Abstractâ In the present work causality property and physics behind it is ... methodologies for checking or enforcing causality in both time and frequency domain ...
Causality and Delay and Physics in Real Systems Mikheil Tsiklauri*, Mikhail Zvonkin*, Jun Fan*, James Drewniak*, Qinghua Bill Chen#,¥ and Alexander Razmadze† *EMC Laboratory, Missouri University of Science and Technology, Rolla, MO, USA, # Yangtze Delta Region Institute of Tsinghua University, Tiaxing, China, ¥ School of Software and Microelectronics, Peking University, Beijing, China, † Altera Corporation, San-Jose, CA, USA Abstract— In the present work causality property and physics behind it is analyzed for physical systems. Different methodologies for checking or enforcing causality in both time and frequency domain are discussed. Causality metric for measuring causality violation is introduced. Physical anomaly of a system with perfectly linear phase is discussed and shown that small perturbation of non-linear portion of the phase can fix the non-causal anomaly. Index Terms— Transfer function; Impulse response; Delay; Pulse response; Kramers-Kronig relations; Causality metric; Linear phase.
impulse is applied, which means that the impulse response should be equal to zero for all t 0 . On Fig.1 there are given three impulse responses. Red curve corresponds to causal impulse response, but green and blue ones are non-causal.
I. INTRODUCTION: CAUSALITY IN TIME AND FREQUENCY DOMAINS
M
ost CAD tools allow system-level simulation for Signal Integrity by computing and connecting together models for the various sub-parts. The success of this model derivation depends on the quality of the network parameters. Different errors may seriously affect the quality of the frequency characterization: frequency-dependent measurement errors, errors due to the numerical simulation and/or discretization. When these errors are large, model assembly and simulation becomes difficult and may even fail. In [1] there are presented different algorithms for S-Parameters quality checking. A very important property for quality check of network parameters is causality. The purpose of the paper is to analyze causality, delay and physics in real systems. The paper is organized as follows: in the introduction causality definition in time domain is given and well-known Kramers-Kronig causality relations in frequency domain in terms of real/imaginary and magnitude phase are derived; in the second section causality metric for non-causality estimation is introduced; in the third section causality enforcing algorithm with real and imaginary parts is discussed; In the fourth section physical anomaly for systems with perfectly linear phase is studied and shown that small perturbation of non-linear part of the phase can fix the non-causal anomaly; In the last section the paper results are summarized and the general conclusions are made. As known, causality is defined in the time domain. A causal system should not respond to the unit impulse before the unit
978-1-4799-5545-9/14/$31.00 ©2014 IEEE
Fig. 1.1. Examples of causal and non-causal impulse responses.
Causality in the time domain is defined by the following formula: h t 0, t 0, (1.1) where h is a causal impulse response. Usually oscillated causality violation near t 0 moment is caused by frequency band limitation or measurement error at high frequencies for corresponding transfer function (see blue curve on Fig. 1), but causality violation similar to the green curve on the Fig. 1 can be caused by wrong simulation model. These topics will be discussed more detailed in the next sections. From (1.1) let us derive well known Kramers-Kronig causality relation between real and imaginary parts of the transfer function. Impulse response h will satisfy (1.1) causality condition if and only if it satisfies the following relation: h t h t sign t , t , . (1.2) Let us assume that H is a transfer function corresponded
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to the impulse response h t ,
In the following sections different algorithms for checking and enforcing causality for non-causal systems will be discussed.
H F h t
h t e
jt
dt ,
(1.3)
II. CAUSALITY METRIC
where F h t is the Fourier transform operator. From (1.3) according to the convolution theorem (see [1], p. 27, eq. (2-74)), it follows that, transfer function H will satisfy the following equation: 1 F h t F sign t 2 (1.4) H ' 1 2 1 H d '. ' 2 j j
H F h t sign t
Here by “ ” is defined the convolution operator. If U and V are respectively the real and imaginary parts of the transfer function H U jV , then (1.4) Kramers-Kronig relations can be rewritten in the following form: 1 V jU H V jU ' d ' j (1.5) j V 1 U d ' d '. ' ' From here if we make equal real and imaginary parts of both sides of the equality, we will obtain well-known Kramers-Kronig relations in terms of real and imaginary parts:
1
V U
1
' d ',
V '
' d ' ,
of one needs not only , but also constant 0 .
proof
of
' 2 '2 d ',
(1.7)
is
H F h t is a transfer function corresponded to h t impulse response then H e j should satisfy (1.6) Kramers-Kronig relations. Hence it follows that delay causality analogue for Kramers-Kronig relations can be written using the following formulas: Re H e j Im H e
' ' 2
based
on
and correspondingly its real
j
1
1
Im H ' e j '
'
Re H ' e j '
'
d ',
(2.3) d '.
