fir, allpass, and iir variable fractional delay digital filter...

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Abstract—This paper presents two-step design methodologies and performance analyses of FIR, allpass, and IIR variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are to be approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense to be solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a globally solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming (QP) problem to be solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to non-variable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency (≥ 0.9π). Filter-examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.

Index Terms— Allpass digital filters, Finite Impulse Response (FIR) digital filters, Infinite Impulse Response (IIR) digital filters, model reduction, polynomial approximation, variable fractional delay (VFD).

I. INTRODUCTION ariable fractional delay (VFD) digital filters are useful in various signal processing applications [1], such as sound synthesis, sampling rate conversion, and digital modem

synchronization. To simplify the notation in this paper, we shall denote VFD digital filters as VFD filters. So far, a number of methods have been developed to design FIR VFD filters [2]-[5], and allpass VFD filters [5]-[7]. References [8]-[9] introduce a special class of fractional delay filters based on B-spline transform. By expressing each filter coefficient as a polynomial function of fractional delay (FD), a VFD filter can be realized using the Farrow’s structure [10]. Since the design problem of an FIR VFD filter can be formulated using matrix notation by

Manuscript received October 20, 2007; revised June 24, 2008; revised October 2, 2008.

The authors are with the Department of Electrical and Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, Canada N9B 3P4 (e-mail: [email protected], [email protected]).

expanding the polynomial functions of filter coefficients in its objective function, closed-form designs [2]-[3] can be obtained. Some other researchers formulate the design problem as a convex optimization problem [4], which can be reliably solved. In contrast, the design of an allpass VFD filter involves nonlinear optimization and the stability constraint needs to be incorporated. However, with the same set of specifications and the same number of coefficients, an allpass VFD filter can achieve a higher design accuracy than the corresponding FIR VFD filter. Since the allpass VFD filters possess unity magnitude responses in the entire frequency band, the objective of an allpass VFD filter design is to minimize the errors between the group delay (or phase) responses of the designed allpass VFD filter and those of a desired VFD filter. However, the group delay (or phase) responses in terms of filter coefficients are neither linear nor quadratic functions. Thus, approximation or iterative techniques often have to be used.

The ideal fractional delay digital filter requires fullband constant group delay (or linear phase) responses achievable by linear phase FIR digital filters and fullband unity magnitude responses achievable by allpass digital filters. In practice, these two ideal properties do not coexist in any FIR, allpass, and IIR digital filter. The study of a general IIR digital filter as a VFD filter [11]-[17] was motivated by the hypothesis [14]-[15] that a general IIR digital filter, which can be considered as a combination of an FIR digital filter and an allpass digital filter, can offer a property that exhibits an overall combined constant group delay and unity magnitude to a greater extent than that offered individually by an FIR digital filter or an allpass digital filter. However, the design of an IIR VFD filter faces more challenges than those of an allpass VFD filter as both its magnitude and group delay (or phase) responses have to be approximated. On the other hand, an IIR VFD filter can achieve a lower group delay than the corresponding allpass VFD filter, within same frequency regions of interest. Recently, the design of IIR VFD filters have been advanced in [11]-[17] and their filter-examples have demonstrated the effectiveness of this new design approach. In [11]-[15], both the denominator and numerator coefficients of a designed IIR VFD filter are variable. In [15]-[17], fixed denominator coefficients are considered. In this paper, low-order fixed denominator IIR VFD filters are considered.

The paper is organized as follows. In Section II, the general design problem of FIR, allpass, and IIR VFD filters is formulated. The design method of FIR FD filters is described in Section III and the allpass FD filter design method is described in Section IV. Then, the IIR FD filter design method is presented in Section V. The second step of the design is to

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

Hon Keung Kwan, Senior Member, IEEE, and Aimin Jiang

V

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approximate each obtained FD filter coefficient (except those fixed denominator coefficients) by a polynomial function of FD, which is described in Section VI. Three sets of filter-examples for FIR, allpass, and IIR VFD filters are designed, analyzed, and presented in Section VII. Based on the results, some conclusions are drawn in Section VIII.

II. PROBLEM FORMULATION The desired frequency response of a VFD digital filter is

given by ( )( , ) [0, ]j j D d

d cH e d e wω ω ω− += ∈ (1)

In (1), D is a positive integer delay, d is a variable fractional delay in the range of [-0.5, 0.5], and wc∈[0, π) is the cutoff frequency. The problem is to design an FIR, a stable allpass or a stable IIR VFD filter, which can best approximate the desired frequency response Hd(ejω, d). The transfer functions of an FIR VFD filter and an IIR VFD filter can be expressed, respectively, by

0

( , ) ( ) ( ) ( )FIR

FIR

Li T

i Li

F z d f d z d z−

== =∑ f φ (2)

0

1

( )( ) ( )( , )( , )

( , ) ( ) ( )1 ( )

Nn

Tnn N

M Tm M

mm

p d zd zP z dH z d

Q z d d zq d z

−

=

−

=

= = =+

∑

∑p φq φ

(3)

where

0( ) ( ) ( )

FIR

T

Ld f d f d⎡ ⎤= ⎣ ⎦f (4)

[ ]0( ) ( ) ( ) TNd p d p d=p (5)

[ ]1( ) 1 ( ) ( ) TMd q d q d=q (6)

1( ) 1Tn

n z z z− −⎡ ⎤= ⎣ ⎦φ (7)

In (2)-(7), the superscript T denotes the transpose operation of a vector, and all the filter coefficients are assumed to be of real-valued. For an allpass VFD filter, the numerator has a specific form

1( , ) ( , ) ( ) ( )M T j

MP z d z Q z d d e ω− −= = q φ (8)

where [ ]1( ) ( ) ( ) 1 T

M Md q d q d−=q (9)

Each of the filter coefficients fi(d), pn(d) and qm(d) in (4)-(6) can be expressed, respectively, as a polynomial function of the fractional delay d, i.e.,

1

0( ) ( ) 0, ,

Kk

i i FIRk

f d a k d i L=

= =∑ (10)

2

0( ) ( ) 0, ,

Kk

n nk

p d b k d n N=

= =∑ (11)

3

0( ) ( ) 1, ,

Kk

m mk

q d c k d m M=

= =∑ (12)

In general, the polynomial orders K1, K2 and K3 can be any positive integer. For the IIR VFD filter design, the denominator coefficients are fixed, therefore the corresponding K3 = 0. For the allpass VFD filter design, there is a mirror symmetric relation between the numerator and the denominator. Consequently, the polynomial orders K2 and K3 are equal.

