Probabilistic Power Analysis
- 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
Editor's Notes
- #21: Another method that is generally more computation efficient involves the use of a more elaborate data structure called Boolean Decision Diagram