ANALYSIS AND COMPUTATION OF OPTIMAL BOUNDS

OF BI-DIRECTIONAL FRAMES

Xiaofang Chen

, Cishen Zhang

and Jingxin Zhang

School of Electrical Electronic Engineering, Nanyang Technological University, Singapore, Republic of Singapore

Faculty of Engineering & Industrial Sciences, Swinburne University of Technology, Hawthorn, Victoria, 3122, Australia

Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3800, Australia

Keywords:

Frames, Linear systems, Mixed causal-anticausal realizations, State space equations.

Abstract:

Frames are mathematical tools which can represent redundancies in many application problems. In the studies

of frames, the frame bounds and frame bound ratio are very important indices characterizing the robustness

and numerical performance of frame systems. In this paper, the frame bounds of a class of frame, which can be

modeled by the bi-directional impulse response of linear time systems, are analyzed and computed. By using

the state space approach, the tightest lower and upper frame bounds can be directly and efﬁciently computed.

1 INTRODUCTION

In the study of vector spaces, frame is a more ﬂex-

ible tool compared with basis. The frame elements

are linearly dependent, so to provide redundancies in

frames, and all elements in the vector space can be

written as a linear combination of the frame elements.

Frames, as a mathematical theory were introduced by

Dufﬁn and Schaeffer (Dufﬁn and Schaeffer, 1952) in

1950s. Since 1986, frames have played an impor-

tant role in signal processing, see (Daubechies et al.,

1986) written by Daubechies, Grossman, and Meyer.

Some particular classes of frames have been exten-

sively studied, for example, Gabor frames, which

are also called Weyl-Heisenberg frames described in

(Heil and Walnut, 1989) and (Casazza, 2001), and

wavelet frames, which are introduced in (Daubechies

et al., 1986), (Daubechies, 1990), and (Daubechies,

1992). Frames, or redundant representations, can

also be found in pyramid coding (Burt and Adel-

son, 1983); source coding (Benedettom et al., 2006);

denoising (Dragotti et al., 2003); robust transmis-

sion (Bernardini and Rinaldo, 2006); CDMA sys-

tems; multiantenna code design; segmentation; clas-

siﬁcation; restoration and enhancement; signal recon-

struction; and so on.

The theory of frames is a powerful means for the

analysis and design of the oversampled uniform ﬁl-

ter banks (FBs). (Vetterli and Cvetkovic, 1996) and

(Cvetkovic and Vetterli, 1998) studied properties of

oversampled FBs. Necessary and sufﬁcient condi-

tions on a FB to implement a frame or a tight frame

in l

(Z) were given in terms of the properties of

the corresponding polyphase analysis matrix. The

frame-theoretic analysis is based on the fact that the

polyphase matrix provides matrix representation of

a frame operator, i.e. S(z) = E

∗

(z)E(z), which can

be found in (Bolcskei et al., 1998) (Bolcskei and

Hlawatsch, 2001) (Mertins, 2003) etc. Recently (Chai

et al., 2007) presented a direct computational method

for the frame-theory-based analysis and design of

oversampled FBs, which employed the state space

representation of the polyphase matrix E(z).

In the studies of frames, the frame bounds and

frame bound ratio are very important indices charac-

terizing the robustness and numerical performance of

frame systems. The quantiﬁcation and computation of

frame bounds have been actively investigated in past

years. The classic approach to obtain frame bounds of

multirate FBs is in the frequency domain, for exam-

ple, (Bolcskei et al., 1998), (Bolcskei and Hlawatsch,

2001), (Mertins, 2003), (Stanhill and Yehoshua Zeevi,

1998). (Bayram and Selesnick, 2009) stated the frame

bounds of iterated FBs making use the wavelet frame

bounds computed in the frequency domain. The fre-

quency approach to compute the frame bounds is an

approximation method which samples the polyphase

matrix of the frame operator over the frequency range

ω ∈ [0,2π) and then performs eigenanalysis on the

sampled matrices. Such sampling approach can be

243

Chen X., Zhang C. and Zhang J..

ANALYSIS AND COMPUTATION OF OPTIMAL BOUNDS OF BI-DIRECTIONAL FRAMES.