Re H e j V cos U sin , Im H e j V sin U cos .
2
fact
NonCausality h
100%.
(2.5)
h t dt 2
that
and imaginary
h t dt 2
d '.
the
(2.4)
Different errors, such as frequency-dependent measurement error, errors due to the numerical simulation and/or discretization, may seriously affect causality property of the physical system. When these errors are large, model assembly and simulation becomes difficult and may even fail. Therefore, it is important to have causality metric to estimate non-causality violation of the physical system. Delay causality is related to the portion of the energy of the signal which comes at the output of the system before the delay time. Because of this, it is natural to define non-causality of the system as a ratio of energy which comes before delay to the total energy:
(1.7)
'
ln H j is analytic on the right half plane,
with time delay, then h t will be causal function and if
where
determined from attenuation , and for the determination
The
From (2.1) it follows that if h t is a delay causal function
Kramers-Kronig relations (1.6) can be established between amplitude and phase of the transfer function (see [2], p. 73, equations (3-23) and (3-24); also see [3],[4]). Let us assume that H e j , then phase can be uniquely
2 0
where is a time delay of the system.
(1.6)
From (1.6) relation it follows that real and imaginary parts of a causal transfer function are not independent and can be reconstructed from each other.
Since physical system has time delay, it should be not only causal, but delay causal system. Delay causality property of the system means that signal should not appear at the output before time delay. In terms of impulse response, delay causality property can be expressed using the following formula. h t 0, t , (2.1)
U '
In this section causality metric for estimation of causality violation will be introduced.
Pictorial explanation of non-causality of the impulse response is given on Fig. 2.1.
parts satisfy (1.6) Kramers-Kronig relations.
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Calculated causality estimation for the differential insertion loss of the given geometry is equal to 89%, but pulse causality estimation is equal to 99.9% (pulse signal was taken with 16ps rise/fall time and 12.5Gbps bitrate). On Fig. 2.3 there are given causality and pulse causality violations for differential insertion loss of the measured S-Parameters.
Fig. 2.1. Pictorial explanation of non-causality metric.
Impulse response includes all frequency components. If the physical system is designed for specific type of signal (with specific rise/fall time and bitrate), then it might be more useful instead of (2.5) definiton of non-causality metric to use the following definition:
r t dt 2
NonCausality hv
100%.
(2.6)
r t dt 2
Fig. 2.4 Causality and pulse causality violations for differential insertion loss for the structure given on Fig.2.2.
Here r t is a pulse response of the system:
r t
h t s v s ds,
(2.7)
where v s is an input pulse signal with specific rise/fall time and bitrate. We call (2.5) equation causality estimation and (2.6) pulse causality equation. Both causality and pulse causality estimation were calculated for measured differential insertion loss of the strucutre, geometry of which is given on Fig.2.2.
From here follows, that 11% causality violation for the insertion is related with high frequency components (more then 12.5Gbps speed signals contain) and if the system is manufactured for the signals less than 12.5Gbps speed, then the system will be almost 100% causal.
III. CAUSALITY CHECKING\ENFORCING BY MEANS OF REAL AND IMAGINARY PARTS OF SYSTEM FUNCTION
In this section delay causality enforcing algorithm in frequency domain will be discussed. Below is given the diagram for the causality enforcement algorithm ( V and U are correspondingly real and imaginary parts of the transfer function after removing delay):
Fig.2.2. Geometry of the DUT structure.
Below, on the Fig. 2.3 there is given magnitude of differential insertion loss for the DUT.
+ Fig. 3.1. Diagram of the causality enforcement algorithm.
Fig. 2.3. Differential insertion loss for the geometry given on Fig. 2.2.
In this algorithm we should calculate Hilbert transform of the real part. The integral in this transform is taken over the infinite
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frequency interval, but because of frequency band limitation for numerical calculation, we can just calculate integral up to a maximum frequency:
V
1
B
U '
' d' Err .
