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Estimation of Probability Density Function of Digital Substrate Noise in Mixed Signal System

  • Manisha Sharma
  • Pawan Kumar Singh
  • Tejbir Singh
  • Sanjay Sharma
Conference paper
Part of the Lecture Notes on Data Engineering and Communications Technologies book series (LNDECT, volume 18)

Abstract

The substrate noise generated in the mixed signal-integrated circuits, which encapsulates the analog, the RF, and the memory parts, is assumed to possess the non-Gaussian cyclostationary nature. This noise creates interference among the various parts of mixed signal circuits and even within the memory circuits itself. To estimate the PDF parameters of non-Gaussian noise, which is modeled by Cauchy’s distribution function (kind of non-Gaussian), the non-Gaussian noise is modeled by the non-Gaussian mixture density. The PDF parameters are estimated using the maximum log likelihood function, and the priori and post priori updates are used for updating the PDF parameters. The substrate noise in a CMOS inverter and in a chain of five CMOS inverters is estimated first, and then this has been considered as an example of non-Gaussian cyclostationary noise for PDF estimation. The probability density function (PDF) of non-Gaussian cyclostationary noise is analytically estimated in this paper.

Keywords

Substrate noise Probability density function Gaussian distribution Non-Gaussian distribution Cyclostationary process Cauchy’s distribution 

1 Introduction

The substrate noise is not possessing the Gaussian nature, i.e., it is a non-Gaussian in noise. Since, the characteristics of substrate noise in time domain in limited time period repeats itself on the full time scale. The substrate noise is then said to be a non-Gaussian and cyclostationary process. The probability density function (PDF) of non-Gaussian nonstationary noise is estimated by modeling it using the Gaussian process parameters in [1]. Also a general algorithm is discussed for the estimation of probability density function of non-Gaussian nonstationary noise process, but the non-Gaussian mixture is defined using the Gaussian mixture density. These methods cannot be applicable for the case of non-Gaussian cyclostationary process. Since the algorithm in [1] is useful for all consideration, distribution may be a kind of Gaussian or non-Gaussian, may be a kind of stationary of nonstationary, and may be a kind of unimodal or multimodal. The algorithm has the capability for estimation of PDF of non-Gaussian nonstationary with zero or nonzero mean. The estimation of probability density function of substrate noise, which is a kind of non-Gaussian cyclostationary noise, is discussed in this paper using the algorithm explained in [1], but the non-Gaussian sample with nonzero mean is modeled using the non-Gaussian mixture densities. The maximum log likelihood function is used to estimate the post-priori parameters of non-Gaussian cyclostationary noise [4, 7, 8]. Further, the priori and post-priori updates are used for the updating the PDF parameters of non-Gaussian sample over the full time scale. The substrate noise in chain CMOS inverters is considered as an example to estimate the PDF using the referred algorithm, and substrate noise is assumed to a kind of cyclostationary Cauchy’s distribution function. Since, the Cauchy’s distribution function can be used to characterize the noise generated in electronic circuits and photo diodes, the digital substrate noise in CMOS inverter is simulated using the gate-level macromodeling technique.

2 Substrate Noise in CMOS Inverter

The CMOS inverter circuit has been shown in Fig. 1 which is integrated over a lightly doped substrate and the substrate noise has been simulated for different design technologies.
Fig. 1

CMOS inverter for macromodel validation

The input waveform is shown in Fig. 2a, which is applied to the input of the CMOS inverter. A SPICE simulator is used to simulate the injected current in substrate at every transition of input for both types either high-to-low or low-to-high, and it is shown in Fig. 2b. The impact ionization and its varying nature are responsible for the generation of large false current at input transitions, and it is positive for most of the cases. The substrate current is also dependent on the integration technology. The higher is the fabrication technology, the greater is the substrate noise current. The glitches and the switching phenomenon in complex circuits generate the high-frequency components.
Fig. 2

(a) Input waveform and (b) resulting substrate current waveform injected into the substrate on different transitions of input

