Systematic Performance Comparison of
Bessel Thomson vs. Padé Approximated
Delay Filters
Sayyed Mahdi Kashmiri, Sandro A. P. Haddad, Wouter A. Serdijn
Electronics Research Laboratory, Faculty of Electrical Engineering, Mathematics and Computer
Science
Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
Phone/Fax: +31-15-278-5922
Emails:[s.m.kashmiri, s.haddad, w.a.serdijn]@ewi.tudelft.nl
Abstract— The frequency domain exponential transfer
function of a delay function cannot be realized with a finite
number of lumped elements. Therefore an approximation of a
rational quotient of polynomials is used. While the all pole
rational realization of the Bessel Thomson approximation is
achieved by the use of Bessel polynomials, a Taylor expansion of
the non-rational transfer function of a delay around one point
results in another type of rational transfer, known as Padé
approximation. Through a series of extensive system level
simulations various performance characteristics of different
approximation methods were investigated. While a Bessel
Thomson approximation results in an overshoot free step
response it has slower response and smaller bandwidth in
comparison to a Padé approximated delay. A new approximation
method through narrowing of a Gaussian time domain impulse
response, in order to approximate the delta time domain
response of an ideal delay is introduced. The Padé approximation
of this system illustrates the least overshooting in time domain
response.
Index Terms—Delay Filters, Bessel Thomson Approximation,
Padé Approximation, Magnitude/Phase Response, Stability, Step
Response, Rise Time.
A
I. INTRODUCTION
delay is a fundamental block in signal processing
systems.
Pure delays should ideally not change the
magnitude or shape of the input signal. Considering a fixed
delay of τ, its frequency domain transfer function is defined
by the exponential: H(s)=exp(-τs), which can not be realized
with a finite number of lumped elements [1]. The best solution
is an approximation with a rational quotient of polynomials as:
G(s)=N(s)/D(s). One common approach to approximate a
delay transfer function is through the use of Bessel
polynomials leading to an all-pole filter approximation called
The authors are with the Electronics Research Laboratory, Faculty of
Electrical Engineering, Mathematics and Computer Science, Delft University
of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. (Phone/Fax:
+31-15-278-5922,Emails:[s.m.kashmiri, s.haddad, w.a.serdijn]@ewi.tudelft.nl
).
Bessel-Thomson realization. Another approximation yielding
a rational expression suitable for implementation through the
use of a Taylor expansion of the frequency domain nonrational transfer function around one point is called the Padé
approximation. By narrowing of a Gaussian impulse response
function with the use of smaller pulse width σ, the delta time
domain impulse response of a delay is approximated as a third
method in this work. The new delay approximation method is
completed through the application of a Padé approximation on
the Laplace transform of the Gaussian function. The
performance of these different approximations of a delay filter
has been systematically compared, in regard to achievable cutoff frequency and time domain characteristics.
Different combinations of numerator and denominator
orders as degrees of freedom were used to investigate their
effect on insertion loss, phase linearity, stability, rise time and
overshoot. It was revealed that our Gaussian approximation of
the time domain transfer function decreased overshooting in
the step response, while keeping the other characteristics
fixed. This paper has six sections. Section I is an introduction
to the concept of this work, while section II is a description of
the mostly used Bessel Thomson delay filter approximation
method. Padé approximation is introduced in section III, and a
system level overview of the delay filter performance is
investigated in this section. A systematic simulation of both
methods using MATLAB® is illustrated in section IV. The
proposed method of narrow Gaussian time domain impulse
response is discussed, simulated, and compared to the other
two methods in section V. A conclusion of the derived results
can be found in section VI.
II. BESSEL THOMSON APPROXIMATION
Bessel Thomson is the most widely used approximation
method of delay filters. This method leads to a family of lowpass all-pole transfer functions T(s), which would give
approximately constant time delay over as large frequency
range as possible. The resultant transfer functions are of the
228
form:
Tn ( s ) =
III. PADE APPROXIMATION
Bn (0)
Bn ( s )
(1)
Where, Bn(s) is a Bessel polynomial of order n. In this
approximation the degree of freedom available is the filter
order, determined by the number of poles. A frequency
domain analysis of these filters illustrates a low frequency
behavior in which the roll-off frequencies are increased by
filter order increase. Fig. 1 illustrates the correspondent
amplitude and delay response of Bessel Thomson filters of
order 1 to 12.
