System and method for testing causality of tabulated S-parameters
A system. The system includes a computing device having a processor, and a causality checker module communicably connected to the processor. The causality checker module is configured to utilize a rational function approximation to a frequency response to determine if a transfer function of a linear time invariant system is causal.
1. A system, comprising:
a computing device having a processor; and
a causality checker module communicably connected to the processor for use in simulating the behavior of a physical system based on data from the physical system, wherein the causality checker module is configured to:
utilize the data from the physical system to perform a rational function approximation to a frequency response to determine if a transfer function of a linear time invariant system is causal; and
calculate a value for the rational function approximation based on the following equation:
H
(
j
ω
)
≈
H
_
(
j
ω
)
=
∑
i
=
1
N
P
r
i
j
ω
-
p
i
+
d
,
(
17
)
wherein the H(jω) is the frequency response at angular frequency ω, H (jω) is a rational function approximation, N p is a number of poles, r i is the i th residue, p i is i th pole, and d is a direct term;
wherein, upon verification of causality of the transfer function, perform a simulation of the physical system using the causal transfer function;
wherein the physical system is built or modified based on results of the simulation of the physical system.
2. The system of claim 1 , wherein the transfer function comprises an s-parameter.
3. The system of claim 1 , wherein the linear time invariant system comprises an electrical system.
4. The system of claim 1 , wherein the causality checker module is further configured to calculate a value for a target fitting error for the rational function approximation.
5. The system of claim 1 , wherein the causality checker module is further configured to calculate a value for a truncation error bound associated with the frequency response.
6. The system of claim 1 , wherein the causality checker module is further configured to calculate a value for a discretization error bound based on the following formula:
{tilde over (D)} n (ω)≈| {tilde over (H)} n μ 1 ( jω )− {tilde over (H)} n μ 2 ( jω )|, (13)
wherein μ 1 and μ 2 are two different interpolating functions.
7. The system of claim 6 , wherein the causality checker module is further configured to calculate values for the expressions {tilde over (H)} n μ 1 (jω) and {tilde over (H)} n μ 2 (jω) based on the following equation:
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(
18
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wherein L H (jω) is the Lagrange interpolation polynomial for H(jω).
8. The system of claim 1 , wherein the causality checker module is further configured
to determine if the condition of the following equation is met:
E n tot (ω k )= T n (ω k )+ {tilde over (D)} (ω k )≦ A ∀k, (12)
where E n (jω) is a truncation error based on H n (jω).
9. The system of claim 1 , wherein the causality checker module is further configured
to determine if the condition of the following inequality is met:
|{tilde over (Δ)} n ( jω k )|> T n (ω k )+ D n (ω k )∃ k, (11)
where D n (jω) is the error in numerically computing H n (jω) numerically with finite-band-width tabulated response.
10. The system of claim 1 , wherein the causality checker module is further configured to determine causality of multi-port tabulated data.
11. The system of claim 1 , wherein the physical system is an electrical network, wherein the electrical network is build or modified based on the simulation of the electrical network.
12. A method, implemented at least in part by a computing device, for testing causality of a transfer function of a linear time invariant system for simulating the behavior of a physical system based on data from the physical system, the method comprising:
determining a value of a truncation error bound for a frequency response associated with the linear time invariant system that is developed using the data from the physical system, wherein the determining is performed by the computing device;
determining a value for an approximation for a discretization error bound for the frequency response utilizing a rational function approximation to the frequency response, wherein the determining is performed by the computing device, wherein determining the value for the approximation for the discretization error bound comprises determining a value for the rational function approximation based on the following equation:
H
(
j
ω
)
≈
H
_
(
j
ω
)
=
∑
i
=
1
N
P
r
i
j
ω
-
p
i
+
d
,
(
17
)
wherein the H(jω) is the frequency response at angular frequency ω, H (jω) is a rational function approximation, N p is a number of poles, r i is the i th residue, p i is i th pole, and d is a direct term;
determining whether the transfer function is causal based on the value for the approximation for the discretization error bound, wherein the determining is performed by the computing device;
wherein, upon verification of causality of the transfer function, a simulation of the physical system is performed using the causal transfer function;
wherein the physical system is built or modified based on results of the simulation of the physical system.
13. The method of claim 12 , wherein determining the value of the truncation error bound comprises determining the value for a frequency response associated with an electrical system.
14. The method of claim 12 , wherein determining the value for the approximation for
the discretization error bound comprises setting amplitude A in the following equation to a predetermined value:
E n tot (ω k )= T n (ω k )+ {tilde over (D)} (ω k )≦ A ∀k, (12)
where E n (jω) is a truncation error based on H n (jω).
15. The method of claim 12 , wherein determining the value for the approximation for the discretization error bound comprises calculating a value for a target fitting error for the rational function approximation.
16. The method of claim 12 , wherein determining the value for the approximation for the discretization error bound further comprises calculating a value for a discretization error bound based on the following equation:
{tilde over (D)} n (ω)≈| {tilde over (H)} n μ 1 ( jω )− {tilde over (H)} n μ 2 ( jω )|, (13)
wherein μ 1 and μ 2 are two different interpolating functions.
17. The method of claim 12 , wherein the causality checker module is further configured to calculate values for the expressions {tilde over (H)} n μ 1 (jω) and {tilde over (H)} n μ 2 (jω) based on the following equation:
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~
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j
ω
)
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,
(
18
)
wherein L H (jω) is the Lagrange interpolation polynomial for H(jω).
18. The method of claim 12 , wherein determining whether the transfer function is causal comprises determining whether an S-parameter is causal.
19. The method of claim 18 , wherein determining whether the s-parameter is causal comprises calculating a value for an approximation of an upper bound of the discretization error based on the following equation:
{tilde over (D)} n (ω)≈| {tilde over (H)} n μ 1 ( jω )− {tilde over (H)} n μ 2 ( j ω)|, (13)
wherein μ 1 and μ 2 are two different interpolating functions.
20. The method of claim 18 , wherein determining whether the S-parameter is causal comprises determining whether the following inequality is met:
E n tot (ω k )= T n (ω k )+ {tilde over (D)} (ω k )≦ A ∀k, (12)
where E n (jω) is a truncation error based on H n (jω).
21. The method of claim 18 , wherein determining whether the s-parameter is causal comprises calculating a value for a reconstruction error based on the following equation:
{tilde over (Δ)} n ( j ω)= {tilde over (H)} n ( j ω)− H ( j ω), (4)
where {tilde over (Δ)} n (jω k ) is the different between H(jω) and a generalized Hilbert transform H n (jω).
22. The method of claim 18 , wherein determining whether the S-parameter is causal comprises determining whether the following inequality is met:
|{tilde over (Δ)} n ( jω k )|> T n (ω k )+ D n (ω k )∃ k, (11)
where D n (jω) is the error in numerically computing H n (jω) numerically with finite-band-width tabulated response.
23. The method of claim 12 , further comprising determining causality of multi-port tabulated data, wherein the determining is performed by the computing device.