A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[7] for a more detailed description. The argument presented in [7] is that the correct current waveform to be used for MTF estimation is one that combines the effects of all possible input waveforms. By considering the set of logical waveforms allowed at the circuit inputs as a probability space [5], the current in any branch of the bus becomes a stochastic process. CREST derives the expected (or mean) waveform (not a time average) of this process, which we call an expected current waveform, E[i(t) This is a waveform whose value at a given time is the weighted average of all possible ....

....t, then we denote by t and t the instances of time immediately before and after the event, respectively. Focusing for now on the output current pulse, its variance waveform starts with a peak of V [I] V [i(t ) at time t and decays linearly to zero at time t . Since V [I] E[I [5], and since CREST already derives the expected pulse peak (E[I] then it will be enough to derive E[I ] Let i p = i p1 i p2 and i n = i n1 i n2 . It s easy to verify that i p1 = i p Theta C n = C p C n ) and i n1 = i n Theta C p = C p C n ) Therefore : t) E[i ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....Signals We will use bold font to represent random quantities. We denote the probability of an event A by PfAg and, if x is a random variable, we denote its mean (or expected value) by E[x] and its distribution function by F x (a) Pfx ag. Let x(t) t 2 ( Gamma1; 1) be a stochastic process [2] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called a 0 1 process (see [3] pp. 38 39) A logic signal x(t) can be thought of as a sample of a 0 1 stochastic process x(t) i.e. x(t) is one of an infinity of possible signals that comprise ....

.... x(t) can be thought of as a sample of a 0 1 stochastic process x(t) i.e. x(t) is one of an infinity of possible signals that comprise the family x(t) A stochastic process is said to be strict sense stationary (SSS) if its statistical properties are invariant to a shift of the time origin [2]. Among other things, the mean E[x(t) of such a process is a constant, independent of time, and will be denoted by E[x] It will be shown below that a logic signal is always a sample of a SSS 0 1 process. Let n x (T ) denote the (random) number of transitions of x(t) in ( Gamma ] If x(t) ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....be applied to compute the power consumed in the combinational block. We will briefly survey the few recently proposed techniques for estimating the power in sequential circuits. All proposed techniques that handle sequential circuits [4 7] make the simplifying assumption that the FSM is Markov [8], so that its future is independent of its past once its present state is specified. Some of the proposed techniques compute only the probabilities (signal and transition) at the flip flop outputs, while others also compute the power. The approach in [4] solves directly for the transition ....

....specified. Some of the proposed techniques compute only the probabilities (signal and transition) at the flip flop outputs, while others also compute the power. The approach in [4] solves directly for the transition probabilities on the present state lines using the Chapman Kolmogorov equations [8], which is computationally too expensive. Another approach that also attempts a direct solution of the Chapman Kolmogorov equations is given in [5] While it is more e#cient, it remains quite expensive, so that the largest test case presented contains less than 30 flip flops. Better solutions are ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....information about this set by considering each transition as an event with a certain probability and then extracting statistical information about the current drawn based on these probabilities. Over this probability space, the supply (or ground) current waveform becomes a stochastic process [7]. An important second order property used to describe a stochastic process is its mean. In this case, the mean is a current waveform whose value at each time point is the expected value or mean of all the values that the actual current can take at that time. It is precisely this mean that the ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York, NY: McGraw-Hill Book Co., 1984.

....as observed in [2] the power is relatively insensitive to the particular distribution, rather it depends mainly on the input transition densities. Our implementation assumes that the distribution is geometric [4] This arises from a simple sufficient condition that an input signal be Markov [5], i.e. that its value after a clock edge depends only on its value before the clock edge, once that value is specified, and not on its values during earlier clock cycles. Under this assumption, we show in appendix B that the pulse widths have a geometric distribution. If 0 and 1 are the mean ....