3 Gate-Level Macromodeling

The gate-level macromodeling is used for the estimation of substrate noise in CMOS inverter. To develop the macromodel for a circuit, a gate-level macromodel is first developed, and then the individual macromodel is combined together for circuit macromodel. This system macromodel is again combined with substrate macromodel for the development of system macromodel, and a circuit simulator is used for the simulation of substrate noise. For more detail, a fundamental digital cell (CMOS inverter) is driven by a same kind of cell, and its circuit macromodel is shown in Fig. 3a, b. The four important parts of this circuit macromodel are first, the switching current IS; second, the capacitance of well CWel; third, the substrate resistance, RSu; and fourth, the capacitance at the output of the cell Cout. The gate-level macromodels are combined, and the circuit macromodel is developed, which is shown in Fig. 3c. The overall values of the elements are given by Eqs. (1), (2), and (3).
$$ {C}_{\mathrm{out}}=\sum {C}_{\mathrm{out}\mathrm{i}} $$
(1)
$$ \frac{1}{R_{\mathrm{Su}}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$ j\omega {C}_{\mathrm{Wel}}$}\right.}=\sum \frac{1}{R_{\mathrm{Su}\mathrm{i}}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$ j\omega {C}_{\mathrm{Wel}\mathrm{i}}$}\right.} $$
(2)
$$ {I}_{\mathrm{S}}=\sum {I}_{\mathrm{S}\mathrm{i}} $$
(3)
Fig. 3

(a) Basic digital gate circuit driven by another gate, (b) noise equivalent macromodel circuit, and (c) simplified macromodel

These equations represent the values after the superposition of individuals, and most importantly the switching currents of individual cells are added at their exact transition time.

The substrate noise will not accurately be estimated without clear information of substrate macromodel. The circuit macromodel shown in Fig. 3c is combined with substrate macromodel which is developed by the substrate resistive macromodel technique for the exact estimation of substrate noise CMOS inverters.

The combined macromodel for the system is shown in Fig. 4. This system macromodel is further used for the simulation of substrate noise in a chain of five CMOS inverters, and the substrate noise waveform is shown in Fig. 5.
Fig. 4

Simplified combined macromodel for the substrate and circuit

Fig. 5

Substrate noise voltage for the chain of five inverters, when ground inductor is 0.6 nH

4 PDF Parameter Estimation Substrate Noise

A Non-Gaussian cyclostationary sequence f(x t ) has been considered which denotes the PDF of x t and is being characterized by Cauchy’s mixture density of size m. The estimation of noise parameter at each time instant is done by using the maximum log likelihood function. Let f(x t ), the PDF of x t at time t and is represented by a Cauchy’s mixture density of size m as
$$ f\left({x}_t\right)=\sum_{i=0}^m{p}_i(t)\ {f}_i\left({x}_t\right) $$
where f i (x t ) is Cauchy’s distribution function and given by
$$ {f}_i\left({x}_t\right)=\frac{\alpha_{i/\pi }}{{\left({x}_t-{\mu}_i\right)}^2+{\alpha}_i^2} $$
$$ \left({x}_t\right)={\sum}_{i=0}^m \ \ {p}_i(t)\ \frac{\alpha_i(t)}{\left[{\left\{{x}_t-{\mu}_i(t)\right\}}^2-{\alpha}_i^2(t)\right]} $$
(4)
where μ i (t) represents the mean and α i (t) represents the spreading nature of Cauchy’s distribution function.
Where f i (x t ) is representing the PDF of x t  for ith Cauchy’s density, μ i (t)=[μ1(t), μ2(t), ……. μ m (t)] and α i (t) = [α1(t), α2(t), ………α m (t)] represent the mean and spreading nature of the distribution function, respectively. The total probability can be given as
$$ {\sum}_{i=0}^m \ {p}_i(t)=1 $$
(5)
It is assumed that ith component of x t  is distributed normally with mean μ i (t) with a spreading \( {\alpha}_i^2(t) \) is denoted by N{α i (t), μ i (t)} at the tth time step which is being related with the event as
$$ {A}_i(t)=\left[x\sim N\left\{{\alpha}_i(t),{\mu}_i(t)\right\}\right] $$
(6)
If the ith component of Cauchy’s distribution is inured on signal \( {x}_t^l, \) then using Baye’s rule, the post prior probability can be given as
$$ P\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}=\frac{p\left\{{x}_t^l \ | \ {f}_i\left({x}_t^l\right)\right\}\ P\left\{{f}_i\left({x}_t^l\right)\right\}}{\sum_{j=1}^mp\left\{{x}_t^l \ | \ {f}_i\left({x}_t^l\right)\right\}\ P\left\{{f}_i\left({x}_t^l\right)\right\}} $$
The conditional probability density function of \( {x}_t^l \) at tth time instant is \( p\left\{{x}_t^l \ | \ {f}_i\left({x}_t^l\right)\right\} \) gives \( {f}_i\left({x}_t^l\right) \) and \( {x}_t^l \)represents the lthspecific data set from total of m data sets and the total probability is given as
$$ {\sum}_{i=1}^mP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}=1 $$
(7)
Here, using the priori updates of parameter and using the maximum log likelihood w.r.t. variables of function p i (t), μ i (t), and α i (t), the parameters p i (t), μ i (t), and α i (t) are being estimated. The exact likelihood function is not used here, but log likelihood function is used because the use of the log likelihood function enables the simpler formulation, and at the same point, it gets maximum like the likelihood function. Using the model parameters, the log of the likelihood function for n independent samples is given as
$$ L(t)={\sum}_{l=1}^n{\log}_e{\sum}_{i=1}^m{p}_i(t). \ {f}_i\left\{{x}_t^l|{\alpha}_i(t),{\mu}_i(t)\right\} $$
(8)
If γ is considered to be Lagrange multiplier, the equation will take a form as
$$ L(t)={\sum}_{l=1}^n{\log}_e{\sum}_{i=1}^m{p}_i(t). \ {f}_i\left\{{x}_t^l|{\alpha}_i(t),{\mu}_i(t)\right\}-\gamma \left\{{\sum}_{i=1}^m{p}_i(t)-1\right\} $$
The parameter μ i (t) can be estimated only when L(t) is assumed to maximum for μ i (t), and it can be obtained by
$$ \frac{\partial L(t)}{\partial {\mu}_i(t)}=0 $$
(9)
The parameter μ i (t) is then estimated in Eq. 12 as
$$ {\mu}_i(t)=\frac{\sum_{l=1}^n{x}_t^l.P\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}}{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}} $$
(10)

