Cumulative Distribution Function for Monte-Carlo Result

Dear illaoi,

illaoi said:

was wondering if there is a way to know the CDF of a random variable through calculator or does it require an SKILL code?

In my case, I have an amplifier and I have simulated its gain at a given frequency across monte-carlo mismatch, and I am interested to know the probability of the gain being between from A to B dB.

If my understanding of your need is correct, an alternative to computing the CDF might be to define a specification for the computed gain as being in the range [A,B]. If you are running a Monte Carlo simulation with N iterations and M of the simulations show a PASS for this new output expression, it suggests the chance that the gain is between A and B inclusive is M/N %. For convenience, if you create an output as a conditional expression for the gain when it is between A and B inclusive, you might assign it a "1" if it is within the inclusive range and a "0" otherwise. Hence the average value of this output expression over your Monte Carlo simulation is an estimate that the gain is between A and B inclusive.I apologize if I am not understanding correctly!

illaoi said:

say I am interested to know the sigma of the input offset voltage (across monte-carlo mismatch), and I have only M1 and M2 as active instances to show variation. Let's assume only Vth1 and Vth2 are random variables. Now, is it fair to state that since in monte-carlo simulation the co-variance of those two random variables is zero, but in real case, with a good layout, they might have non-zero covariance, the monte-carlo offset result is pessimistic (worst case), since the co-variance can reduce the overall sigma?

I do not believe this is an accurate statement illaoi. Why? The statistical model for threshold voltage variation is defined as that for random mismatch and is based on Pelgrom's law (see references [1] and [2]). Any correlation between the variation is a function of the layout and not included in the statistical model. Therefore, an excellent layout can provide matching that approaches that predicted by Pelgrom and anything less than an excellent layout can only increase the variation. I hope my explanation makes some sense illaoi.

Shawn

1. A. Sheikholeslami, "Process Variation and Pelgrom's Law [Circuit Intuitions]," in IEEE Solid-State Circuits Magazine, vol. 7, no. 1, pp. 8-9, Winter 2015,

2. M. J. M. Pelgrom, A. C. J. Duinmaijer and A. P. G. Welbers, "Matching properties of MOS transistors," in IEEE Journal of Solid-State Circuits, vol. 24, no. 5, pp. 1433-1439, Oct. 1989,