In the first step of the two-step design, the fractional delay d is assumed to be uniformly sampled within [-0.5, 0.5], i.e.,

0.5 0, , lld l LL

= − + = (13)

For simplicity, in the later discussion, the symbols fl, pl, ql, and

lq for l = 0, 1, …, L are to be used to represent the filter coefficients f(d), p(d), q(d) of (4)-(6) and ( )dq of (9) of the l-th corresponding FD filter with an FD value dl.

III. FIR FRACTIONAL DELAY DESIGN In this section, the design problem of FIR FD filters are

formulated using the PCWLS method [18] and solved using the PLS algorithm introduced in [19]. For designing FIR FD filters, a complex error function for each FIR FD filter is first defined as

( ) ( , ) ( , ) 0,1, ,j j

l d l le H e d F e d l Lω ωω = − = (14)

The error function in the weighted least-squares (WLS) sense can be expressed as

2

1, 10( ) ( )

2 constant 0,1, ,

cw

l l

T Tl l l l l

e W e d

l L

ω ω ω=

= − + =∫f V f f v

(15)

In (15), W1(ω) is a specified nonnegative weighting function, and

{ }10

( ) Re ( ) ( )c

FIR FIR

w j H jl L LW e e dω ωω ω= ∫V φ φ (16)

{ }( )10( ) Re ( )c

l

FIR

w j D d jl LW e e dω ωω ω+= ∫v φ (17)

The superscript H in (16) denotes the combined operations of complex conjugation and transposition. As seen from (15), the error function e1,l is a quadratic function of fl. Under the WLS criterion, the magnitude responses of the optimal design obtained by minimizing (15) have large ripples near the cutoff


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frequency wc. In order to control the peaks of such ripples, peak constraints are applied to the FIR FD filter design as

min 2subject to | ( )|

0,1, , , [0, ]

T Tl l l l l

l

c

el L w

ω δω

−≤

= ∀ ∈

f V f f v (18)

where δ is a prescribed peak error limit. These imposed constraints in (18) are essentially in a quadratic form. Since the PLS algorithm can only cope with linearly constrained quadratic programming (QP) problems, the constraints imposed in (18) need to be linearized. An approximation technique is applied here [18], [20]-[21], which utilizes a (2Γ)-vertex regular polygon to approximate a circle as

{ } { } cos Re ( ) sin Im ( )

cos2

0, , 1, 0,1, , , [0, ], 0, ,

l i l i

i c

e e

l L w i J

γπ γπ⎛ ⎞ ⎛ ⎞ω + ω⎜ ⎟ ⎜ ⎟Γ Γ⎝ ⎠ ⎝ ⎠π⎛ ⎞≤ δ ⎜ ⎟Γ⎝ ⎠

γ = Γ − = ω ∈ =

(19)

With a sufficient large Γ, a close approximation can be obtained. Using the linearized constraints defined in (19), the design problem formulated in (18) can now be solved by the PLS algorithm.

IV. ALLPASS FRACTIONAL DELAY DESIGN Since an allpass digital filter has fullband unity magnitude

responses, only the group delay (or phase) responses need to be addressed. Due to the existence of a denominator, an allpass FD filter design has to incorporate stability constraints. For designing an allpass FD filter, a set of estimation error functions in the least squares (LS) sense is defined as

'

2

2, 0

2( )

0

( , ) ( , ) ( , )

( , ) ( , )

' and 0,1, ,

c

cl

w jM j j jl l l d l

w j D dj jl l

e e Q e d Q e d H e d d

Q e d Q e d e d

D D M l L

ω ω ω ω

ωω ω

ω

ω

− −

− +−

= −

= −

= − =

∫

∫ (20)

In the filter-examples, the integer delay D of an allpass VFD filter is set equal to the filter order M. Note that it cannot be guaranteed that minimizing e2.l of (20) can lead to a globally optimal solution in the LS sense, but it provides a simple and efficient way to design an allpass FD filter satisfying given specifications. Substituting Q(z, d) of (3) into (20), the error function e2,l can be reformulated as a quadratic function of the denominator coefficients:

2, 0,1, ,T

l l l le l L= =q Ψ q (21)

where

{ }{ }'

0

( )

0

2 Re ( ) ( )

2 Re ( ) ( )

0,1, ,

c

cl

w Hl M M

w j D dTM M

d

e d

l L

ω

ω ω ω

ω ω ω− +

=

−

=

∫∫

Ψ φ φ

φ φ (22)

In order to guarantee the stability of the designed allpass FD

filter, a set of positive-realness based stability constraints [20] is applied as follow:

{ } { }Re ( , ) Re ( )

0,1, , , [0, ]

j T jl l MQ e d e

l L

ω ω ε

ω π

= ⋅ ≥

= ∈

q φ (23)

In (23), ε is a prescribed small positive number. An approximate way to incorporate these constraints into the design is to impose them on a set of equally-spaced grid points over [0, π]. Obviously, (23) are linear inequality constraints with respect to ql. Then, the design problem of the l-th (l = 0, 1, …, L) allpass FD filter can be formulated as a linearly constrained QP problem:

[ ]{ }

minsubject to 1 0 0 1

Re ( ) [0, ]

Tl l l

l

T jl M e ω ε ω π

⋅ =

⋅ ≥ ∈

q Ψ qq

q φ

(24)

The design problem formulated in (24) can also be solved using the PLS algorithm.

V. IIR FRACTIONAL DELAY DESIGN For IIR VFD filter design, L+1 FIR FD filters satisfying the

specifications are to be first designed using the PCWLS method described in Section III. Then L+1 IIR FD filters with the fixed denominator are to be designed to best approximate these L+1 FIR FD filters in the WLS sense. The approximation error can be expressed as

2

3, 21 ( ) ( , ) ( , )

20,1 ,

j jl l le W F e d H e d d

l L

π ω ω

πω ω

π −= −

=

∫ (25)

In (25), W2(ω) is a prescribed nonnegative weighting function. In this section, a model reduction technique is applied to obtain the L+1 IIR FD filters.


