Broadband transfer function synthesis using orthonormal rational bases
View Patent ↗In order to generate a broadband transfer function of complex characteristics of a linear time-invariant (LTI) system, data characterising properties of the system are acquired. A set of poles in the complex plane are defined to characterize the system, and then an iterative process is performed to: define a set of orthonormal rational basis functions incorporating the defined poles, use the orthonormal rational basis functions to estimate transfer function coefficients, and derive revised values for the complex poles, until a desired level of accuracy of the transfer function coefficients is attained. The revised complex poles are used to determine parameters of the broadband transfer function.
1. A method of generating a broadband transfer function of complex characteristics of a linear time-invariant system within an apparatus, comprising:
using a user input/output interface unit to acquire measured data characterising properties of the system;
using a processing unit having a central processing unit and a memory to define a set of poles in the complex plane to characterise the system;
using the processing unit to iteratively perform the following:
define a set of orthonormal rational basis functions incorporating the defined poles,
use the orthonormal rational basis functions to estimate transfer function coefficients, and
derive revised values for the complex poles,
until a desired level of accuracy of the transfer function coefficients is attained;
using the revised complex poles to determine parameters of the broadband transfer function with the processing unit; and
using the broadband transfer function to identify a mapping between inputs and outputs of the time invariant system.
2. The method of claim 1 , wherein the revised values of the complex poles are derived by constructing a state-space representation (A, B, C, D), and solving for the eigenvalues of the expression (A−B*C), wherein A represents a state matrix, B represents an input matrix, C represents an output matrix and D represents a feedforward term.
3. The method of claim 2 , wherein the state matrix A is adjusted with the processing unit for the presence of complex conjugate poles.
4. The method of claim 2 , wherein the state-space representation comprises the following components:
A
P
×
P
=
[
-
z
1
,
d
(
t
-
1
)
0
0
…
0
2
ℜe
(
-
z
1
,
d
(
t
-
1
)
)
-
z
2
,
d
(
t
-
1
)
0
…
0
2
ℜe
(
-
z
1
,
d
(
t
-
1
)
)
2
ℜe
(
-
z
2
,
d
(
t
-
1
)
)
-
z
3
,
d
(
t
-
1
)
…
0
2
ℜe
(
-
z
1
,
d
(
t
-
1
)
)
2
ℜe
(
-
z
2
,
d
(
t
-
1
)
)
2
ℜe
(
-
z
3
,
d
(
t
-
1
)
)
…
0
…
…
…
…
…
2
ℜe
(
-
z
1
,
d
(
t
-
1
)
)
2
ℜe
(
-
z
2
,
d
(
t
-
1
)
)
2
ℜe
(
-
z
3
,
d
(
t
-
1
)
)
…
-
z
P
,
d
(
t
-
1
)
]
B
1
×
P
=
[
1
1
…
1
]
,
C
P
×
1
=
[
d
~
1
(
t
)
2
ℜe
(
z
1
,
d
(
t
-
1
)
)
d
~
2
(
t
)
2
ℜe
(
z
2
,
d
(
t
-
1
)
)
…
d
~
P
(
t
)
2
ℜe
(
z
P
,
d
(
t
-
1
)
)
]
T
D
1
×
1
=
1
where P is the number of poles used in defining the basis functions, Re represents the real part of a complex value, z (t) d ={−z (t) 1,d , . . . , −z (t) P,d } are elements of an array of complex zeroes at iteration step t, and {tilde over (d)} p (t) are transfer function coefficients.
5. The method of claim 4 , wherein the state matrix A is adjusted for the presence of complex conjugate poles with the processing unit by replacing
(
-
z
p
,
d
(
t
-
1
)
0
2
ℜe
(
-
z
p
,
d
(
t
-
1
)
)
-
z
p
+
1
,
d
(
t
-
1
)
)
in the state matrix A by
(
ℜe
(
-
z
p
,
d
(
t
-
1
)
)
ℜe
(
-
z
p
,
d
(
t
-
1
)
)
-
z
p
,
d
(
t
-
1
)
ℜe
(
-
z
p
,
d
(
t
-
1
)
)
+
z
p
,
d
(
t
-
1
)
ℜe
(
-
z
p
,
d
(
t
-
1
)
)
)
.
6. The method of claim 1 , wherein the data characterising properties of the system are frequency-domain data.
7. The method of claim 1 , wherein the data characterising properties of the system are time-domain data.
8. The method of claim 1 , wherein some of the data characterising properties of the system are frequency-domain data and additional data characterising properties of the system are time-domain data.
9. The method of claim 1 , wherein the system is an electromagnetic system.
10. The method of claim 1 , wherein the system comprises an individual component.
11. An apparatus comprising:
a memory storing software program instructions for generating a broadband transfer function of complex characteristics of a linear time-invariant system;
a processing unit having a central processing unit and the memory, the processing unit configured to implement procedures for generating the broadband transfer function of complex characteristics of the linear time-invariant system according to the stored software program instructions; and
a user input/output interface unit configured for acquiring measured data characterising properties of the linear time-invariant system;
the processing unit further configured to, according to the stored software program instructions:
define a set of poles in the complex plane to characterise the system with a processing unit; and
iteratively:
define a set of orthonormal rational basis functions incorporating the defined poles,
use the orthonormal rational basis functions to estimate transfer function coefficients, and
derive revised values for the complex poles,
until a desired level of accuracy of the transfer function coefficients is attained;
use the revised complex poles to determine parameters of the broadband transfer function; and
use the broadband transfer function to identify a mapping between inputs and outputs of the time invariant system.