Wavelet Frames Generated by Bandpass Prolate
Functions
Jeffrey A. Hogan
Joseph D. Lakey
School of Mathematical and Physical Sciences
University of Newcastle
Callaghan NSW 2308 Australia
Email: jeff.hogan@newcastle.edu.au
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003–8001, USA
Email: jlakey@nmsu.edu
Abstract—We refer to eigenfunctions of the kernel corresponding to truncation in a time interval followed by truncation in a
frequency band as bandpass prolates (BPPs). We prove frame
bounds for certain families of shifts of bandpass prolates, and
we numerically construct dual frames for finite dimensional
analogues. In the continuous case, the corresponding families
produce wavelet frames for the space of square-integrable functions.
I. I NTRODUCTION
For T > 0 and Ω > 0 fixed, denote by QT the operator that
multiplies f ∈ L2 (R) by 1[−T,T ] and set PΩ = F −1 QΩ/2 F
R∞
where (Ff )(ξ) = fb(ξ) = −∞ f (t)e−2πitξ dt denotes the
Fourier transform. Abbreviate Q = Q1 . The eigenfunctions of
PΩ Q are the (zero order) prolate spheroidal wave functions,
e.g., [1]. When ordered by the eigenvalues of PΩ Q, which
is compact and self-adjoint, the prolate ϕn (t) has n zeros in
[−1, 1]. The sequence {ϕn } forms an orthonormal basis for
the Paley–Wiener space PWΩ = {PΩ f : f ∈ L2 (R)}, and is
a complete orthogonal family in L2 [−1, 1].
More generally, (abusing notation slightly) one can define
(QS f )(t) = f (t) 1S (t) and PΣ f = (QΣ fb)∨ where f and
Σ are measurable. When S and Σ are both compact, PΣ QS
is compact and it is self-adjoint on PWΣ = {PΣ f : f ∈
2
L
R (R)}. The operator PΣ QS is given by (PΣ QS f )(t) =
ρ (t − s) f (s) ds where ρbΣ = 1Σ . We will be particularly
S Σ
interested in the case in which Σ = {ξ : Ω0 ≤ |ξ| ≤ Ω}.
In this case we denote the corresponding operator PΣ by
PΩ0 ,Ω = P2Ω − P2Ω0 . The following is a special case of a
more general fact observed by Slepian and Pollak [2, (9) and
p. 59] for symmetric kernels.
Theorem 1: The eigenfunctions of PΩ0 ,Ω Q form an orthonormal basis for PWΩ0 ,Ω . Their restrictions to [−1, 1]
a complete orthogonal family in L2 [−1, 1] with
Rform
1
|ψ(t)|2 dt = λ when PΩ0 ,Ω Qψ = λψ.
−1
The eigenvalues of PΩ0 ,Ω Q are not necessarily nondegenerate, as are those of PΩ Q. However, just as with baseband
prolate eigenfunctions of PΩ Q, the eigenfunctions of PΩ0 ,Ω Q
can be taken as either symmetric or antisymmetric and we
index them as ψn where ψ2n is even and ψ2n+1 is odd, and
their eigenvalues λn satisfy λn+2 ≤ λn .
In what follows we are interested in frames for PWΩ0 ,Ω
generated by shifting the first several eigenfunctions. We are
ultimately interested in using prolates to analyze discrete
sample data. As such, we consider bandpass analogues of finite
dimensional versions of the prolates. The finite dimensional
analogue of PWΩ0 ,Ω is the space of (K 0 , K)-bandlimited
vectors in CN whose discrete Fourier transforms are supported
in frequencies between K 0 and K (mod N ). The baseband prolate theory for this case was originally worked out by Chamzas
and Xu [3]; see also [1, pp. 32-33], which also discusses other
contributions to the finite theory. Fix a finite dimension N
and think of vectors x = [x(0), . . . , x(N − 1)]T ∈ CN =
`2 (ZN ) as one period of a periodic sequence. Fix K such
that 2K + 1 ≤ N . Define the Toeplitz matrix A = AK
with Ak` = ak−` = sin ((2K + 1)(k − `)π/N )/(N sin((k −
`)π/N ), k, ` = 0, . . . , N − 1. Vectors in the image of A are
said to be K-bandlimited, as the DFTs of the columns of A
vanish at indices m ∈ ZN such that m > K mod N (or
|m| > K when index values are taken in −N/2, . . . , N/2,
N even). Denote by AM = AK
M the principal M × M
minor of A and consider the eigenproblem AM s = λs,
s = [s(0), . . . , s(M − 1)]T which can be regarded as a finite
dimensional analogue of the problem QPΩ Qϕ = λQϕ. In [3]
it is proved that when M + 2K < N and M > 2K, as we
will assume here, the ordered eigenvalues λn of AM satisfy
1 > λ0 > λ1 > · · · > λ2K , that is, they are nondegenerate.
