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

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.

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

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

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

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

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

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

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

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

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

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

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

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