DESIGN OF ALLPASS FILTERS WITH SPECIFIED DEGREES OF

FLATNESS AND EQUIRIPPLE PHASE RESPONSES

Xi Zhang

Department of Information and Communication Engineering, The University of Electro-Communications

1–5–1 Chofugaoka, Chofu-shi, 182-8585, Tokyo, Japan

Keywords:

IIR allpass ﬁlter, Flatness, Equiripple approximation, Remez exchange algorithm.

Abstract:

This paper proposes a new method for designing allpass ﬁlters having the speciﬁed degrees of ﬂatness at the

speciﬁed frequency point(s) and equiripple phase responses in the approximation band(s). First, a system of

linear equations are derived from the ﬂatness conditions. Then, the Remez exchange algorithm is used to

approximate the equiripple phase responses in the approximation band(s). By incorporating the linear equa-

tions from the ﬂatness conditions into the equiripple approximation, the design problem is formulated as a

generalized eigenvalue problem. Therefore, we can solve the eigenvalue problem to obtain the ﬁlter coefﬁ-

cients, which have the equiripple phase response and satisfy the speciﬁed degrees of ﬂatness simultaneously.

Furthermore, a class of IIR ﬁlters composed of allpass ﬁlters are introduced as one of its applications, and it is

shown that IIR ﬁlters with ﬂat passband (or stopband) and equiripple stopband (or passband) can be designed

by using the proposed method. Finally, some examples are presented to demonstrate the effectiveness of the

proposed design method.

1 INTRODUCTION

Allpass ﬁlters possess constant magnitude response at

all frequencies and are a basic scalar lossless building

block (Mitra and Kaiser, 1993), (Regalia et al., 1988).

Interconnections of allpass ﬁlters have found numer-

ous applications in many practical ﬁltering problems

such as low-sensitivity ﬁlter structures, wavelet ﬁlter

banks, and so on (Mitra and Kaiser, 1993), (Shenoi,

1999), (Regalia et al., 1988), (Laakso et al., 1996),

(Lang, 1998), (Selesnick and Burrus, 1998), (Se-

lesnick, 1999), (Zhang and Iwakura, 1999). In many

applications, it is necessary to design an allpass ﬁl-

ter both satisfying the speciﬁed degrees of ﬂatness at

the speciﬁed frequency point(s) and having equiripple

phase response in the approximation band(s). For ex-

ample, in the allpass-sum structure (Selesnick, 1999),

the phase response of the allpass sub-ﬁlter is required

to be ﬂat in the band(s) where the corresponding ﬁl-

ter has the ﬂat magnitude response, and is equiripple

in other band(s) to get the equiripple magnitude re-

sponse. Many methods have been proposed for the

phase design of allpass ﬁlters: the maximally ﬂat

design (Thiran, 1971), least squares design (Laakso

et al., 1996), (Lang, 1998), and equiripple design

(Zhang and Iwakura, 1999), (Tseng, 2003).

However, the approximation of allpass ﬁlters with

both the speciﬁed degrees of ﬂatness and equiripple

phase responses in the approximation band(s) is still

open.

In this paper, we propose a new method for de-

signing allpass ﬁlters which have both the speciﬁed

degrees of ﬂatness at the speciﬁed frequency point(s)

and equiripple phase responses in the approximation

band(s). First, we derive a system of linear equations

from the ﬂatness conditions of the phase response at

the speciﬁed frequency point(s). Then, we apply the

Remez exchange algorithm to obtain the equiripple

reponse in the approximation band(s). By incorpo-

rating the linear equations from the ﬂatness condi-

tions into the equiripple approximation, we formulate

the design problem as a generalized eigenvalue prob-

lem (Zhang and Iwakura, 1996), (Zhang and Iwakura,

1999). Therefore, we can obtain the ﬁlter coefﬁcients

by iteratively solving the eigenvalue problem. The

resulting allpass ﬁlters have the equiripple phase re-

sponses and satisfy the speciﬁed degrees of ﬂatness si-

multaneously. Furthermore, as one of the applications

of allpass ﬁlters, we introduce a class of IIR ﬁlters

composed of allpass ﬁlters (Regalia et al., 1988), (Se-

lesnick, 1999), whose design problem can be reduced

to the phase approximation of the allpass sub-ﬁlter.

205

Zhang X. (2010).

DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE RESPONSES.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 205-210

DOI: 10.5220/0002934102050210

Copyright

c

SciTePress

Thus, we can design the ﬁlters with ﬂat passband (or

stopband) and equiripple stopband (or passband) by

using the proposed method. Finally, some design ex-

amples are presented to demonstrate the effectiveness

of the proposed design method.

2 ALLPASS FILTERS

It is well-known that the transfer function of an all-

pass ﬁlter A(z) is deﬁned by

A(z) = z

−N

N

∑

n=0

a

n

z

n

N

∑

n=0

a

n

z

−n

, (1)

where N (∈ Z) is ﬁlter degree, and a

n

(∈ R) are real

coefﬁcients and a

0

= 1.

It can be seen that A(z) in Eq.(1) has unit mag-

nitude response at all frequencies, and its phase re-

sponse θ(ω) is given by

θ(ω) = −Nω + 2tan

−1

N

∑

n=0

a

n

sinnω

N

∑

n=0

a

n

cosnω

. (2)

Let θ

d

(ω) be the desired phase response. The dif-

ference θ

e

(ω) between θ(ω) and θ

d

(ω) is

e

jθ

e

(ω)

= e

j{θ(ω)−θ

d

(ω)}

=

N

∑

n=0

a

n

e

j{(n−

N

2

)ω−

θ

d

(ω)

2

}

N

∑

n=0

a

n

e

− j{(n−

N

2

)ω−

θ

d

(ω)

2

}

,

(3)

and

θ

e

(ω) = 2tan

−1

N

∑

n=0

a

n

sin{(n−

N

2

)ω−

θ

d

(ω)

2

}

N

∑

n=0

a

n

cos{(n−

N

2

)ω−

θ

d

(ω)

2

}

.

(4)

Therefore, the design problem of allpass ﬁlters

is the phase approximation of θ(ω) to θ

d

(ω) in the

approximation band(s), that is, the minimization of

the phase error θ

e

(ω) in Eq.(4) in the speciﬁed cri-

terion, e.g., in the least squares, and/or Chebyshev

(minimax), and/or maximally ﬂat sense. In the fol-

lowing, we discuss the design of allpass ﬁlters hav-

ing equiripple phase responses in the approximation

band(s) while satisfying the speciﬁed degrees of ﬂat-

ness at the speciﬁed frequency point(s).

3 ALLPASS FILTER DESIGN

In this section, we describe the design method of all-

pass ﬁlters with both the speciﬁed degrees of ﬂatness

and equiripple phase responses in the approximation

band(s). Firstly, we consider the ﬂatness condition of

the phase response at the frequency point ω

p

. It is re-

quired that the derivatives of θ(ω) in Eq.(2) are equal

to that of θ

d

(ω) at ω = ω

p

, that is,

∂

r

θ(ω)

∂ω

r

ω=ω

p

=

∂

r

θ

d

(ω)

∂ω

r

ω=ω

p

(r = 0, 1, ··· , K − 1),

(5)

where K (∈ Z) is a parameter that controls the degree

of ﬂatness. It is seen that to satisfy the speciﬁed de-

grees of ﬂatness, the ﬂatness conditions in Eq.(5) be-

come

∂

r

θ

e

(ω)

∂ω

r

ω=ω

p

= 0 (r = 0, 1, ··· , K − 1). (6)

From Eq.(4), we have

tan

θ

e

(ω)

2

=

N(ω)

D(ω)

, (7)

where

N(ω) =

N

∑

n=0

a

n

sin{(n−

N

2

)ω−

θ

d

(ω)

2

}

D(ω) =

N

∑

n=0

a

n

cos{(n−

N

2

)ω−

θ

d

(ω)

2

}

. (8)

Therefore, it is proven that the condition in Eq.(6) is

equivalent to

∂

r

N(ω)

∂ω

r

ω=ω

p

= 0 (r = 0, 1, ··· , K − 1). (9)

By substituting N(ω) in Eq.(8) into Eq.(9), we can

derive a system of linear equations as follows,

N

∑

n=0

∂

r

sin{(n−

N

2

)ω−

θ

d

(ω)

2

}

∂ω

r

ω=ω

p

a

n

= 0, (10)

for r = 0, 1, ··· , K − 1. For example, if a linear phase

is required, that is, θ

d

(ω) = −τω, then Eq.(10) is re-

duced to

SIGMAP 2010 - International Conference on Signal Processing and Multimedia Applications

206

N

∑

n=0

(n−

N−τ

2

)

r

sin{(n−

N−τ

2

)ω

p

}a

n

= 0 (even r)

N

∑

n=0

(n−

N−τ

2

)

r

cos{(n−

N−τ

2

)ω

p

}a

n

= 0 (odd r)

.

