Interpolatory frames in signal space∗

Amir Z. Averbuch and Valery A Zheludev

School of Computer Science, Tel Aviv University

69978, Tel Aviv, Israel

Abstract

We present a new family of frames, which are generated by perfect reconstruction filter banks

consisting of linear phase filters. The filter banks are designed on the base of the discrete interpola-

tory splines and are related to the Butterworth filters. Each filter bank comprises one interpolatory

symmetric low-pass filter and two high-pass filters, one of which is also interpolatory and symmet-

ric. The second high-pass filter may be symmetric or antisymmetric. These filter banks generate

the analysis and synthesis scaling functions and pairs of framelets. We introduced the concept of

semi-tight frame. While in the tight frame all the analysis waveforms coincide with their synthesis

counterparts, in the semi-tight frame we can vary the second framelets making them different for

the synthesis and the analysis cases. By this means we can switch the vanishing moments from the

synthesis to the analysis framelets or to add smoothness to the synthesis framelet. We constructed

dual pairs of frames, where all the waveforms are symmetric and all the framelets have the same

number of vanishing moments. Although most of the designed filters are IIR, they allow fast imple-

mentation via recursive procedures. The waveforms are well localized in the time domain despite

their infinite support. The frequency response of the designed filters are flat.

Introduction

Recently frames or redundant expansions of signals attracted a considerable interest of researchers

working in signal processing although one particular class of frames, the Gabor frames, is being applied

and investigated since 1946 [9]. As the requirement of one-to-one correspondence between the signal

and its transform coefficients is dropped, there is more freedom in the design and implementation

of the frame transforms. The frame expansions of signals demonstrate resilience to the quantization

noise and to the coefficients losses [10, 11, 12]. Thus, frames may serve as a tool for error correction by

the transmission of signals through lossy channels. Additional adaptation abilities of the overcomplete

representation of signals has a potential to succeed in feature extraction and identification of signals.∗The research was partially supported by STRIMM Consortium (2003-2004) administrated by the Ministry of Trade

of Israel.

1

An important class of frames are the frames generated by filter banks [5, 2, 12]. Actually, under

some relaxed conditions, a perfect reconstruction oversampled filter bank produces the frame expan-

sion. In this paper we use filter banks as an engine to construct a new family of frames possessing a few

properties, which are attractive for signal processing: symmetry, interpolation, flat spectra, combined

with fine time-domain localization, efficient implementation.

Infinite iteration of the frame filter banks results in limit functions, the so-called framelets. The

framelets, which are derived in the paper, are smooth, symmetric, interpolatory and may have any

number of vanishing moments. Non-compactness of their support is compensated by the exponential

decay as the time tends to infinity.

We consider in the paper the 3-channel analysis and synthesis filter banks comprising one low-

pass and two high-pass filters each. Downsampling factor N = 2 and the transfer functions of all

filters are rational functions. The low-pass filter and one of the high-pass filters in each filter bank

are interpolatory, whereas the even polyphase components of the other high-pass filters are zero.

Our approach to the design of interpolatory perfect reconstruction filter banks is, to some extent,

similar to the approach, which we used for the construction of the biorthogonal wavelet transforms

[1]. For example, the output of the low-pass component of the analysis filter bank is the sum of the

even polyphase component of the input signal and the approximation of the even component by the

values of the discrete spline of order 2r, which interpolates the odd samples of the signal. Such a

procedure is equivalent to the application to the signal of the filter, whose transfer function is the

squared magnitude of the transfer function of the half-band low-pass Butterworth filter of order r,

followed by downsampling. By using this approach we construct a diverse family of tight, semi-tight

and bi-frames.

The paper is organized as follows. In the introductory Section 1 we recall some facts concerning

filter banks and frames, which are necessary for the rest of the presentation. In Section 2 we describe

how to construct a tight frame and a bundle of semi-tight frames starting from arbitrary low-pass

filter. Having a pair of interpolatory low-pass filters, we construct a set of bi-frames. In Section 3

we present the derivation of the interpolatory filters from discrete splines and explain the relation

between the designed filters and the Butterworth filters. In addition, we establish some properties

of these filters and their corresponding waveforms. Section 4 is devoted to the construction of tight,

semi-tight and bi-frames using the designed filters. We call these frames the Butterworth frames. We

provide numerous examples supplied by graphical illustrations. In the concluding section we compare

a frame transform with the related biorthogonal wavelet transform and discuss the obtained results.

2

1 Preliminaries: filter banks and frames

1.1 Filter banks

We call the sequences x ∆= {xk}, k ∈ Z, which belong to the space l1, (and, consequently, to l2)

discrete-time signals. The z-transform of a signal x is defined as X(z) ∆=∑

k∈Z z−k xk. Throughout

the paper we assume that z = ejω.

The input xn and the output yn of a linear discrete time shift-invariant system are linked as

yn =∑

k∈Z fn−kxk. Such a processing of the signal x is called digital filtering and the sequence {fn}is called the impulse response of the filter f. Its z-transform F (z) ∆=

∑n∈Z z−nfn is called the transfer

function of the filter. Usually, a filter is designated by its transfer function F (z). The function

F (ω) = F (e−jωn is called the frequency response of the digital filter.

The set of filters{F k(z) =

∑n∈Z z−nfk

n

}K

k=1, which, being time-reversed and applied to the input

signal x, produce the set of decimated output signals {yk}Kk=1:

ykl =

∑

n∈Zfk

n−Nl xn, k = 1, . . . ,K,

is called the K−channel analysis filter bank. Here N ∈ N is the downsampling factor. The set of

filters{F k(z) =

∑n∈Z z−nfk

n

}K

k=1, which, being applied to the set of upsampled with the factor N

input signals {yk}Kk=1 produce the output signal x:

xl =K∑

k=1

∑

n∈Zfk

l−Nn ykn, k = 1, . . . , K,

is called the K−channel synthesis filter bank. If the number K of channels is equal to the downsampling

factor N then the filter bank is said to be critically sampled. If K > N then the filter bank is

oversampled.

In this paper we consider only 3-channel filter banks comprising one low-pass and two high-pass

filters each, whose transfer functions are rational functions and downsampling factor N = 2. The

analysis and synthesis low-pass filters we denote by H(z) and H(z), respectively, and the high-pass

filters are denoted as Gr(z), r = 1, 2 and Gr(z), r = 1, 2. We denote the output signals from the

analysis filter bank as s, dr, r = 1, 2. These signals are used as the input for the synthesis filter bank.

Then the analysis and synthesis formulas are:

sl = 2∑

n∈Zhn−2l xn ⇔ S(z2) = H(1/z)X(z) + H(−1/z)X(−z), (1.1)

drl = 2

∑

n∈Zgrn−2l xn ⇔ Dr(z2) = Gr(1/z)X(z) + Gr(−1/z)X(−z), r = 1, 2, (1.2)

xl =∑

n∈Zhl−2n sn +

2∑

r=1

∑

n∈Zgrl−2n dr

n ⇔ X(z) = H(z)s(z2) +2∑

r=1

Gr(z)dr(z2), r = 1, 2. (1.3)

3

Polyphase representation of filtering: Denote

Fe(z) ∆=∑

k∈Zz−k f2k, Fo(z) ∆=

∑

k∈Zz−k f2k+1,

E(z) ∆=∑

k∈Zz−k x2k, O(z) ∆=

∑

k∈Zz−k x2k+1.

We have

F (z) = Fe(z2) + z−1Fo(z2), X(z) = E(z2) + z−1O(z2) ⇒ Y (z) = F (z)X(z)

=(Fe(z2)E(z2) + z−2Fo(z2))O(z2)

)+ z−1

(Fo(z2)E(z2) + Fe(z2)O(z2)

).

Hence the z−transforms of the even and odd subarrays of the array y = {yk} are

Ye(z) = Fe(z)E(z) + z−1Fo(z)O(z), Yo(z) = Fo(z)E(z) + Fe(z)O(z),

respectively.

We introduce the analysis P(z) and the synthesis P(z) polyphase matrices , respectively:

P(z) ∆=

He(z) Ho(z)

G1e(z) G1

o(z)

G2e(z) G2

o(z)

, P(z) ∆=

He(z) G1

e(z) G2e(z)

Ho(z) G1o(z) G2

o(z)

.