....of the probability of transitioning at the clock edge. Again, the distribution of the pulse widths is arbitrary, and can be specified by the user. Our implementation is based on a Markov assumption, so that the length of time between successive transitions is a random variable with an exponential [5] distribution. The length of time a signal stays in the low (high) state has mean 0 ( 1 ) From this information, the waveform is easily generated using an exponential random number generator. Additionally, when running asynchronously the simulator requires a setup period. This is a waiting ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....which represents a combination of logical waveforms to be applied at the circuit inputs 1 ; this is the range of inputs over which the expected current waveform is to be derived. If certain probabilities are assigned to the elements of Omega Gamma then we can think of it as a probability space [8]. Associated with each element of Omega is an actual current waveform that the circuit would draw if subjected to that combination of inputs. This association (or mapping) defines a stochastic process i(t) whose mean E[i(t) is the expected current waveform to be derived. Likewise, every input ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York, NY: McGraw-Hill Book Co., 1984.

....for FSMs of moderate size. One technique, given in [28] completely ignores this problem and assumes that all states (of the FSM) are equally probable, which is not true in practice. Other techniques [40 43] have been proposed that are based on the simplifying assumption that the FSM is Markov [34] (so that its future is independent of its past once its present state is specified) This assumption is somewhat restrictive because it is only true when the sequence of input vectors at the FSM primary inputs are independent. Some of these techniques compute only the probabilities (signal and ....

....inputs are independent. Some of these techniques compute only the probabilities (signal and transition) at the FF outputs, while others also compute the power. The approach in [40] solves directly for the transition probabilities on the present state lines using the Chapman Kolmogorov equations [33, 34], which is computationally too expensive. Another approach that also attempts a direct solution of the ChapmanKolmogorov equations was given in [41] While it is more efficient, it remains quite expensive, so that the largest test case presented contains less than 30 FFs. Better solutions are ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....We will use bold font to represent random quantities. We denote the probability of an event A by PfAg and, if x is a random variable, we denote its mean (or expected value) by E[x] and its distribution function by F x (a) 4 = Pfx ag. Let x(t) t 2 ( Gamma1; 1) be a stochastic process [2] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called a 0 1 process (see [3] pp. 38 39) A logic signal x(t) can be thought of as a sample of a 0 1 stochastic process x(t) i.e. x(t) is one of an infinity of possible signals that ....

.... x(t) can be thought of as a sample of a 0 1 stochastic process x(t) i.e. x(t) is one of an infinity of possible signals that comprise the family x(t) A stochastic process is said to be strict sense stationary (SSS) if its statistical properties are invariant to a shift of the time origin [2]. Among other things, the mean E[x(t) of such a process is a constant, independent of time, and will be denoted by E[x] It will be shown below that a logic signal is always a sample of a SSS 0 1 process. Let n x (T ) denote the (random) number of transitions of x(t) in ( Gamma T 2 ; T ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....we will denote its mean (or expected value) by E[x] In this section, we will briefly review the concept of a companion process from [1] Let x(t) t 2 ( Gamma1; 1) be a logic signal, i.e. it is a function of time that takes the values 0 or 1. A 0 1 stochastic process is a stochastic process [2] that takes only 0 1 values. We associate with x(t) a 0 1 stochastic process x(t) called its companion process, defined as x(t) 4 = x(t ) where is uniform over the whole real line [1] It was shown in [1] that x(t) is strict sense stationary (for brevity : stationary) and mean ergodic. ....

....make use of the intuitive property that the values of a logic signal at widely separated time points are relatively independent. If we extend this property to the point that the future value of the signal is independent of its past, once its present value is specified, then it is said to be Markov [2] and its high and low pulses are known to be exponentially distributed. In the absence of any other information, therefore, it seems that the exponential distribution is a reasonable assumption. We will come back to this point later in this section, after we ve considered the effect of these ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....the paper, the propagation of probabilities is performed at the switch level, but this is not essential to the approach. The simplest way to propagate probabilities is to work with a gate level description of the circuit. Thus if y = AND(x 1 ; x 2 ) then it follows from basic probability theory [34] that P s (y) P s (x 1 )P s (x 2 ) provided x 1 and x 2 are (spatially) independent. Similarly, other simple expressions can be derived for other gate types. Once the signal probabilities are computed at every node in the circuit, the power is computed by making use of (1) and (2) based on the ....