DOI: 10.5220/0003437802430250

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 243-250

ISBN: 978-989-8425-74-4

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

very tedious when the frequency grid is dense and

the polyphase matrix is nondiagonal and of inﬁnite

impulse response. Moreover, the error due to the

frequency domain sampling of the polyphase matrix

cannot be precisely quantiﬁed and predicted by the

density of the frequency grid for generic oversam-

pled FBs. The existing literature shows that wavelet

frame bounds are computed in the frequency do-

main see for example (Daubechies, 1992) and (Chris-

tensen, 2003). The frequency approach to approxi-

mate wavelet frame bounds is even more complex and

more tedious since it involves much denser frequency

grids.

Frame bounds can also be obtained in the time

domain via linear matrix inequality (LMI) technique,

see (Chai et al., 2008), which is an application of the

KYP lemma stated in (Rantzer, 1996). This method

avoids the frequency-domain sampling and approx-

imation, but is only applicable to causal ﬁlter bank

(FB) frames in the forward direction.

In this paper, motivated by the limitation of ex-

isting techniques, we propose a direct state space ap-

proach to the analysis and computation of the frame

bounds of a more general class of frames. This class

of frames can be bi-directional with mixed causal-

anticausal realizations and may not necessarily be in

the form of multirate FBs. A state space approach

is presented to the modeling of the bi-directional

frames. The LMI solution, based on the state space

model, can then provide accurate and efﬁcient com-

putation of the optimal frame bounds.

The rest of the paper is organized as follows.

Section 2 presents notations followed by the fun-

damentals of frames. Different representations of

mixed causal-anticausal LTI systems are introduced,

including the state space representation and the trans-

fer function representation in Section 2. Section 3

presents a direct state space approach to compute the

frame bounds of a class of frame which can be mod-

eled as mixed causal-anticausal linear systems. Ex-

amples are given in Section 4 to illustrate how to ob-

tain the bounds of frames that are modeled as stable

mixed causal-anticausal LTI systems. The paper is

concluded by Section 5.

2 PRELIMINARIES

2.1 Notations

R(C) denotes the set of real (complex) numbers,

q×p

) denotes the set of real (complex) ma-

trix with size q× p. Let Z denote the set of integer

numbers. H denotes the Hilbert space, which is a real

or complex inner product space. Let (.)

denote the

transpose of a matrix or vector, and (.)

∗

denote the

Hermitian (conjugate) transpose of a matrix or vec-

tor or function, which is also known as the adjoint of

(.), (.)

−1

denote the inverse of (.), and (.)

†

the left

pseudo-inverse of (.). l

(Z) is the space of square

summable scalar sequences with an index set Z de-

ﬁned as

(Z) :=

(

∈ C,m ∈Z|

∑

m∈Z

)

The l

norm is a vector norm deﬁned for a complex

column vector x =







−1







kxk =

√

∗

The l

(Z) space is a Hilbert space with respect to the

inner product

< x,y >= x

∗

y = y

∗

2.2 Fundamental of Frames

A bi-directional frames is deﬁned as:

Deﬁnition 1. A sequence {f

∈R

∞×1

}

k∈Z

of elements

in H is a frame for H if there exist constants α,β > 0

such that

αkfk

≤

∑

k∈Z

| < f, f

> |

≤ βkfk

,∀f ∈ H.

The constants α and β are called frame bounds.

A vector space can be represented in terms of

frames and elements in such vector space can be writ-

ten as a linear combination of frame elements. In this

paper, we consider a class of bi-directional inﬁnite di-

mensional frames.

The optimal lower frame bound is the supremum

over all possible lower frame bounds, and the opti-

mal upper frame bound is the inﬁmum over all possi-

ble upper frame bounds. Note that the optimal frame

bounds are called the frame bounds in short in the rest

of the paper. If the frame bounds satisfy α = β, the

frame is called a tight frame.

For a frame {f

}

k∈Z

in H, the pre-frame operator

or the synthesis operator is given by

T : l

(Z) → H, T{c

}

k∈Z

∑

k∈Z

The adjoint of the pre-frame operator is given by

∗

: H → l

(Z),T

∗

f = {< f, f

k∈Z

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

244

By composing T with its adjoint T

∗

, we obtain the

frame operator

S : H → H,S f = TT

∗

f =

∑

k∈Z

< f, f

> f

The frame operator S is bounded by the frame bounds

α and β, invertible, self-adjoint, and positive.

If {f

}

k∈Z

is a frame for H, the pre-frame oper-

ator (synthesis operator) can be shown as an inﬁnite

dimensional matrix

T =



··· f

−1

··· f

···



i.e. the inﬁnite dimensional matrix has the vectors f

as columns. The adjoint of the pre-frame operator can

be shown as

∗







−1







i.e. the inﬁnite dimensional matrix T

∗

has the vectors

as rows.