Then

S(z)

D1(z)

D2(z)

= 2P(1/z) ·

E(z)

O(z)

,

E(z)

O(z)

= P(z) ·

S(z)

D1(z)

D2(z)

.

Here E(z) and O(z) are the z-transforms of the even and odd components of the output signal

x, respectively. If the signal x = x then the analysis and synthesis filter banks form a perfect

reconstruction filter bank. Analytically this property is expressed via the polyphase matrices as:

P(z) · P(1/z) =12I, (1.4)

where I denotes the 2× 2 identity matrix. Thus, the synthesis polyphase matrix must be left inverse

of the analysis matrix (up to the factor 1/2). Obviously, if such a matrix exists, it is not unique.

1.2 Frames

Definition 1.1 A system Φ ∆= {φj}j∈Z of signals form a frame of the signal space if there exist

positive constants A and B such that for any signal x = {xl}l∈Z

A‖x‖2 ≤∑

j∈Z|〈x, φj〉|2 ≤ B‖x‖2.

If the frame bounds A and B are equal to each other then the frame is said to be tight.

4

If a system Φ is a frame then there exist another frame Φ ∆= {φi}i∈Z of the signals space such that any

signal x can be expanded into the sum x =∑

i∈Z〈x, φi〉φi. The frames Φ and Φ can be interchanged.

Together they form the so-called bi-frame. If a frame is tight then Φ = cΦ.

Given the analysis H(z), G1(z) G2(z) and the synthesis H(z), G1(z) G2(z) filter banks, we denote:

ϕ1 ∆= {ϕ1(n) ∆= 2h(n)}, ψr,1 ∆= {ψr,1(n) ∆= 2gr(n)},ϕ1 ∆= {ϕ1(n) ∆= 2h(n)}, ψr,1 ∆= {ψr,1(n) ∆= 2gr(n)}, r = 1, 2 n ∈ Z.

Then the analysis and synthesis formulas (1.1) and (1.2) can be presented in the following way:

s1l = 〈x, ϕ1(· − 2l)〉, dr,1

l = 〈x, ψr,1(· − 2l)〉, r = 1, 2, l ∈ Z,

x =12

∑

l∈Zs1l ϕ

1(· − 2l) +12

2∑

r=1

∑

l∈Zdr,1

l ψr,1(· − 2l).

If the given set of filters form a perfect reconstruction filter bank then we have

x =12

∑

l∈Z〈x, ϕ1(· − 2l)〉ϕ1(· − 2l) +

12

2∑

r=1

∑

l∈Z〈x, ψr,1(· − 2l)〉ψr,1(· − 2l). (1.5)

Following is the condition for the Eq. (1.5) to yield the frame expansion of the signal x (Bolskei et al.

[2]).

Proposition 1.1 ([2]) Let the impulse response {h(n)}, {gr(n)}, r = 1, 2 of the analysis filters H,

Gr, r = 1, 2 belong to l1. Then the perfect reconstruction filter bank H, G1 G2 and H, G1 G2

implements a frame expansion of a signal x ∈ l2 if and only if the polyphase analysis matrix P (z) has

a full rank 2 on the unit circle |z| = 1.

For FIR filter banks this condition was established by Cvetkovic and Vetterli [5]. Obviously, if the

condition (1.4) is fulfilled then the matrix P (z) has a full rank.

Corollary 1.1 Let the impulse response {h(n)}, {gr(n)}, r = 1, 2 of the analysis filters H, Gr, r =

1, 2 belong to l1. Then the perfect reconstruction filter bank H(z), G1(z) G2(z) and H(z), G1(z) G2(z)

implements a frame expansion of a signal x ∈ l2 and the set of two-sample shifts of the signals ϕ1,

ψr,1, ϕ1, ψr,1, r = 1, 2 generate a bi-frame of the signal space..

One solution to (1.4) is the parapseudoinverse of P:

P(z) = P+(z) ∆=12

(PT (z) · P(1/z)

)−1 · PT (z). (1.6)

The synthesis frame corresponding to the polyphase matrix P+(z) is dual to the analysis frame. If

P(z) = PT (z) then the signals ϕ1 and ψr,1, r = 1, 2 generate a tight frame.

5

1.3 Multiscale frame transforms

The iterated application of the analysis filter bank to the signal s1 = {sk} produces the following three

signals:

s2l =

∑

n∈Zhn−2l s1

n, =∑

n∈Zhn−2l

∑

m∈Zhm−2n xm = 〈x, ϕ2(· − 4l)〉,

dr,2l =

∑

n∈Zgrn−2l s1

n, =∑

n∈Zgrn−2l

∑

m∈Zhm−2n xm = 〈x, ψr,2(· − 4l)〉,

where ϕ2(l) ∆= 2∑

n∈Zhn ϕ1(n− 2l), ψr,2(l) ∆= 2

∑

n∈Zgrn ϕ1(n− 2l), r = 1, 2.

Then the signal s1 is restored as

s1l =

12

∑

n∈Zhl−2n s2

n +12

2∑

r=1

∑

n∈Zgrl−2n dr,1

n , r = 1, 2,

and the signal x is expanded into the following sums:

x =14

∑

l∈Z〈x, ϕ2(· − 4l)〉ϕ2(· − 4l) +

14

2∑

r=1

∑

l∈Z〈x, ψr,2(· − 4l)〉ψr,2(· − 4l)

+12

2∑

r=1

∑

l∈Z〈x, ψ1

r (· − 2l)〉ψr,1(· − 2l),

where ϕ2(l) ∆= 2∑

n∈Zhn ϕ1(n− 2l), ψr,2(l) ∆= 2

∑

n∈Zgrn ϕ1(n− 2l), r = 1, 2.

Thus, provided the condition (1.4) is fulfilled, the sets of four-sample shifts of signals ϕ2, ψr,2, ϕ2,

ψr,2, r = 1, 2 and of two-sample shifts of signals ψr,1, ψr,1, r = 1, 2 generate a new bi-frame of the

signal space. If P(z) = cP(z) then the signals ϕ2 and ψr,2, ψr,1, r = 1, 2 generate a tight frame.

Subsequent iterations lead to the following expansion of the signal x:

x = 2−N∑

l∈Z〈x, ϕN (· − 2N l)〉ϕN (· − 2N l) +

2∑

r=1

N∑

ν=1

2−ν∑

l∈Z〈x, ψr,ν(· − 2ν l)〉ψr,ν(· − 2ν l),

where ϕN (l) ∆= 2∑

n∈Zhn ϕN−1(n− 2l), ψr,ν(l) ∆= 2

∑

n∈Zgrn ϕν−1(n− 2l),

ϕN (l) ∆= 2∑

n∈Zhn ϕN−1(n− 2l), ψr,ν(l) ∆= 2

∑

n∈Zgrn ϕν−1(n− 2l), r = 1, 2.

Thus we have a new bi-frame consisting of the shifts of the signals ϕN , {ψr,ν} and ϕN , {ψr,ν}, r =

1, 2, ν = 1, . . . , N .

1.4 Scaling functions and framelets

It is well known [6] that under certain conditions the filter bank H(z), G1(z), G2(z) generates the

continuous scaling function ϕ(t) and two framelets ψ1(t) and ψ2(t). Suppose that H(1) = 1. If the

6

infinite product

limN→∞

N∏

ν=1

H(ej2−νω) (1.7)

converges to a function Φ(ω) ∈ L2(R) then the inverse Fourier transform of this function Φ(ω) is the

scaling function ϕ(t) ∈ L2(R), which is a solution to the refinement equationϕ(t) = 2∑

k∈Z hk ϕ(2t−k).

If G1(−1) = G2(−1) = 1 then the functions

ψr(t) = 2∑

k∈Zgrk ϕ(2t− k) , r = 1, 2, (1.8)

are the framelets. There established in [6] a simple sufficient condition of the existence of a smooth

scaling function.

Proposition 1.2 ([6]) Let the transfer function H(z) be factorized as H(z) =(

1+z−1

2

)pK(z), where

K(z) is a rational function such that K(1) = 1. If the condition κ∆= sup|z|=1 |K(z)| < 2p−1−m

is fulfilled then there exists the scaling function ϕ(t) ∈ L2(R), which is continuous together with its

derivatives up to the order m.