....are equally probable, which is not true in practice. To simplify the discussion, we will assume that the sequential circuit implements a finite state machine (FSM) with a connected state space. Another simplifying assumption that has been made by most researchers is to say that the FSM is Markov [34] (so that its future is independent of its past once its present state is specified) If the signal and transition probabilities at the present state inputs of the FSM (i.e. the latch outputs) are known, then, with some approximation, any of the above combinational circuit techniques can be used ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....the paper, the propagation of probabilities is performed at the switch level, but this is not essential to the approach. The simplest way to propagate probabilities is to work with a gate level description of the circuit. Thus if y = AND(x 1 ; x 2 ) then it follows from basic probability theory [34] that P s (y) P s (x 1 )P s (x 2 ) provided x 1 and x 2 are (spatially) independent. Similarly, other simple expressions can be derived for other gate types. Once the signal probabilities are computed at every node in the circuit, the power is computed by making use of (1) and (2) based on ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....for MTF estimation. The argument presented in [1] is that the desired current waveform in any branch of the bus is one that combines the effects of all possible waveforms at the circuit primary inputs. By considering the set of logical waveforms allowed at the circuit inputs as a probability space [5], the current in any branch of the bus becomes a stochastic process. CREST derives the expected (or mean) waveform (not a time average) of this process, which we call an expected current waveform, E[i(t) This is a waveform whose value at any given time is the weighted average of all possible ....

....let n T = bT=t 0 c. If J k , k = 1; N are defined as follows : J k 4 = 1 t 0 t 0 Z 0 f(j k )dt; 2:4) then : J eff = lim T 1 1 T T Z 0 f(j)dt = lim n T 1 N X k=1 J k n k (T ) n T = N X k=1 J k lim n T 1 h n k (T ) n T i : 2:5) By the law of large numbers [5], lim n T 1 [n k (T ) n T ] P k , which leads to : J eff = N X k=1 1 t 0 t 0 Z 0 f(j k )dt P k = 1 t 0 t 0 Z 0 h N X k=1 f(j k )P k i dt (2:6) and, finally : J eff = 1 t 0 t 0 Z 0 E Theta f(j) dt (2:7) where E[ denotes the expected value operator. ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....for estimating the power in sequential circuits. To simplify the discussion, we will assume that the sequential circuit implements a non decomposable finite state machine. All proposed techniques that handle sequential circuits [3 6] make the simplifying assumption that the FSM is Markov [7], so that its future is independent of its past once its present state is specified. ACM IEEE Design Automation Conference, 1995. Some of the proposed techniques compute only the probabilities (signal and transition) at the latch outputs, while others also compute the power. The approach in [3] ....

....1995. Some of the proposed techniques compute only the probabilities (signal and transition) at the latch outputs, while others also compute the power. The approach in [3] solves directly for the transition probabilities on the present state lines using the Chapman Kolmogorov equations [7], which is computationally too expensive. Another approach that also attempts a direct solution of the ChapmanKolmogorov equations is given in [4] While it is more efficient, it remains quite expensive, so that the largest test case presented contains less than 30 latches. Circuit Combinational ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....of the intuitive property that the values of a logic signal at widely separated time points are relatively independent. If we extend this property to the point that the future value of a signal is independent of its past, once its present value is specified, then the signal is said to be Markov [2] and its high and low pulses are known to be exponentially distributed. In the absence of any other information, therefore, we will assume an exponential distribution. 0 100 200 300 Input density (1e6) 0.2 0.4 0.6 0.8 1.0 Output density Input density logsim (exponential) densim (exponential) ....

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....T centered at , rather than at 0. We can then talk of the random power of x i (t) over the interval ( Gamma T 2 ; T 2 ] to be denoted by : P T = V 2 dd 2 m X i=1 C i n x i (T ) T (3) where n x i (T ) is now a random variable. It was shown in [8] that x i (t) is stationary [12] so that, for any T , the expected average number of transitions per second is a constant : D(x i ) E n x i (T ) T = lim T 1 n x i (T ) T (4) where E[ Delta] denotes the expected value (mean) operator. In [5] and [8] D(x i ) was called the transition density of x i (t) it is the ....