Similarly, the parameter α i (t) can be estimated only when L(t) is assumed to maximum for α i (t), and it can be obtained by

$$ {\alpha}_i^2(t)=\frac{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}.{\left\{{x}_t^l-{\mu}_i(t)\right\}}^2}{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}} $$
(11)
Now, the parameter p i (t) can be estimated only when L(t) is assumed to maximum for p i (t), and it can be obtained by
$$ {p}_i(t)=\frac{1}{\gamma }{\sum}_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\} $$
(12)
Now, the total probability for m independent samples can be written as
$$ \sum_{i=1}^m{p}_i(t)=\sum_{i=1}^m\frac{1}{\gamma}\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\} $$
Then the parameter γ is being estimated as
$$ ={\sum}_{l=1}^n{\sum}_{i=1}^mP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\} $$
(13)
It is clear from Eq. 7, that total probability equal to 1, using this condition in Eq. 13,
$$ \gamma =n $$
Using the value of γ, the parameter p i (t) is now
$$ {p}_i(t)=\frac{1}{n}{\sum}_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\} $$
(14)

Putting the value of parameters p i (t), μ i (t), and α i (t),

$$ {\fontsize{8}{10}\selectfont{\begin{aligned} f\left({x}_t\right)&&=\sum_{i=0}^m\frac{1}{n}\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}\ \frac{{\left[\frac{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}.{\left\{{x}_t^l-{\mu}_i(t)\right\}}^2}{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}}\right]}^{1/2}/\pi }{\left[{\left\{x-{\frac{\sum_{l=1}^n{x}_t^l.P\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}}{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}}}_i\right\}}^2-\frac{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}.{\left\{{x}_t^l-{\mu}_i(t)\right\}}^2}{\sum_{l=1}^nP\left\{{f}_i\left({x}_t^l\right)|{x}_t^l\right\}}\right]} \end{aligned}}} $$
(15)

The parameters and weight of non-Gaussian cyclostationary noise, which is assumed to have Cauchy’s distribution, i.e., μ i (t), α i (t), and p i (t), are estimated using the discussed algorithm in two stages of priori updates and post priori updates. The priori update is used to set the values of parameters of Cauchy’s function at each time instant, and the final values are estimated using the post priori updates as given in Eqs. 10, 11, and 12. The priori values of parameters are set as.