In the time domain the relation (1.7) corresponds to the infinite iteration of the subdivision scheme

whose symbol is equal to 2H(z) and the initial data is the delta sequence. The limit function of this

scheme is the scaling function ϕ(t). This method is called the cascade algorithm [6]. Therefore for

the analysis of the convergence of the cascade algorithm and the regularity of the scaling functions

methods of the subdivision theory can be exploited. Sometimes these methods provide more accurate

estimation of the regularity than the above Fourier transform method. The following proposition is a

direct consequence of a result by Dyn, Gregory, Levin in [7].

Proposition 1.3 Let the transfer function H(z) be factorized as H(z) =(

1+z−1

2

)pK(z), where K(z)

is a rational function such that K(1) = 1. If the subdivision scheme whose symbol is 2K(z), converges

to a continuous function then there exists the scaling function ϕ(t) ∈ L2(R), which is continuous

together with its derivatives up to the order p.

Under certain relaxed conditions on the low-pass filter, whose transfer function H(z) is a rational

function, the generated scaling function ϕ(t)) decays exponentially. We cite the following sufficient

conditions.

Proposition 1.4 ([19]) Let the transfer function H(z) be factorized as H(z) =(

1+z−1

2

)M(z), where

M(z) =∑

i∈Z z−imi is a rational function, which has no poles on the unit circle |z| = 1. If the

inequality

max{∑

i∈Z|m2i|,

∑

i∈Z|m2i+1|} < 1

holds then there exists the continuous scaling function ϕ(t) and positive numbers A and g such that

|ϕ(t)| ≤ Ae−g|t|.

7

If the scaling function ϕ(t) decays exponentially and the filters G1(z), G2(z) have no poles on the

unit circle |z| = 1 then their impulse response gri , r = 1, 2, decay exponentially. Thus the framelets

ψ1(t) and ψ2(t) defined by (1.8) also decay exponentially.

It is said that a framelet ψr(t) has p vanishing moments if∫ ∞

−∞tsψr(t) dt = 0, s = 0, . . . p− 1.

The number of its vanishing moments of the framelet ψr(t) is equal to the multiplicity of zero of the

filter Gr(z) at z = 1 [17].

2 Interpolatory frames

2.1 Bi-frames

If the even polyphase component of a filter F (z): Fe(z) = 1/2 then the filter is called interpolatory. In

the rest of the paper we deal exclusively with the filter banks, whose low-pass filters are interpolatory:

H(z) =1 + z−1U(z2)

2, H(z) =

1 + z−1U(z2)2

. (2.1)

We assume that U(z) and U(z) are rational functions that have no poles on the unit circle |z| = 1,

U(1) = U(1) = 1 and the following symmetry conditions hold

z−1U(z2) = zU(z−2), z−1U(z2) = zU(z−2). (2.2)

If an interpolatory low-pass filter generates the scaling function ϕ(t) then this scaling function is

interpolatory that is ϕ(n) = δn, n ∈ Z.

The polyphase matrices for a filter bank using the interpolatory low-pass filters H(z) and H(z)

are

P(z) ∆=

1/2 U(z)/2

G1e(z) G1

o(z)

G2e(z) G2

o(z)

, P(z) ∆=

1/2 G1

e(z) G2e(z)

U(z)/2 G1o(z) G2

o(z)

.

Then the perfect reconstruction condition (1.4) leads to the following equation:

Pg(z) · Pg(1/z) = Q(z), (2.3)

where

Pg(z) ∆=

G1

e(z) G1o(z)

G2e(z) G2

o(z)

, Pg(z) ∆=

G1

e(z) G2e(z)

G1o(z) G2

o(z)

,

Q(z) ∆=

1/4 −U(z−1)/4

−U(z)/4 (2− U(z)U(z−1))/4

.

8

We can immediately obtain a solution to (2.3) with interpolatory filters G1(z) and G1(z):

G1e(z) = G1

e(z) =12, G1

e(z) = − U(z)2

, G1e(z) = −U(z)

2, G2

e(z) = G2e(z) = 0.

As for the odd components of the filters G2(z) and G2(z), they are derived from the factorization

v(z)v(z−1) = V (z), where V (z) ∆=1− U(z)U(z−1)

2(2.4)

The filters

G2(z) = z−1v(z2), G2(z) = z−1v(z2). (2.5)

Note that the filters

G1(z) =1− z−1U(z2)

2= H(−z), G1 =

1− z−1U(z2)2

= H(−z) (2.6)

are interpolatory. They are the high-pass filters because of U(1) = U(1) = 1. Due to (2.2) the filters

G1(z) and G1(z) are symmetric about inversion z −→ z−1.

Proposition 2.1 Let the rational functions U(z) and U(z) have no poles on the unit circle. Then

the the perfect reconstruction filter bank H(z), G1(z) G2(z) and H(z), G1(z) G2(z) defined by Eqs.

(2.1), (2.6), (2.5) implements a frame expansion of a signal x ∈ l2.

Proof: Since the functions U(z) and U(z) have no poles on the unit circle, the impulse response

{h(n)}, {gr(n)}, r = 1, 2 of the analysis filters H(z) Gr(z), r = 1, 2 belong to l1. The minors of the

analysis polyphase matrix

P(z) ∆=

1/2 U(z)/2

1/2 −U(z)/2

0 v(z)

are m1(z) = U(z)/2, m2(z) = v(z)/2. Due to (2.4) they can not vanish simultaneously. Thus the

matrix P(z) has full rank and the assertion follows from Proposition 2.1.

The rational function V (z2) can be represented as

V (z2) =1− z−1U(z2) · zU(z−2)

2= V (z−2).

Thus a rational symmetric or antisymmetric factorization is possible. The trivial rational symmetric

factorizations are v(z) = 1, v(z) = V (z) or v(z) = 1, v(z) = V (z) . Since V (1) = 0, at least one of the

filters G2(z) and G2 is high-pass and the corresponding framelet has vanishing moments. We conclude

the section with the analysis and synthesis formulas for the above interpolatory filter banks:

S(z) = E(z) + U(z−1)O(z), D1(z) = E(z)− U(z−1)O(z), D2(z) = 2v(z−1)O(z),

E(z) =S(z) + D1(z)

2, O(z) = U(z)

S(z)−D1(z)2

+ v(z)D2(z).

9

2.2 Tight and semi-tight frames

If we take the filter U(z) = U(z) then we get H(z) = H(z), G1(z) = G1(z) and

V (z) = (1− |U(z)|2)/2, V (z2) = 2H(z)H(−z). (2.7)

Provided the inequality

|U(z)| ≤ 1 as |z| = 1 (2.8)

holds, the function V (z) can be factored as V (z) = v(z) v(1/z). The factorization is not unique. Due

to the Riesz’s lemma [6], a rational factorization is possible. Then we have G2(z) = G2(z). Thus the

synthesis filter bank coincides with the analysis filter bank and generates the tight frame. Note that,

due to (2.7), the (anti)symmetric rational factorization is possible if and only if all roots and poles of

the function H(z) have even multiplicity. Provided H(z) has the root of multiplicity 2m by z = 1,

the filter G2(z) has the roots of multiplicity m by z = 1 and z = −1. The corresponding framelet

ψ2(t) has m vanishing moments, if exists. Chui and He [3] presented a similar construction of the

tight frame based on a family of interpolatory symmetric FIR filters. However, the filter G2(z) in

their construction lacks the symmetry.

If the condition (2.8) is not fulfilled we still are able to devise the frames, which are very close to

the tight frame. Namely,

H(z) = H(z) = (1 + z−1U(z2))/2, G1(z) = G1(z) = (1− z−1U(z2))/2,

G2(z) = z−1v(z2), G2(z) = z−1v(z2), v(z) v(1/z) = V (z) = (1− |U(z)|2)/2. (2.9)

It is natural to refer to such a frame as to the semi-tight frame. Note that, due to the symmetry of

V (z), an (anti)symmetric) factorization of type (2.9) is always possible. Therefore, even when (2.8)

holds, sometimes it is preferable to construct a semi-tight rather than the tight frame. For example,

it was proved in [14] that the compactly supported interpolatory symmetric tight frame is possible

only with the low-pass filter H(z) = 1/2 + (z + 1/z)/4. In this case the scaling function and the

framelets are piece-wise linear. The framelets ψ1(t) and ψ2(t) have two and one vanishing moments,

respectively. However, it is possible to construct a variety of compactly supported interpolatory

symmetric semi-tight frames with smooth framelets.

