2.
Monte Carlo Simulation
Simulation based power analysis require a set of
simulation vectors
The switching activity information is collected and
applied to appropriate power model
Each simulation vector causes some energy
dissipation and total power dissipation is derived by
summing up energy of each vector and dividing over
simulation time.
3.
Monte Carlo Simulation
How many input vectors are required for correct
estimation of Power dissipation??????
How much extra accuracy can be achieved by
simulating million vectors vs thousand vectors.
Stopping Criteria for simulation
4.
Consider Basic sample period T in which a single
power dissipation value is observed.
E.g. T may be several vectors or several clock cycles.
The estimated power of circuit under
simulation is given by average value of the
samples
𝑃 =
𝑝1+𝑝2+𝑝3+⋯……..𝑝𝑛
𝑛
classical mean estimation
Statistical Estimation of mean
5.
In statistics we draw N samples from a large
population and try to find mean of population.
For small value of N, P is not truthful and
for large N unnecessary computation will be
performed without gaining meaningful accuracy.
Statistical Estimation of mean
The
stopping
criteria
Determine
the sample
size N
6. Let pi are random variables following unknown
probability density function. The distribution of pi depends
on the circuit, simulation vectors and sample intervals.
Let µ and σ2 mean and variance of pi, Now the question is
how accurate is P in estimating µ with N samples?
According to the well-known central limit theorem in
statistics, the sample mean P approaches the normal
distribution for large N regardless of the distribution of Pi'
Assume that the samples Pi have normal distribution.
Basic statistical theory states that the average of normally
distributed random variables also has normal distribution.
The mean of P is exactly µ and its variance is
𝜎𝜌
2=
𝜎2
𝑁
(1)
7. The normal distribution curve for P
To quantify the accuracy of the sample mean P, a maximum error tolerance
term is used, Given , find what is the probability that P is within the error
range of
the true mean
what is the probability for the condition??
0 ≤
𝑃−𝜇
𝜇
≤ 𝜀 (2)
8. The normal distribution curve for P
0 ≤
𝑃−𝜇
𝜇
≤ 𝜀 (2)
# If this probability is high, trust the estimate P; otherwise, increase the
sample size N to gain more confidence.
# The probability can be obtained by integrating the normal distribution
curve p(x).
# The probability is more conveniently expressed by a confidence
variable
The confidence level is defined as 100 (1 - )%. A confidence level of 100%
( = 0) means that P is absolutely within the error tolerance of e.
* Typically, the confidence level is set to more than 90% to be
meaningful.
9. To explore the relationships among , and N, define a variable
z /2 such that the area between µ- z /2 crp and µ+ z /2 crp under the nonnal
distribution curve p(x) is (1 - ).
To Ensure the error condition
(3)
Using (1) and (3)
The value of z distribution is typically obtained
from a mathematical table known as the z-
distribution
function.
10. # It is actually not very practical because actual mean and variance are
unknown quantities dependent on the circuit, simulation vectors and the
sample interval.
# For limited sample size find the sample average and sample variance
# The variables P and S2 are quantities that can be directly computed from
the observed N samples
# To get the confidence for measurement within desired level we would
change the z-distribution to the t-distribution. Thus to achieve a confidence
level of (1 - ) and an error tolerance of , the number of samples required is:
11. The procedure is summarized as follows:
1. Simulate to collect one sample Pi'
2. Evaluate sample mean P and variance S2 using Equation
3. Check if the inequality is satisfied; if so stop, else repeat from Step 1
12.
1. Simulate to collect samples Pi‘
2. Evaluate sample mean P and variance S2
3. Check if the inequality is satisfied; if so stop, else
repeat from Step 1.
Monte carlo procedure