Probabilistic Power Analysis for Low Power VLSI Design

1. Probabilistic Power Analysis
By
Dr. Gargi Khanna
Department of Electronics &
Communication Engg.
NIT Hamirpur

2. Introduction
 A different concept in power analysis, namely
probabilistic analysis.
 The primary reason for applying probabilistic analysis
is computation efficiency.
 The application of probabilistic power analysis
techniques has mainly been developed for gate level
abstraction and above.

3. Intro…
 A logic signal is viewed as a random zero-one
process with certain statistical characteristics.
 We no longer know the exact event time of each
logic signal switching.
 prescribe or derive several numerical statistical
characteristics of the signal.
The power dissipation of the circuit is then derived
from the statistical quantities.
Only a few statistical quantities need to be computed at
a given
node of the circuit as opposed to thousands of events
during simulation.
The biggest drawback of the probabilistic approach is
the loss in accuracy

4. Random Logic Signals
 The modeling of zero-one logic signals is crucial
to the understanding of probabilistic analysis .
 By capturing only a few essential statistical
parameters of a
signal, we can construct a very compact
description of the signal and analyze its effect on a
circuit.

5. Characterization of logic signals
 A logic signal only consists a waveform with
zero-one voltage levels
 The most precise way to logic signal is to record
all transitions of the signal at the exact times the
transition occur.
 To represent the signal, we write down the
initial state of the signal (state 1) and the time
value when each transition occurs (5, 15, 20,
35, 45).

6. Characterization of logic signals
 To compute the frequency of the signal, count how
many times the signal changes state and divide the
number by the observation period.
 This exact characterization of the signal gives
the full details of the signal history, allowing
precise reconstruction of the signal.
 For some purposes, the exact characterization of
the signal is too cumbersome, inefficient and
results in too much computation resource.
 E.g. To know the frequency of the signal, there is
no need to know the initial state and the exact
switching times; the number of switches should
be sufficient.

7.  number of transitions per unit time.
 By describing only the frequency of the signal, we
can reduce the computation requirements

8. Probability and Frequency
 Because of importance of the P=CV2f,
switching frequency is a very important characteristic in
the analysis of a digital signal.
 Regardless of the continuous or discrete signal model,
the switching frequency f of a digital signal is defined
as half number of transistors per unit time
N(T) is the number of logic transitions in the period T.
In the continuous random signal model, the
observation period is often not specified.
T
T
N
f
2
)
(


9. Static probability and Frequency
 The static probability of a digital signal is the ratio of the
time it remain in logic 1 (t1) to the total observation time t0+t1
expressed in a probability value between zero and one
 The static probability and the frequency of a digital
signal are related.
 If the static probability is zero or one, the frequency of
the signal has to be zero because if the signal makes a
transition, the ratio of logic 1 to logic 0 has to be strictly
between zero and one.
1
0
1
t
t
t
p



10. Static probability and Frequency
 The probability that the state is logic 1 is p1 =
P
 the probability that it is logic 0 is p0 = (1 - p).
 Suppose that the state is logic 1, the
conditional probability that the next state is
also logic 1
 PII = P and the conditional probability that the
next state is logic 0 is p10 = (1-p).
 memoryless assumption.
)
1
( p
p
f 


11.  The probability T that a transition occurs at a
clock boundary is the probability of a
 zero-to-one transition T01plus the probability of a
one-to-zero transition T10

12.  Expected frequency and static probability of
discrete random signals.

13. Conditional Probability and Frequency
 We define p01(p11) to be the conditional probabilities
that the current state will be logic 1, given that the
previous state was logic 0(logic 1).
 The four variables are not independent but related by
the following equations
1
1
10
11
00
01




p
p
p
p

14. The static probability p1(t) of the current state t is
dependent on the static probability of the previous
state p1 (t - 1) by
When the zero-one sequence is time homogeneous
01
10
01
1
p
p
p
p



15.  The equation for the static probability of the signal to the
conditional probabilities of transition is

16. Probabilistic Power Analysis
Techniques
 The random logic signals at the primary inputs are
expressed by some statistical quantities.
 From the primary inputs, we propagate the statistical
quantities to the internal nodes and outputs of the circuit.
 The propagation of the statistical quantities is done
according to a probabilistic signal propagation model.