4

A. Fixed Denominator Design For designing IIR VFD filters with a fixed denominator, an

average FIR filter Favg(z) is first determined in the time-domain from the L+1 FIR FD filters by taking the average impulse responses of the L+1 FIR FD filters as

0

1( ) ( , )1

L

avg ll

F z F z dL =

=+ ∑ (26)

The above time-domain averaging method to obtain an average FIR filter is simpler but offers an equivalent solution obtained by the frequency-domain averaging method adopted in [16]-[17]. Then, a model reduction method [22]-[23] is utilized to obtain the fixed denominator Q(z). The advantages of this model reduction method include: (a) The method is applicable to any IIR digital filter (including high-order) with a small numerical error since calculation is performed directly on filter coefficients without involving a transformation. (b) All the roots of the designed denominator will lie inside the unit circle if the procedure converges, which guarantees the stability of the designed IIR FD filter. (c) The model reduction method first determines the denominator. Except the numerator order N, the denominator design does not need any other information of the numerator, which greatly facilitates the design procedure.

To apply the LS design method [22] to solve the WLS model reduction problem, a maximum-phase polynomial G(z) in z-1 with an order I (≥ N0 = M–N–1) is introduced. On the unit circle, the squared magnitude of G(z) is equal to the specified W2(ω), i.e., W2(ω) = |G(ejω)|2. Since the fixed denominator is determined by the FIR average filter Favg(z) defined in (26) instead of F(z,dl), a new WLS error function using Favg(z) should be defined and adopted in the fixed denominator design as

2

4

2

( )

1 ( )( ) ( )2 ( )

0,1 ,

j

jj j

avg j

e e

P eG e F e dQ e

l L

ω

ωπ ω ωωπ

ωπ −

= Δ

⎡ ⎤= −⎢ ⎥

⎣ ⎦=

∫ (27)

where ∆(z) = G(z)[Favg(z)–P(z)/Q(z)], and ||·|| denotes the L2-norm of a function whose analytical region contains the unit circle. Note that for clarity a polynomial P(z) in z-1 with the order N is introduced in (27). However, P(z) is actually not to be designed since only the fixed denominator Q(z) is required, and the denominator design does not need any other information of P(z) except the numerator order N. According to the Theorem 3 of [23], if I ≥ N0 = M–N–1, the optimal design can be achieved for ∆(z) = zN0-IA(z)R(z), where A(z) is the transfer function of an allpass filter shown in Fig. 1, and R(z) is an unknown FIR filter of order LFIR+N0. Hence, from (27), we have

0

0

22 ( )

2

2

0

1( ) = ( ) ( )21 ( )

2

0,1 ,

FIR

j N Ij j j

j

L N

nn

e e A e R e d

R e d

r

l L

π ωω ω ωπ

π ω

π

ωπ

ωπ

−

−

−

+

=

Δ

=

=

=

∫

∫

∑

(28)

where rn for n = 0 to LFIR+N0 represent the impulse responses of R(z). The derivation in (28) makes use of |A(ejω)|2 = 1 in the second equality and the third equality is derived using the Parseval’s theorem. The design problem is then equivalent to determining R(z) for any given Favg(z) and G(z).

In order to determine the optimal R(z) which results in the minimum e4, we shall refer to the allpass system shown in Fig. 1 with a specific input signal X(z) = z-IG*(z-1)z-LFIRFavg(z-1) where G*(z-1) is obtained by replacing the coefficients of G(z-1) with their complex conjugates. Then, according to the Theorem 4 of [23], the optimal design can be attained if rn = [uLFIR+N0–n]* (n = 0, 1, …, LFIR+N0), where un is the output signal of the allpass system in Fig. 1, and the superscript * denotes the complex conjugate. Therefore, the optimal denominator can also be obtained by minimizing the energy of un (n = 0, 1, …, LFIR+N0), i.e., ∑n|un|2. However, due to the existence of the denominator Q(z) in A(z), it is difficult to directly minimize ∑n|un|2. In order to overcome this obstacle, an iterative strategy is adopted, where at the s-th iteration the denominator Q(s)(z) in A(s)(z) is replaced by Q(s-1)(z) obtained at the previous iteration, such that the transfer function A(s)(z) is reduced to z-MQ(s)(z-1)/Q(s-1)(z). Consequently, U(s)(z) can be expressed as a linear function of the current denominator coefficients, and ∑n|un

(s)|2 can be easily evaluated. The iterative scheme is given below:

1) Set s = 0, and choose Q0(z) = 1. 2) Set s = s+1. Then, calculate X(s)(z), which is defined as

1 1

*( ) ( )( 1)

0

( ) ( )( )

( )

FIRLIavgs s n

nsn

z G z z F zX z x z

Q z

−− − − ∞−

−=

= =∑ (29)

3) In order to calculate the output of the approximate allpass

filter A(s)(z), i.e., u(s) = [u0(s), u1

(s), …, uLFIR+N0(s)]T, construct

the matrix A(s) and the vector b(s), which are defined by

Fig. 1. Allpass filter system.


5

0

( )0( ) ( )1 0

( )( ) ( )

1 0

( ) ( )

0 0

FIR FIR

s

s s

ss s

M

s sL N L N

xx x

x x

x x

−

+ −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A (30)

( ) ( ) ( )00 0

FIR

Ts s sL Nx x −⎡ ⎤= − ⎣ ⎦b (31)

According to the derivation given in [22], it can be shown that ( ) ( ) ( ) ( )s s s s

M= −u A q b , where ( )sMq = [qM

(s), qM-1(s), …,

q2(s), q1

(s)]T. Then, by minimizing 2( ) ( ) ( ) ( ) ( ) ( ) ( )

2

Ts s s s s s sM M⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦u A q b A q b (32)

we can obtain

( )sMq . In (32), ||·||2 denotes the l2-norm of a

vector. 4) Repeating Steps 2 and 3 for a sufficient number of

iterations, a minimum ||u(s)||2 can be reached, and the entries of

( )sMq are chosen as the final denominator

coefficients. It has been proved in [22] that the roots of the resulting

denominator always locate inside the unit circle if the iterative procedure converges.

