Discrete prolate spheroidal wave functions and interpolation.
Citation for published version (APA):
Delsarte, P., Janssen, A. J. E. M., & Vries, L. B. (1985). Discrete prolate spheroidal wave functions and interpolation. SIAM Journal on Applied Mathematics, 45(4), 641-650. https://doi.org/10.1137/0145037
DOI:
10.1137/0145037
Document status and date: Published: 01/01/1985
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SIAM.J.APPL. MATH. Vol. 45, No. 4,August1985
(C)1985Society for IndustrialandApplied Mathematics 008
DISCRETE
PROLATESPHEROIDAL WAVE FUNCTIONS
AND INTERPOLATION*
PH. DELSARTE?, A. J. E. M. JANSSEN: AND L. B. VRIES:
Abstract. Wedescribe analgorithm for the interpolation of bursterrorsin discrete-time signals that can bemodelled as being band-limited. The algorithmcorrectlyrestores amutilatedsignal that is indeed band-limited. The behavior of the algorithm when applied to signals containing noise orout-of-band components can be analysed satisfactorily with the aid ofasymptotic properties of the discreteprolate
spheroidal sequences and wave functions. The effect of windowingcan also be described conveniently in
termsofthese sequences and functions.
1. Introduction.
In
this paperwe consider discrete-time, real or complex-valued signalssT=(s(k))kZ.
Letting m=1, 2,..., we assume thats(k)
is given for allintegers kexceptk =0, 1,.
,
m-1.The vectorzT
[S(0),
,
s(m-
1)]
ofunknownsamples is estimated according to the following principle.
Assume
thatk0,,.-.,m-
]s(k)l
2<
c. Define the Fourier transform S ofs by(1.1)
S(0):=
E
s(k)
e’kand let 0
<
a<
1. Then we choose z so asto minimize(.)
f
Is(0)l
dO./1o/2
This is a finite-dimensional least squares
problem
in z, and the z minimizing(1.2)
isgiven by
(.3)
=
(1-
Mo)-’(Mt
+
Mt),
where
t(=
(s(k))<o,
t=
(s(k))>m_
andMo, M,
M2
aresubmatricesof the low-passmatrix M:
(sin
w(
kl)a)
(.4)
M
:[MllMoIM]:
k-
t)
:o,,,...,m-,,-<,< of whichthe columns with index l= 0, 1,..., m- constituteMo.
We
observe thatwe can write down the formula(1.3)
for for any s for whichMt
andM2te
are well defined; forthis to be the case it is not necessary thatl/
[s(o)[
ao
<
.
-1/2If e.g.
0o
/2, s(k)
exp(2ikOo)
forkZ,
thenMt
andM:t:
arewell defined asconditionally convergent series.
The motivation for choosing as in
(1.3)
is as follows. The signal s is called band-limited toa/2
if its Fourier transformS(O)
vanishes fora/2]O].
For
such an s the of(1.3)
equalsz=[s(0),..., s(m-1)]
r since wehave(1.5)
z MsM
t
+
MoZ
+
Mat,
*Receivedbythe editorsMay 17, 1983,and in final revised form November 16, 1984. tPhilipsResearchLaboratory, Brussels,Belgium.
PhilipsResearchLaboratories, P.O. Box80.000,5600JA Eindhoven,the Netherlands. 641
so that indeed z
=(I-Mo)-(Mltl
+
m2t2).
The purpose of this paper is to find out how stable formula(1.3)
is under imprecision ofdata,
such as additive white noise,orthepresenceofout-of-bandcomponents in thesignal. Wearefurthermore interested in the magnitude and nature of the interpolation error
z-:?
for such signals. Theinterpolationmethod is sufficiently stable only forrather modest values ofma, such as ms -<5. Thiscanbe seenfrom
(2.6)
whichshows thatthere are band-limited signals of unitenergy whose energyoutsidetheset{0,
1,,
m}
isoftheorder exp(-rna).
Slepian has derived in
[6]
asymptotic properties, as rn-
,
ofthe eigenvalues andeigenvectors of
M0
in connection with discreteprolate spheroidal wave functions and sequences. These asymptotic properties are known to be quite accurate, already for valuesofrnassmallas4 or5,and can therefore be usedfor analysing the interpolationmethod for modest values ofms.
