An Optimum Linear Phase Approximation With Small Delay Obtained by The Manipulation of

All-Pass Padé Approximants

Douglas David Baptista de Souza and Sidnei Noceti Filho LINSE – Circuits and Signal Processing Laboratory

Department of Electrical Engineering Federal University of Santa Catarina Florianópolis, SC, 88040-900, Brazil

{douglas, sidnei}@linse.ufsc.br

Abstract—This paper proposes a new rational approximation of a symmetric impulse response. The proposed technique uses both the Padé method to obtain an all-pass approximation of e sT− and specific zeros in order to achieve good phase linearity characteristics. The obtained functions, when compared with other classical ones, such as Bessel, for instance, present better linear phase characteristics with a smaller delay. Another advantage of this approach is that the transfer functions are provided in explicit forms. Finally, the effect of varying parameters and results from simulations are shown.

I. INTRODUCTION Approximation functions with different attenuation phase

and time characteristics are sought by researchers due to the varying requirements in signal processing. When a linear phase and a consequent constant group delay is desired over the largest possible frequency band, the well-known Bessel (BS) [1], Ulbrich-Piloty (UP) [2] and Filanovsky (FI) [3] functions are referred for presenting a relative small overshoots and delays in the step response.

It is shown in [4] that for low-pass functions a symmetric impulse response implies in a linear phase characteristic. Then, this symmetry would be a reasonable start point to achieve a linear phase characteristic.

In this paper a new approximation function developed by the manipulation of all-pass Padé approximants is proposed. The obtained function has minimum phase and it is compared with other functions which may be found in the literature. This function presents a smaller delay over a larger frequency band. The function has also a more symmetric impulse response and a smaller overshoot in the step response as well.

II. OBTAINING THE TRANSFER FUNCTION

Let us consider a hypothetical odd periodic function ( )g t defined for 0t ≥ , with period 2T as shown in Fig. 1(a). We obtain ( )g t T− in Fig. 1(b) by moving forward ( )g t by its half-period. Now, considering ( ) ( ) ( )h t g t g t T= + − , we note that the negative and positive lobes cancel each other. As result we have ( )h t as shown in Fig. 1(c), which has just the first lobe. The function ( )h t is adopted as an approximation of the symmetric impulse response [5].

1

0

-1

gt()

0 T 2T 3T 4T

gt-T(

)

0 T 2T 3T 4T

1

0

-1

ht()

1

00 T 2T 3T 4T

(a)

(b)

(c) Fig. 1. (a) Odd periodic function ( )g t . (b) Function ( )g t T− . (c) Function

( ) ( ) ( )h t g t g t T= + − .

Note that ( )h t can be rewritten as

[ ]( ) ( ) ( ) ( )h t g t t t T= ∗ δ + δ − . (1)

For convenience, a normalized delay of 1sT = is adopted. Taking the Laplace transform of both sides of (1) and considering 1sT = , we obtain

( )( ) ( ) 1 e sT s G s −= + . (2)

Now, we utilize the Padé method to approximate the exponential e s− in (2) as an all-pass function. The Padé approximation is a generic case of the Taylor series and it may still work where the Taylor series does not converge [6].

Depending on the order, the Padé function may be unstable. We guarantee stable approximations for e s− , by using Padé all-pass functions, which present poles in the left-half plane for any order. Considering an even order approximation to the complex exponential, the function ( )T s in (2) can be rewritten as

978-1-4244-9474-3/11/$26.00 ©2011 IEEE 2265

1

1 1 01

1 1 0( ) ( ) 1

n nn n

n nn n

a s a s a s aT s G sa s a s a s a

−−

−−

⎡ ⎤− + − += +⎢ ⎥+ + + +⎢ ⎥⎣ ⎦

. (3)

In the sequel, we proceed with the following steps:

( )

11 1 0

11 1 0

11 1 0

11 1 0

( ) ( )n n

n nn n

n n

n nn n

n nn n

a s a s a s aT s G sa s a s a s a

a s a s a s a

a s a s a s a

−−

−−

−−

−−

⎡ + + + += +⎢+ + + +⎢⎣

⎤− + − +⎥⎥+ + + +⎦

(4)