(11)

It is known that the phase response θ(ω) is an odd

function with respect to ω = 0 and π. If ω

p

= 0 or π,

then the equations with even r are satisﬁed without

any conditions, and thus the number of the condi-

tions reduces about a half, that is, L = ⌊

K

2

⌋, where ⌊x⌋

means the largest integer not greater than x. When

ω

p

6= 0 and π, then L = K.

When the above-mentioned conditions are im-

posed at several frequency points ω

pi

(i = 1, 2, ··· ,

M), the total number of the conditions is L =

∑

M

i=1

L

i

,

where L

i

= ⌊

K

i

2

⌋ if ω

pi

= 0 or π, and L

i

= K

i

if

ω

pi

6= 0 and π. Note that K

i

is a parameter that con-

trols the degree of ﬂatness at ω

pi

. Therefore, if L = N,

we can solve a system of linear equations as shown

in Eq.(10) to obtain a set of ﬁlter coefﬁcients, which

has the maximally ﬂat phase response and satisﬁes the

speciﬁed degrees of ﬂatness at the speciﬁed frequency

point(s) ω

pi

.

Next, we consider the case of L < N. Besides sat-

isfying the ﬂatness conditions in Eq.(5), we want to

obtain an equiripple phase response in the approxi-

mation band(s) by using the remaining degree of free-

dom. We apply the Remez exchange algorithm in the

approximation band(s). Let ω

i

(i = 0, 1, ··· , N − L)

are the extremal frequencies in the approximation

band(s), we formulate θ

e

(ω) as

tan

θ

e

(ω

i

)

2

=

N

∑

n=0

a

n

sin{(n−

N

2

)ω

i

−

θ

d

(ω

i

)

2

}

N

∑

n=0

a

n

cos{(n−

N

2

)ω

i

−

θ

d

(ω

i

)

2

}

= (−1)

i

δ,

(12)

where δ (∈ R) is an error. We incorporate Eq.(10) into

Eq.(12), and formulate the design problem as a gener-

alized eigenvalue problem. Then we rewrite Eqs.(10)

and (12) in the matrix form as

PA = δQA, (13)

where A = [a

0

, a

1

, ··· , a

N

]

T

, and the elements of the

matrices P and Q, for example, when the ﬂatness

condition in Eq.(5) is imposed at only one frequency

0 0.1 0.2 0.3 0.4 0.5

−8

−6

−4

−2

0

NORMALIZED FREQUENCY

PHASE RESPONSE ( )

K=7

K=9

K=11

π

Figure 1: Phase responses of allpass ﬁlters.

point ω

p

(6= 0 and π), are given by

P

ij

=

∂

i

sin{( j −

N

2

)ω−

θ

d

(ω)

2

}

∂ω

i

ω=ω

p

(i = 0, 1, ··· , L− 1)

sin{( j −

N

2

)ω

(i−L)

−

θ

d

(ω

(i−L)

)

2

}

(i = L, L+ 1, ··· , N)

,

(14)

Q

ij

=

0 (i = 0, 1, ··· , L− 1)

(−1)

(i−L)

cos{( j −

N

2

)ω

(i−L)

−

θ

d

(ω

(i−L)

)

2

}

(i = L, L+ 1, ··· , N)

.

(15)

Once the design speciﬁcation: the ﬁlter degree N,

the desired phase response θ

d

(ω), the degree of ﬂat-

ness K

i

, the speciﬁed frequency point(s) ω

pi

, and the

extremal frequencies ω

i

in the approximation band(s)

are given, the elements P

ij

and Q

ij

of the matrices P

and Q can be computed by Eqs.(14) and (15). There-

fore, it should be noted that Eq.(13) corresponds to

a generalized eigenvalue problem, i.e., δ is an eigen-

value, and A is a corresponding eigenvector. In or-

der to minimize the error δ, we must ﬁnd the absolute

minimum eigenvalue by solving the eigenvalue prob-

lem, so that the corresponding eigenvector gives a set

of ﬁlter coefﬁcients a

n

. To obtain an equiripple phase

response, we make use of an iteration procedure so

that the optimal ﬁlter coefﬁcients is easily obtained.