We
referto Figs. 3-7 of[6].
Let
us summarize the contents of this paper. The largest eigenvalue ofMo
isusuallymuch closerto than the others. This is apparentfrom
[6],
Figs. 5-7(observe
that a 2W)
from which one can see that 1-Ao
is more than 10(50)
times smallerthan 1-
A
for values of ma as small as 2(3).
Hence,
there is a strong tendency forz-Y
(=
(I-Mo)-(z-Ms)
according to(1.3)
and(1.5))
to be a multiple of theeigenvector of
Mo
corresponding to the largest eigenvalue. We show in 3 that the components ofthiseigenvectorare all of onesign(also
see[7,
III]
and[5]).
Hence,
the interpolation errortendstobe of one sign;dependingontheparticular applicationthis property must be considered as a ratherunfavourable feature of the algorithm.
In
4 we studythe interpolationerrorz- whenthesignals contains white noise orout-of-bandcomponents.We
also present a qualitativeanalysis showingtheapproxi-mate location of the column index of the largest columns of the infinite matrix
(I-Mo)-[MIIM2]
as well as the slowdecayto zero ofthecolumns when the column index tends to.
The reason to study this matrix is formula(1.3),
expressing the estimate interms of the known samples by means ofthis matrix. In order that thecolumns of thismatrix tendto zero faster one could consider perturbations
M
ofthe matrixM
in(1.4).
These perturbations should have the effect that the columns of(I-o)-’[/r,
I/2]
tend tozerofaster,
while for signals band-limitedto/3/2
(with/3
slightly smaller than
c)
still good interpolation results are obtained. The stability of the perturbed method is still largely determined by the distance to of the largest eigenvaluesof/Qo.
Hence
we give in 5 approximating formulas and plots for theperturbed eigenvalues.
The interpolation methoddiscussedherehas been tested ondigital audiosignals,
and the performance ofthe method forthese signals is reported on in
[2].
2. Discreteprolatespheroidalsequencesand wave functions.
We
presentsomefactsfrom
[6]
in a slightly different notation.We
letMo
denote the Toeplitzmatrixsin.
"a’(
k-l)a)
r(k
l)
,:o,,...,,-’
and we order its eigenvalues Ak
k(m, a)
decreasingly so that>
Ao>/1>">
A,,-i
>
0.We
denote the corresponding eigenvectors, normalized as in[6],
by Vk(Vk(r))r=o,,...,,,-.
As
in[6],
Vk(r)
is defined for r<0,r>
m-1 bym- sin
r(r
I)
a(2.1)
Akv,(r)
2
vk(l),
:o
r(r-l)
DISCRETE SPHEROIDAL WAVE FUNCTIONS AND INTERPOLATION 643
The discreteprolate spheroidal wavefunctions
Uk(O)= Uk(m,
a;O)
aregivenbym--1
(2.2)
gk(O)-"ekVk(r)
e-’rri(m-l-2r)O,
r=0
where ek is or according as k is even or odd
(k
0, 1,...,m-1).
Among
themany formulas satisfied bythe
Uk
and /)k(see
[6,
2.1-2.3),
we mention(2.3)
vk(r)=--e-A-
f,/2<__lol<_/2
Uk(O)
e=i(m-l--r)dOe-lA-Wk(
m-1)2
rforr<0, r>m-1, where
Wk(’):=,/2<=lol<=/2
Uk(O)
exp(27ri’0)
dO.Further properties of the Vk and
Uk
canbe described in terms of the tridiagonalmatrix
D,
defined as(2.4)
Dnl
A
B2
B2
Am_2Bm
B.,_
withAk
((m--
1)/2--k)
2cos7ra,
Bk
=1/2k(m-k).
ThisD
hasreal,
distincteigenvaluestXk
Ik(m,
a).
If we order themas/Zo>/x >. >/x,,_, we have(2.5)
d477.2
--[(cos
dO 2vr0- cos77"o)Uk(O)]q-[J(m
2-
1)
cos2"rrO--tXk]Uk(O):0
for k 0, 1,...,m-
1,101-<_
.