The frame {g

}

k∈Z

= {S

−1

}

k∈Z

is called the

canonical dual frame of {f

}

k∈Z

, which satisﬁes

f =

∑

k∈Z

< f,g

> f

,∀f ∈ H.

If α,β are the optimal bounds for {f

}

k∈Z

, then

the bounds β

−1

,α

−1

are optimal for the canonical

dual frame {S

−1

}

k∈Z

. Hence, the lower bound of a

frame is equivalent to the inverse of the upper bound

of the canonical dual frame.

2.3 Fundamental of LTI Systems

State space equations of causal and anticausal linear

time-invariant (LTI) systems are given as

m+1

= Ax

+ Bu

′

m−1

= A

′

+ B

′

= Cx

′

+ (D+ D

′

(1)

where x

∈ R

is the forward state variable, x

′

∈

is the backward state variable, u

∈ R

the system input, y

∈ R

is the system output.

A ∈ R

×n

,B ∈ R

×p

,C ∈ R

q×n

,D ∈ R

q×p

, A

′

∈

×n

′

∈ R

×p

′

∈ R

q×n

and D

′

∈ R

q×p

are

the state space matrices of the causal-anticausal sys-

tem. We use

(z) =



A B

C D



= D+C(zI −A)

−1

to represent the causal system and

(z) =



′



= D

′

−1

I −A

′

)

−1

′

to represent the anticausal system. Hence the mixed

causal-anticausal LTI system has transfer function

matrix

E(z) =



A B

C D





′



= D+ D

′

+C(zI −A)

−1

′

−1

I −A

′

)

−1

′

For consistency of the symbols, we use E to de-

note the system operator for the causal-anticausal LTI

system (1), which gives

y = Eu.

In the above equation,

u =







−1







, y =







−1







, (2)

and

E =







··· D+ D

′

···

··· CB D+ D

′

···

··· CAB CB D+ D

′

···







. (3)

The entries of the system operator E (i

row and j

column, i, j ∈ Z) can be shown as







i−j−1

B, i > j,

D+ D

′

, i = j,

′

)

j−i−1

′

, i < j.

An LTI system is causal if the system operator E is

left lower-triangular and anticausal if E is right upper-

triangular.

The mixed causal-anticausal LTI system (1) is sta-

ble if and only if

|ρ(A)| < 1 and |ρ(A

′

)| < 1,

where ρ(.) indicates the largest eigenvalue of (.),

which means that E(z) has no poles on the unit cir-

cle.

The operator norm (induced l

norm) of stable

mixed causal-anticausal systems is deﬁned by:

kEk = sup

u∈l

(Z),u6=0

kEuk

kuk

ANALYSIS AND COMPUTATION OF OPTIMAL BOUNDS OF BI-DIRECTIONAL FRAMES

245

It is equivalent to the square of the upper frame bound

if E implements a frame.

The following lemma from (Shu and Chen, 1996)

is very useful to obtain the state space description of

the cascaded causal-anticausal discrete time LTI sys-

tems. It is essential in the state space analysis of LTI

systems.

Lemma 1. Assume the compatibility of the opera-

tors. We have the state space matrices of the cascaded

causal-anticausal discrete time systems as



′









′





′

+ A

′









′





′

+ A

′





′



where X,Y are given by the Sylvester equations:

′

−X + B

′

= 0,

′

−Y + B

′

= 0.

(4)

Let E

be a causal-anticausal LTI system such

that E

(z) =







′



, and E

be another causal-anticausal LTI system such that

(z) =







′



Lemma 2 below presents the necessary and sufﬁ-

cient condition for an LTI system to be a frame. Its

proof can be found in (Cvetkovic and Vetterli, 1998).

Lemma 2. Let E(z) =



A B

C D





′



∈

q×p

be the transfer function of a stable causal-

anticausal LTI system. Then for all u ∈ l

, there exist

constants α and β such that

αkuk

≤ kEuk

≤ βkuk

if and only if E(e

jω

) is full column rank on [0,2π).

The frame bounds of frames that modeled as

causal LTI systems are computed in (Chai et al.,

2008), and this result is stated in lemma 3.

Lemma 3. Given a causal LTI system E with state

space representation



A B

C D



, the problem of

ﬁnding the minimum β and maximum α over ω ∈

[0,2π) is equivalent to

min

subject to:



PA−P+C

C A

PB+C

PA+ D

C B

PB+ D

D−βI



≤ 0,

P = P

,β > 0.

and

max

subject to



PA−P+C

C A

PB+C

PA+ D

C B

PB+ D

D+ αI



≤0,

Q = Q

,α > 0.

