f(t) =
dx,
(6)
(7) I$f(t~,.~a(t~,“~~~~~t~af ); m+l
m,n
J,,,(at,,,,) = 0, n = 1,2,. . . ; tm,” = 5.
For our development, we may note here that the sampling functions {S(t, t,)} in (5) constitute an orthogonal set for integral transforms (3) with Fourier-type symmetric inverse (4):
F(x) =
1dt)Wt,
x)f(t) dt.
(8)
This is the case for the Hankel transform. f(t) =
IrnxJ,(xt)F(x)
dx,
(9)
0
F(x) = Ix tJ,,,(tx)f(t)
dt.
(10)
0
The truncation
error ET(t) is defined as
%-w=fW - fhw = ,“& f(tnxw9 t”). One of the earliest, but elementary, truncation for the Shannon sampling expansion (2)
lez-(t)l5 where complex 324
4M dN(
contour
1 - r)’
(11)
error bounds was derived (5)
M = Maxlf(t)l for all t; - UJ< t < 00,
integration
was employed
(12)
and a smaller finite band Journal of the
Franklin Institute Pergamon Press Ltd.
Truncation Error for the Generalized Bessel Type Sampling Series
limit ra ; 0 < r < 1, is used instead of the required a in (1). Numerous improved truncation and other relevant error bounds for (2) can be found in (2), Sec. VI. The only known attempt for a truncation error of the generalized sampling expansion (4) is due to Yao (6), whose expansion was shown to be a realization of the abstract reproducing kernel Hilbert space. His treatment and notation can be greatly simplified if we realize that the sampling functions {S(t, t,)} in (4-5) constitute an orthogonal set for transforms with Fourier type symmetric inverse (4), which is the case for the transform we are to consider; i.e. the Hankel transform (9-10). Let C;=I(S(
7 ti)iI:=/
S(t, ti)S(t, ti)dt;
thus (4) takes the form, f(t) = lim N+-
F
fCti)Ci4Ct9
(13)
ti).
/n aN
We also have ((4) Eq. (17)), Ci = IlS( , ti)ll = l/IjK( , ti)ll. The truncation error (11) now can be handled with the help of the usual Schwartz inequality, Parseval’s equality, and the orthogonality of S(t, ti) = C;+(t, ti), to obtain
5 p,
f2(ti)c~]“z[~,
5 [fob p(t)f2(t) -f
dt
-2
4*(t, tj)]“2
lK( 3ti)II~2f2(ti)]“2 *[labP(X)K2(& t) dx
S2(t9ti)llK( 9ti)l12]“2*
(14)
i=l
We note that, in contrast I(K(- , ti)ll, we have this plicates the evaluation of K(x, t) = J,,,(xt) with a = lET(
to the Shannon sampling expansion and its constant norm to evaluate for each transform, which comthe indicated series in (14). For example, the case of 1 becomes
5 [I,’ tf’(t) dt - 2 2 J;;i;i)]“2.
[$J:(t)
+ J:+,(t)}-
2 $ $i~~(;;]“2. m.1 (15)
In comparison with existing bounds for the Shannon sampling expansion, which are given as more simple and direct functions of the truncation limit N, Vol. 314, No. 5, pp. 323-328, Printed in Great Britain
November
1982
325
A. J. Jerri and I. A. Joslin
this bound is very complicated. Thus, a more computationally-feasible is needed, which is the subject of the next section. II. Truncation
Error Bound for the &-Bessel
bound
Sampling Expansion
In the following we present a more computationally-feasible bound for the &,-Bessel sampling expansion. This can be improved and the method extended to other sampling expansions,
(16) where the sampling series (7) is truncated by ra, O< r< 1, and .Il(j&)
at N, with the band limit a replaced
= 0, s = 1,2,. . . ,
If(z)/ 5 K =
I0
I($$*,
(17)
’ x*F*(x) dx.