Furthermore,Mo
and D commute and havethereforethe same eigenvectors. Also, Vk is the eigenvector of
D
with eigenvalueIn addition,we have the following asymptoticresults, for m- ooand fixed k"
(2.6)
--/kCOS 7FO
/
2)
]r
(r)_J
(2.7)
la,k=1/4m2-(k+1/2)m
sin7r+
O(1),
(2.8)
U(O)---
aJo
mL-
-l---c
---c
JbkR(O)
cosbk(0),
-<0_-<0,.
2--(o.<-
o<_-).
Here
Jo
is the Bessel function of 0thorder,
0,,1/27r
arccos(cos
7ra m-3/2)
(2.9)
ak(_
)[k/2][
vrm(
tk)
l
1/2cos 7ra
/
2)
bk
2(--
1)tk/21
(1
Ak)
sin7r
(2.10)
R
0(cos
7ra cos27r0 cos 27r0]-
/4,
(2.11)
Chk(O)
7r+
(2.12)
F(0-)
mk+l/2
=
arcsinq,(0-),
Gk(0-)
=-
arcsinq2(0-),
(2.13)
cos7/’0"2
g,,(0")=-I
+2coir-72)]
cos(’c/2)
1+ 1-sinThese asymptotic formulas are known to be accurate, also formoderate values of m
(see
the figuresin[6],
and Figs. and 2 of the presentpaper).
1.2,10-4 1.0
1-XoI’Y
l
0.608 0.2 0.0 0.2 0.4 0.6 0.g 1.0.10-2FIG. 1. 1-1o(m,
"
) as afunction of for m=6, =0.6 (solid line)" the dashedline is the linear approximation obtainedfrom (5.11 ).3. Positivity of the zeroth eigenvector of
Mo.
We
shall show in this section that Vo haspositiveelementsonly.We
present in addition some observations that concern theconjecture that
(I-M0)
-thas positive elementsonly; one ofthe authors learnedthis conjecture from F. A. Grfinbaum in 1981.
Vo has positive elements. In
2
it was noted that vk is the eigenvector of D correspondingto the kth eigenvalue. Consider the three-term recurrence relation(3.1)
Bk+lPk+l(X
(X-
A,)p,(x)
fork 0, 1,..., m-2,withthe initialization
p_l(x)=
0,po(x)=
1.This generatesthepolynomials
po(x),
pt(x),""’, p,,_(x)
withp(x)
ofdegree k and limx_pk(x)>0. In addition, set(3.2)
pro(X)
(X-
Am_l)Pm_l(X)
Bm_lPm_2(X).
It
is easy to show from(3.1)
and(3.2)
that the eigenvalues /-k ofD are the zeros of the polynomialp,,(x),
and that the corresponding eigenvectors vg of D equalVk--0"k[Po(lZk),
p(tXk),’’’, p,,-l(/Zk)]
r for some 0"kR,
0"k #0. In particular,the zeros ofp,,(x)
arereal anddistinct,and this also holds for thepl(X)
with<
m.Moreover,
the zeros ofpl(x)
strictly separate those ofpl+(x).
This is easily proved from(3.1)
and(3.2).
Hence,
all zeros of allpl(x)
with 1-<l<=m-1 lie between /x,,_ and /Xo, theextremal zerosof
p,(x).
Sincelimx_p(x) >
0for <-=<
m wehavep(/xo) >
0for 1-<<=
m-1.In
view ofvo(l)=
0"opt(iXo) and the normalization of Vo we see that allDISCRETE SPHEROIDAL WAVE FUNCTIONS AND INTERPOLATION 645 1.1.10-2 1.o Xo(.y}
l
0.9-08 0.7 0.6 1.6104 1.51-X
(’y) 1.4 1.3 1.2 1.10.0 0.2 O.L 0.6 0.8 1.0 10 0.0 0.2 O.L 0.6 O.8 1.0
O-2
FIG. 2. l--Ak(m a"3/) as afunctionofTform--8, a=0.3,k=0, (solid lines)" thedashed linesare
the linear approximations obtainedfrom(5.11).