17. Conti….
 The basic idea is to treat each transistor as a
switch controlled by its gate signal probability.
 The signal probability is propagated from the
source to the drain of a transistor, modulated by
the gate signal probability.
 In this way, the signal probabilities of all nodes in
the circuit are computed and the switching
frequencies of the nodes can be derived .
The power dissipation
 P = CV2f

18. Propagation of static probability in logic
circuits
 If we can find a propagation model for static
probability, we can use it to derive the frequency of
each node of a circuit, resulting in efficient power
analysis algorithm.
 Two input AND gate
 If the static probabilities of the inputs are p1 and p2
respectively and the two signals are statistically
uncorrelated, the output static probability is
p1p2because the AND-gate sends out a logic 1 if
and only if its inputs are at logic 1.

19. Propagation of static probability in logic
circuits
 Let y=f(x1,…… xn) be an n-input Boolean function.
Applying Shannon's decomposition w.r.t xi,
 The static probabilities of the input variables be
P(xl ), •.• , P(xn).
 Since the two sum terms in the decomposition
cannot be at logic 1 simultaneously, they are
mutually exclusive. We can simply add their
probabilities
i
i x
i
x
i f
x
f
x
y 


20. Conti….
 The new Boolean functions fxi , and , do not
contain the variable xi. The probabilities
 P are computed from the recursive application of
Shannon's
decomposition to the new Boolean functions.
At the end of the recursion,
P(y) will be expressed as an arithmetic function of
the input probabilities P(x).
The static probability can also be obtained from the truth table of
the Boolean function
by adding the probabilities of each row where the output is 1.

21. Example
 Boolean Function
 P(a)=0.1, P(b)=0.3 and P(c)= 0.2
 P(y)= 0.1*0.3+0.2-0.1*0.3*0.2
 =0.03+0.2-0.006= 0.224

22. Transition density signal model
 Probabilistic analysis of the gate-level circuit.
 A logic signal is viewed as a zero-one stochastic
process characterized by two parameters:
1. Static Probability: the probability that a signal is
at logic 1
2. Transition Density: the number of toggles per
unit time.

23.  Static probabilities and transition densities of
logic signals
In the lag-one model, when the static probability p and transition
density
T are specified, the signal is completely characterized

24. Propagation of Transition density
 For Boolean function y= f(x1,….xn), the static
probability P(xi) and transition density D(xi) of the
input variables are known.
 Find the static probability P(y) and transition
density D(y) of the output signal y.
 Zero gate delay model
 In other words, the exclusive-OR of the two
functions has to be 1, i.e.,
1


i
i x
x f
f

25. Propagation of Transition density
 Find the conditions in which an input
transition at Xi triggers an output
transition.
 Shannon's decomposition equation
 Xi makes a transition from 1 to 0.
 According to the Shannon's decomposition
equation,
 when Xi = 1, the output of y is

26.  If a 1-to-0 or 0-to-1 transition in Xi were to trigger
a logic
change in y,
. and must have different values.
There are only two possible scenarios
The exclusive-OR of the two functions has to be 1, i.e.,
1


i
i x
x f
f

27. Contd…….
 The exclusive-OR of the two functions is called the
Boolean difference of y w.r.t xi,
 Input transition at xi propagates to the output y if
and only if
dy / dxi = 1.
Let P(dy/dxi) = static probability that the Boolean
function dy/dxi evaluates to logic 1,and let D(xi) be the
transition density of xi.
The output has transition density of:::::
i
i x
x
i
f
f
dx
dy


uncorrelated inputs assump

28. Contd…….
The total transition density of the output y as



n
i
i
i
x
D
dx
dy
p
y
D
1
)
(
)
(
)
(
uncorrelated inputs assump

29. Example

31. Gate Level Power Analysis Using
Transition Density

32. Disadvantage
 Accuracy.
 Assumption of the analysis is that the logic gates
of the circuits have zero delay.
 The signal glitches and spurious transitions are
not properly modeled.
 The signal correlations at the primary inputs have
been ignored.
 transition density analysis is only works well on
combinational circuits.
computation

33. Signal Entropy
 Entropy is a measure of the randomness carried by
a set of discrete events observed over time.
 The average information content of the system is
the weighted sum of the information contents of Ci
by its occurrence probability pi. This is also called
the entropy of the system



m
i
p
i
i
p
H
1
1
2
log

34. END