( ) ( )

( ) ( )

11 11

1 1 0

1 1 01

1 1 0

2( ) ( )

2

n nn n n

n nn n

n nn n

a s a a sT s G s

a s a s a s a

a a s aa s a s a s a

−− −−

−

−−

⎡ + − += +⎢

+ + + +⎢⎣⎤+ − +⎥

+ + + + ⎥⎦

(5)

( ) ( ) ( ) ( )2

01

1 1 0

2 ( ) 0 0( )

n nn

n nn n

G s a s s s aT s

a s a s a s a

−

−−

⎡ ⎤+ + + +⎣ ⎦=+ + + +

(6)

( ) ( )

( ) ( ) ( )

22 01

1 1 0

2 ( )( )

n nn n n

n nn n n n

G s s a a s a aT s

s a a s a a s a a

−−−

−

⎡ ⎤+ + +⎣ ⎦=+ + + +

. (7)

Note that the numerator of (7) has only even powers of .s

If we had chosen an odd degree approximation to e s− , we would still obtain a numerator with only even powers of s . According to this characteristic, one can rewrite the numerator of ( )T s as a product of 2

js k+ terms

( )( ) ( ) ( )nn

nnn

n

n

jj

aasaasaas

kssGsT

011

1

2

1

2)(2)(

++++

+= −

−

=∏

. (8)

Now, we have to choose ( )G s in such way that ( )T s would be stable and realizable. We defined ( )G s as

21

,( ) 2m

j

j j

kG s m n

s k=

⎛ ⎞⎜ ⎟= ≤⎜ ⎟+⎝ ⎠

∏ . (9)

It is easy to verify that the 2js k+ terms of the numerator

of ( )T s in (8) are cancelled by the terms in the denominator of ( )G s presented in (9). The function ( )T s obtained after the cancellation is realizable and stable. This procedure can be verified in (10), (11) and (12).

( )

( )( ) ( ) ( )

2

1

/22

11

1 1 0

( )

2

jm

jj

n

jj

n nn n n n

kT s

s k

s k

s a a s a a s a a

=

=−

−

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦

⎡ ⎤+⎢ ⎥

⎢ ⎥× ⎢ ⎥+ + + +⎢ ⎥⎢ ⎥⎣ ⎦

∏

∏

(10)

12 2

1

2 2 2 21 1 /2

11 1 0

( )( ) ( )

2( ) ( )( ) ( )( / ) ( / ) ( / )

m

m

m m nn n

n n n n

k kT s

s k s k

s k s k s k s ks a a s a a s a a

+−

−

⎡ ⎤= ⎢ ⎥

+ +⎢ ⎥⎣ ⎦⎡ ⎤+ + + +× ⎢ ⎥

+ + + +⎢ ⎥⎣ ⎦

(11)

2 2

1 1 /21

1 1 0

2( )( ) ( )( )

( ) ( ) ( )m m n

n nn n n n

k k s k s kT s

s a a s a a s a a+

−−

+ +=

+ + + +. (12)

Considering 1m = and 2m = in (9), we obtain the respectively Laplace transforms pairs:

( ) ( )tkks

k1

12

1 sin⇔+

, (13)

( )( )( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−⇔

++ 21

12

21

21

22

12

1 sinsinkk

tkkkk

tkk

ksks

k

(14)

At first ( )g t was considered as an odd periodic function. It is shown in (13) and (14) that the inverse Laplace transforms of ( )G s for 1=m and 2m = meets this condition. Considering the values of m presented in this paper, one can show that the obtained function in time domain is a sum of sinus. Therefore, ( )g t is an odd function and the initial condition is attended.

So, after obtaining the expression for ( )T s shown in (10),

it is only necessary to choose the 2js k+ terms from the

numerator that will be cancelled by the same terms of the denominator of ( )G s . Thus, one just needs to choose which quadratic terms will be presented in the numerator or, in a simpler way, the pure imaginary zeros of the transfer function.