Vk(0),""" Vk(m--1)
has exactly k changes of sign and at most kzeros.) We
were kindly informedby oneof the refereesthataproofof these results can also be derived from[1,
7.11].
Conjecture:
(I-Mo)
-
has positive elements.We
have extensive numerical evidence that the matrix(I-Mo)
-
has positive elements only. Once this conjecturehas beenproved, positivityof all components of Vofollowsfrom the Perron-Frobenius theorem.
Since I-
Mo
is asymmetricToeplitz matrix, itmakes sense to find a formulation of the positivity condition for(I-Mo)
-
in terms of the quantities that appear in the Levinson-Durbinalgorithmforthefast solutionofaToeplitz system of linearequations.This can be done as follows.
Denote
rk sinrka/Trk,
and letR
(p)--(
Okl l’k_l k,l=O,l,...,p_ r(p)-[--rl,’’’
--rp]
Tfor p 1,.,
m.It
followsfrom elemen-tary matrixtheory and[4,
C],
that(I-
Mo)
-
has positive elements only,if and onlyif
a(p,p)<O
for p= 1,..., m.Here
a(p,p)
is the pth component of the vector[RP]-rP).
While it is known thatla(p,
P)I
<
for all p(cf. [4]),
we have numerical evidencethata p,p)
-sin
7ra/ 2)
as p-
.
(This
formulais accurate for small valuesof p
already.)
In
terms oftheinterpolation algorithm the conjectureadmitsthefollowingformula-tion: if
s(k)=0, k-l,
0,..., m-l,s(k)
is unknown fork=0,
1,..., m-l, ands(-1)
1, thenm_=-a(m,
m)>0.
4. Application to the interpolation algorithm.
We
analyse the interpolation algorithm of 1, employing the asymptotic properties of the eigenstructure ofMo
(accurate
as they are for small values ofmaalready). In
the following subsections we consider the effect of additive white noise, and ofthe presence of out-of-bandcomponents, and westudy the asymptotic behavior of the norm of the columns of the
4.1. Band-limited signals corrupted by white noise.
We
first show how additive white noise affects theinterpolation.It
turns outthat the interpolation errormustbeexpected to bepulse-shaped.
THEOREM4.1.
Assume
thats(k)
x(k)+ n(k),
where x(x(k))kZ
isband-limitedto
a/2
andn(
k))k
isadditive whitenoisewithvariancetr2.
Therearerandom variables Pk, kO,
1,,
m with[Pk]
0,[PkPl]
O’2Ak(1
Ak)
-l6kl
such that Zo-.
m--l T
k=O
pkVk, where Zo=Ix(0), x(1),... ,x(m-1)].
Proof.
We may assumethat Zo 0. We let T=[trl0
t]
so thatM
t
+ M2
t2--Mt,
m-1
=(I-Mo)-’Mt=
(1--Ak)-’(Mt,
Vk)Vk.
Whenqk:=
(Mt, Vk)
it canbeshown fromthe definition of the Vk and theorthogonalityrelations satisfied by the Vk
(over
the ranges{0,
1,...,m-1}
and{...,-2,-1}t_J
{m,m+
1,...},
see[6])
that [qk]=0,[qkql]=tr2Ak(1--Ak)6k.
Hence,
the theorem is proved withPk--(1
Ak)
-1qk.
Note.
In view ofthe asymptotic properties of theAk
as given in(2.6)
it follows that usuallyAo
is much smaller than A1, A is much smaller than 2, andm--1
so on.
Hence,
inthe sumk=O
pkVk thetermwith k 0is usuallydominant. Itis now clearfrom the result of 3thataddingwhite noisetoband-limitedsignalsoften results in one-sided interpolation errors.Compare
Fig. 2 in[2].
4.2. Signals containing out-of-band components. The following theorem can be
used to determine the effect of out-of-band components on the interpolation result.
As
in the previous subsection, the interpolation errorsmust be expected to be pulse-shaped.7-=
(exp
(27rikO))k be partitioned THEOREM 4.2. Let101
<--
1/2,
101
!2,
and let Som--1
in the usual wayas
[to[zoltfo].