The result makes use of the KYP lemma stated

in (Rantzer, 1996).

3 FRAMES IN LTI STATE SPACE

REALIZATIONS

Let {h(m) ∈ R

q×p

} be the impulse response of a sta-

ble causal-anticausal LTI system E. Then the output

signal of E is given by

y = Eu.

The bi-directional inﬁnite sequence h

= {h

(m)} can

be represented by the state space matrices as

(m) =







k−m−1

B, k > m,

D+ D

′

, k = m,

′

)

m−k−1

′

, k < m,

(5)

resulting in

∑

m∈Z

(m)u

For a frame whose elements {f

(m)} can be writ-

ten as the bi-directional inﬁnite impulse responses of

the linear system E that maps the input u to the out-

put y, i.e. {f

(m) = h

(m)}, we say that the frame

can be modeled as a mixed causal-anticausal LTI sys-

tem with the state space realization. For such frames,

its canonical dual frame is actually the pseudo-inverse

system. Hence the lower frame bound can be found

as the inverse of the upper frame bound of the canon-

ical dual frame, which is constructed by the impulse

response of the pseudo-inverse system.

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

246

Theorem 1. Let {f

}

k∈Z

({h

}

k∈Z

) be a frame im-

plemented by the impulse response {h

}

k∈Z

of a sta-

ble mixed causal-anticausal LTI system E. Then its

canonical dual frame is given by the impulse response

of E

†

, the pseudo inverse system of E. The square

of the operator norm of E

†

equals the inverse of the

lower frame bound of {f

}

k∈Z

Proof: There exists a dual frame {g

}

k∈Z

for

given frame {f

}

k∈Z

for which

u =

∞

∑

k=−∞

< u,g

> f

∞

∑

k=−∞

< u, f

> g

. (6)

The frame element represents the bi-directional in-

ﬁnite impulse responses of a linear system, hence

= h

results in E = T

∗

, where T

∗

is the adjoint

of the synthesis operator of the frame {f

}

k∈Z

. The

linear system operator E can be factorized into:

E = Q

∗

where Q is inner, U

∗

U = I and VV

∗

= I and R

the left and right outer. More details of the inner

factor and the outer factor can be found in (Dewilde

and Van Der Veen, 1998). E has a Moore-Penrose

(pseudo-) inverse

†

= V

∗

−1

∗

The pre-frame operator for {g

}

∞

k=−∞

is denoted

T. The canonical dual frame is obtained by

}

∞

k=−∞

= {S

−1

}

∞

k=−∞

which implies

T = S

−1

T = (TT

∗

)

−1

hence

∗

= I. The canonical dual frame has the

minimum l

norm among the dual frames.

Since E = T

∗

, the pre-frame operator of the

canonical dual frame can be obtained:

T = (TT

∗

)

−1

= (V

∗

−1

∗

= V

∗

−1

∗

U(R

∗

)

−1

∗

−1

∗

= V

∗

−1

∗

(7)

We can see that

T = E

†

Thus

∗

= I and E

†

E = I.

Since E

†

equals the pre-frame operator

T of the

dual frame, the square of the operator norm of E

†

equivalent to the upper frame bound of the dual frame,

following the deﬁnition of the operator norm of a lin-

ear system given in Section 2.3. The inverse of the

lower bound of the frame is equivalent to the upper

bound of the canonical dual frame, which is equiv-

alent to the square of the operator norm of the left

pseudo-inverse system E

†

Lemma 3 presents the LMI approach to obtain the

frame bounds of the frames modeled as causal LTI

systems. In this paper, we propose a direct method to

obtain the frame bounds of frames modeled as mixed

causal-anticausal LTI systems. This method avoids

a large amount of computations to convert the sta-

ble mixed causal-anticausal realization into unstable

causal realization.

Theorem 2. Given a frame that can be modeled as

a stable mixed causal-anticausal LTI system E with

E(e

jω

) being full column rank on ω ∈ [0,2π), the op-

timal upper frame bound is the inﬁmum (minimum) of

all possible β satisfying

kEuk

≤ βkuk

The problem of ﬁnding the minimum β is equivalent

to the following optimization problem:

min

P,Q,X

subject to:







PA−P+C

C A

X −XA

′

A−A

′

C A

′

−Q+C

′

PA+ D

C−B

′

+ D

′

+ B

PB+C

D−XB

′

D+X

PB+ B

′

+ D

D−βI







≤ 0,

where P = P

> 0, Q = Q

> 0 and X is a general

matrix.