Also, λ2K+1 = · · · = λM −1 = 0, which follows from the
fact that the range of AK has dimension 2K + 1. When
M + 2K ≥ N one will have some eigenvalues equal to one
(loc. cit.). On the other hand, one can ask which, and how
many, eigenvalues can be close to one (when M + 2K < N ).
Here it is possible to argue as in [4] that the number of
eigenvalues larger than 1/2 cannot exceed the normalized
time–bandwidth product defined as (2K + 1) × M/N . Direct
numerical eigenspace decomposition of AM is problematic for
large M and K. However, AM commutes with the symmetric
tridiagonal matrix TM defined by
π
π
Tkk = − cos (2k + 1 − M ) cos (2K + 1),
N
N
Tk,k+1 = sin
π
π(k + 1)
sin (M −1−k),
N
N
k = 0, . . . , M −1
and Tk−1,k = Tk,k+1 and Tk` = 0 if |k − `| > 1. Finite
prolate (FPS) sequences sn = [sn (0), . . . , sn (M − 1)]T ,
n = 0, . . . , 2K can readily be computed as eigenvectors of TM
corresponding to the nonzero eigenvalues of AK
M . Extensions
to `2 (ZN ) are obtained by applying the N ×N Toeplitz matrix
A to the vectors [sn (0), . . . , sn (M − 1), 0, . . . , 0] ∈ CN .
These extensions lie in the space of K-bandlimited vectors.
They posses the same double orthogonality properties of the
baseband prolates and form a complete family for the space
of K-bandlimited vectors.
II. E IGENFUNCTIONS OF TIME AND FREQUENCY LIMITING
In this section we prove a general identity for eigenfunctions
of operators of the form PΣ QS where QS denotes multiplication by 1S and PΣ = F −1 QΣ F. We assume that S and
Σ are compact and that Σ = −Σ. The latter implies that
the kernel of PΣ is real and symmetric. That PΣ QS is selfadjoint on PWΣ , PΣ is a projection on PWΣ and QS is a
projection on L2 (R) whose range has zero intersection with
that of PΣ then imply that the eigenvalues {λn } of PΣ QS
satisfy 1 > λ0 ≥ λ1 ≥ · · · > 0.
Lemma 2: Under the conditions above, the restrictions
to S of the eigenfunctions ψn of PΣ QS , normalized to
kψn kL2 (R) = 1, form a complete orthogonal family in L2 (S).
P
In addition,
λn |ψbn (ξ)|2 = |S|, ξ ∈ Σ.
Proof 1 (of Lemma 2): Completeness and orthogonality
follow from arguments in Slepian and Pollak [2]. In particular,
orthogonality in L2 (S) follows from symmetry of the kernel
of PΣ , cf., [2, (20) and p. 59]. Now one has
X
X 1
|(PΣ QS ψ n )∧ (ξ)|2
λn |ψbn (ξ)|2 =
λn
Z
Z
∞
X
1
=
1Σ (ξ) ψn (t) e−2πitξ dt ψn (s) e2πisξ ds
λ
S
S
n=0 n
Z X
Z
∞
1
= 1Σ (ξ)
e2πisξ ψ n (s) ds ψn (t) e−2πitξ dt
λn S
S
Z n=0
Z
2πitξ −2πitξ
= 1Σ (ξ)
e
e
dt = 1Σ (ξ) 1S = 1Σ (ξ) |S| .
S
Here
used the orthonormal expansion e2πitξ =
P 1 we2πi·ξ
he
, ψn iL2 (S) ψn (t) of the exponential in L2 (S) for
n λn
P
ξ ∈ Σ. Thus
λn |ψbn (ξ)|2 = |S| on Σ.
0
When Σ = {Ω ≤ |ξ| ≤ Ω} and S = [−1, 1] the convergence
can be shown to be uniform in ξ. Details will be provided
elsewhere. An illustration of the corresponding fact for standard prolates is foundPin [5]. As such, given A < |S| there is
N
an N such that A ≤ n=0 λn |ψbn (ξ)|2 ≤ |S| for all ξ ∈ Σ.