B. Numerator Design After determining the fixed denominator Q(z), L+1

numerators of the IIR FD filters can then be designed. Note that given the fixed denominator Q(z), G(z)H(z,dl) (l = 0, 1, …, L) can be represented as a linear combination of the basis functions defined as

( )( ) 0,1, ,( )

z G zz NQ z

λ

λη λ−

= = (33)

Thus, the numerator design problem can be regarded as a linear approximation problem. According to the orthogonality principle [24], for a given Q(z), e3,l achieves its minimum value if and only if G(z)[F(z,dl)–H(z,dl)] is orthogonal to the basis functions, i.e.,

*( , )( ) ( , ) ( ) 0

( )0,1, , 0,1, ,

jj j jl

l j

P e dG e F e d e dQ e

N l L

ωπ ω ω ωλωπ

η ω

λ

−

⎡ ⎤− =⎢ ⎥

⎣ ⎦= =

∫ (34)

By solving the linear equation (34), we can obtain the optimal numerator for a given Q(z).

VI. VFD COEFFICIENT POLYNOMIAL APPROXIMATION Based on (10)-(12), the polynomial fitting error functions are

defined as

1

2

5,0 0

( ) ( ) 0, ,KL

ki i l i l FIR

l k

e a k d f d i L= =

⎡ ⎤= − =⎢ ⎥

⎣ ⎦∑ ∑ (35)

2

2

6,0 0

( ) ( ) 0, ,KL

kn n l n l

l k

e b k d p d n N= =

⎡ ⎤= − =⎢ ⎥

⎣ ⎦∑ ∑ (36)

3

2

7,0 0

( ) ( ) 1, ,KL

km m l m l

l k

e c k d q d m M= =

⎡ ⎤= − =⎢ ⎥

⎣ ⎦∑ ∑ (37)

In (35), e5,i is a quadratic function of ai(k), a set of linear

equations can be obtained by setting to zero the partial derivative of e5,i with respect to each coefficient of the coefficient vector ai = [ai(0), ai(1), …, ai(K1)]T as

1

0 0

1

2 ( ) ( ) =0

0, , , 0,1, ,

KLk jl i l i l

l j

FIR

d a j d f d

i L k K= =

⎡ ⎤−⎢ ⎥

⎣ ⎦= =

∑ ∑ (38)

From (38), the optimal polynomial fitting coefficients can be obtained by solving

0, ,i i FIRi L= =Ca y (39)

In (39), C is a Hankel matrix, and each entry C(i, j) (i, j = 0, 1, …, K1) of C is defined by

0

( , )L

i jl

lC i j d +

==∑ (40)

The vector yi in (39) is composed by

1

0( ) ( ) 0, , , 0, ,

Lk

i l i l FIRl

y k d f d i L k K=

= = =∑ (41)

It can be proved that the matrix C is positive definite. Note that C only depends on the FD dl (l = 0, …, L) and the polynomial order K1. This suggests that C can be computed in advance (An efficient way to compute C-1 is to perform the Cholesky decomposition, i.e., C = C1

TC1 where C1 is an upper triangular matrix). The computations of the polynomial coefficients bn(k) and cm(k) in (36) and (37) can be performed in a similar way as described above for ai(k).

VII. SIMULATIONS In this section, three sets of filter-examples for the design of

FIR, allpass, and IIR VFD filters are presented. To evaluate their performances, the maximum absolute error (MAE) and the L2 error (L2E) measured on magnitude (Mag) and group delay (GD) are adopted and defined by


6

{ }MA, abs,max ( , ) [0, ], | | 0.5 1, 2j j ce e d w d jω ω= ∈ ≤ = (42)

1

20.5 2L2, abs,0 0.5

( , ) 1, 2cw

j je e p dpd jω ω−

⎡ ⎤= =⎢ ⎥⎣ ⎦∫ ∫ (43)

where

abs,1( , ) ( , ) ( , )j jde d H e d H e dω ωω = − (44)

( )abs,2 ( , ) ( , )e d d D dω τ ω= − + (45)

In (45), τ(ω, d) denotes the group delay of a designed VFD filter. In all the filter-examples, the polynomial orders K1, K2, and K3 are set to K = 6. The total number of FD filters designed in the first step is always chosen as 11, i.e., L = 10. The peak error limit δ used in (19) is set to 10-1 in all the filter-examples. For the PCWLS FIR FD filter design, S in (19) is always chosen as 32.

A. FIR and Allpass VFD Filters The FIR and allpass VFD filter specifications are

summarized in Table I. Three different cutoff frequencies wc within the same range are employed in each set of filter-examples. In Table I, LFIR and MAP denote, respectively, the FIR and allpass VFD filter orders, and DFIR and DAP represent the integer group delays used in FIR and allpass VFD filter designs. Note that in each set of filter-examples, DFIR and DAP are always set to LFIR/2 and MAP, respectively, to achieve good quality VFD filter design. For a fair comparison, the FIR and allpass VFD filters are specified to have the same number of non-unity coefficients, that is, (K3+1)MAP = (K1+1)(LFIR+1), which implies LFIR = MAP−1 for K1 = K3 = K. The parameter ε used in the positive-realness based stability constraint (23) is always set to 10-3.

To evaluate the interpolation (or generalization) ability of each of the two design methods, rL+1 (for r = 2 and 16) FD filters are interpolated from each designed VFD filter with polynomial-function represented coefficients given in (10) and (12). The magnitude and group delay responses of FD filters with r = 2 are shown in Figs. 2-7. The corresponding MAE (eMA) and L2E (eL2) measurements of the designs are summarized in Tables III and IV. Since allpass VFD filters always have fullband unity magnitude responses, only the MAE and L2E measurements of group delays are required for the allpass VFD

filters (see Table IV). Results show that the performances are similar and consistent for different FIR and allpass FD filters for r = 2 and 16 interpolated from the corresponding designed VFD filters which confirm the interpolation ability of these VFD filters. In general, with the same filter order and integer delay, a lower cutoff frequency wc yields reduced MAE and reduced L2E values on the magnitude and group delay responses for both FIR and allpass VFD filters. The only exception exists for the allpass VFD filters in which wc = 0.775π yields lower L2E values in group delay responses than those of wc = 0.75π for r = 1, 2, and 16. Compared with the FIR VFD filters, the allpass VFD filters always yield reduced MAE and L2E values on the magnitude and group delay responses using the same wc and the same number of non-unity coefficients.