Proof: The class of frames modeled by the mixed

causal-anticausal LTI system has the rational transfer

function as

E(z)

= D+C(zI −A)

−1

B+C

′

−1

I −A

′

)

−1

′



A B

C D





′



and kEuk

≤ βkuk

implies E

∗

(z)E(z) ≤ βI.

∗

(z)E(z)







′





A B

C D





′









A B

C D





′





′





′





A B

C D









′



ANALYSIS AND COMPUTATION OF OPTIMAL BOUNDS OF BI-DIRECTIONAL FRAMES

247

There are four terms in the above expression, each

term is a cascaded system. By using lemma 1, we can

represent the ﬁrst term as:







A B

C D





A B

C+ B

PA D

D+ B





D+ A



where A

PA − P + C

C = 0, yielding B

−1

I −

)

−1

PA −P + C

C)(zI −A)

−1

B = 0. The sec-

ond term can be rewritten as:



′





′





′





′



where A

′

−Q + C

′

= 0, yielding B

′

(zI −

′

)

−1

′

− Q + C

′

)(z

−1

I − A

′

)

−1

′

= 0.

The third term can be rewritten as:



′





A B

C D







A 0 B

′

C A

′

0 B

′









A 0 B

0 A

′

D+UB

−B

′

U B

′





where UA − A

′

U + C

′

C = 0, yielding B

′

(zI −

′

)

−1

(UA −A

′

U + C

′

C)(zI − A)

−1

B = 0. The

fourth term can be rewritten as:







′







′

0 A

′









0 −VB

′

0 A

′

+ B

V 0





where A

V −VA

′

+ C

′

= 0, yielding B

−1

I −

)

−1

V −VA

′

)(z

−1

I −A

′

)

−1

′

= 0. The

third term condition UA − A

′

U + C

′

C = 0 and

fourth term condition A

V −VA

′

+ C

′

= 0 result

in V = U

,U = V

Hence we can represent E

∗

(z)E(z) −βI as

∗

(z)E(z) −βI

= (D

D+ B

PB+ B

′

−βI)+ (D

PA)(zI −A)

−1

B+ B

−1

I −A

)

−1

PB) + B

−1

I −A

)

−1

PA−P+C

×(zI −A)

−1

B+ B

′

(zI −A

′

)

−1

′

)

+(B

′

)(z

−1

I −A

′

)

−1

′

+ B

′

(zI −A

′

)

−1

×(A

′

−Q+C

′

)(z

−1

I −A

′

)

−1

′

−B

′

×(zI −A)

−1

B+ B

′

(zI −A

′

)

−1

′

D+UB)

′

(zI −A

′

)

−1

(UA−A

′

U +C

′

C)(zI −A)

−1

×B−B

−1

I −A

)

−1

′

+ (D

′

+ B

×(z

−1

I −A

′

)

−1

′

+ B

−1

I −A

)

−1

×(A

V −VA

′

)(z

−1

I −A

′

)

−1

′

We can further rewrite E

∗

(z)E(z) −βI ≤ 0 as:

∗

(z)E(z) −βI

−1

I −A

)

−1

′

(zI −A

′

)

−1







PA−P+C

C A

V −VA

′

UA −A

′

U +C

′

C A

′

−Q+C

′

C+ B

PA−B

′

U B

′

+ D

′

+ B

D+A

PB−VB

′

D+UB

D+B

PB+ B

′

−βI













(zI −A)

−1

I −A

′

)

−1

′







≤ 0

Thus the matrix







PA−P+C

C A

V −VA

′

UA −A

′

U +C

′

C A

′

−Q+C

′

C+ B

PA−B

′

U B

′

+ D

′

+ B

D+A

B−VB

′

D+UB

D+B

PB+ B

′

−βI







≤ 0

is a negative deﬁnite matrix. Letting X = V, which

means that X

= U, we prove the theorem.

Theorem 2 can be used to obtain the upper bound

of the causal LTI system by setting A

′

= 0,B

′

0,C

′

= 0,D

′

= 0, resulting X = 0. In this case, theo-

rem 2 can be found in lemma 3. The theorem can also

be used to obtain the upper bound of the anticausal

LTI system by setting A = 0,B = 0,C = 0,D = 0,

hence X = 0.

Similarly, we have the lower bound computed by

the following theorem.