We now specialize to the case in which S = [−1, 1] and
Σ = {ξ : Ω0 ≤ |ξ| ≤ Ω}, letting {ψn } be a complete set
of eigenfunctions of PΩ0 ,Ω Q with kψn k = 1. Recall that a
family {fn } in a Hilbert space H is a frame for H if there
are constants 0 < A ≤ B < ∞ such that for each x ∈ H,
X
Akxk2H ≤
|hx, fn i|2 ≤ Bkxk2H .
n
The frame is tight if one can√take A = B.
∞,∞
Theorem 3: The family { λn ψn (t − k/(2Ω))}n=0,k=−∞
forms a tight frame for PWΩ0 ,Ω with bound 4Ω.
Proof 2 (of Theorem 3): The bandpass prolates ψn are concentrated in [−1, 1] and frequency supported in Ω0 < |ξ| < Ω.
Using Plancherel’s theorem we have
∞ D
XX
p
` E2
f, λn ψn · −
2Ω
` n=0
∞
D
E2
XX
=
λn fb, eπi` ·/Ω ψbn (·) n=0
`
=
∞
XX
`
Z
λn 2
fb(ξ) e−πi`ξ/Ω ψbn (ξ) dξ −Ω
n=0
= 2Ω
Ω
∞
X
Z
Ω
λn
fb(ξ) ψbn (ξ)2 dξ
−Ω
n=0
Ω
Z
|fb(ξ)|2
= 2Ω
−Ω
∞
X
2
λn ψbn (ξ) dξ
n=0
Z
Ω
= 4Ω
−Ω
|fb(ξ)|2 = 4Ω
Z
Ω0 <|ξ|<Ω
|fb(ξ)|2
where we have used the fact that if S = [−1, 1] then |S| = 2
in Lemma 2 and that f ∈ PWΩ0 ,Ω , that is, fb(ξ) = 0 unless
Ω0 < |ξ| < Ω.
In [5] the corresponding
√ tight frame property was proved for
the baseband prolates { λn ϕn }, so Theorem 3 is an extension
of that result to the case of bandpass prolates. However, the
frame bound here depends solely on Ω and not on Ω0 . In a
sense, the frame is more redundant
Ω0 → Ω. As in [5], the
PN −1 as b
2
0
monotone convergence of
n=0 λn |ψn (ξ)| to 2 on Ω <
|ξ| < Ω implies the following corollary of Lemma√2.
Corollary 4: For N sufficiently large, the shifts { λn ψn (t−
N −1,∞
k/(2Ω))}n=0,k=−∞
form a (non-tight) frame for PWΩ0 ,Ω .
III. F INITE DIMENSIONAL BANDPASS PROLATES
In this section we discuss how to construct numerically
bandpass prolates for the finite dimensional analogue of
PWΩ0 ,Ω , namely, the space of (K 0 , K)-bandlimited vectors
in CN whose discrete Fourier transforms are supported in
frequencies between K 0 and K (mod N ).
Numerically computed bandpass prolates were used in [6] to
study synchrony in sampled EEG signals. Here we numerically
compute finite dimensional bandpass prolates (FBPs). In this
setting it is possible to compute numerically the canonical dual
elements for frames generated by certain shifted FBPs.
For K 0 < K we regard the (K 0 , K)-bandpass prolates
concentrated in CM as particular elements of the span of the
M -concentrated and K-bandlimited finite prolates that happen
to be orthogonal to the M -concentrated and K 0 -bandlimited
prolates. In the continuous case, the (Ω0 , Ω) bandpass prolates
are defined as eigenfunctions of (PΩ − PΩ0 )Q. We then
define the M -concentrated and (K 0 , K)-bandlimited finite
dimensional bandpass prolates as the unique K-bandlimited
K0
extensions to CN of the eigenvectors of AK
M − AM . In
turn, these eigenvectors can be expressed in terms of their
coefficients when expanded in eigenvectors of AK
M.
Lemma 5: Let sn (M, K) denote the nth eigenvector of
K0
AK
Let σ be an eigenvector of AK
M , 0 ≤ n ≤ 2K. P
M − AM
expressed as σ =
n αn sn (M, K). Then the coefficients
α = [α0 , . . . , α2K ]T of σ form an eigenvector of the matrix ΛK − T T (M, K, K 0 ) where ΛK is the diagonal matrix whose entries are the nonzero eigenvalues of AK
M and
0
T
(T (M, K, K 0 ))k,` = AK
s
(M,
K)
s
(M,
K),
0
≤
k, ` ≤
`
M k
2K.