The design algorithm is shown as follows.

DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE

RESPONSES

207

4 DESIGN ALGORITHM

Procedure. {Allpass Filter Design Algorithm.}

Begin

1) Read N, θ

d

(ω), K

i

, and ω

pi

.

2) Select initial extremal frequencies Ω

i

(i =

0, 1, ··· , N − L) equally spaced in approximation

band(s).

Repeat

3) Set ω

i

= Ω

i

(i = 0, 1, · · · , N − L).

4) Compute P and Q by using Eqs.(14) and (15), and

ﬁnd the absolute minimum eigenvalue δ to obtain

a set of ﬁlter coefﬁcients a

n

.

5) Search the peak frequencies Ω

i

(i = 0, 1, · · · , N −

L) of θ

e

(ω) in approximation band(s).

Until

Satisfy the following condition for a prescribed small

constant ε (for example, ε = 10

−8

):

|ω

i

− Ω

i

| < ε (for all i)

End.

5 IIR FILTERS COMPOSED OF

ALLPASS FILTERS

Many methods for designing IIR ﬁlters have been pro-

posed in (Mitra and Kaiser, 1993), (Regalia et al.,

1988), (Zhang and Iwakura, 1996), (Lang, 1998),

(Hegde and Shenoi, 1998), (Selesnick and Burrus,

1998), (Selesnick, 1999). These design methods have

considered the maximally ﬂat and/or equiripple mag-

nitude responses. It is required in some applications

that the magnitude response of the ﬁlters is ﬂat in

passband(s) and equiripple in stopband(s) (Darling-

ton, 1978), (Vaidyanathan, 1985), (Selesnick and Bur-

rus, 1996), (Hegde and Shenoi, 1998). In this section,

we discuss the design of IIR ﬁlters with ﬂat pass-

band(s) and equiripple stopband(s), which are com-

posed of two allpass ﬁlters.

It is known in (Regalia et al., 1988), (Lang, 1998),

(Selesnick and Burrus, 1998) and (Selesnick, 1999)

that a parallel interconnection of two allpass ﬁlters

(allpass-sum) has many advantages, such as low-

sensitivity structures, low-complexity structures with

low roundoff noise behavior, and so on. The classical

digital (Butterworth, Chebyshev, and elliptic) ﬁlters

can be realized as an allpass-sum structure. In addi-

tion, the allpass-sum structure can realize a more gen-

eral class of transfer functions. Here, we consider this

class of IIR ﬁlters whose transfer function is given by

H(z) =

1

2

[z

−J

A

1

(z) + A

2

(z)], (16)

0 0.1 0.2 0.3 0.4 0.5

−0.1

−0.05

0

0.05

0.1

NORMALIZED FREQUENCY

PHASE ERROR ( )

K=7

K=9

K=11

π

Figure 2: Phase errors of allpass ﬁlters.

where A

1

(z), A

2

(z) are two causal stable allpass ﬁlters

of degree N

1

, N

2

, and J (∈ Z) is a nonnegative integer.

Eq.(16) can be rewritten to

H(z) =

1

2

A

1

(z)[z

−J

+ A(z)], (17)

where

A(z) =

A

2

(z)

A

1

(z)

, (18)

whose degree is N = N

1

+ N

2

. Note that A(z) needs

not be causal stable. The magnitude response of H(z)

is given by

|H(e

jω

)| = |cos

θ(ω) + Jω

2

|, (19)

where θ(ω) is the phase response of A(z). It is clear

that the phase difference between A(z) and z

−J

must

be 2nπ in the passband(s) of H(z), and (2n + 1)π in

the stopband(s), where n ∈ Z. Therefore, the desired

phase response of A(z) is

θ

d

(ω) =

−Jω+ 2nπ (in passband)

−Jω+ (2n+ 1)π (in stopband)

,

(20)

then the design problem of H(z) becomes the phase

approximation of A(z). The conventional design

methods, for example, the maximally ﬂat design (Thi-

ran, 1971), equiripple design (Zhang and Iwakura,

1999), (Tseng, 2003) and so on, can be used in the

design. However, these methods cannot design all-

pass ﬁlters with ﬂat and equiripple phase response in

passband(s) and stopband(s), respectively. By using

the design method proposed in the preceding section,

we can obtain easily the ﬂat passband(s) and equirip-

ple stopband(s) of H(z).