B. IIR VFD Filters The specifications of the IIR VFD filters are shown in Table

II. The same values for passband cutoff frequency wc and integer delay D, and the closest but less number of independent coefficients used as each pair of FIR and allpass VFD filters are adopted in designing the corresponding IIR VFD filter. This implies (K2+1)(N+1)+M ≤ (K3+1)MAP and M ≤ (K+1)(MAP−N−1) for K2 = K3 = K. For each of the three sets of LFIR2 (FIR FD prototype filter order) and D values, five pairs of filter orders (i.e., N and M) and three different wc are employed (see Table II) and the designed results are summarized in each respective set of three tables (see Tables V-VII, VIII-X, XI-XIII). In the filter-examples, D = ⎣N/2⎦ + 1, where ⎣x⎦ denotes the maximum integer not more than x. Each set of FIR FD prototype filters of specified LFIR2, D, and dl (for l = 0 to L) values and different wc values is to be approximated by corresponding IIR FD filters using model reduction technique. The weighting function W1(ω) in (15) is chosen as W1(ω) = 1 within [0, wc] and W1(ω) = 0 within (wc, π). The weighting function W2(ω) in (25), which is used to determine the G(z) in the allpass filter system shown in Fig. 1, is chosen as W2(ω) = 1 within [0, wc] and W2(ω) = 10-20 within (wc, π). After obtaining the polynomial coefficients, rL+1 (for r = 2, and 16) FD filters are obtained using (11) to examine the interpolation ability of the proposed method.

The magnitude and group delay responses of IIR FD filters with r = 2 are shown in Figs. 8-13. The obtained MAE and L2E values of magnitude and group delay responses are summarized in Tables V-XIII. It can be observed that the performances for FD filters with different r (= 1, 2, 16) are similar and consistent which confirms the interpolation ability of such IIR VFD filters. Design results confirm that all the designed IIR VFD filters are stable. In the first set of filter-examples (with LFIR2 = 80 and D = 21), as wc decreases from 0.95π to 0.9π, the designed IIR VFD filters can achieve reduced MAE and L2E values of magnitude and group delay responses. Also, the best combination of (N, M) for wc = 0.95π, 0.925π, 0.9π are, respectively, (41, 6), (41, 4), and (40, 10). In the second set of filter-examples (with LFIR2 = 60 and D = 16), as wc decreases from 0.875π to 0.825π, the designed IIR VFD filters with wc = 0.85π achieve the best performances with (N, M) = (30, 10). The best combination of

TABLE I FIR AND ALLPASS VFD FILTER SPECIFICATIONS

wc / π LFIR , DFIR MAP , DAP [0.95, 0.925, 0.9] 42, 21 43, 43 [0.875, 0.85, 0.825] 32, 16 33, 33 [0.8, 0.775, 0.75] 22, 11 23, 23

TABLE II

IIR VFD FILTER SPECIFICATIONS wc/ π LFIR2 D (N, M)

[0.95, 0.925, 0.9] 80 21 (40, 10), (40, 8) (41, 6) (41, 4) (41, 2) [0.875, 0.85, 0.825] 60 16 (30, 10) (30, 8) (31, 6) (31, 4) (31, 2) [0.8, 0.775, 0.75] 40 11 (20, 10) (20, 8) (21, 6) (21, 4) (21, 2)


7

(N, M) for wc = 0.875π is (31, 6) whereas the best combinations of (N, M) for wc = 0.825π are (30, 8) for reduced L2E in magnitude and group delay responses and (31, 4) for reduced MAE in magnitude and group delay responses. In the third set of filter-examples (with LFIR2 = 40 and D = 11), as wc decreases from 0.8π to 0.75π, the MAE and L2E values of magnitude and group delay responses increase, and the best combination of (N, M) among these three wc values is (21, 6). The increasing trends of the MAE and L2E values of magnitude and group delay responses of the first and third sets of filter-examples are reverse of each other while that of the second set of filter-examples exhibit minimum MAE and L2E values when the wc value is at the middle. From the results, it could be observed that: (a) For LFIR2 = 80 and D = 21, the best overall performances can be achieved for (N, M) = (40, 10) and wc = 0.9π. (b) For LFIR2 = 60 and D = 16, the best overall performances can be achieved for (N, M) = (30, 10) and wc = 0.85π. (c) For LFIR2 = 40 and D = 11, the best overall performances can be achieved for (N, M) = (21, 6) and wc = 0.8π.

C. Comparisons of VFD Filters For direct comparisons among all the VFD filters, identical

number of coefficients, and identical order of polynomial function for each variable coefficient have been used for FIR and allpass VFD filters. For each corresponding IIR VFD filter, the closest but less number of independent coefficients has been used. Comparing the MAE and L2E performances in magnitude and group delay responses of the FIR, allpass, and IIR VFD filters, some general observations are summarized as below: (a) The IIR VFD filters are better than the FIR VFD filters in

the MAE and L2E performances in magnitude and group delay responses for r = 1, 2, 16 except when wc = 0.825π (for LFIR = 32, DFIR = 16 but excluding GD eMA) and wc = 0.75π (for LFIR = 22, DFIR = 11). In other words, the MAE and L2E performances in magnitude and group delay responses of the FIR VFD filters improve as cutoff frequency wc is reduced even when filter order LFIR is also reduced.

(b) The IIR VFD filters are better than the allpass VFD filters in the MAE performances in group delay responses for r = 1, 2, 16 and for wc ≥ 0.9π (including 0.9π, 0.925π, 0.95π) and in the L2E performances in group delay responses for r = 1, 2, 16 and for wc ≥ 0.95π. In other words, the most preferred choice of VFD filters is IIR VFD filters for wc ≥ 0.9π and for the lowest achievable group delay performance.

(c) The allpass VFD filters are better than the FIR VFD filters in the MAE and L2E performances in magnitude and group delay responses for r = 1, 2, 16 and for all wc.

(d) The allpass VFD filters possess prefect magnitude responses at the cost of having the highest group delay value equal to its filter order, DAP = MAP.

(e) For all the FIR, allpass, and IIR VFD filters, a higher passband cutoff frequency wc requires a higher filter order or equivalently a larger number of filter coefficients.

(f) The interpolation ability of the variable coefficients

represented by polynomial functions exhibits similar and consistent MAE and L2E performances in magnitude and group delay responses for r = 1, 2, 16 in each type of FIR, allpass, and IIR VFD filters.