The priori probability is set for m independent samples as
$$ {p}_{0i}(t)=\frac{1}{m} $$
(16)
The priori value for the α i (t) is set as
$$ {\alpha}_{0i}^2(t)={\alpha}_{01}^2\left(t-1\right) $$
(17)
whereas the priori value of mean μ i (t) is set for m independent samples as.
When m is odd,
$$ {\mu}_{0i}(t)=\mu \left(t-1\right)\left(1+\left(\frac{m+1}{2}-i\right)\frac{1}{m\ }\right) $$
(18)
And for even value of m and when i = 1 to m/2,
$$ {\mu}_{0i}(t)=\mu \left(t-1\right)\left(1+\left(\frac{m+2}{2}-i\right)\frac{1}{m\ }\right) $$
(19)
For even value of m and when i = m/2 to 1,
$$ {\mu}_{0i}(t)=\mu \left(t-1\right)\left(1+\left(\frac{m}{2}-i\right)\frac{1}{m\ }\right) $$
(20)
where
$$ \mu \left(t-1\right)={p}_1\left(t-1\right){\mu}_1\left(t-1\right)+\dots {p}_m\left(t-1\right){\mu}_m\left(t-1\right) $$
(21)

5 Result and Discussion

The probability density function of the substrate noise using the discussed algorithm is estimated. The priori update values are set, and post priori update values for the each parameter are evaluated; on the basis of these updates, the PDG is estimated. The substrate noise shown in Fig. 5 is considered in the estimation of probability density function.

The substrate noise is a non-Gaussian cyclostationary noise and assumed to have Cauchy’s distribution, hence modeled by a Cauchy’s density function. For simulation setup, total time is divided into 5 equal bins, and each time bin is containing 1000 samples. The parameter μ i (t), α i 2(t) and p i (t) have been estimated for each time bin. For example, considered in Fig. 5, the priori update values of μ i (t), α i 2(t), and p i (t) for different time bins and a particular time instant (say fifth time instant) are set for i = 1 to 5 (so, m = 5) and given as μ0i(t) = \( \left[0.3629\ 0.3110\ 0.2592\ 0.2074\ 0.1555\right],{\alpha}_{0i}^2(t) \) = [0.45 0.4 0.3 0.35 0.25] and p0i(t) = [0.2, 0.2, 0.2, 0.2, 0.2]; these priori updated values are again used for estimation of post priori updated values. The PDF of substrate noise for the mentioned updates at the fifth time instant is estimated and shown in Fig. 6.
Fig. 6

Probability density function of substrate noise at fifth time instant

The same example of substrate noise is now considered to have a Gaussian distribution, the same priori update values are taken, and the post priori updates are evaluating as per the Gaussian assumption. The probability density function obtained from modeling non-Gaussian cyclostationary substrate noise using Gaussian distribution function and modeling of same using Cauchy’s distribution function is plotted together and shown in Fig. 7.
Fig. 7

Comparison of probability density function obtained by Gaussian and non-Gaussian modeling

The probability density function of substrate noise by modeling this non-Gaussian cyclostationary (substrate noise) using Gaussian distribution and Cauchy’s distribution is estimated and compared in Fig. 7. The comparison of PDF by both modeling is suggested that if substrate noise will be modeled by the Gaussian distribution, the probability density function will be bimodal instead of unimodal. Therefore, the algorithm shows its capability that it’s an efficient way to estimate the probability density function of non-Gaussian cyclostationary noise.

6 Conclusion

The algorithm developed in this paper for the estimation of substrate noise has been validated using the simulation. In this proposed algorithm, the substrate noise has been assumed to be a kind of non-Gaussian cyclostationary noise and is being modeled by the Cauchy’s distribution function. The proposed model has also been compared with when the noise is modeled using a Gaussian distribution. The result has been established that the proposed algorithm has produced a better result. The PDF obtained by using the proposed algorithm is smooth PDF of the substrate noise.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Manisha Sharma
    • 1
  • Pawan Kumar Singh
    • 1
  • Tejbir Singh
    • 1
  • Sanjay Sharma
    • 2
  1. 1.ECED, SRM University Delhi-NCRSonepatIndia
  2. 2.ECED, Thapar UniversityPatialaIndia

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