2.3 Dual frame

Let

P(z) ∆=

1/2 U(z)/2

1/2 −U(z)/2

0 v(z)

10

be the polyphase matrix of an interpolatory filter bank, which generates the analysis frame. The

dual synthesis frame is generated by the filter bank, whose polyphase matrix is the parapseudoinverse

P+(z) of P(z) (see (1.6)). Denote

U(z) ∆=U(z)

|U(z)|2 + 2|v(z)|2 , v(z) ∆=v(z)

|U(z)|2 + 2|v(z)|2 . (2.10)

Since v(1) = 0, we have U(1) = 0. Obviously, U(z) has no poles on the unit circle and z−1U(z2) is

symmetric about inversion z → 1/z. The function v(z) has zero of the same multiplicity as v(z) at

z = 1. It is readily verified that

P+(z) =

1/2 1/2 0

U(z)/2 −U(z)/2 v(z)

.

We see that the matrix P+(z) has the same structure as PT (z). The filters

H(z) ∆= 1/2 + z−1U(z2)/2 and G1(z) ∆= 1/2− z−1U(z2)/2 (2.11)

are interpolatory. The product v(z)v(1/z) = (1−U(z)U(1/z))/2. Since U(1) = 1, we have G1(1) = 0.

Thus the framelet ψ2(t) has vanishing moments.

Theorem 2.1 Let an interpolatory low-pass symmetric filter H(z) = 1/2 + z−1U(z2)/2 have zero of

multiplicity m by z = −1 and no poles on the unit circle |z| = 1. Then it generates an invertible

analysis filter bank H(z), G1(z), G2(z) such that the high-pass filters G1(z), G2(z) are symmetric

and have zero of multiplicity m by z = 1. In addition, the filter G1(z) is interpolatory and the even

polyphase component of the filter G2(z) is zero. The filter G2(z) and has zero of multiplicity m by

z = −1. The dual synthesis filter bank H(z), G1(z), G2(z) has exactly the same properties.

Proof: Obviously, the high-pass filters G1(z) = 1/2− z−1U(z2)/2 is interpolatory, symmetric and has

zero of multiplicity m by z = 1. To obtain the filter G2(z) we choose v(z) ∆= V (z) = (1− |U(z)|2)/2.

Then the filter G2(z) ∆= z−1v(z2) = 2z−1G1(z)H(z) is symmetric and has zero of multiplicity m by

z = 1 and so also by z = −1. The dual filter G2(z) ∆= z−1v(z2), where v(z) is defined in (2.10), has

the same properties. The dual high-pass filter G1(z) is defined in (2.11). We have |U(z)|2 +2|v(z)|2 =(1 + |U(z)|4

)/2. Then the symmetric interpolatory filter

G1(z) ∆=12

(1− z−1U(z2)

)=

12

(1− z−1 U(z2)

|U(z2)|2 + 2|v(z2)|2)

=1 + |U(z2)|4 − 2z−1U(z2)

2(1 + |U(z2)|4) =1 + |U(z2)|4 − 2z−1U(z2)

2(1 + |U(z2)|4)

=

(1− z−1U(z2)

)+

(z−2U2(z2) z2U2(z−2)− z−1U(z2)

)

2(1 + |U(z2)|4)

11

=

(1− z−1U(z2)

)+ z−1U(z2)

(z−3U3(z2)− 1

)

2(1 + |U(z2)|4)

=(1− z−1U(z2)

) 1− z−1U(z2)− z−2U2(z2)− z−3U3(z2)2(1 + |U(z2)|4) .

Finally we get

G1(z) = 2G1(z)G1(z)− z−2U2(z2)H(z)

2(1 + |U(z2)|4) .

Hence G1(z) has zero of multiplicity m by z = 1. By the similar calculations we derive

H(z) = 2H(z)H(z)− z−2U2(z2)G1(z)

2(1 + |U(z2)|4) .

Hence H(z) has zero of multiplicity m by z = −1.

Remark. All four framelets ψ1(t), ψ2(t), ψ1(t), ψ2(t) generated by the above filter bank H(z),

G1(z), H(z), G1(z), G2(z) have m vanishing moments.

3 Design of interpolatory filters

In order to generate reasonable frames, the interpolatory rational transfer function H(z) = (1 +

z−1U(z2))/2 must have zero of a multiplicity m > 0 by z = −1. Then G1(z) = (1 − z−1U(z2))/2

has zero of multiplicity m by z = 1. On the other hand, the filter with the transfer function G1(z)

eliminates sampled polynomials p up to the degree m − 1 and the filter with the transfer function

H(z) restores such polynomials. To achieve this, the filter U(z), being applied to the even subarray of

the signal p, must produce exactly the odd subarray and the filter z−1U(z), being applied to the odd

subarray, must produce the even subarray. In other words, the filtered half-array of p must exactly

predict another half-array. As a source for the design of such filters we employ the so-called discrete

splines. We will show that the devised filters are intimately related to the Butterworth filters, which

are commonly used in signal processing [13].

3.1 Discrete splines

We outline briefly the properties of discrete splines, which will be needed for further constructions.

For a detailed description of the subject, see [15, 16]. The discrete splines are defined on the grid {k}and present a counterpart to the continuous polynomial splines.

The signal

b1,n = {b1,nk } ∆=

1, as k = 0, . . . , 2n− 1

0, otherwise⇐⇒ B1,n(z) =

1− z2n

1− z,

12

is called the discrete B-spline of first order.

We define by recurrence the higher order B-splines via the discrete convolutions:

bp,n = b1,n ∗ bp−1,n ⇐⇒ Bp,n(z) =

(1− z2n

1− z

)p

.

In this paper we are interested only in the case when p = 2r, r ∈ N and n = 1. In this case

we have B2r,1(z) = (1 + z−1)2r. The B-spline b2r,1 is symmetric about the point k = r where it

attains its maximal value. We define the central B-spline q2r of order 2r as the shift of the B-spline:

q2r ∆= {q2rk = b2r,1

k+r}, Q2r(z) = zrB2r,1(z) = zr(1 + z−1)2r. The discrete spline a2r = {a2rk }k∈Z of

order 2r on the grid {2k} is defined as a linear combination with real-valued coefficients of shifts of

the central B-spline:

a2rk

∆=∞∑

l=−∞cl q

2rk−2l ⇐⇒ A2r(z) = C(z2)Q2r(z) = C(z2)

(υ2r(z2) + z−1θ2r(z2)

),

υ2r(z2) ∆= Q2re (z2) =

12

(zr

(1 + z−1

)2r+ (−z)r

(1− z−1

)2r)

,

θ2r(z2) ∆= Q2ro (z2) =

z

2

(zr

(1 + z−1

)2r − (−z)r(1− z−1

)2r)

.

Our scheme to design prediction filters using the discrete splines consists of the following. We construct

the discrete spline a2r , which interpolates even samples {ek = x2k+1} of a signal x ∆= {xk}k∈Z:

a2r2k} = ek, k ∈ Z. Then we use the values a2r

2k+1 for the prediction of odd samples {ok = x2k+1}.The z-transform of the even component of the spline a2r

A2re (z) = C(z)υ2r(z) = E(z) =⇒ C(z) = E(z)/υ2r(z).

Then the z-transform of the odd component of the spline a2r

A2ro (z) = C(z)θ2r(z) = U2r(z)E(z), where U2r(z) ∆=

θ2r(z)υ2r(z)

. (3.1)

Thus, in order to predict the odd samples of the signal x, we conduct filtering the even subarray of x

by the filter U2r(z).

3.2 Properties of designed filters

In this section we prove that the designed filters can serve as a source for the construction of frames.

Denote

χ2r(z) ∆=12

(1 + z−1U2r(z2

), γ2r(z) ∆=

12

(1− z−1U2r(z2)

).

Proposition 3.1 The rational functions U2r(z) defined in (3.1) have the following properties:

P1. No poles on the unit circle |z| = 1.

13

P2. U2r(1) = 1.

P3. Symmetry: z−1U2r(z2) = zU2r(z−2).