(18)
The method employs complex contour integration, which parallels that for deriving the error bound (12) for the Shannon sampling expansion. The present result (16), however, required the derivation of the lower bound for .I,,,(jA,,+ iy) at any j& (7), the zeros of Jo, (19) where I,,,(y) is the modified Bessel function of order m. This allows us to consider a truncation error for any N which is not necessarily large, the condition for the existing asymptotic lower bounds ((8) p. 584),
where z is on the line joining A, - im to AN + im and provided that N exceeds a value dependent on V. The choice of j;,, is to maximize J,,,(jA,,) and hence improve the bound (19). In deriving the truncation error (11) for the sampling expansion (7), we consider a rectangular contour C = C, + C2 + C, + C,, where C, and C, are the vertical left and right sides that extend from j6N+1 - im to C, and C, are the j,k,N+,+ im, and -jA,,N+1- im to -j&.,+1 + im, respectively. horizontal lines. For f(t) in (6), we consider the contour integral
dz I II(2 - OJAZ) f(z)
(21)
c
326
Journal
of the Franklin Institute Pergamon Press Ltd.
Truncation
Error for the Generalized
Bessel Type Sampling Series
which vanishes along C, and C, as y + 7 03, since it can be shown (9), using the Schwartz inequality, that for f(z) in (6), with band limit ra instead of a, we have
If(t)15
K[$$]‘;‘,0 < r < 1
(22)
and that jJm(z)l is bounded as in (19). Now it is easy to see that the truncation error (11) can be expressed in terms of the integrals along C, and C, as
I4t)l = If(t) - Mt)l = ~lJ_(‘)IIj-+c3 (x”‘t;;~z)~,
(23)
which, after evaluating the residues at z = t, z = t,,,, and using the bounds (17) and (19), yields the desired truncation error bound (16).
III. Truncation Error for Other Sampling Expansions
The method of the last section can be extended to other sampling series, provided, of course, that a lower bound for the kernel K(x, t) is available. Another possibility is to derive the truncation error bound when the samples f(t,) and their derivatives f’(tJ are involved (9). It is expected that such error bound may have the usual N-’ dependence. It should be stressed that the work here represents only a starting point in finding practical truncation error bounds, and efforts should be continued to derive more efficient and tighter bounds, as has been done for the Shannon sampling expansion ((2) Sec. VI). Our present efforts, aimed toward improving the convergence of the generalized sampling series (13), and introducing a tighter truncation error bound, consist of convolving the sampling function 4(t, ti) with higher degree weight of related form. In the frequency domain, this is equivalent to multiplying the gate function Pa(X) =
I1, 0,
Ixl
Id> a
by a high order general hill function (10)associated with the general kernel K(x, t), instead of the known hill functions (B-spline) associated with the Fourier transform. We are continuing to develop and study the properties of these very new but useful “generalized hill functions”. References (1) C.
E. Shannon, “Communications in the presence of noise”, Proc. IRE, Vol. 37, pp. lo-21,1949. (2) A. J. Jerri, “The Shannon sampling theorem-its various extensions and applications: A tutorial review”, Proc. IEEE, Vol. 65, pp. 1565-1596, 1977.
Vol. 314, No. 5, pp. 32M28, Printed in Great Britain
November
,982
327
A. J. Jerri and I. A. Joslin (3)H. P. Kramer, “A generalized sampling theorem”, J. math. Phys., Vol. 38, pp. 68-72, 1959. (4) A. J. Jerri, “Application of the sampling theorem to time varying systems”, J. Franklin Inst., Vol. 293, pp. 53-58, 1972. (5) H. D. Helms and J. B. Thomas, “On truncation error of sampling theorem expansion”, Proc. IRE, Vol. 50, pp. 179-184, 1962. (6) K. Yao, “Application of reproducing kernel Hilbert spaces-band-limited signal models”, Inf. Contr., Vol. 11, pp. 427-444, 1967. (7) A. J. Jerri, “A lower bound for Bessel functions”, to be submitted. (8) G. N. Watson, “A Treatise on Theory of Bessel Functions”, Cambridge University Press, London, 1964. (9) A. J. Jerri and D. W. Kreisler, “Sampling expansions with derivatives for finite Hankel and other transforms”, Siam J. math. Anal., Vol. 6, pp. 262-267, 1975. (10)A. J. Jerri, “On general transforms hill functions”, Appl. Analysis, 1982, in press.
328
Journal
of the Franklin Institute Pergamon Press Ltd.