P2K
Proof 3 (of Lemma 5): Let x = n=0 αn sn (M, K). Then
0
0
K
K
K
(AK
M − AM )x = (AM − AM )
=
2K
X
2K
X
αn sn (M, K)
n=0
2K
X
λn αn sn (M, K) −
n=0
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0
αn AK
M sn (M, K) .
n=0
0
Since the range of AK is contained in that of AK
and the vectors {sn (M, K)}, 0 ≤ n ≤ 2K extend
uniquely to a basis
range of AK , one can write
P for Kthe
0
K0
sn (M, K) = k hAM sn (M, K), sk (M, K)i sk (M, K)
AMP
0 T
=
the sums
k (T (M, K, K ) )n,k sk (M, K). Rearranging
P
one finds that the coefficients βn of
βn sn (M, K) =
n
K0
(AK
M − AM )x are given by
K
0.2
0
-0.2
0 T
β = (Λ − T (M, K, K ) )α
where α = [α0 , . . . , α2K ]. In particular, if x is an eigenK0
vector of AK
M − AM then the coefficient vector α of x
in the basis {sn (M, K)} is an eigenvector of the matrix
ΛK − T (M, K, K 0 )T .
IV. N UMERICAL FBP S AND SHIFT FRAMES
Theorem 3 really says that shifts of all BPPs by multiples
of two over the time–bandwidth product result in a tight frame
for PWΩ0 ,Ω , though the frame has infinite redundancy. In that
case, the number of shifts per unit duration (D = 2) is equal to
the time–bandwidth product. Shifting at any integer multiple
of this rate also results in a tight frame in the continuous
case. In the FBP case, the analogous result is that taking
2KM/N shifts by multiples of N/(2K) of each FBP per unit
duration M results in a tight frame. As the space of (K 0 , K)bandlimited vectors has dimension 2(K − K 0 ), using all 2K
of the shifts of 2(K − K 0 ) independent FBPs would result in
a frame having redundancy 2K. In [5] we proved that one can
still obtain non-tight baseband prolate-shift frames for PWΩ
by shifting each of the first several prolates at a lower rate,
with the minimal rate needed for a (nonredundant) Riesz basis
being one prolate shift per unit time–bandwidth per unit time,
cf., also [7]. Any higher rate results in a redundant frame. The
motivation was that, in contrast to expansions in shifted sinc
functions, most of the energy of a bandlimited signal over a
given duration could be captured in terms of the prolate shifts
most concentrated in that duration. In what follows we will
illustrate this point numerically for bandpass prolates in the
finite setting. Figure 1 illustrates the first several FBPs and
their shifts in the particular case N = 128, M = 64, K = 16,
K 0 = 8 using the four most concentrated FBP generators.
Fig. 1. Eigenvectors corresponding to the four largest eigenvalues of (AK
M−
0
0 = 8. The first plot also shows
AK
)
for
N
=
128,
M
=
64,
K
=
16
and
K
M
a circular shift of the first FBP. The normalized time–bandwidth product and
the number of eigenvalues larger than 1/2 in this case is 8. The eigenvalues
for the four modes plotted are each at least 0.998.
4
2
0
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
4
2
0
4
2
4
2
Fig. 2. Moduli of the centered DFTs of the 128-point FBPs plotted in Fig. 1.
V. BANDPASS WAVELET FRAMES
Walter and Shen [7] proposed use of a cosine-modulate
of the first baseband prolate as a generator of a wavelet
frame. Besides not being optimally localized compared to
bandpass prolates, using a single generator does not provide as
much frequency resolution as using the first several bandpass
prolates. When Ω = 2Ω0 , a 1/Ω-dilated BPP has Fourier
support 1/2 ≤ |ξ| ≤ 1 and is time-concentrated in [−Ω, Ω].
As indicated in the discussion following Theorem 3, the shifts
by multiples of 1/(2Ω) of the first several dilated BPPs
D1/Ω ψn (t) = √1Ω ψn Ωt will then form a frame for PW1/2,1
Ω
of functions of the form {ψn,k
(t) = ψnΩ (t − k/2) : k ∈
Z, n = 0, . . . , N − 1} where ψnΩ = D1/Ω ψn . By the same
token, the dilates by 2j of ψnΩ are frequency supported in
2−j−1 ≤ 2−j and their translates by half-integers will form a
frame for PW2−j−1 ,2−j . Since the union of these multiresolution spaces is dense in L2 (R), it follows that the functions
Ω
= 2j/2 ψn (2j t − k/2), j, k ∈ Z, n = 0, . . . N − 1
ψn,j,k
form a wavelet frame for L2 (R) when N is sufficiently large.