The amplitude of Bessel Thomson Filters of orders 1 to 12
0
10
-2
N=1
10
N=2
-4
10
N=3
N=4
-6
Amplitude
10
N=5
N=6
-8
10
N=7
N=9
N=10
N=11
N=12
-12
10
-1
0
10
1
10
2
10
(2)
Where Pm(x) and Qn(x) are two polynomials as below:
Pm ( s ) = p 0 + p1 s + p 2 s 2 + ... + p m s m
(3)
2
n
Q n ( s ) = q 0 + q1 s + q 2 s + ... + q n s
The unknown coefficients p0 ... pm and q0 ... qn of Rm,n(s)
can be determined from the condition that the first (m+n+1)
terms vanish in the Maclaurin series:
(4)
Substituting the two polynomials into this expression and
equating the coefficients leads to a system of m+n+1 linear
homogeneous equations, which can be expressed in matrix
form.
-14
10
P (s)
Rm , n ( s ) = m
Qn (s)
P (s)
A( s ) − m
=0
Qn ( s)
N=8
-10
10
Padé approximation is a method of approximating a rational
transfer function by expanding a function as a ratio of two
power series [2]. The Padé approximation of order (m,n) for a
function A(s) is defined to be a rational function Rm,n(s)
expressed in a fractional form:
10
Frequency rad/Sec (Normalized)
(a)
The phase of Bessel Thomson Filters of orders 1 to 12
0.012
The amplitude of Pade Approximated Delay Filters
1
10
0.01
0
m=4, n=4
10
0.006
0.004
Amplitude
Phase (Degrees)
0.008
N=12
m=4, n=5
m=4, n=6
N=11
N=10
N=9
N=8
N=7
N=6
N=5
N=4
N=3
N=2
N=1
0.002
0
0
2
4
6
8
10
Frequency rad/Sec (Normalized)
12
14
-1
m=4, n=7
10
m=4, n=8
m=4, n=9
16
-2
10
(b)
0
5
10
15
Frequency rad/Sec (Normalized)
Fig. 1. (a) Bessel Thomson filter amplitude responses for orders 1 to 12 (b)
Phase response of the same filters
Fig.3. Amplitude response of Padé approximated delay filters with m=4 and
n= 4..9
Fig. 2 illustrates the step response of different orders of
Bessel Thomson delay filters. The most important
characteristic of the Bessel Thomson filter, which is an over
shoot free step response, in addition to the resultant decrease
of rise time due to filter order increase can also be viewed
here.
Fig. 3 illustrates that in a Padé approximated delay filter,
for an increase of denominator order the 3-dB cut off
frequency of amplitude response increases, while peaking
occurs for increase of order difference between numerator and
denominator.
A delay response simulation of Padé approximated delay
filters for different order differences between numerator and
denominator order (n-m), while increasing the denominator
order (n), from 5 to 12 is illustrated in Fig. 4. This figure
illustrates that for higher orders of denominator order, higher
delay roll-off frequency is achieved, while the peaking effect
near the roll-off frequency increases for higher numerator and
denominator order differences. More accurate delay response
is achieved for smaller order difference and higher
denominator order.
Step Response
1.4
1.2
1
Amplitude
0.8
N=1
N=2
0.6
N=3
N=4
N=5
N=6
0.4
N=7
N=8
N=9
N=10
N=11
N=12
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
Time (sec)
Fig. 2. Step response of different order Bessel Thomson filters.
229
The rise time of Pade Approximated delay filters
The Delay of Pade Approximated Delay Filters
0.35
0.02
n-m=4
n=m+3
Increase of Order
0.018
n=m+2
n-m=3
n=5 ~ n=12
n=m+1
0.3
n-m=2
0.016
n-m=1
n-m=0
rise time (Seconds)
0.014
Delay
0.012
0.01
0.25
0.2
0.008
0.15
0.006
0.004
0.1
0.002
0
0
5
10
15
Frequency rad/Sec (Normalized)
20
25
30
Fig.4. Delay response of Pade approximated delay for different order
difference and different denominator order.
A series of time domain simulations on the step response of
Padé approximated delay filters illustrates that the rise time
decreases with the increase of the denominator order (Fig. 5),
while zero time oscillations are generated when the order
difference between the denominator and numerator decreases
(Fig. 6). These zero time oscillations are not favorable and
might cause instability in some applications.
Step Response
1.2
1
0.8
4
5
6
7
8
m (numinator order)
9
10
11
12
Fig.7. Rise time comparison for different combinations of numerator and
denominator orders.