Theorem 3. For a frame whose elements are the

bi-inﬁnite dimensional impulse responses of a stable

mixed causal-anticausal LTI systems E with E(e

jω

)

being full column rank in ω ∈ [0,2π), the optimal

lower frame bound is the supermum (maximum) of all

possible α satisfying

αkuk

≤ kEuk

The problem of ﬁnding the maximum α is equivalent

to the following optimization problem

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248

min

P,Q,X

−α

subject to:







PA−P+C

C A

X −XA

′

A−A

′

C A

′

−Q+C

′

PA+ D

C−B

′

+ D

′

+ B

PB+C

D−XB

′

D+X

PB+ B

′

+ D

D−αI







≥ 0,

where P = P

> 0, Q = Q

> 0 and X is a general

matrix.

The proof is analogue to the proof of theorem 2.

4 EXAMPLES

Example 1. We make use of the IIR Butterworth ﬁl-

ters shown in (Herley and Vetterli, 1993). The low

pass ﬁlter H(z) is given as

H(z) =

∑

i=0





−i

√

∑

(N−1/2)

l=0





−2l

where N is the order of the ﬁlter and the high pass

ﬁlter G(z) is given as:

G(z) = z

2n−1

H(−z

−1

We give an example with order N = 7 and n = 0:

H(z) =

1+7z

−1

+21z

−2

+35z

−3

+35z

−4

+21z

−5

+7z

−6

−7

√

2(1+21z

−2

+35z

−4

+7z

−6

)

G(z) = z

−1

1−7z

+21z

−35z

+35z

−21z

+7z

−z

√

2(1+21z

+35z

+7z

)

The causal-anticausal state space representations the

low pass ﬁlter H(z) and high pass ﬁlter G(z) are

shown as:

H(z) =







0 −0.2319 0 1

1 0 0 0

0 1 0 0

0.3499 0 0.0234 0.7071













0 −0.6881 0 −0.0331 0

1 0 0 0 −6.6787

0 1 0 0 0

0 0 1 0 16.3918

0 0 0 0.0331 0







and

G(z) =







0 −0.6881 0 −0.0331 0 1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0.7071 −0.5431 0.4865 −0.1524 0.0234 −0.3499













0 −0.2319 −0.0404

1.001 0 0

−0.001 −1.4273 0







The frame bounds of the Butterworth FB frame are

shown in table 1. The FB constructs a approximate

tight frame. The dual frame is realized by:

H(z) = H(z

−1

G(z) = G(z

−1

Table 1: Frame bounds of Butterworth FB frame.

Decimation Factor 1 2

β 2 1.0008

α 1.9985 0.9984

Example 2. The scaling ﬁlter H(z) and wavelet ﬁlter

G(z) are given as

H(z) =

√



−1

−2



G(z) = 6

1313



−

z+ 1−

−1



The state space representations of the low pass and

high pass ﬁlters are given as

H(z) =





0 0 0.5

1 0 0

0.7071 0.1768 0.5303









0 0 0.5

1 0 0

0.7071 0.1768 0





and

G(z) =



0 1

−0.6927 1.3854





0 1

−0.6927 0



The frame bounds of the FB frame are given in table

Table 2: Frame bounds of FIR FB frame.

Decimation Factor 1 2

β 7.6773 3.8385

α 1.1090 0

5 CONCLUSIONS AND FUTURE

WORK

A class of frames, with elements in the form of bi-

directional inﬁnite impulse responses of an LTI sys-

tem, can be equivalently modeled as mixed causal-

anticausal LTI systems. This paper presents a direct

state space approach to the analysis and computation

of optimal frame bounds of this class of frames. It

is shown that the lower frame bound is also equal to

the inverse of the square of the operator norm of the

left pseudoinverse system which achieves perfect re-

construction. Accurate and efﬁcient computation of

ANALYSIS AND COMPUTATION OF OPTIMAL BOUNDS OF BI-DIRECTIONAL FRAMES

249

the frame bounds has been achieved using the LMI

technique and the obtained results are demonstrated

by examples.

The results obtained in this paper are applicable

to a class of frames which are governed by exponen-

tial type perfoamance hahavior and can be modeled

by LTI system responses in the time and frequency

domains. Currently, the authors are extending the LTI

state space approach presented in this paper to lin-

ear time varying (LTV) state space modeling, analy-

sis and computation of frames. This study will en-

able deeper understanding and more efﬁcient evalua-

tion of a more general class of frames which may not

be properly analyzed and evaluated in the conventi-

noal frequency domain.

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