P4. |U2r(z)| ≤ 1.

P5. The function χ2r(z) has the root of multiplicity 2r by z = −1 and the function γ2r(z) has the

root of multiplicity 2r by z = 1.

Proof: We substitute z = ejω into z−1U2r(z2). We have

z−1U2r(z2) =eirω(1− e−jω)2r + (−1)reirω(1− e−jω)2r

eirω(1 + e−jω)2r + (−1)reirω(1− e−jω)2r=

(cos ω

2

)2r − (sin ω

2

)2r

(cos ω

2

)2r +(sin ω

2

)2r .

Hence P1 – P4 follow. The function

χ2r(z) =(cos ω

2

)2r

(cos ω

2

)2r +(sin ω

2

)2r =(1 + z−1

)2r

(1 + z−1)2r + (−1)r (1− z−1)2r , (3.2)

γ2r(z) =(sin ω

2

)2r

(cos ω

2

)2r +(sin ω

2

)2r =(−1)r

(1− z−1

)2r

(1 + z−1)2r + (−1)r (1− z−1)2r . (3.3)

Eqs. (3.2) and (3.3) imply P5.

Remark. It is seen from (3.2) and (3.3) that the functions χ2r(z) and γ2r(z) coincide with the

squared magnitudes of the frequency response of the low- and high-pass digital Butterworth filters of

order r, respectively. For details we refer to [13].

Proposition 3.2 The filter χ2r(z) generates the scaling function Φ2r(t) ∈ L2(R) such that

Φ2r(ω) = limN→∞

N∏

ν=1

χ2r(ej2−νω), Φ2r(t) = 2∑

k∈Zχ2r

k Φ2r(2t− k).

The scaling function Φ2r(t) is continuous together with its derivatives up to the order r−1 (belongs to

Cr−1). The filter γ2r(z) generates the framelet Ψ2r(t) ∈ L2(R) with the same smoothness as Φ2r(t),

such that

Ψ2r(t) = 2∑

k∈Zγ2r

k Φ2r(2t− k). (3.4)

The framelet Ψ2r(t) has 2r vanishing moments.

14

Proof: Due to (3.2), the function χ2r(z) can be factorized as

χ2r(z) =

(1 + z−1

2

)p

K(z), K(ejω) =e−jrω

(cos ω

2

)2r +(sin ω

2

)2r .

The function K(1) = 1 and the following estimate is true: κ∆= sup|z|=1 |K(z)| = 2r−1 < 22r−1−(r−1).

Then Proposition 1.2 implies that there exists the scaling function Φ2r(t) ∈ L2(R), which belongs to

Cr−1.

The rational function γ2r(z) has no poles on the unit circle |z| = 1. Therefore its impulse response

{γ2rk }k∈Z decays exponentially as k → ∞. Therefore the function Ψ2r(t) ∈ L2(R) defined in (3.4)

exists and has the same smoothness as Φ2r(t). The multiplicity of zero of the filter γ2r(z) at z = 1 is

2r. Therefore the framelet Ψ2r(t) has 2r vanishing moments.

Using Propositions 1.3 and 1.4 we established improved evaluations of smoothness for a few scaling

functions and framelets [19].

Proposition 3.3 The filters χ2r(z), r = 2, 3, 4 generate the scaling functions Φ2r(t), which decay

exponentially as t →∞. In addition Φ4(t) ∈ C2, Φ6(t) ∈ C4, Φ8(t) ∈ C5.

4 Butterworth frames

The above considerations suggest that the filters U2r(z), χ2r(z), γ2r(z), which originate from the

discrete splines, can be useful for the construction of frames in the signal space. To be specific, we

choose U(z) = U2r(z), H(z) = χ2r(z), U(z) = U2p(z), H(z) = χ2p(z), where r and p are some

natural numbers, which, in particular, may coincide with each other. Because of the relation of the

filters to the Butterworth filters we call the corresponding frames the Butterworth frames. We denote

ρ(z) ∆= z + 2 + z−1. Thus ρ(−z) = −z + 2− z−1.

4.1 Tight frames

We define the filters

H(z) = H(z) ∆= χ2r(z) =1 + z−1U2r(z2)

2=

ρr(z)ρr(z) + ρr(−z)

,

G1(z) = G1(z) = γ2r(z) =ρr(−z)

ρr(z) + ρr(−z).

Due to P4 we get a tight frame as soon as we factorize the function

V 2r(z) =12

(1− |U2r(z)|2

)= vr(z)vr(1/z).

15

From (2.7) we have

V 2r(z2) = 2H(z) H(−z) =2(−1)rz−2r

(1− z2

)2r

(zr (1 + z−1)2r + (−z)r (1− z−1)2r

)2

= vr(z2)vr(z−2), vr(z2) ∆=√

2(1− z2

)r

ρr(z) + ρr(−z). (4.1)

Note that if r = 2n is an even number then we can define the function vr(z2) in a slightly different

manner:

vr(z2) ∆=√

2(z − z−1

)2n

ρ2n(z) + ρ2n(−z).

Hence the three filters H(z) = χ2r(z), G1(z) = H(−z) = γ2r(z) and G2(z) ∆= z−1vr(z2) generate a

tight frame in the signal space. The scaling function ϕ(t) and the framelet ψ1(t) are symmetric, whereas

the framelet ψ2(t) is symmetric when r is even and antisymmetric when r is odd. The framelet ψ1(t)

has 2r vanishing moments and the framelet ψ2(t) has r vanishing moments. The frequency response

of the filter H(z) is maximally flat. The frequency response of the filter G1(z) is mirrored version of

the former one. The frequency response of the filter G2(z) is symmetric about ω = π/2 and vanishes

at the points ω = 0 and ω = π.

Examples:

The simplest case, r = 1: We have

U2(z) =1 + z

2, H(z) =

1 + z−1U2(z2)2

=z−1 + 2 + z

4(4.2)

G1(z) = H(−z) =−z−1 + 2− z

4, G2(z) =

√2(1− z2)

4z.

The filter U2(z) is FIR and, therefore, the scaling function ϕ(t) and the framelets ψ1(t) and

ψ2(t) are compactly supported. The framelet ψ1(t) has two vanishing moments. The framelet

ψ2(t) is antisymmetric and has one vanishing moment.

Cubic discrete spline, r = 2:

U4(z) = 41 + z

z + 6 + z−1, H(z) =

(z + 2 + z−1)2

2 (z−2 + 6 + z2)(4.3)

G1(z) =(z − 2 + z−1)2

2 (z−2 + 6 + z2), G2(z) =

√2z−1(z − z−1)2

2 (z−2 + 6 + z2).

The framelet ψ1(t) has four vanishing moments. The framelet ψ2(t) is symmetric and has two

vanishing moments.

16

Discrete spline of sixth order, r = 3: We have

U6(z) =(z + 14 + z−1)(1 + z)

6z−1 + 20 + 6z, H(z) =

(z−1 + 2 + z)3

2 (6z2 + 20 + 6z−2)(4.4)

G1(z) =(−z−1 + 2− z)3

2 (6z2 + 20 + 6z−2), G2(z) =

√2z−1(1− z2)3

2 (6z2 + 20 + 6z−2).

The framelet ψ1(t) has six vanishing moments. The framelet ψ2(t) is antisymmetric and has

three vanishing moments.

Discrete spline of eighth order, r = 4:

U8d (z) =

8(1 + z)(z−1 + 6 + z)z−2 + 28z−1 + 70 + 28z + z2

H(z) =(z−1 + 2 + z)4

2 (z−4 + 28z−2 + 70 + 28z2 + z4)(4.5)

G1(z) =(z−1 − 2 + z)4

2 (z−4 + 28z−2 + 70 + 28z2 + z4)G2(z) =

√2z−1(z − z−1)4

2 (z−4 + 28z−2 + 70 + 28z2 + z4).

The framelet ψ1(t) has eight vanishing moments. The framelet ψ2(t) is symmetric and has four

vanishing moments.

4.2 Semi-tight frames

Unlike tight frames, a symmetric factorization of type (2.9) of the function V 2r(z) is possible for either

of even and odd values of r:

V 2r(z2) =2

(2− z−2 − z2

)r

(ρr(z) + ρr(−z))2= v2p,s(z2)v2(r−p),2−s(z−2),

v2p,s(z2) ∆=√

2(2− z−2 − z2

)p

(ρr(z) + ρr(−z))s , (4.6)

v2(r−p),2−s(z2) ∆=√

2(2− z−2 − z2

)r−p

(ρr(z) + ρr(−z))2−s .

