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where n is a positive integer, b n is either logic high or low, and x n are the distinct mutually independent<br />

primary inputs of node y.<br />

Activity Calculation Using Boolean Difference<br />

Activity a(y) at node y in a circuit is given by [1]<br />

© 2002 by CRC Press LLC<br />

(20.3)<br />

where a(xi) represents switching activity at input xi, while P(∂y/∂xi) is the probability of sensitizing input<br />

xi to output y.<br />

Equation (20.3) does not take simultaneous switching of the inputs into account. To consider the<br />

simultaneous switching, the following modifications have to be made:<br />

⊥ ⊥<br />

• P(∂y/∂xi) is modified to P(∂y/∂x i xi ), where xi denotes that input xi is switching.<br />

• a(xi) is modified to {a(x i) Π (1 – a(x j))}<br />

j≠i,1≤ j ≤ n<br />

Example<br />

For a Boolean expression y with three primary inputs x 1, x 2, x 3, the activity a(y) is given by the sum of<br />

three cases, namely,<br />

• when only one input is switching:<br />

i=1<br />

• when two inputs are switching:<br />

3<br />

∑<br />

a( y)<br />

P ∂y<br />

= ⎛------- ⎞<br />

∑ ⋅ a( x<br />

⎝ ⎠ i)<br />

• and when all three inputs are switching simultaneously:<br />

n<br />

i=1<br />

The activity calculation using Boolean difference can now be readily extended to the general case of n<br />

inputs.<br />

Activity Calculation Using Signal Probability<br />

The calculation of the activity of a node using the Boolean difference is computationally intensive. The<br />

complexity and computation time grow exponentially with the number of inputs. Hence, an alternative,<br />

and more efficient method to compute the activity using signal probability can be used instead.<br />

Let P(y(t)) be the signal probability at time t. The probability of a given node y is not switching at time<br />

1<br />

1<br />

t is P(y(t − T)y(t)) = P(y(t)) − --a(y)<br />

= P(y) − --a(y).<br />

Hence, a(y) = 2(P(y) − P(y(t − T)y(t))). To calculate<br />

2<br />

2<br />

the activity a(y) from the pre-computed signal probability P(y), we must first calculate P(y(t − Τ )y(t)).<br />

∂x i<br />

P ∂y ⎛------| ⎞⎛ ⊥ a( xi)<br />

1 a( x<br />

⎝ xi ⎠ ∏ ( – j)<br />

) ⎞<br />

⎝ ⎠<br />

∂x i<br />

j ≠1,1≤<br />

j ≤ n<br />

1<br />

∂<br />

-- P<br />

2<br />

2 ⎛ ⎞ ∂<br />

⎜-------------- y| 00 ⎟ P<br />

⎝∂xi∂xj⊥⎠ 2<br />

⎛ ⎛ ⎛ ⎞⎞<br />

⎜ + ⎜-------------- y| 01 ⎟⎟<br />

⎛a( xi)a(<br />

xj) ( 1 – a( xk) ) ⎞<br />

⎞<br />

⎜ ∑ ⎝ ⎝∂x ∏<br />

⎝ i∂x j<br />

⊥⎠⎠<br />

⎝ ⎠<br />

⎟<br />

⎠<br />

1≤i

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