VIII. CONCLUSIONS In this paper, the methodologies and relative performances of

FIR, allpass, and IIR VFD filters designed using a two-step scheme and realizable via the Farrow’s structure have been described, analyzed, and presented. The work is the first study of its kind on the relative performance analyses among the FIR, allpass, and IIR VFD filters. Based on the relative performances of the FIR, allpass, and IIR VFD filters, the following conclusions can be drawn: (a) FIR VFD filters offer the simplest and optimal design option without a stability concern but their MAE and L2E performances in magnitude and group delay responses are the poorest except for designs requiring a sufficient low passband cutoff frequency and a sufficient low order filter. (b) Allpass VFD filters offer the prefect fullband magnitude responses but have the highest latency requirement in term of group delay value. Also, its MAE and L2E performances in magnitude responses deteriorate for sufficiently high passband cutoff frequency (equal or higher than 0.9π in the filter-examples). Moreover, the variability nature of each denominator coefficient in an allpass VFD filter can create undesirable transients [25]. (c) IIR VFD filters with a low-order fixed denominator offer the following advantages: (i) Better MAE and L2E performances in magnitude and group delay responses than those of a corresponding FIR VFD filter except for some FIR VFD filters with sufficiently low passband cutoff frequency and low filter order. The improved performances are due to a more precise control by the denominator over its passband but the required denominator filter order is low (and within M = 4 to 10 in the filter-examples). (ii) Better combinations of the MAE and L2E performances in group delay responses than those of the corresponding allpass VFD filters for high passband cutoff frequency (equal or higher than 0.9π in the filter-examples). (iii) IIR VFD filters can be designed to have the lowest group delay value compared to those of the FIR and allpass VFD filters. (iv) A fixed denominator IIR VFD filter can be easily designed to be stable by model reduction of a FIR obtained through simple time-domain averaging of the L+1 FIR FD filter coefficients; and does not introduce undesirable transients as in the case of a variable denominator IIR VFD filter. (v) IIR VFD filters offer a flexible combination of the numbers of variable numerator and fixed denominator coefficients tailored to any specified set of passband cutoff frequency wc and MAE and L2E values in magnitude and group delay responses.


8

Fig. 2. Magnitude responses of FIR VFD filters (LFIR = 42, wc = 0.9π, r = 2).



Fig. 5. Group delay responses of FIR (left) and allpass (right) VFD filters (LFIR = 42, MAP = 43, wc = 0.9π, r = 2).

Fig. 6. Group delay responses of FIR (left) and allpass VFD (right) filters (LFIR = 32, MAP = 33, wc = 0.85π, r = 2).

Fig. 7. Group delay responses of FIR (left) and allpass (right) VFD filters (LFIR = 22, MAP = 23, wc = 0.8π, r = 2).


9

Fig. 8. Magnitude responses of IIR VFD filters (LFIR2 = 80, N = 40, M = 10, wc = 0.9π, r = 2).



Fig. 11. Group delay responses of IIR VFD filters (LFIR2 = 80, N = 40, M = 10, wc = 0.9π, r = 2).




10

TABLE III

PERFORMANCES OF FIR VFD FILTERS (DFIR = LFIR/2) r LFIR wc/π Mag eMA Mag eL2 GD eMA GD eL2

1

42 0.95 -27.1940 -53.4110 1.1435 6.8899e-2

0.925 -38.5108 -67.0582 3.3302e-1 1.7632e-2 0.9 -55.0526 -83.0475 7.3549e-2 3.4611e-3

32 0.875 -53.7928 -79.8375 7.1593e-2 3.9538e-3 0.85 -62.8238 -91.0989 2.5600e-2 1.3051e-3

0.825 -77.3437 -103.6817 6.9193e-3 3.3055e-4

22 0.8 -61.9587 -86.5744 2.5566e-2 1.6313e-3

0.775 -68.3415 -94.9551 1.2212e-2 7.3563e-4 0.75 -77.6018 -103.7048 5.1414e-3 2.9646e-4

2

42 0.95 -27.1940 -53.4264 1.1553 6.9086e-2

0.925 -38.5108 -67.0699 3.3623e-1 1.7688e-2 0.9 -55.0526 -83.0509 7.3707e-2 3.4739e-3

32 0.875 -53.7928 -79.8614 7.1593e-2 3.9698e-3 0.85 -62.8238 -91.0709 2.5600e-2 1.3109e-3

0.825 -76.1054 -103.4142 6.9193e-3 3.3208e-4

22 0.8 -61.9587 -86.6122 2.5566e-2 1.6386e-3

0.775 -68.3415 -94.9656 1.2212e-2 7.3903e-4 0.75 -77.2873 -103.6691 5.1414e-3 2.9785e-4

16

42 0.95 -27.1940 -53.4314 1.1631 6.9146e-2

0.925 -38.5108 -67.0735 3.3840e-1 1.7707e-2 0.9 -54.9642 -83.0540 7.4377e-2 3.4780e-3

32 0.875 -53.7659 -79.8698 7.2026e-2 3.9750e-3 0.85 -62.6492 -91.0666 2.5670e-2 1.3127e-3

0.825 -75.9781 -103.3958 6.9239e-3 3.3257e-4

22 0.8 -61.9294 -86.6254 2.5566e-2 1.6410e-3

0.775 -68.2259 -94.9716 1.2221e-2 7.4013e-4 0.75 -77.0810 -103.6720 5.1556e-3 2.9830e-4

TABLE IV PERFORMANCES OF ALLPASS VFD FILTERS (DAP = MAP)

r MAP wc/π GD eMA GD eL2

1

43 0.95 8.7792e-1 1.8523e-2

0.925 4.9473e-2 7.8993e-4 0.9 4.5526e-3 5.1213e-5

33 0.875 2.9432e-3 4.2400e-5 0.85 4.6279e-4 5.2770e-6

0.825 1.8638e-5 1.5519e-6

23 0.8 6.0118e-4 8.3038e-6

0.775 5.4363e-5 8.6373e-7 0.75 2.4749e-5 3.1647e-6

2

43 0.95 8.7792e-1 1.7910e-2

0.925 4.9473e-2 7.6203e-4 0.9 4.5526e-3 4.9737e-5

33 0.875 2.9432e-3 4.0967e-5 0.85 4.6279e-4 5.1472e-6

0.825 3.7477e-5 1.8952e-6

23 0.8 6.0118e-4 8.0465e-6

0.775 5.4363e-5 8.3710e-7 0.75 4.2498e-5 3.8353e-6

16

43 0.95 8.7792e-1 1.7700e-2

0.925 4.9473e-2 7.5248e-4 0.9 4.5526e-3 4.9224e-5

33 0.875 2.9432e-3 4.0478e-5 0.85 4.6279e-4 5.1007e-6

0.825 3.7477e-5 1.9081e-6

23 0.8 6.0118e-4 7.9593e-6

0.775 5.4363e-5 8.2739e-7 0.75 4.3705e-5 3.8909e-6

TABLE V

PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 80, D = 21, wc = 0.95π) r (N, M) Mag eMA Mag eL2 GD eMA GD eL2