We observed that the best results are obtained by cancelling the imaginary zeros with the lower absolute value. Table I shows the transfer functions obtained by cancelling determined zeros, so the results have good linear phase characteristics. These are the Baptista-Noceti functions (BN). All the functions were normalized in order to present the same attenuation of 3dB at the frequency of 1 rad/s.

2266

TABLE I TRANSFER FUNCTIONS FOR 1n = UNTIL 10n =

1n =

1( )1

T ss

=+

2n = 2

20.5 2.3971( )

0.5 1.8962 2.3971sT s

s s+=

+ +

3n = 2

3 27.64106 30.9812( )

7.64106 24.3274 30.9812T s

s s ss +=

+ ++

4n = 2

4 3 20.7166 8.8466( )

5.38762 13.0619 16.4203 8.84664sT s

s s s s+=

+ + + +

5n = 2

5 4 3 25.69152 41.7109( )

8.03628 30.1381 64.5863 77.8549 41.7109sT s

s s s s s+=

+ + + + +

6n = 2

6 5 4 3 21.30589 36.7485( )

8.196376 31.99075 74.91664 109.650994.1537 36.74853

sT ss s s s s

s

+=+ + + +

+ +

7n = 2

7 6 5 4 3

2

13.09603 170.6708( )10.79123 56.14588 180.3224 382.2306

530.3237 442.8394 170.6708

sT ss s s s s

s s

+=+ + + +

+ + +

8n = 4 2

8 7 6 5 4

3 2

1.1893 106.0467 954.2718( )13.817 92.799 391.776 1127.719

2250.641 3023.2632 2486.3992 954.2727

s sT ss s s s s

s s s

+ +=+ + + +

+ + + +

9n = 4 2

9 8 7 6 5

4 3 2

20.536 857.13 6228.058( )17.271 145.836 783.630 2932.4602

7878.548 15119.322 19895.80816226.938 6228.058

s sT ss s s s s

s s ss

+ +=+ + + +

+ + ++ +

10n = 6 4 2

10 9 8 7 6

5 4 3

2

1.189 225.786 7204.786 45418.788( )21.109 218.756 1455.32 6842.490

23636 60478.391 112745.314

146046 118336.345 45418.788

s s sT ss s s s s

s s s

s s

+ + +=+ + + +

+ + +

+ + +

III. COMPARISONS BETWEEN THE FUNCTIONS BN, BS, AND UP. RESULTS FROM SIMULATION

The group delay, impulse response and magnitude of the BN, BS, FI and UP functions of order 10=n are shown in Fig. 2(a), (b) and (c), respectively. The functions were normalized in such way that they present the same attenuation of 3dB at the frequency of 1 rad/s. In addition, the UP and FI functions have different varying parameters, which are respectively the ripple amplitude at the group delay response

( ),δ and the delay-to-rise-time ratio ( ).ρ In order to have a flatter group delay we chose 0.01δ = s and 1.96ρ = .

In Fig. 2, one can observe that for the same order and attenuation at 1 rad/s, the BN function presents a group delay as flat as the BS function. Also, the smallest delay, which is the initial value of the group delay [ (0)],gτ is achieved by the BN approximation. Furthermore, notice that the band in which the group delay is flat is wider for the BN functions than it is for the other ones.

BNUPFIBS

0

1

2

3

4

5

6

7

010 210Frequency (rad/s)

Gro

up d

elay

(s)

(a)

Am

plitu

de

BNUPFIBS

0

0.2

0.4

0.6

1.0

0.8

0 5 10

Time (s) (b)

BNUPFIBS

Frequency (rad/s)

010 210110

-250

-200

-150

-100

-50

0

Mag

nitu

de (d

B)

(c)

Fig. 2. Functions with 10n = presenting a 3dB attenuation at 1 rad/s. (a) Group delay. (b) Impulse Response. (c) Magnitude.