We can get an antisymmetric factorization choosing an odd p:

v2p,s(z2) ∆= −√

2(−z2)−r(1− z2

)p

(ρr(z) + ρr(−z))s (4.7)

v2(r−p),2−s(z2) ∆=√

2(−z2)p−2r(1− z2

)2r−p

(ρr(z) + ρr(−z))2−s , s ∈ Z.

By means of such a factorization we can vary number of vanishing moments in framelets ψ2(t) and

ψ2(t). One option is for one of the filters G2(z) = z−1vp,s(z2) or G2(z) = z−1v2r−p,2−s2 (z2) to have a

finite impulse response. It is achieved if s ≤ 0 or s ≥ 2.

Examples:

17

The simplest case, r = 1: We have

U2(z) =1 + z

2, H(z) =

z−1 + 2 + z

4G1(z) =

−z−1 + 2− z

4.

Increase the number of vanishing moments in the analysis framelet ψ12 to two at the cost of lack

of vanishing moments in the synthesis framelet ψ12

G2(z) = 2z−1 G2(z) =z−1(−z2 + 2− z−2)

4. (4.8)

The framelets are symmetric. The synthesis framelet ψ2(t) = 4ϕ(2t).

Cubic discrete spline, r = 2:

U4(z) = 41 + z

z + 6 + z−1, H(z) =

(z + 2 + z−1)2

2 (z−2 + 6 + z2), G1(z) =

(z − 2 + z−1)2

2 (z−2 + 6 + z2).

1. Increase the number of vanishing moments in the analysis framelet ψ2 to four at the cost

of lack of vanishing moments in the synthesis framelet ψ2

G2(z) =√

2z−1

2 (z−2 + 6 + z2)G2(z) =

√2z−1(z − z−1)4

2 (z−2 + 6 + z2). (4.9)

2. The synthesis filter G2(z) is FIR. Both the synthesis and analysis framelets are symmetric

and have two vanishing moments.

G2(z) =√

2z−1(z − z−1)2

2G2(z) =

√2z−1(z − z−1)2

2 (z−2 + 6 + z2)2. (4.10)

3. Antisymmetric factorization. Increase the number of vanishing moments in the analysis

framelet ψ2 to three at the cost of reducing the number of vanishing moments in the

synthesis framelet ψ2 to one.

G2(z) = −√

2z−1z−4(1− z2)2 (z−2 + 6 + z2)

G2(z) =√

2z−1(−z2)−3(1− z2)3

2 (z−2 + 6 + z2)(4.11)

Discrete spline of sixth order, r = 3: We have

U6(z) =(z + 14 + z−1)(1 + z)

6z−1 + 20 + 6z, H(z) =

(z−1 + 2 + z)3

2 (6z2 + 20 + 6z−2), G1(z) =

(−z−1 + 2− z)3

2 (6z2 + 20 + 6z−2),

1. Symmetric factorization. Increase the number of vanishing moments in the analysis framelet

ψ2 to four whereas the synthesis framelet ψ2 has two vanishing moments.

G2(z) =√

2z−1(2− z2 − z2)2 (6z2 + 20 + 6z−2)

G2(z) =√

2z−1(2− z2 − z2)2

2 (6z2 + 20 + 6z−2). (4.12)

2. Antisymmetric factorization. Increase the number of vanishing moments in the analysis

framelet ψ2 to five whereas the synthesis framelet ψ2 has only one vanishing moment.

G2(z) = −√

2z−1(−z2)−3(1− z2)2 (6z2 + 20 + 6z−2)

G2(z) = −√

2z−1(−z2)−5(1− z2)5

2 (6z2 + 20 + 6z−2). (4.13)

18

3. The synthesis filter G2(z) is FIR. Both the synthesis and analysis framelets are antisym-

metric and have three vanishing moments.

G2(z) =√

22

z−1(1− z2

)3G2(z) =

√2z−1(1− z2)3

2 (6z2 + 20 + 6z−2)2. (4.14)

4.3 Bi-frames

We take U(z) = U2r(z), U(z) = U2p(z). Then we have

H(z) ∆=ρr(z)

ρr(z) + ρr(−z), H(z) ∆=

ρp(z)ρp(z) + ρp(−z)

,

G1(z) ∆=ρr(−z)

ρr(z) + ρr(−z), G1(z) ∆=

ρp(−z)ρp(z) + ρp(−z)

,

G2(z) ∆= z−1v(z2), G2(z) ∆= z−1v(z2).

where

2v(z2)v(z−2) = 1− U2r(z2)U2p(z−2) = 1− z−1U2r(z2)z−1U2p(z2)

= 1− (ρr(z)− ρr(−z)) (ρp(z)− ρp(−z))(ρr(z) + ρr(−z)) (ρp(z) + ρp(−z))

= 2ρr(z)ρp(−z) + ρp(z)ρr(−z)

(ρr(z) + ρr(−z)) (ρp(z) + ρp(−z)).

Suppose p < r. Then we have

v(z2)v(z−2) =(−1)p

(z − z−1

)2p (ρr−p(z) + ρr−p(−z))(ρr(z) + ρr(−z)) (ρp(z) + ρp(−z))

.

A possible way to (anti)symmetrically factorize this function is

v(z2) =(1− z2

)p (ρr−p(z) + ρr−p(−z))ρp(z) + ρp(−z)

, v(z2) =(1− z2

)p

ρr(z) + ρr(−z).

If p = 2n then the symmetric factorization is possible

v(z2) =(z − z−1

)2n (ρr−p(z) + ρr−p(−z))ρp(z) + ρp(−z)

, v(z2) =(z − z−1

)2n

ρr(z) + ρr(−z).

Examples:

Case p = 2, r = 1:

H(z) =z−1 + 2 + z

4, H(z) =

(z + 2 + z−1)2

2 (z−2 + 6 + z2), (4.15)

G1(z) =−z−1 + 2− z

4, G1(z) =

(z − 2 + z−1)2

2 (z−2 + 6 + z2),

G2(z) ∆= z−1v(z2), G2(z) ∆= z−1v(z2).

where

v(z2)v(z−2) =− (

z − z−1)2

2 (z−2 + 6 + z2).

The

19

1. Antisymmetric factorization:

v(z2) =1− z2

2, v(z2) =

1− z2

z−2 + 6 + z2. (4.16)

All the synthesis filters are FIR . Consequently the synthesis scaling function ϕ(t) and

the framelets ψ1(t) and ψ2(t) are compactly supported. The analysis framelet ψ1(t) has

four vanishing moments, the synthesis framelet ψ1(t) has two vanishing moments, both

the synthesis and the analysis framelets ψ2(t) and ψ2(t) are antisymmetric and have one

vanishing moment.

2. Symmetric factorization:

v(z2) =1

z−2 + 6 + z2, v(z2) =

− (z − z−1

)2

2. (4.17)

The analysis filter G2(z) is FIR, the analysis framelet ψ2 is symmetric and have two van-

ishing moments. The synthesis framelet ψ2 is symmetric and does not have vanishing

moments.

3. A trivial factorization:

v(z2) = 1, v(z2) =− (

z − z−1)2

2(z−2 + 6 + z2). (4.18)

The analysis framelet ψ2 is symmetric and have two vanishing moments. The synthesis

framelet ψ2(t) = 2ϕ(2t) is compactly supported.

Case p = 2, r = 3:

H(z) =(z−1 + 2 + z)3

2 (6z2 + 20 + 6z−2), H(z) =

(z + 2 + z−1)2

2 (z−2 + 6 + z2), (4.19)

G1(z) =(z − 2 + z−1)2

2 (z−2 + 6 + z2), G1(z) =

(−z−1 + 2− z)3

2 (6z2 + 20 + 6z−2),

G2(z) ∆= z−1v(z2), G2 ∆= z−1v(z2).

where

v(z2)v(z−2) =(z − z−1

)4

(z−2 + 6 + z2) (6z2 + 20 + 6z−2).

The analysis framelet ψ1(t) has four vanishing moments, the synthesis framelet ψ1(t) has six

vanishing moments.