1

(40, 8) -44.1375 -71.4829 5.5449e-1 1.9125e-2 (41, 6) -47.1362 -75.6506 4.3022e-1 1.3522e-2 (41, 4) -46.2599 -75.4279 4.8952e-1 1.4743e-2 (41, 2) -34.5005 -62.9553 9.5841e-1 3.3295e-2

2

(40, 8) -44.0769 -71.4867 5.5449e-1 1.9153e-2 (41, 6) -47.1362 -75.6483 4.3022e-1 1.3544e-2 (41, 4) -46.2599 -75.4231 4.8952e-1 1.4766e-2 (41, 2) -34.5005 -62.9640 9.5841e-1 3.3345e-2

16

(40, 8) -44.0733 -71.4882 5.5619e-1 1.9162e-2 (41, 6) -47.1362 -75.6485 4.3190e-1 1.3551e-2 (41, 4) -46.2599 -75.4226 4.8990e-1 1.4773e-2 (41, 2) -34.4986 -62.9669 9.5995e-1 3.3361e-2

TABLE VI PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 80, D = 21, wc = 0.925π)

r (N, M) Mag eMA Mag eL2 GD eMA GD eL2

1

(40, 10) -64.7957 -92.4824 1.3368e-1 3.0306e-3 (40, 8) -64.3880 -94.9257 1.3947e-1 3.2421e-3 (41, 6) -66.5209 -96.9777 1.1461e-1 2.6056e-3 (41, 4) -68.6203 -95.7852 4.2777e-2 1.1943e-3 (41, 2) -47.4973 -74.5520 2.4868e-1 9.3600e-3

2

(40, 10) -64.7957 -92.3241 1.3368e-1 3.0259e-3 (40, 8) -64.3880 -94.6661 1.3947e-1 3.2376e-3 (41, 6) -66.5087 -96.5685 1.1461e-1 2.6019e-3 (41, 4) -68.5795 -95.4905 4.3990e-2 1.1983e-3 (41, 2) -47.4973 -74.5629 2.4867e-1 9.3852e-3

16

(40, 10) -64.7563 -92.3063 1.3368e-1 3.0243e-3 (40, 8) -64.3734 -94.6400 1.3947e-1 3.2361e-3 (41, 6) -66.4574 -96.5279 1.1461e-1 2.6007e-3 (41, 4) -68.5795 -95.4650 4.3990e-2 1.1996e-3 (41, 2) -47.4973 -74.5669 2.4868e-1 9.3934e-3

TABLE VII PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 80, D = 21, wc = 0.9π)


1

(40, 10) -87.7696 -102.2183 3.9384e-3 8.8310e-5 (40, 8) -82.9076 -106.4218 1.2091e-2 3.1072e-4 (41, 6) -85.3328 -108.9769 6.0270e-3 1.3936e-4 (41, 4) -82.0335 -105.3612 1.4467e-2 3.7204e-4 (41, 2) -64.7001 -86.0334 6.8403e-2 2.3146e-3

2

(40, 10) -83.6132 -101.4313 3.9384e-3 8.8137e-5 (40, 8) -79.9168 -104.6632 1.2306e-2 3.1021e-4 (41, 6) -81.1364 -106.1820 6.0953e-3 1.3968e-4 (41, 4) -79.3343 -103.9386 1.4768e-2 3.7146e-4 (41, 2) -64.7001 -86.0086 6.8979e-2 2.3213e-3

16

(40, 10) -83.5590 -101.3679 3.9384e-3 8.8068e-5 (40, 8) -79.9111 -104.5559 1.2313e-2 3.1004e-4 (41, 6) -81.0709 -106.0270 6.1101e-3 1.3977e-4 (41, 4) -79.3343 -103.8516 1.4770e-2 3.7127e-4 (41, 2) -64.7001 -86.0044 6.9272e-2 2.3234e-3


11

TABLE VIII PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 60, D = 16, wc = 0.875π)


1

(30, 10) -70.3375 -102.4819 2.4706e-2 6.7839e-4 (30, 8) -69.8369 -101.9814 2.5421e-2 7.0794e-4 (31, 6) -72.0740 -104.2735 1.9215e-2 5.1934e-4 (31, 4) -70.2840 -98.7472 3.8849e-2 1.1821e-3 (31, 2) -50.7810 -80.2103 9.0793e-2 4.1001e-3

2

(30, 10) -69.8392 -101.7473 2.5087e-2 6.7764e-4 (30, 8) -69.3940 -101.3092 2.5699e-2 7.0714e-4 (31, 6) -71.3617 -103.2584 1.9215e-2 5.1880e-4 (31, 4) -70.2840 -98.4639 3.8849e-2 1.1795e-3 (31, 2) -50.7810 -80.2484 9.0793e-2 4.1163e-3

16

(30, 10) -69.8312 -101.6387 2.5103e-2 6.7740e-4 (30, 8) -69.3816 -101.2071 2.5743e-2 7.0687e-4 (31, 6) -71.3617 -103.1244 1.9326e-2 5.1862e-4 (31, 4) -70.2331 -98.4300 3.8849e-2 1.1786e-3 (31, 2) -50.7810 -80.2627 9.0793e-2 4.1215e-3

TABLE IX PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 60, D = 16, wc = 0.85π)


1

(30, 10) -83.3572 -112.0867 2.2948e-3 7.4372e-5 (30, 8) -79.8963 -110.6772 5.0451e-3 1.4754e-4 (31, 6) -77.7958 -109.0799 7.4821e-3 2.0766e-4 (31, 4) -78.9496 -106.2390 1.5502e-2 4.4310e-4 (31, 2) -60.2234 -89.4779 3.9248e-2 1.6355e-3