0.1
0
-0.1
20
40
60
80
100
120
0.1
0
-0.1
0.05
0
-0.05
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
4
20
40
60
80
100
120
0.2
0
-0.2
2
0
20
40
60
80
100
120
20
40
60
80
100
120
0.2
0
-0.2
Fig. 3. Dual frame generators for the frame generators plotted in Fig. 1.
The FBP frame for the bandpass space consists of eight circular shifts by
multiples of 16 of the four generators, and the duals are shifts by the same
amount of the dual generators. The number of frame shifts is 4 × 8 = 32
whereas the dimension of the bandpass space is 16, so the redundancy factor
is two. The dual generators were computed numerically up to a pointwise
error of 10−6 by applying the frame conjugate gradient method [8], [9] to
the FBP generators. The duals are not as concentrated within M -points of
the center as are the primal generators, although dual generators tend to be
better localized when there is more redundancy (not shown).
1.5
1
0.5
20
40
60
80
100
120
1.5
1
0.5
4
2
0
-2
×10-16
Fig. 5. Frame reconstruction of a bandpass vector using the FBP-shift frame
corresponding to N = 128, M = 64, K = 16 and K 0 = 8, with the first
four FBPs each shifted by multiples of 16 units. The first plot shows the
real (solid) and imaginary (dash-dot) parts of a signal obtained by randomly
generating DFT values from a uniform distribution in [0, 1] supported in the
(K 0 , K)-band. The second plot shows the DFT values (solid) plotted under
the DFT modulus of the first FBP. The third plot shows the reconstruction
error when coefficients of the FBP shifts are taken as inner products with
corresponding shifts of the dual frame generators. The relative pointwise error
is on the order of 10−14 , consistent with the precision imposed in estimating
the dual frame generators.
bandwidth–of the first few BPPs that would then generate a
wavelet Riesz basis for L2 (R).
R EFERENCES
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
5
4
3
2
1
5
4
3
2
1
Fig. 4. Centered DFT moduli of the dual frame generators plotted in Fig. 3.
Orthogonality across scales follows simply from Plancherel’s
theorem and the fact that dilates of the BPPs by different
powers of two have disjoint frequency supports. Typically such
a wavelet frame will be redundant and not tight. Nonetheless,
the two-scale dilates of the BPP-shift dual frame for PW1/2,1
will form a dual frame for L2 (R), so frame expansions are
not complicated once the duals are computed.
VI. C ONCLUSION
We have established frame bounds for certain bandpass
prolate shift families, numerically generated such frames with
low redundancy for the analogous finite dimensional case,
and outlined a method to generate wavelet frames by dilating
and translating the first several (most concentrated) bandpass
prolates. It is an open problem to prove frame bounds for nonredundant families of shifts–one per unit time per unit time–
[1] J. Hogan and J. Lakey, Duration and Bandwidth Limiting. Prolate
Functions, Sampling, and Applications. Boston, MA: Birkhäuser, 2012.
[2] D. Slepian and H. Pollak, “Prolate spheroidal wave functions, Fourier
analysis and uncertainty. I,” Bell System Tech. J., vol. 40, pp. 43–63,
1961.
[3] W. Xu and C. Chamzas, “On the periodic discrete prolate spheroidal
sequences,” SIAM J. Appl. Math., vol. 44, pp. 1210–1217, 1984.
[4] H. Landau, “On the density of phase-space expansions,” IEEE Trans.
Inform. Theory, vol. 39, pp. 1152–1156, 1993.
[5] J. Hogan and J. Lakey, “Frame properties of shifts of prolate
spheroidal wave functions,” Appl. Comput. Harmonic Anal., 2014,
http://dx.doi.org/10.1016/j.acha.2014.08.003.
[6] J. Hogan, J. Kroger, and J. Lakey, “Time and bandpass limiting and an
application to EEG,” Sampling Thy. Signal Image Process., vol. 13, pp.
296–313, 2014.
[7] G. Walter and X. Shen, “Wavelets based on prolate spheroidal wave
functions,” J. Fourier Anal. Appl., vol. 10, pp. 1–26, 2004.
[8] K. Gröchenig, “Acceleration of the frame algorithm,” IEEE Trans. Signal
Process., vol. 41, no. 12, pp. 3331–3340, Dec 1993.
[9] P. Casazza, G. Kutyniok, and F. Philipp, “Introduction to finite
frame theory,” in Finite Frames, ser. Appl. Numer. Harmon. Anal.
Birkhäuser/Springer, New York, 2013, pp. 1–53.