1. A symmetric factorization.

v(z2) =(z − z−1

)2

z−2 + 6 + z2v(z2) =

(z − z−1

)2

6z2 + 20 + 6z−2. (4.20)

Both the synthesis and the analysis framelets are symmetric and have two vanishing mo-

ments.

20

2. Another symmetric factorization with maximal number of vanishing moments in the anal-

ysis framelet.

v(z2) =1

z−2 + 6 + z2v(z2) =

(z − z−1

)4

6z2 + 20 + 6z−2. (4.21)

3. Antisymmetric factorization.

v(z2) =1− z2

z−2 + 6 + z2v(z2) =

(−z2)3 (

1− z2)3

6z2 + 20 + 6z−2. (4.22)

Both the synthesis and the analysis framelets are antisymmetric. The synthesis framelet

has one vanishing moment, whereas the analysis one has three vanishing moments.

4.4 Dual frames

We put U(z) = U2r(z), v(z) = (1− |U2r(z)|2)/2. Then, due to (2.10),

z−1U(z2) =2z−1U2r(z2)1 + |U2r(z2)|4 , v(z2) =

1− |U2r(z2)|21 + |U2r(z2)|4 .

We have

1 + |U2r(z2)|4 = 1 +(

ρr(z)− ρr(−z)ρr(z) + ρr(−z)

)4

= 2ρ4r(z) + 6ρ2r(−z)ρ2r(z) + ρ4r(−z)

(ρr(z) + ρr(−z))4,

1− |U2r(z2)|2 = 1−(

ρr(z)− ρr(−z)ρr(z) + ρr(−z)

)2

= 4ρr(z)ρr(−z)

(ρr(z) + ρr(−z))2.

Thus

z−1U(z2) =(ρ2r(z)− ρ2r(−z)

)(ρr(z) + ρr(−z))2

ρ4r(z) + 6ρ2r(−z)ρ2r(z) + ρ4r(−z), (4.23)

v(z2) = 2ρr(z)ρr(−z) (ρr(z) + ρr(−z))2

ρ4r(z) + 6ρ2r(−z)ρ2r(z) + ρ4r(−z). (4.24)

Theorem 2.1 implies that the filter bank H(z) = 1/2 + z−1U(z2)/2, G1(z) = 1/2 − z−1U(z2)/2,

G2(z) = z−1v(z2)/2 is dual to the filter bank H(z) = 1/2 + z−1U(z2)/2, G1(z) = 1/2− z−1U(z2)/2,

G2(z) = z−1v(z2)/2. All framelets have 2r vanishing moments.

Example: Case r = 1:

U2(z) =1 + z

2, H(z) =

z−1 + 2 + z

4, G1(z) =

−z−1 + 2− z

4, (4.25)

v(z) = (1− |U(z)|2)/2 =−z−1 + 2− z

8.

Using Eqs. (4.23) and (4.24), we derive

v(z) =4

(−z + 2− z−1)

z2 + 4z + 22 + 4z−1 + z−2, U(z) = 16

z + 1z2 + 4z + 22 + 4z−1 + z−2

(4.26)

21

H(z) =1 + z−1U(z2)

2=

(z + 2 + z−1

) z3 − 2z2 + 7z + 4 + 7z−1 − 2z−2 + z−3

2 (z4 + 4z2 + 22 + 4z−2 + z−4)

G1(z) =1− z−1U(z2)

2=

(z − 2 + z−1

) z3 + 2z2 + 7z − 4 + 7z−1 + 2z−2 + z−3

2 (z4 + 4z2 + 22 + 4z−2 + z−4)G2(z) = z−1v(z2).

The analysis filters are FIR and the analysis scaling function and framelets are compactly supported.

All framelets have two vanishing moments.

4.5 Illustrations

In this section we display the designed filters and the framelets generated by these filters. Correspond-

ing formulas are given in Sections 4.1–4.4.

Figure 1: We present the tight frames, which originate from the discrete splines of second and eight

order. The plots in the left picture from bottom to top display the scaling function ϕ(t), and the

framelets ψ1(t) and ψ2(t), which are generated by the filters H(z), G1(z) and G2(z), respectively.

The filters are defined by Eq. (4.2). They are FIR and the waveforms are compactly supported.

The framelet ψ1(t) is symmetric and has two vanishing moments and ψ2(t) is antisymmetric

and has one vanishing moment. The frequency response of the filters are displayed in the second

left picture. The waveforms and the filters (Eq. (4.5)), which stem from the eight order discrete

splines are displayed in a similar manner in the third and fourth pictures from the left. In

this case the framelet ψ1(t) has eight vanishing moments and ψ2(t) has four. Both framelets

are symmetric. We observe that the frequency response of the filters H(z) and G1(z) have

near-rectangle shape. They are mirrored versions of each other.

ψ2

ψ1

φ

G2

G1

H

ψ2

φ

ψ1

G2

G2

H

Figure 1: Left picture from bottom to the top: the scaling function ϕ(t), and the framelets ψ1(t)

and ψ2(t) for the tight frame originated from the second order discrete spline. Second from the left

– corresponding filters. Third and fourth pictures: the same for the tight frame related to the eight

order discrete spline.

Figure 2: We present the tight and semi-tight frames originated from the fourth order (cubic) discrete

spline. The left pair of pictures displays the waveforms and filters related to the tight frame

22

(Eq. (4.3)). The framelet ψ1(t) has four vanishing moments and ψ2(t) has two. Both framelets

are symmetric. The next two pairs of pictures illustrate three ways of factorization of the function

V (z), which are given by Eqs. (4.9), (4.10) and (4.11). The bottom row in these four pictures is

related to Eq. (4.9). There are depicted subsequently the synthesis framelet ψ2(t), the synthesis

filter G2(z), analysis framelet ψ2(t) and the analysis filter G2(z). All four vanishing moments

are assigned to the analysis framelet. Both ψ2(t) and ψ2(t) are symmetric. The filter G2(z) is

all-pass. The central row depicts the same objects related to Eq. (4.10). Here both the analysis

and synthesis framelets have two vanishing moments and are symmetric. The filter G2(z) is

FIR. The upper row is related to Eq. (4.11). The framelets here are antisymmetric, ψ2(t) has

three vanishing moments, and ψ2(t) has only one. Note that the analysis framelets and filters

can be interchanged with the synthesis ones.

ψ2

ψ1

φ H

G1

G2

ψ2 G2 tilde ψ2 tilde G2

Figure 2: Tight and semi-tight frames originated from the fourth order discrete spline. Left hand pair

of pictures: waveforms and filters for the tight frame. Central pair: synthesis framelets ψ2(t) and

filters G2(z) for the various modes of factorization of V (z). Right hand pair: corresponding analysis

framelets ψ2(t) and filters G2(z).

Figure 3: We present the tight and semi-tight frames originated from the sixth order discrete spline.

The left pair of pictures displays the waveforms and filters related to the tight frame (Eq. (4.4)).

The framelet ψ1(t) is symmetric and has six vanishing moments, the framelet ψ2(t) is antisym-

metric and has three vanishing moments. The next two pairs illustrate three ways of factorization

of the function V (z), which are given by Eqs. (4.12), (4.13) and (4.14). The bottom row in these

four pictures is related to Eq. (4.12). There are depicted subsequently the synthesis framelet

ψ2(t), the synthesis filter G2(z), analysis framelet ψ2(t) and the analysis filter G2(z). Both

ψ2(t) and ψ2(t) are symmetric. The synthesis framelet ψ2(t) has two vanishing moments and

the analysis framelet ψ2(t) has four. The central row depicts the same objects related to Eq.

(4.13). Here both the analysis and synthesis framelets are antisymmetric. The analysis framelet

ψ2(t) has five vanishing moments and only one is left for the synthesis framelet. The upper

row is related to Eq. (4.14). The framelets here are antisymmetric and have three vanishing

moments. The filter G2(z) is FIR.

23

ψ2

ψ1

φ

G2

G1

H

ψ2

G1 tilde ψ2

tilde G2

Figure 3: Tight and semi-tight frames originated from the sixth order discrete spline. Left hand pair

of pictures: waveforms and filters for the tight frame. Central pair: synthesis framelets ψ2(t) and

filters G2(z) for the various modes of factorization of V (z). Right hand pair: corresponding analysis

framelets ψ2(t) and filters G2(z).