Then we havezo-,o=O
orEk--O
Ck(O)Vk
accordingas101</2
or> /2,
whereCk( O)
(1--
Ak)--le-
exp Tri(m--1)O) Uk( O).
Proof
We
have(see
1)
(Mso)(r)=so(r)=zo(r)
or 0 according as[01<a/2
or>
a/
2.Hence
:?0
Zoor-(I
Mo)
-1Mozo
accordingas101
<
/
2 or>
/
2.For0[
>
/
2m--i
we have
Zo--Zo=k=O(1--Ak)
1(Zo,
Vk)Vk,
and by(2.2)
we have(Zo, Vk)
e
exp(Tri(m-1)O)Uk(O).
This proves the theorem.Note.
Itfollows from(2.8)
thatUk
asymptotically(for
smallk)
israpidlyoscillat-ing in the set
a/2<-IOl<-_,
with a slowly varying amplitudeIbkR(O)l
on a large part of that set. Sincebk
is proportionalto(1
--Ak)
1/2,
we seethat themost importanttermm--l
in thesum k=O Ck
(O)
Vkisusuallythe one with k 0. This implies that the interpolationerror Zo-
.o
tends to be one-sided.Compare
Fig. 3 in[2].
4.3. Asymptoticbehaviourof the columns of the interpolationmatrix. We conclude this section by indicating roughly the behaviourof the columns
(I-Mo)-ly
<r> of thematrix
(i
Mo)-1[M,
M],
wherey(r)
(sin.
7r(r-l)ce)
7-k
"/T(
U
/:O,1,...,m--By
the definition of theVk’S
we havem-1
(4.1)
(I-Mo)-ly
(r=(1--Ak)-lAkvk(r)vk.
k=O
DISCRETE SPHEROIDAL WAVE FUNCTIONS AND INTERPOLATION 647
of the kthterm in this sum equals
(4.2)
(
r>m--I
)
(see
[6]),
and this isclearly largestfor small k.Hence,
we mustfind informationabout the asymptotic behavior ofvk(r)
for k small and r<
0 or r>
m-1.To
that end weuse the expression
(2.3)
ofvk(r)
in terms ofThe asymptotic behavior of
Wk(’)
as rn gets large can be determined from the asymptotic behavior ofUk(O)
in the range c/2_-<0_-<1/2
which is given by(2.8)
and(2.6).
Ignoring the contribution ofthe set c/2_-<0-<_0,, we obtain an approximation ofWk(’)
by consideringZk(-)+(--1)kZk(’),
where(4.3)
Zk(’)
bk
I
eZ=iR(O)
cos&k(0)
dOc/2_<--0 1/2
with
bk,
R and0k
given by(2.9)-(2.13).
Theintegral in
(4.3)
is fit for application of thestationaryphasemethod. We can writethe formula(2.11)
for&k(0)
as(4.4)
bk(0)
=+
(F’(o-)
+
G(o-))
do-.c/2
By
employing some trigonometric identities it is seen that-mr sinro"
r(k+ 1/2)
sin(-c/2)/sin
o(4.5)
F’(r)
/sin2
rr-sin(ra/2)’
G,(o’)
/sin
rr-sin(rc/2)
Let
->
0. Accordingto the stationaryphase principle the largest contribution to theintegral
(4.3)
comes from the 0-region where,(0)
is close to +2r’, i.e. whereF’(O)+ G’(O)
is close to +2rr. Now when k is small compared to m,we may ignoreGk.
It turns out thatF’
increases on[c/2,1/2]
from -oe to-mr/(cos (rc/2)),
andF"(1/2)
0.Hence,
it canbe expected thatIZk(r)l
is close to its maximum valuewhen"
is such thatF’(1/2)
=-2r’, i.e. r=m/(2
cos(rc/2)).