SIGMAP 2010 - International Conference on Signal Processing and Multimedia Applications

208

0 0.1 0.2 0.3 0.4 0.5

−80

−60

−40

−20

0

NORMALIZED FREQUENCY

MAGNITUDE RESPONSE (dB)

K=7

K=9

K=11

Figure 3: Magnitude responses of IIR lowpass ﬁlters.

For example, if we want H(z) to be a lowpass ﬁl-

ter, the desired phase response is given by

θ

d

(ω) =

−Jω (0 ≤ ω ≤ ω

1

)

−Jω± π (ω

2

≤ ω ≤ π)

, (21)

where ω

1

and ω

2

are the cutoff frequencies of the

passband and stopband, respectively. Note that in

this case, the ﬁlter degrees N

1

and N

2

must satisfy

N

2

− N

1

= J ∓ 1. If we set N

1

= 0 and N = N

2

=

J ∓1, then the ﬁlter will have an approximately linear

phase response also (Laakso et al., 1996), (Zhang and

Iwakura, 1999).

We use the proposed method to design the allpass

ﬁlter A(z), whose phase response θ(ω) satisﬁes Eq.(5)

at ω

p

= 0. Note that K should be an odd number, be-

cause θ(ω) is an odd function with respect to ω = 0.

Thus, the resulting lowpass ﬁlter H(z) has a ﬂat mag-

nitude response at ω

p

= 0, and the degree of ﬂatness

is 2K.

6 DESIGN EXAMPLES

In this section, we present some examples to

demonstrate the effectiveness of the proposed design

method.

First, we consider the design of allpass ﬁlter of de-

gree N = 8 with the desired phase response θ

d

(ω) =

−7ω in [0, 0.3π] and θ

d

(ω) = −7ω − π in [0.5π, π].

The degree of ﬂatness is required to be K = 9 at

ω

p

= 0, then L = 4. Since the remaining degree of

freedom is N − L = 4, we have selected initial ex-

tremal frequencies 0.5π = ω

0

< ω

1

< ·· · < ω

4

< π

equally spaced in [0.5π, π], and obtained the optimal

0 0.1 0.2 0.3 0.4 0.5

−8

−6

−4

−2

0

NORMALIZED FREQUENCY

PHASE RESPONSE ( )

K=7

K=9

K=11

π

Figure 4: Phase responses of IIR lowpass ﬁlters.

ﬁlter coefﬁcients a

n

by using the design algorithm de-

scribed in the section IV. The resulting phase response

and phase error are shown in the solid line in Fig.1 and

Fig.2, respectively. It is clear in Fig.2 that the phase

response is ﬂat at ω = 0 and equiripple in [0.5π, π]. In

Fig.1 and Fig.2, the phase responses of two allpass ﬁl-

ters with K = 7 and K = 11 are shown also. It is seen

that the degree of ﬂatness K can be arbitrarily spec-

iﬁed. It is found that these allpass ﬁlters are causal

stable since all poles are within the unit circle (Zhang

and Iwakura, 1999).

Next, we use the obtained allpass ﬁlters to con-

struct IIR lowpass ﬁlters: H(z) =

1

2

[z

−7

+ A(z)]. The

magnitude and phase responses of the IIR ﬁlters are

shown in Fig.3 and Fig.4, respectively. It is seen in

Fig.3 and Fig.4 that these lowpass ﬁlters have the ﬂat

passband and equiripple stopband responses, while

the phase responses are approximately linear.

7 CONCLUSIONS

In this paper, we have proposed a new method for de-

signing allpass ﬁlters which have both the speciﬁed

degrees of ﬂatness and equiripple phase responses

in the approximation band(s). Firstly, a system of

linear equations have been derived from the ﬂatness

conditions of the phase responses, then the Remez

exchange algorithm is used to get the equiripple re-

sponses in the approximation band(s). The design

problem has been formulated as a generalized eigen-

value problem by incorporating the ﬂatness condi-

tions into the equiripple approximation, thus, a set of

ﬁlter coefﬁcients can be easily obtained by solving the

eigenvalue problem. Furthermore, as one application

DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE

RESPONSES

209

of allpass ﬁlters, a class of IIR ﬁlters composed of two

allpass ﬁlters has been discussed also. Finally, some

examples have been presented to demonstrate the ef-

fectiveness of the proposed design method.

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