Frequency-domain interpolation of long structures for system-level ...

Frequency-domain interpolation of long structures for System-level Signal Integrity Analysis Mikheil Tsiklauri, Mikhail Zvonkin, Nana Dikhaminjia, Jun Fan and James L. Drewniak EMC Laboratory, Missouri University of Science and Technology, Rolla, MO, USA  Abstract—System-level modeling requires cascading frequency domain characteristics of the subparts of the system. It can only be done if the data is given in the same frequency samples. Since often it is not the case, interpolation has to be done to reduce the data to the same frequency points. Incorrect interpolation might create nonsense artifacts, especially if the model includes a long structure. In the present work interpolation problems of a frequency response data are studied. The practical solution for these problems is suggested and different measured test cases are investigated. Index Terms— Transfer function; Cascading; response; Frequency domain interpolation.

The interpolation given by the formula (1) works pretty well for short structures, but if the structure is long, then the interpolation can create artificial effects in the time domain. Linear interpolation using formula (1) was done for frequency responses of measured 3m differential cable and differential 1.2in microstrip with 2.4mm sma connectors and 6in cables at each side. The detailed geometry model of the microstrip is shown on Fig. 1. Fig. 2 shows the magnitudes and phases of frequency responses of the measured microstrip and 3m cable.

Impulse

I. INTRODUCTION

Fig.1. Geometry model of the microstrip

S

ystem-level simulation for Signal Integrity usually requires connecting together models of the various subparts. Different measurements and/or simulations might be a source of frequency characteristic data corresponding to those subparts. The frequency characteristics can be measured or simulated for different frequency samples. To obtain the whole system, it is necessary to cascade the frequency characteristics corresponding to the subparts of the model. Frequency characteristics can be cascaded only if they are given at the same frequency samples. Therefore, interpolation of such frequency responses is necessary to reduce the data to common frequency samples. Different interpolation methodologies can be used to reduce all frequency responses to the same frequency samples (e.g. see [1], [2]). The simplest frequency domain interpolation method is a linear interpolation of real and imaginary parts of the given frequency responses. Let us assume that original frequency response H Zk , k 1,2,...,N is given at frequency samples

Z1 , Z2 ,...,Z N and we need to interpolate it for the frequency samples Z1 , Z2 ,...,ZM . The linear interpolation of real and imaginary parts of the transfer function can be done using the following formula: Zi  Z k 1 Z  Zk ReH Zi ReH Z k  i ReH Z k 1 , Z k  Z k 1 Z k 1  Z k (1) Zi  Z k 1 Zi  Z k ImH Zi ImH Z k  ImH Z k 1 , Z k  Z k 1 Z k 1  Z k

Fig.2. Magnitudes/phases of Sdd21 of and 3m cable and microstrip

The microstrip with connectors was measured up to 30GHz and 3m Cable was measured up to 14GHz. Original frequency responses for the both structures are given on frequency samples with 10MHz step. Interpolation was done for 10.2MHz frequency step (original frequency samples changed just by 2%). The impulse response is obtained by the Inverse Fast Fourier Transform (IFFT) and the step response is obtained by the numerical integration of the impulse response. Effectively, it corresponds to the pulse/step responses with the rise/fall time equal to one time step. In the presented cases the time step is equal to 35ps for the 3m cable and 18ps for the microstrip.

where Zi  >Zk , Zk 1 @ .

978-1-4799-1993-2/15/$31.00 ©2015 IEEE

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transfer function: H  Z

H * Z . Therefore, the imaginary

part of the transfer function has to be 0 at DC and the real part can be interpolated as a quadratic function, using know values at the samples Z1 , Z2 , Z1 , and Z2 . Note that this extrapolation cannot be responsible for the observed time domain artifacts, for those artifacts only appear after the interpolation, but after the DC extrapolation and before the interpolation no artifacts are present.

Fig.3a. Impulse response before and after interpolation for 3m cable

Fig.4b. Step responses before and after interpolation for microstrip

The artifacts shown on Fig. 3a and 3b are related with the spectral leakage phenomena originated by the lack of coherence [3]. Fig.3b. Step response before and after interpolation for 3m cable

Fig. 3a and 3b demonstrate that even the small change in frequency step can cause artificial effects on impulse response of 3m cable after interpolation. It can be seen that the energy of the impulse response has got redistributed over the time interval, which also caused the decrease of the amplitude of the main pulse. However, Fig. 4a and 4b show that the same effect is not observed for the microstrip.