Furthermore there are no sol-utions 0 ofthe equationF’(0)=-2rr
when 0<r<m/(2
cos(rc/2)),
hence it mustbe expected that
IZk(’)l
is small for r in the vicinity of 0. Finally, whenm/(2cos (rc/2)),
the contribution to(4.3)
of the 0-region whereF’(O)
is close to -2r" is of the orderR(Oo)lF"(Oo)1-1/2,
where0o
is such thatF’(0o)=
-2r’.Usingtheexplicit formulas for
R
and F and noting thatOo-c/2=O(mr-1),
we find thatIZk(’)l
O(bkml/2"-1)
when"
gets largeand k is small compared to m.This provides sufficient information for getting an idea how
vk(r)
behaves for small k and r<0 or r>m-1.As
aresult we see from(4.1)
thatII(I-Mo)-ly(r)ll
canbe expected to be largest for
]r-(m-1)/2l.--m/(2cos(rc/2)).
It was found bycomputer simulations that this estimate is quite accurate, also for moderate values of
m. Furthermore, it is seen that the decay rate of
vk(r)
is not impressive: it can beexpected that
vk(r)=
O(r
-1)
as5. Perturbing thelow-pass matrix
M.
Inthis section wereplacethe matrix M of(1.4)
byawindowed versionof it so astoobtainaninterpolation matrix whose columnstendtozerofaster
(see
3rd subsectionof4).
Thiswindowingis achievedbyreplacing the numbersrk=sin ’ka/rk
in(1.4)
by rk(y)rkW(yl/2k),
where W is a smooth evenfunction onE withW(x)
-
0when x->o,
and y>
0 issmall.One should choose Wand ysuchthat theinterpolationresults are satisfactory for allsignalsband-limitedto/3
/
2,where/3
isonly slightlysmaller thana.To
judgethestabilityofthe interpolation methodthusobtained, oneshouldknow howthe eigenvaluesof the windowed version of the matrixMo
behave asafunction of y.It
turns out that for GaussianW
one canderive,for thelargest eigenvalues, perturbationseriesthe first 2termsof which provide
an accurate approximation inthe relevant y-range. We startwith the followingobservation.
THEOREM 5.1.
Assume
k=-
Is(k)l
<" LetQ
(q(k-
l))k=O,l,...,m_l,_c<l<c
where
q(-k)=q(k)
for
k2;’,
o
=_lq(k)[
<
andQo
(q(k--l))k,l=O,,...,m_
ispositive
definite.
PartitionQ=[Q,IQolQ=3,
sT=[tlzlt],
and letp(0)=
k---
q(k)
exp(-27rikO).
Then3:=
-Q(Qlt +
Q2t)
minimizes the integral 1/2(5.1)
I(a):=
p(O)
2
(k)
e2ik dO-1/2
k=-as a
function
of
a=[a(0),
a(1),.
.,
a(m- 1)]
7,
whereg(k)=
a(k)
ors(k)
according as O<-k<=
m-1 or not.Proof
Insert
in the integralforI(a)
the definitionofp and write out theintegrandas a triple sum. Performing the integration and using
1_/1
2exp(27rikO)dO=
6kO
forinteger
k,
one getsI(a)=(Qoa,
a)+2 Re [(a, Qlt,+Qzt2)]+c,
where C is a constantdetermined by
t
and t2. Theproofis easily completed now.Now
considerthe windowed matrix(5.3)
M(
T)
M,(
y)IMo(
,)IM2(
/)]
(rk-l(
Y)
)k=O,,,...,,,-,-<l<.
The function
p(O)=-p(O; y)
ofTheorem 5.1 isgiven by(5.4)
p(1 -X)
*where
X(0)=0
(lOl<=a/2),
x(O)=l(a/2<-101<-1/2)
the asterisk denotes convolution forperiodicfunctions of period 1,and(5.5)
(0)
2
W(y’/Zk)
e2rik.
According to Szeg6’s limit
theorem,
the eigenvalue distribution ofthe matrixQo
in Theorem 5.1 as m is asymptoticallyequal tothe value distribution of the function p.Hence,
the largest eigenvalues ofMo(y)
in(5.3)
tend to decrease with increasingy
>
0(providedthat0). Moreover
Irk(7)l
isadecreasingfunctionof7>
0.Hence,
there is definitely a tendency for the columns of
(1-Mo(y))-[M(y)lM2(7)]
todecrease in magnitude when y increases and also to tend to zero faster when the column indextendsto
.