....If (i) the consecutive times between identical transitions of x i (t) are independent (which, using renewal theory (see [11] pp. 62 63) means that nx i (T ) T is normally distributed for large T ) and (ii) the nx i (T ) T are independent (so that they are also jointly normal (see [12], p. 126) for large T ) then P T is normal for large T (see [12] p. 144) To the extent that these conditions are approximately met in practice, the power should be approximately normal. We have found that for a number of benchmark digital circuits [13] the normality assumption is very good, as ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....[7] for a more detailed description. The argument presented in [7] is that the correct current waveform to be used for MTF estimation is one that combines the effects of all possible input waveforms. By considering the set of logical waveforms allowed at the circuit inputs as a probability space [5], the current in any branch of the bus becomes a stochastic process. CREST derives the expected (or mean) waveform (not a time average) of this process, which we call an expected current waveform, E[i(t) This is a waveform whose value at a given time is the weighted average of all possible ....

....and t the instances of time immediately before and after the event, respectively. Focusing for now on the output current pulse, its variance waveform starts with a peak of V [I] V [i(t ) at time t and decays linearly to zero at time t . Since V [I] E[I 2 ] Gamma E[I] 2 [5], and since CREST already derives the expected pulse peak (E[I] then it will be enough to derive E[I 2 ] Let i p = i p1 i p2 and i n = i n1 i n2 . It s easy to verify that i p1 = i p Theta C n = C p C n ) and i n1 = i n Theta C p = C p C n ) Therefore : E[i 2 (t) E[i 2 p ....

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....from probability theory, present a model of random logic signals, define the transition density, and basically lay down the foundations for the rest of the paper. Throughout this paper, we will use bold font to represent random quan tities. Let x(t) t 6 ( oo, oo) be a stochastic process [1] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called 0 3 process (see [2] pp. 38 39) A stochastic process is said to be sirict sense tionarF (SSS) if its statistical properties e invariant to a shift of the time origin [1] Specifically, ....

....process [1] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called 0 3 process (see [2] pp. 38 39) A stochastic process is said to be sirict sense tionarF (SSS) if its statistical properties e invariant to a shift of the time origin [1]. Specifically, the mean of such a process is a constant, independen of time. If a constant mean process x(t) h finite variance and s such hag x(t) and x(t r) become uncorrelaed r , hen x(t) is mea ergodic (see [1] pp. 245248) These conditions, which are satisfied for mos regular processes ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1084.

....from probability theory, present a model of random logic signals, define the transition density, and basically lay down the foundations for the rest of the paper. Throughout this paper, we will use bold font to represent random quantities. Let x(t) t 2 ( Gamma1; 1) be a stochastic process [1] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called a 0 1 process (see [2] pp. 38 39) A stochastic process is said to be strict sense stationary (SSS) if its statistical properties are invariant to a shift of the time origin [1] ....

....process [1] that takes the values 0 or 1, transitioning between them at random transition times. Such a process is called a 0 1 process (see [2] pp. 38 39) A stochastic process is said to be strict sense stationary (SSS) if its statistical properties are invariant to a shift of the time origin [1]. Specifically, the mean of such a process is a constant, independent of time. If a constant mean process x(t) has finite variance and is such that x(t) and x(t ) become uncorrelated as 1, then x(t) is mean ergodic (see [1] pp. 245 248) These conditions, which are satisfied for most ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

....the MTF. The argument presented in [4, 5] is that the correct current waveform to be used for MTF estimation is one that combines (in some sense) the effects of all possible logic input waveforms. By considering the set of logic waveforms allowed at the circuit inputs as a probability space [6], the current in any branch of the bus becomes a stochastic process [6] CREST derives the mean waveform (not a time average) of this process, which we call an expected current waveform. This is a waveform whose value at a given time is the weighted average of all possible current values at that ....

....waveform to be used for MTF estimation is one that combines (in some sense) the effects of all possible logic input waveforms. By considering the set of logic waveforms allowed at the circuit inputs as a probability space [6] the current in any branch of the bus becomes a stochastic process [6]. CREST derives the mean waveform (not a time average) of this process, which we call an expected current waveform. This is a waveform whose value at a given time is the weighted average of all possible current values at that time, as shown in Fig. 1. CREST uses statistical information about the ....

[Article contains additional citation context not shown here]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2 nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, Second Edition. New York, NY: McGraw-Hill Book Co., 1984.

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A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd Edition. New York, NY: McGraw-Hill Book Co., 1984.

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