1. Missouri University of Science and Technology
Stable Recursive Convolution for Channel Response Calculation with Causality Enforcement
Mikheil Tsiklauri, Mikhail Zvonkin, Nana Dikhaminjia, Jun Fan and James Drewniak
Missouri University of Science and Technology
December ??, 2015

2. Causality Definition 5 10 15 Time (ns)
Time (ns)
A causal system should not respond to the unit impulse before the unit impulse is applied, which means that the impulse response should be 0 for all t < 0
Definition of causality is given in time domain. Causal system should not respond to the unit impulse before the unit impulse is applied, which means that the impulse response should be 0 for all t < 0. On the plot you see 3 impulse responses. Read is causal, but green and blue are non-causal. Blue has some oscillations before zero and it is usually depended on frequency bend limitation and green has symmetrical shape with center at zero. This usually means that we have something fundamentally wrong in the model.

3. Causality Definition: Time and frequency domain
Causality in the time domain
Causality in frequency domain Kramers-Kronig relation:
Causality in frequency domain can be expressed thorough Kramers-Kronig relation. Which say that real and imaginary parts of causal system can not be independent and one can be calculated from another.
Where U and V are correspondingly real and imaginary parts of transfer function

4. Causality metric
Non causality of the transfer function can be defined as a portion of energy which comes before the delay.

5. Causality Enforcement in Time Domain
Impulse Response
Enforce causality in time domain
Take non-causal part of impulse response
Reflect symmetrically to the time delay
Add reflected non-causal part to the causal one 5

6. Correspondence with frequency domain
Causality Enforcement in Time Domain
Correspondence with frequency domain
Time domain enforcement 6

7. Stability is guaranteed if
Link Path Analysis - Simulation Procedure
Stability is guaranteed if 7

8. This part is still stable
Link Path Analysis - Simulation Procedure
But even if we have stable poles, time domain causality enforcement will introduce unstable poles as well.
After time domain causality enforcement we get the following impulse response
This part is still stable
Real part of the pole becomes positive and standard recursive convolution algorithm is no longer stable 8

9. Recursive Convolution Idea
Recursive convolution idea is to calculate channel response value y(tk) at sample tk based on previous value y(tk-1)
If absolute value of the coefficient ak is less than one then the recursive formula is stable
coefficient ak is related with that’s why we need to have poles with negative real part
h1(t) is related with stable recursive convolution formula, but h2(t) does not 9

10. Stable Recursive Convolution
Channel response part related with h1(t) will be calculated with standard recursive convolution algorithm …
Calculation direction for h1(t)
For h2(t) we have unstable recursive convolution:
We can make the algorithm stable if we will change calculation direction
For Channel response part related with h2(t) will are changing calculation direction to make it stable …
Calculation direction for h2(t) 10

11. Link Path Analysis - Simulation Procedure
The following application example illustrates the proposed recursive convolution algorithm with time domain causality enforcement.
S-parameters of the stripline structure were measured and used to simulate the response of a 10 Gbps pulse with a 30 ps rise time.
Magnitude and phase of the measured S21 component. 11

12. Link Path Analysis - Simulation Procedure
S-parameters of the real system must be causal;
however, because of the measurement errors they still have to be checked before use.
For this example, less than 0.1% of the pulse response energy has come before the delay, thus this data can be considered causal.
Further, the transfer function was multiplied on , where was a random variable. That introduced some random nonlinear distortion to the transfer function’s phase.
Since magnitude was not modified and causal functions should satisfy Kramer-Kronig relations between magnitude and phase, the modified S-parameters become non-causal
Then the modified S-parameters were used to obtain the pulse response both with and without time domain causality enforcement
Figure shows the original causal pulse response, pulse response of the modified non-causal S-parameters and pulse response of the modified S-parameters with enforced causality 12

13. Simulation Example
Pulse response after enforcing causality in time domain was calculated with standard recursive convolution and proposed algorithm;
Using standard recursive convolution algorithm (red curve), we are getting unstable pulse response;
Proposed algorithm can fix the problem and we are getting stable pulse response (blue curve) 13

14. Thank you for your attention Any Questions?
Stable Recursive Convolution
Thank you for your attention
Any Questions? 14