2267

Increasing the order has the effect of enlarging the band in which the group delay is flat. This characteristic is shown in Fig. 3 for the orders 8,n = 9,n = and 10n = . Notice that increasing the order has the effect of enlarge the band, as the delay remains the same. The functions were normalized to present an attenuation of 3dB at 1 rad/s.

n=10n=9n=8

110Frequency (rad/s)

0.5

1.0

1.5

2.0

2.5

Gro

up d

elay

(s)

Fig. 3. Group Delay of the BN functions with 8n = , 9n = and 10n = presenting a 3dB attenuation at 1 rad/s.

In Table II, different characteristics of the BN, BS, UP and FI functions of orders 9n = and 10n = are compared. The functions were normalized in order to present an attenuation of 3dB at 1 rad/s. The varying parameter of the functions UP and FI were kept as 0.01δ = s and 1.96ρ = .

As can be seen, the BN approximation presents a smaller delay [ (0)]gτ . The biggest values of (0.1)B are from the UP and BN functions. This measure refers to the frequency band in which the group delay does not exceed 10% of its initial value. The smallest raise time ( RT ) is achieved by the FI approximation, followed by the BN one. It also can be seen in Fig. 4, which shows the step response of the functions of order 10n = from Table II.

TABLE II COMPARISONS BETWEEN BN, BS, UP AND FI FUNCTIONS

9n = Functions (0)gτ RT (0.1)B

BN 2.602 2.167 2.683 BS 3.391 2.172 2.062 UP 3.676 2.169 2.714 FI 3.737 2.122 1.141

10n =

BN 2.603 2.1661 3.002 BS 3.590 2.1663 2.198 UP 3.899 2.173 2.914 FI 3.997 2.138 2.757

Am

plitu

de

0

0.2

0.4

0.6

1.0

0.8

BNUPFIBS

Time (s)1 3 52 4 6

Fig. 4. Step Response of the BN, BS, UP and FI functions with 10n = and normalized to present a 3dB attenuation at 1 rad/s.

IV. CONCLUSIONS

This paper presented a new function with good phase linearity characteristics and smaller delay for a wider band, when compared with the well-known Bessel, Ulbrich-Piloty, and Filanovsky functions. The approach consists in manipulating all-pass Padé functions in order to obtain an optimum approximation of an ideal impulse response. First, we considered the periodic and the oddness features of a generic function aiming to develop a set of realizable transfer function by applying the Padé method. Then, we compared the obtained functions with other classical ones. It is shown that BN functions do not only provide a smaller delay, but also present a good linear phase characteristics, such a relatively flat group delay response over a wider band and high symmetric impulse response. Moreover, another advantage lies in the fact that BN functions are presented in explicit forms, which makes it easier to work with.

ACKNOWLEDGMENT

This work was supported in part by the Committee for Postgraduate Courses in Higher Education (CAPES).

REFERENCES [1] L. D. Paarmann, Design and Analysis of Analog Filters: A Signal

Processing Perspective. Boston: Kluwer Academic Publishers, 2001. [2] E. Ulbrich and H. Piloty, “Uber den entwurf von allpässen, tiefpässen,

und bandpässen mit einer in Tschebyscheffschen–Sinne approximierten konstanten gruppenlaufzeit,” Arch. Elekt. Ubertragung, vol. 14, pp. 451-467, Oct. 1960.

[3] I. M. Filanovsky, “One class of transfer functions with monotonic step response”, in Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), Bangkok, Thailand, May 2003, vol. 1, pp. 1389-1392.

[4] H. J. Blinchikoff and A. I. Zverev, Filtering in the Time Frequency Domains. Gresham: Noble Publishing Associates, 2001.

[5] B. Liu, “A time domain approximation method and its application to lumped delay lines,” IRE Trans. Circuit Theory, vol. 9, no. 3, pp. 256-261, Sep. 1962.

[6] K. L. Su, Time-Domain Synthesis of Linear Networks. Englewood Cliffs, NJ: Prentice-Hall, 1971.

2268