Figure 4: This figure corresponds to the bi-frames generated by the pair of low-pass filters: synthesis

H(z), which stems from the second order discrete spline and analysis H(z), which stems from the

fourth order discrete spline. These filters and the related high-pass filters G1(z) and G1(z) are

defined in Eq. (4.15). The frequency response of the filters H(z) and G1(z) and the generated

waveforms ϕ(t) and ψ1(t) are displayed in Figure 1. The filters H(z) and G1(z) and waveforms

ϕ(t) and ψ1(t) are displayed in Figure 2. We depict in Figure 4 the filters G2(z) and G2(z) and

the framelets ψ2(t) and ψ2(t), which result from different ways of factorization of the function

V (z) ( Eqs. (4.16), (4.17) and (4.18)). The bottom row is related to Eq. (4.16). Both ψ2(t)

and ψ2(t) are antisymmetric and have one vanishing moment. The synthesis filter G2(z) is FIR

and ψ2(t) is compactly supported. The central row illustrates the symmetric factorization of Eq.

(4.17). The analysis framelet ψ2(t) has two vanishing moments at the expense of ψ2(t), which

has none. In the trivial factorization of Eq. (4.18), which is illustrated in the upper row, ψ2 is

symmetric and has two vanishing moments. The synthesis framelet ψ2(t) = 2ϕ(2t) is compactly

supported.

ψ2

G2

tilde ψ2

BiFrame:1 11 2

tilde G2

Figure 4: Filters and framelets for the bi-frames resulting from the pair of discrete splines of second

and fourth order. The left hand pair of pictures: synthesis framelets ψ2(t) and filters G2(z) for the

various modes of factorization of V (z). Right hand pair: corresponding analysis framelets ψ2(t) and

filters G2(z).

Figure 5: This figure corresponds to the bi-frames generated by the pair of low-pass filters: synthesis

24

H(z), which stems from the sixth order discrete spline and analysis H(z), which stems from the

fourth order discrete spline. These filters and the related high-pass filters G1(z) and G1(z) are

defined in Eq. (4.19). The frequency response of the filters H(z) and G1(z) and the generated

waveforms ϕ(t) and ψ1(t) are displayed in Figure 3. The filters H(z) and G1(z) and waveforms

ϕ(t) and ψ1(t) are displayed in Figure 2. We depict in Figure 5 the filters G2(z) and G2(z)

and the framelets ψ2(t) and ψ2(t), which result from different factorizations of the function

V (z) ( Eqs. (4.20), (4.21) and (4.22)). The bottom row is related to Eq. (4.20). Both ψ2(t)

and ψ2(t) are symmetric and have two vanishing moments. The central row illustrates the

symmetric factorization of Eq. (4.21). The analysis framelet ψ2(t) has four vanishing moments

at the expense of ψ2(t), which has none. In the antisymmetric factorization of Eq. (4.22),

which is illustrated in the upper row, ψ2 has three vanishing moments. The synthesis framelet

ψ2(t) = 2ϕ(2t) has one.

ψ2

G2

tilde ψ2tilde G2

Figure 5: Filters and framelets for the bi-frames resulting from the pair of discrete splines of sixth

and fourth order. The left hand pair of pictures: synthesis framelets ψ2(t) and filters G2(z) for the

various modes of factorization of V (z). Right hand pair: corresponding analysis framelets ψ2(t) and

filters G2(z).

Figure 6: We display the filters and the waveforms for the pair of dual frames, which is defined by

Eqs. (4.25) and (4.26). All waveforms are symmetric, all framelets have two vanishing moments.

The analysis filters are FIR and the waveforms are piece-wise linear and compactly supported.

Note that, unlike all above examples, the frequency response of the synthesis filters H(z) and

G1(z) are not localized in the half-bands.

5 Discussion

5.1 Comparison with a wavelet transform

We used similar filters to construct biorthogonal wavelet transforms. It is interesting to compare a

wavelet and a frame transforms based on the same Butterworth filter. As an example we use the tight

frame originated from the fourth order discrete spline, which is defined by Eq. (4.3) and displayed in

25

φ

ψ1

ψ2

BiFrame:1 11 2

G2

G1

H

tilde ψ2

tilde ψ1

tilde φ

tilde G2

tilde G1

tilde H

Figure 6: Filters and wavelets associated with the dual pair of frames. Left hand pair of pictures:

synthesis waveforms and filters Right hand pair: corresponding analysis waveforms and filters.

Figure 2. For the biorthogonal wavelet transform we have the synthesis low-pass filter and the analysis

high-pass filter

Hw(z) =(z + 2 + z−1)2

2 (z−2 + 6 + z2)and Gw(z) = z−1 (z − 2 + z−1)2

2 (z−2 + 6 + z2), (5.27)

respectively. They obviously coincide with the filters H(z) and G1(z) of the frame transform (up to

the factor z−1. The analysis low-pass filter and the synthesis high-pass filter have more complicated

structure:

Hw(z) = H(z) + V (z2) and Gw(z) = z−1(G1(z) + V (z2)

), (5.28)

where V (z2) =(z − z−1)4

2 (z−2 + 6 + z2)2= G2(z)G2(1/z).

The frequency response of the wavelet filters and the corresponding waveforms are displayed in Fig-

ure 7. The analysis waveforms belong to C1, whereas the synthesis waveforms and the framelets belong

φw

ψw

G

w

Hw

tilde ψw

tilde φw

tilde Gw

tilde Hw

Figure 7: Filters and wavelets associated with the dual pair of frames. Left hand pair of pictures:

synthesis waveforms and filters Right hand pair: corresponding analysis waveforms and filters.

to C2. We can conclude from Eqs. (5.27) and (5.28) that the change from the biorthogonal wavelet

transform to the frame transform simplifies the structure of filters and enhances the smoothness of

waveforms. The additional filter G2(z) compensates the difference between the analysis and synthesis

wavelet filters. Comparing Figure 7 with Figure 2, where the tight frame is displayed we see that the

”bumps”, which present at the displays of the frequency response of the wavelet filters Hw(z) and

Gw(z) are removed by introducing the filter G2(z). In the tight frame case the frequency response

of the filters H(z) and G1(z) are mirrored versions of each other. The scaling function ϕ(t) and the

26

framelet ψ1(t) are smoother than their wavelet counterparts ϕw(t) and ψw(t). Loosely speaking, while

changing from the wavelet to the frame transforms we split the complex filters Hw(z) and Gw(z) and

wavelets ϕw(t) and ψw(t) into simple components. By this means we gain an additional flexibility in

capturing characteristic features of the processed signal.

5.2 Conclusion

We presented a new family of frames, which are generated by perfect reconstruction filter banks

consisting of linear phase filters. The filter banks are designed on the base of the discrete interpolatory

splines and are related to the Butterworth filters. Note that a similar scheme of the filter design is

possible on the base of the continuous interpolatory and quasi-interpolatory splines. Each designed

filter bank comprises one interpolatory symmetric low-pass filter and two high-pass filters, one of which

is also interpolatory and symmetric. The second high-pass filter may be symmetric or antisymmetric.

These filter banks generate the analysis and synthesis scaling functions and pairs of framelets. The

scaling function and one of the framelets in either of the analysis and synthesis sets are symmetric,

whereas the second framelet is symmetric or antisymmetric. One step of the framelet transform of

a signal of length N produces 1.5N coefficients. Thus, the full transform of this signal consisting of

J = log 2(N) steps produces 2N coefficients.

We introduced the concept of semi-tight frame. While in the tight frame all the analysis waveforms

coincide with their synthesis counterparts, in the semi-tight frame we can vary the second framelets

making them different for the synthesis and the analysis cases. By this means we can, for example,

switch the vanishing moments from the synthesis to the analysis framelets or to add smoothness to

the synthesis framelet. We constructed dual pairs of frames, where all the waveforms are symmetric

and all the framelets have the same number of vanishing moments.

Although most of the designed filters are IIR, they allow fast implementation via recursive pro-

cedures. The waveforms are well localized in the time domain despite their infinite support. The

frequency response of the designed filters are flat, due to their Butterworth relationship.

We anticipate a wide field of signal processing applications for this new family of transforms.

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