An investigation of the pole zero constellations for different
order combinations of Padé approximated delay filters
illustrated that the increase of the numerator and denominator
order above a specific level would lead to instability in high
order delay filters. Table I illustrates the unstable order
combinations in bold.
TABLE I
STABLE AND UNSTABLE ORDER COMBINATIONS
1/12
1/11
1/10
1/9
1/8
1/7
1/6
1/5
1/4
1/3
1/2
1/1
2/12
2/11
2/10
2/9
2/8
2/7
2/6
2/5
2/4
2/3
2/2
3/12
3/11
3/10
3/9
3/8
3/7
3/6
3/5
3/4
3/3
4/12
4/11
4/10
4/9
4/8
4/7
4/6
4/5
4/4
512
5/11
5/10
5/9
5/8
5/7
5/6
5/5
6/12
6/11
6/10
6/9
6/8
6/7
6/6
7/12
7/11
7/10
7/9
7/8
7/7
8/12
8/11
8/10
8/9
8/8
9/12
9/11
9/10
9/9
10/12
10/11
10/10
11/12
11/11
12/12
Num/Denom
Unstable
Stable
0.6
Amplitude
m=2,n=5
IV. COMPARISON OF BESSEL THOMSON AND PADÉ
m=3,n=6
m=4,n=7
0.4
m=5,n=8
APPROXIMATED DELAY FILTERS
m=6,n=9
m=7,n=10
0.2
0
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
Fig.5. Rise time decrease with filter order increase
Step Response
1.2
1
0.8
In this section a comparison of the frequency domain and
time domain performance metrics of Bessel Thomson and
Padé approximated delay filters are illustrated. According to
amplitude and delay responses sketched for different order
combinations of Bessel Thomson and Padé delay filters, in
Fig. 8 and Fig.9 respectively, the lower orders of Padé
approximation can always achieve higher frequencies than the
highest reasonable Bessel Thomson approximations.
m=5,n=10
0.6
Comparison of The amplitude of Pade Delay Filters and Bessel Thomson Delay Filters
Amplitude
m=6,n=10
1.4
m=7,n=10
0.4
m=8,n=10
m=9,n=10
1.2
Bessel Thomson 12
m=10,n=10
0.2
Pade 2/4
1
0
0.8
Pade 10/12
Amplitude
-0.2
0.6
-0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
0.4
Fig.6. Increase of zero time oscillations with the decrease of denominator and
numerator order difference.
0.2
Bessel Thomson 4
0
Fig. 7 illustrates that the filter order increase results in a
faster step response while filters with larger numerator and
denominator order difference provide smaller rise times.
0
5
10
15
Frequency rad/Sec (Normalized)
20
25
30
Fig.8. Amplitude comparison of Bessel Thomson delay filters of orders 4 to
12 vs. Padé approximated delay filters of order 2/4 to 10/12.
230
The Delay of Pade 6/8 Approximated Delay vs. 8th Order Bessel Thomson
40
0.012
Pade 4/6
35
0.01
Sigma=0.01
Pade 10/12
30
Mean = 1
0.008
Delay
25
20
0.006
15
0.004
10
Bessel Thomson 4
Sigma=0.1
0.002
5
Bessel Thomson 12
0
0
5
10
15
Frequency rad/Sec (Normalized)
20
25
Fig.9. Delay comparison of Bessel Thomson delay filters of orders 4 to 12 vs.
Pade approximated delay filters of order 4/6 to 10/12.
A comparison of step response of both approximations (Fig.
9) illustrates that Padé approximated delay filters always
introduce more overshoot and oscillations in comparison to
Bessel Thomson counterparts, while the former has smaller
rise times (Fig. 10).
Step Response
1.2
1
Pade 6/8
0.8
0
0.6
30
Bessel 8
Amplitude
0.6
0.4
0.7
0.8
0.9
1
1.1
1.2
1.3
Fig.11. A Gaussian function’s response vs. the change of σ
1
e
σ 2π
h (t ) =
− (t − µ ) 2
2σ 2
(5)
In this method a Padé approximation is applied to the
Laplace transform of a Gaussian impulse response. Figure 12
illustrates that the amplitude response of the Gaussian impulse
response delay filter becomes more near to Bessel Thomson
approximation for larger values of σ, decreasing the roll-off
frequency, while Fig. 13 illustrates that the effect on the delay
response is minimum.