In the present work the nature of the artificial effects shown on Fig. 3a and Fig. 3b is studied and a practical solution to avoid this problem is suggested. The paper is organized as follows: the second section is dedicated to the study of nature of artifacts in time domain response after frequency domain interpolation; in the third section the practical solution is suggested; the last section of the paper summarizes the results.

II. ARTIFACTS AFTER FREQUENCY DOMAIN INTERPOLATION To analyse the nature of artifacts shown on Fig. 3a,b we need to look at the interpolated magnitude and phase of both frequency responses. Fig. 5a and 5b show magnitudes and phases of the original and interpolated frequency responses correspondingly for 3m cable and the microstrip.

Fig.4a. Impulse responses before and after interpolation for microstrip

Fig. 4a and 4b show the impulse and step responses before and after interpolation for the microstrip. For this case no distortion in time domain responses after the interpolations are observed. Note, that to apply FFT, the frequency samples have to start from 0 and be evenly spaced. The measured data starts from 10MHz and has step of 10MHz. Thus, extrapolation at DC has to be made. It was done using the Hermitian property of the

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Fig.5a. Magnitude of Sdd21 of 3m cable before and after interpolation

The corrected magnitude of the Sdd21 of the 3m cable interpolated using formula (2) is shown on the Fig. 7.

Fig.5b. Phase of Sdd21 of 3m cable before and after interpolation

Fig. 5a shows that the magnitude interpolation is very inaccurate. The reason is that the real and imaginary parts of frequency responses for long structures are highly oscillated and the linear interpolation of real/imaginary parts cannot give a good result because of insufficient frequency samples per period. On the other hand, as shown on Fig. 6 and 7, the same interpolation method gives accurate result for a short structure.

Fig.7. Sdd21 of 3m cable using the corrected magnitude interpolation

From the Fig. 7 and Fig. 5b it is seen that the magnitude of the interpolated frequency reponse is almost the same but as seen on the Fig. 8, the corresponding impulse and step responses still have artificial effects from both sides of the main pulse. The difference from real/imaginary parts interpolation is that the small copies of the main impulse from both sides have different signs (The signs of the artifacts are the same in case of the real/imaginary parts interpolation).

Fig.6a. Magnitude of Sdd21 of the microstrip before and after interpolation

Fig.8. Impulse and step response of 3m cable before and after magnitude interpolation

Fig.6b. Phase of Sdd21 of the microstrip before and after interpolation

However, the reason of the artifacts on Fig. 3a,b is not the magnitude oscillation of the interpolated frequency response shown on the Fig. 5a. These artifacts in the time domain will not disappear if the magnitude is corrected with the logarithmic interpolation using the following formula:

log10 H Zi

Zi  Zk 1 log10 H Z k Zk  Zk 1 Z  Zk log10 H Z k 1 ,  i Zk 1  Zk

Zi  >Zk ,Zk 1 @

(2)

The artifacts seen on the Fig. 8 remind of the aliasing effect in the time domain [5]. Similarly to the frequency domain aliasing, the effect occurs when the data is undersampled. For the undersampled time domain data its frequency response gets corrupted and in the same way time domain response gets corrupted if the frequency response is undersampled. However, if the step is small enough this phenomenon does not occur. On contrary, the considered artifacts appear even if the interpolation step is smaller than the original step for which no artifacts are observed. To understand the reason behind the artifacts seen on the Fig.3a,b and Fig 8, the non-linear part of the phase before and after the interpolation should be observed. The non-linear part of the phase can be obtained by the following procedure: linear part of the unwrapped phase is defined as a straight line connecting the first and the last points of the given unwrapped phase; non-linear part of the unwrapped phase is the difference between the unwrapped phase and its linear part. Formulas (3)

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can be used for the non-linear part extraction (as it is discussed above, the first frequency point is assumed to be 0):

M Z

W

phase ^H Z `

M ZN O Z WZ ZN

T  Z M Z  O Z

(3)

phase ^H Z e jWZ ` ,

where M Z is the unwrapped phase of the given transfer

function H Z , O Z is its linear part, and T Z is its nonlinear part. The transformation H Z o H Z e jZW is equivalent to the shift of the corresponding impulse response in the time domain by W [4]. Therefore, it can be shown that this procedure effectively removes the delay of the channel. Fig. 8 shows that the non-linear part of the phase after interpolation oscillates around the original phase of 3m cable. The oscillation amplitude is pretty small – 0.05 radians (less than 3 degree), but as we see on Fig.8 it is enough to create small copies on the both sides of the main impulse.