2

(30, 10) -80.5832 -109.4628 2.3308e-3 7.4311e-5 (30, 8) -77.8525 -108.5534 5.0642e-3 1.4725e-4 (31, 6) -76.3620 -107.4720 7.4821e-3 2.0770e-4 (31, 4) -78.9496 -105.4120 1.5502e-2 4.4202e-4 (31, 2) -60.2234 -89.5133 3.9248e-2 1.6423e-3

16

(30, 10) -80.5030 -109.2570 2.3413e-3 7.4277e-5 (30, 8) -77.8036 -108.3567 5.0929e-3 1.4716e-4 (31, 6) -76.3466 -107.3062 7.5290e-3 2.0771e-4 (31, 4) -78.7978 -105.3392 1.5502e-2 4.4167e-4 (31, 2) -60.2234 -89.5308 3.9318e-2 1.6445e-3

TABLE X PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 60, D = 16, wc = 0.825π)


1

(30, 10) -71.6441 -89.4985 1.5476e-2 4.9332e-4 (30, 8) -71.9538 -89.5758 1.0309e-2 4.1834e-4 (31, 6) -76.8405 -88.8834 1.4005e-2 5.6386e-4 (31, 4) -73.7257 -88.3907 5.4881e-3 4.5771e-4 (31, 2) -69.2245 -87.8679 1.5442e-2 6.7230e-4

2

(30, 10) -71.6441 -89.7054 1.5476e-2 4.7276e-4 (30, 8) -71.9538 -89.7716 1.0309e-2 4.0588e-4 (31, 6) -76.6006 -89.0363 1.4051e-2 5.5704e-4 (31, 4) -73.7257 -88.5933 6.0267e-3 4.4909e-4 (31, 2) -69.2245 -88.0785 1.5442e-2 6.5341e-4

16

(30, 10) -71.6441 -89.8039 1.5476e-2 4.6297e-4 (30, 8) -71.9481 -89.8675 1.0309e-2 3.9894e-4 (31, 6) -76.5370 -89.1155 1.4217e-2 5.5361e-4 (31, 4) -73.7258 -88.6879 6.0730e-3 4.4534e-4 (31, 2) -69.2245 -88.1761 1.5442e-2 6.4516e-4

TABLE XI PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 40, D = 11, wc = 0.8π)


1

(20, 10) -79.0154 -95.7289 4.2083e-2 7.1585e-4 (20, 8) -85.8341 -112.8241 4.0489e-3 1.0208e-4 (21, 6) -88.8399 -113.9009 3.6108e-3 9.4648e-5 (21, 4) -76.6034 -100.0873 1.5551e-2 5.0474e-4 (21, 2) -58.0804 -82.9811 3.9266e-2 2.0607e-3

2

(20, 10) -78.5749 -96.3340 4.2083e-2 6.6264e-4 (20, 8) -83.6992 -111.6100 4.0409e-3 1.0080e-4 (21, 6) -86.5317 -112.3385 3.6108e-3 9.4160e-5 (21, 4) -76.5961 -99.9930 1.5551e-2 5.0330e-4 (21, 2) -58.0804 -82.9946 3.9489e-2 2.0684e-3

16

(20, 10) -78.5749 -96.5698 4.2083e-2 6.4387e-4 (20, 8) -83.6692 -111.5389 4.0409e-3 1.0037e-4 (21, 6) -86.5317 -112.2349 3.6108e-3 9.3996e-5 (21, 4) -76.4750 -99.9813 1.5551e-2 5.0283e-4 (21, 2) -58.0804 -82.9993 3.9667e-2 2.0708e-3

TABLE XII PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 40, D = 11, wc = 0.775π)


1

(20, 10) -71.8415 -89.4656 4.3467e-2 8.6542e-4 (20, 8) -86.5819 -102.2047 9.3969e-3 1.9467e-4 (21, 6) -88.6186 -109.8133 3.9284e-3 1.1965e-4 (21, 4) -83.6327 -103.6919 8.7873e-3 2.7590e-4 (21, 2) -65.9425 -88.1153 2.1316e-2 1.0612e-3

2

(20, 10) -71.8415 -89.7025 4.3467e-2 8.1271e-4 (20, 8) -85.0915 -102.2160 9.3969e-3 1.8911e-4 (21, 6) -86.5044 -109.4242 3.9284e-3 1.1888e-4 (21, 4) -83.5457 -103.5625 8.7873e-3 2.7528e-4 (21, 2) -65.9425 -88.1159 2.1790e-2 1.0646e-3

16

(20, 10) -71.8207 -89.7853 4.3467e-2 7.9427e-4 (20, 8) -85.0915 -102.2385 9.3969e-3 1.8723e-4 (21, 6) -86.5044 -109.4012 3.9284e-3 1.1862e-4 (21, 4) -83.4238 -103.5450 8.7873e-3 2.7508e-4 (21, 2) -65.9425 -88.1164 2.1794e-2 1.0657e-3

TABLE XIII PERFORMANCES OF IIR VFD FILTERS (LFIR2 = 40, D = 11, wc = 0.75π)


1

(20, 10) -26.9761 -54.9252 1.5784 7.2898e-2 (20, 8) -58.0737 -88.2786 1.5590e-2 8.4098e-4 (21, 6) -75.2627 -92.6654 9.9047e-3 4.3371e-4 (21, 4) -73.1467 -92.4597 1.1834e-2 4.3275e-4 (21, 2) -63.1711 -89.6293 1.7379e-2 8.6948e-4

2

(20, 10) -26.9761 -55.0139 1.5808 7.2960e-2 (20, 8) -58.0737 -88.4182 1.5590e-2 8.4037e-4 (21, 6) -75.2627 -92.7954 9.9047e-3 4.3470e-4 (21, 4) -72.5967 -92.5654 1.1834e-2 4.3131e-4 (21, 2) -63.1711 -89.7189 1.7567e-2 8.6735e-4

16

(20, 10) -26.9761 -55.0435 1.5847 7.2980e-2 (20, 8) -58.0737 -88.4697 1.5748e-2 8.4010e-4 (21, 6) -75.2627 -92.8513 9.9481e-3 4.3491e-4 (21, 4) -72.5745 -92.6118 1.1862e-2 4.3061e-4 (21, 2) -63.1711 -89.7554 1.7567e-2 8.6658e-4


12

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