0.2
Comparison of The amplitude of Pade Delay Filter of order 6/8 and a Bessel Thomson of order 8
1.4
0
1.2
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
Pade of Delay order 8/10
4
Time (sec)
1
Pade of Gaussian order 8/10, sigma=0.01
Fig.9. Comparison of step response in both approximations
Amplitude
0.8
The rise time of Pade Approximated delay filters
1.1
Pade of Gaussian order 8/10, sigma=0.1
0.6
Pade m,n=m+3
1
Pade m,n=m+2
Pade m,n=m+1
0.9
Bessel Thomson Order m
0.4
rise time (Seconds)
0.8
Bessel Thomson order 32
0.2
0.7
Bessel Thomson order 10
0.6
0
0
5
10
0.5
0.4
20
25
30
Fig.12. The amplitude response of the Gaussian impulse response delay vs. σ
in comparison to normal Padé approximated delay and Bessel Thomson
responses
0.3
0.2
0.1
15
Frequency rad/Sec (Normalized)
4
5
6
7
8
m (numinator order)
9
10
11
12
The Delay of Pade Approximated Delay vs. 8th Order Bessel Thomson, and Gaussian Pade Approximated 8/10
0.012
Fig.10. Rise time comparison
Gaussian Pade 5/7 & sigma: 0.01~0.1
Gaussian Pade 8/10 & sigma: 0.01~0.1
0.01
In order to have the higher frequency and faster time
domain response of Padé approximation, together with less
overshoots as the Bessel Thomson approximation achieves, a
new approximation method is introduced in the following
section.
Delay
0.008
Delay Pade 4/6
Delay Pade 10/12
0.006
0.004
Bessel Thomson 4
0.002
V. GAUSSIAN TIME DOMAIN IMPULSE RESPONSE METHODE
Bessel Thomson 12
0
The time domain impulse response of a delay function is a
delta function. A delta function can be approximated by
narrowing of a Gaussian time domain transfer function with
the use of smaller pulse width σ (depicted by equation 5, and
Fig. 11).
0
5
10
15
Frequency rad/Sec (Normalized)
20
25
30
Fig.13. The delay response of the Gaussian impulse response delay vs. σ in
comparison to normal Padé approximated delay and Bessel Thomson
responses
An investigation of the Gaussian impulse response delay
approximation delay filter’s step response proves that the
231
increase of σ, making the impulse response wider, leads to a
decrease of the zero time oscillations and overshoots, while it
slightly increases the rise time (Fig. 14). As a result of these
simulations it is revealed that using a time domain Gaussian
impulse response gives the possibility of decreasing the
overshoots and zero time ringings in a Padé approximation of
delay by increasing σ, while the price paid is an increase of
insertion loss in the amplitude response and a slightly slower
rise time in the system step response.
Fig.15 compares the step response of a 6/8 Padé
approximated delay filter with the same order Gaussian
impulse response delay filter with σ value equal to 0.05. As
illustrated by this graph, for the same nominator and
denominator order, the Gaussian impulse response delay has
less overshooting.
REFERENCES
[1]
[2]
Rolf Schaumann, Mac Van Valkenburg, "Design of Analog Filters",
Oxford University Press, ISBN: 0-19-511877-4
Haddad, S.A.P.; Verwaal, N.; Houben, R.; Serdijn, W.A,"Optimized
dynamic translinear implementation of the Gaussian wavelet transform",
IEEE International Symposium on Circuit and Systems, May 2004
Step Response
1.2
1
m=8 and n=8
0.8
sigma=0.01
sigma=0.02
0.6
Amplitude
sigma=0.03
sigma=0.04
0.4
sigma=0.05
sigma=0.06
sigma=0.07
0.2
sigma=0.08
sigma=0.09
0
sigma=0.10
-0.2
-0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec)
Fig.14. The step response of the Gaussian impulse response delay vs. σ
1.2
Pade approximation
1
Gaussian Impulse response approximation
Sigma = 0.05
0.8
0.6
0.4
0.2
0
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig.15. The step response of the Gaussian impulse response delay for a value
of σ = 0.05 and filter order of 6/8, vs. the same filter order Padé approximated
delay filter
VI. CONCLUSION
While Bessel Thomson is the most widely used method
of delay filter approximation, higher frequency, and faster
delay responses can be achieved through the use of Padé
approximation at the expense of more overshooting, while
different order combinations would result in responses, with
different stability conditions. The here introduced method of
Gaussian time domain impulse response leads to less
oscillation and overshooting, at the expense of slower rise
time and lower roll off frequency in amplitude response
compared to the previously mentioned methods.
232