(3.1)
calculation
B
The error will be equal to the integral outside the frequency band: Err
U ' U ' d ' d ' . ' ' B
B
(3.2)
If we assume that we have a passive system, which means that the norm of the real part (as well as of the imaginary part) of the transfer function should be less than one for all the frequency samples, then error can be estimated by the following inequality (in this estimation we have also used that U U ):
B Err ln , B
n
V LV
q
q 1
U ' LU d ' , (3.4) n ' B ' q B
q 1
where LV and LU are Lagrange polynomials for the imaginary and real parts of the transfer function, respectively. If we move Lagrange polynomial LV from the left to the right hand side, we get the following formula for the imaginary part of the transfer function: n
V LV
q
q 1
Vk
the
following
subtraction
points
an error caused by limitation of frequency band (see [5]). At the frequency points near DC the error is too small and small number of subtractions will be enough, but near the maximum frequency where the error becomes large we will have already calculated enough values for the imaginary part and we can use as many subtraction points as it will be necessary for error reduction. There are two sources of errors in formula. The first one is caused by limitation of the frequency band and, as explained before, it can be reduced by increasing the number of subtractions. The other error is in Lagrange polynomial LV and k
(3.3)
From this formula, we see that the error is equal to zero for frequency 0 , is small near DC and tends to infinity when B . It means that the calculations near maximum frequency will be inaccurate. The integration error can be reduced by using so-called subtractions (see [5]). The idea of subtractions is that Kramers-Kronig relation is applied not for real and imaginary parts but for the real part minus its Lagrange polynomial and the imaginary part minus its Lagrange polynomial:
of
0,1, 1,...,k 1, k 1 can be used. Subtractions can reduce
For numerical reconstruction of V1 V 1 , formula (3.5) with one subtraction at the point 0 can be used. After obtaining V1 we can reconstruct V2 V 2 , using formula (3.5) with 3 subtractions at frequency samples 1 ,0, 1 Thus for
U ' LU d ' , (3.5) B n ' ' q B
q 1
This formula is used for causality verification in [5]. However, it cannot be used for causality enforcement because the construction of Lagrange polynomial LV requires the
is caused by using approximate values already calculated for the imaginary part. This error becomes larger with increasing number of subtractions. From here follows that increasing number of subtractions reduces one error but amplifies the other error and vice versa. So it is important to find the optimal number of subtractions for formula imaginary part reconstruction, but this problem is not solved in the present paper and remains open. IV. CAUSALITY VIOLATION WITH PERFECTLY LINEAR PHASE In this section we will analyze causality anomaly for the system with perfectly linear phase. Delay causal system can’t have perfectly linear phase, if magnitude is frequency depended. Let us consider transfer function with linear phase and frequency depended magnitude: H A e j , 0, (4.1) where A is magnitude and is a delay of the system. To obtain real impulse response, for negative frequencies transfer function should be defined by the following way: H H * A e j , 0. (4.2) Impulse response of (4.1) transfer function is calculated using inverse Fourier transform: 1 jt (4.3) h t H e d. 2 From here using (4.2) formula we will obtain: 0
h t
H e jt d H e jt d 0
0
0
values of unknown imaginary part of the transfer function at some frequency samples. Below we propose an algorithm for causality enforcement of the transfer function on the basis of formula (3.5).
H e j t d H e jt d
The imaginary part of the causal transfer function is an odd function and is equal to zero at the zero frequency sample. It means that the value of the reconstructed imaginary part at the first frequency sample is already known. (3.6) V0 0
In this formula let us replace H and H by (4.1) and
0
0
(4.4)
H e jt d H e jt d .
(4.2) formulas, we will obtain
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0
0
h t A e j e jt d A e j e jt d
A e 0
j t
e
j t
(4.5)
d .
Then we will calculate impulse responses for all three cases and compare to each other. As we see from Fig. 4.2-4.3, all three cases have absolutely the same amplitudes and maximal difference between phases is equal to 0.2%.
Using formula (4.5) let us calculate h t - we need just replace t by t in (4.5) formula:
j t j t h t A e e d 0
A e
jt
e
jt
(4.6)
d .
0
By the same way let us calculate h t :
j t j t h t A e e d 0
A e
jt
e
jt
d .
(4.7)
0
From (4.6) and (4.7) we see that h t h t , which means that (4.3) impulse response is symmetric regarding and if we have some response after , we should have the same response before . If (4.3) impulse response is delay causal than h t should be equal to zero, from here and symmetricity of the impulse response follows that h t also should be equal to zero. It is possible if and only if we have a delta function (shifted by ), which is impossible if a magnitude of the transfer function is frequency depended. Below we will show that a small perturbation of the phase for network parameters can significantly change impulse response. For this purpose let us construct three different insertion losses: The first SDD1 is original differential insertion loss for the geometry given on Fig. 2.2 (blue curve on Fig 4.1 and Fig. 4.2); The second SDD2 insertion loss has the same magnitude as original one, but the phase is perfectly linear (red curve on Fig. 4.1 and Fig. 4.2); The third SDD3 insertion loss has the same amplitude as original one but the phase is equal to perfectly linear phase plus reversed non-linear part of original one (green curve on Fig. 4.1 and Fig. 4.2).