Theprice to be paidisthat when 7is toolargethe functionp is too large around 0 0 which results inunsatisfactory restoration of signals with
significantlow-frequencycomponents.
More
specificnumbers andfiguresarepresented atthe end of this section forthe case ofa Gaussian windowWe
shall now use some peurbation theory to find approximations forAk(m,
a; y),the kth eigenvalue ofMo(y).To
that end we assume that Whasapowerseries expansion around 0,
(.6)
W(x)=.=o2
DISCRETE SPHEROIDAL WAVE FUNCTIONS AND INTERPOLATION 649
It
is arather immediate consequence ofthetheory in[3,
Chap.II,
2] that,
forsmall 3,>0,(5.7)
Ak(y)
E
ck,,y’,
n=0
where the coefficients ck,, can be expressed in terms ofthe unperturbed
Ak
and thenumbers
(mo’)V,
vi),
where(5.8)
Mo
)W(2)(0)
((k
2l)
rk_l)k,l=O,1,...,m_
1.(2v)!
For
example,the first three terms ofthe expansion for ,k(y) are(5.9)
hk+
’Y(M(o’)vk,
Vk)+
2[
(M(o2)Vk,
Vk)--jk
(M
l)vj’
Vk)12]"
)kj )kk
It
is possibleto express the coefficients ck,, intermsof theUk’s.
Indeed,
it is not hardto show that
2W(2)(0)
(_1)
dO2-’UUi
(5.10)
(Mo)Vj,
vi)=(47rZ)(2v)!
0, j- even, j-i odd.In
(5.10)
the derivatives of theUk’s
ata/2
canbeexpressedintermsof/xk
andUk(a/2)
by evaluatinganddifferentiating
(2.5)
repeatedlyat 0a/2
(where
cos 27r0-cos7ra0).
We
thus findthe firstorder approximation(k
small comparedtom)
for(5.11)
mTW(2)(O)
2 sin rra
/ 2)
cos ,n’a/ 2)
-_(m -1)sin rr-7- k+ msin
rr-7+-
(1
4
We finally present some resultsforthe window function
W(x)
exp(-x2).
Withrespectto the choice of3’ we note a trade-off between faster decrease of the columns
of the interpolation matrix and deterioration of the interpolation results for in-band
signalswithspectral energyclosetothebounds ofthe interval
[-a/2, a/2].
Forvalues ofmainthe range[2,
4]
wenoticedthatmakingthe elementsmk(3’)
oftheinterpolationmatrix
(I-Mo(3"))-M(3")
vanish up to 3 decimal places fork=0,
1,...,m-l, _-<-100 or=>
200+
rn- requireda 3’ in the range[10
-3,
10-2].
Taking 3’largerthan10-2resultedinunsatisfactoryinterpolation results forexponentials
s(k)
exp(2
wilcO)with/3/2
<-I01-<
a/2
with/3
up to over20% smallerthan a. Figures 1, 2 showgraphsof 1-
hk(m,
a;3’)
as a function of 3’ in the range[0,
10-2].
In
this range and for the values of rn and a as chosen in the figures, the linear approximation(5.11)
turns out to be quite convincing. This further underlines the accuracy of Slepian’s asymptoticresults.
REFERENCES
[1] J. N.FRANKLIN, MatrixTheory, Prentice-Hall, EnglewoodCliffs,NJ, 1968.
[2] A. J. E. M. JANSSENANDL. B. VRIES, Interpolationofband-limited discrete-timesignalsbyminimizing
out-of-bandenergy, Proc.IEEE Internat.Conf. ASSP,1984.
[4] J.MAKHOUL,Linearprediction:atutorial review,Proc. IEEE,63(1975),pp. 561-580.
[5] A.PAPOULISANDM. S.BERTRAN, Digitalfiltering and prolatefunctions,IEEE Trans.CircuitTheory, CT-19(1972),pp. 674-681.
[6] D.SLEPIAN, Prolate spheroidalwavefunctions,Fourieranalysis, anduncertainty-V: The discretecase,
Bell Syst.Tech.J.,57(1978),pp. 137 l- 1429.