Fig. 10 Impulse responses before and after 11MHz step interpolation

Fig.9. Non-linear part of 3m cable phase before and after interpolation Fig. 11 Impulse responses before and after 14.9MHz step interpolation

As it was shown in [6] non-linear part of the phase plays critical role in the causality property of the transfer function. Thus, the non-causal effects, observed in the impulse response after the interpolation, are expected. And therefore the noncausality in this case is not a reason of the time domain artifacts but another result of the incorrect interpolation. As it is shown on Fig. 10-12, these artifacts are smaller copies of the original impulse response and they are moving when the interpolation step is changing. The artifacts are near the main pulse for 10.2MHz interpolation step (Fig. 3a); for 11MHz interpolation step (Fig. 10) distance from the main pulse and artifacts increases on the both sides; for 14.9MHz interpolation step (Fig. 11) the left and the right side artifacts are approaching each other and for 15MHz interpolation step (Fig. 12) the left and the right side artifacts meet each other. For real/imaginary interpolation these artifacts have the same sign and are summed, while in case of magnitude correction, the artifacts have opposite signs and they are canceling each other. This phenomenon is periodic and the length of the period is 5MHz (50% of the original frequency step).

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Fig. 12 Impulse responses before and after interpolation with 15MHz frequency step

III. SOLUTION The problem of interpolation shown on Fig. 10-12 lies in the fact that real and imaginary parts of 3m cable are highly oscillated and in case of 10MHz frequency step there is no sufficient number of frequency samples per period. Interpolation procedure of this type of data is very sensitive.

As it is shown on Fig. 14 the linear phase extraction before interpolation gives accurate approximation of non-linear part of the phase and therefore, obtained impulse and step responses are without the artifacts (see Fig. 15, Fig. 16).

Fig. 16 Step responses before and after interpolation, using linear phase extraction before interpolation

Fig. 13 Real parts of the frequency response of 3m cable before and after 10.2MHz frequencys step interpolation

On Fig. 13 there is given real part of frequency response of 3m cable before and after 10.2MHz step interpolation (The similar picture we have for the imaginary part). This problem can be solved by removing the linear part of the phase before interpolation and return it after the interpolation is done.

From here follows that interpolation will be much more accurate if the linear portion of phase will be removed before interpolation and returned after interpolation, especially for long structures.

IV. DISCUSSIONS AND CONCLUSION There are other interpolation methods that help to avoid the interpolation problem for long structures. One of them is the interpolation of the magnitude and the unwrapped phase. However, for long structures the slope of the unwrapped phase is too steep while the number of samples per period might be low. In this case there might be a problem in the phase unwrapping procedure and ʹߨ jumps between periods can occur. On Fig. 17 there is shown the unwrapped phase of the simulated 7m cable. Standard Matlab function was used to unwrap the phase.

Fig. 14 Non-linear part of the phase of the frequency response of 3m cable using interpolation with and without removing the linear part

Fig. 17. Unwrapped phase with ૛࣊jumps.

Fig. 15 Impulse responses before and after interpolation, using linear phase extraction before interpolation

Alternatively, interpolation can be done by the vector fitting [7]. Vector fitting approximates a transfer function with analytical rational function using poles and residues. These poles and residues correspond to the elements of a circuit model. After an analytical representation is obtained, the value

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at any frequency point is known and can be used for the interpolation. Although this method gives good results, the algorithm is much harder to implement and it is more time consuming than the proposed approach, especially for the long, highly oscillating data. It has been demonstrated that linear interpolation of real and imaginary parts of the given frequency response can create artifacts in time domain, especially if the structure is long. Practical solution to avoid this type of artifact was suggested: removing linear portion of the phase can substantially improve interpolation quality. It is possible to avoid unwanted artifacts in time domain responses by using this methodology. ACKNOWLEDGEMENT This material is based upon work supported partially by the National Science Foundation under Grant No. 0855878. REFERENCES [1]

[2]

[3]

[4] [5] [6]

[7]

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