Fig. 4.3 Unwrapped phases for SDD1, SDD2 and SDD3 insertion losses.
On Fig. 4.4 there are given impulse responses for all three cases. Blue curve corresponds to the original insertion losss - SDD1; red curve corresponds to insertion loss with perfectly linear phase – SDD2 (Non-linear part of the phase is removed from SDD1); and green curve corresponds to SDD3 insertion loss (Non-linear part of the phase in SDD1 is reversed).
Fig. 4.4 Impulse responses corresponded to SDD1 (Original insertion loss), SDD2 (Insertion loss with linear phase) and SDD3 (Insertion loss with reversed non-linear part of the phase).
Fig. 4.2 Magnitudes for SDD1, SDD2 and SDD3 insertion losses.
From Fig. 4.2-4.3 we see that magnitudes for all three cases are absolutely the same and difference between phases are less than 0.2%, but From Fig. 4.4 we see that corresponding impulse responses are absolutly different: original impulse response is physical, the causality is equal to 89%; impulse responses with linear phase and reversed non-linear part of the phase are non-physical. The causality of the impulse response with linear phase is equal to 50% and causality of the impulse response with reversed non-linear part of the phase is equal to 11%. From here follows that if we have causality violation for network parameters, it is possible to make a small perturbation in the phase that improves causality of the system function
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[5]
without changing the magnitude. Reconstruction of the phase from magnitude is possible from the first equation of formula (1.7). Because of frequency band limitation the similar problem as for real/imaginary (see section III) will occur. Solution of this problem also needs to find optimal number of subtractions for reducing numerical error caused by frequency band limitation. This problem is not resolved in the proposed paper and remains open. V. CONCLUSION Checking, estimating and enforcing causality for the transfer function is very important, as using non-causal transfer function for link path analysis leads to nonsense: we will get the response of physical channel before actual delay. In the proposed paper there are discussed different aspects of delay causality property and physics behind it. A natural metric for causality estimation for impulse response of the system is proposed. There is shown that the system with frequency depended magnitude and linear phase cannot be physical. Impulse response of such system is symmetric and the center of the symmetry is a system delay. From here follows that everything what happens after delay, should happen before delay, which violates causality property. Many commercial tools have implemented causality check and enforce algorithms, but they are related with Kramers-Kronig relation between real and imaginary parts. This type of enforcement will change magnitude of the original frequency response and therefore significantly modify original data. In the paper is shown that small perturbation of the phase can dramatically change causality property. From here follows that the best way to enforce causality for non-causal system is a small perturbation of the phase. We are going to develop algorithm for phase reconstruction from magnitude using formula (1.7). The main challenge of the algorithm is to reduce maximally numerical error caused because of frequency band limitation of the measured or simulated frequency characterization. ACKNOWLEDGEMENT This material is based upon work supported partially by the National Science Foundation under Grant No. 0855878, and partially by Jiaxiang S&T project (2011AZ1013), Qianjiang talent project (2013R10082) of Zheijiang Province and NSF (BK2010137) of Jiangsu Province, China. REFERENCES [1]
[2] [3] [4]
Y. Shlepnev, Quality of S-parameter models, Asian IBIS Summit, Yokohama, November 18, 2011, http://www.simberian.com/Presentations/Shlepnev_Japan_IBIS_Summit 2011.pdf N.M. Nussenzveig, “Causality and dispersion relation”, Academic Press, 1972. J. Bechhoefer, Kramers–Kronig, Bode, and the meaning of zero, American Journal of Physics, Volume 79, Issue 10, pp. 1053-1059 (2011). K. N. van Dalen, E. Slob and Ch. Schoemaker, Generalized minimum-phase relations for memory functions associated with wave phenomena, Geophysical Journal International, Volume 195, Issue 3, p.1620-1629 (2013).
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Triverio, P.; Grivet-Talocia, S., "A robust causality verification tool for tabulated frequency data," Signal Propagation on Interconnects, 2006. IEEE Workshop on , vol., no., pp. 65,68, 9-12 May 2006, doi: 10.1109/SPI.2006.289191.
