The name is absent



d — [≠ι, Φ2, ∙ ∙ ∙, Φn]t ∙

Then, we need to solve following system of linear equations to find the gate
variations.

p = Ad.                            (3.4)

Since there are N unknown variables (φi,i — 1... N), N independent mea-
surements are needed to describe completely the solution of the linear system in
Equation 3.4. In the presence of power measurement noise, we can least square.

min∣∣Ad-p∣∣∣.                          (3.5)

We call this method the ^-minimization method.

Note that each input vector bj, based on the topology of the circuit, determines
a row of the measurement matrix A (power vector). It may be that the rows of
the measurement matrix are not necessarily independent, making it impossible
to find the variation of all gates by optimization as in Equation 3.5.

Multi-voltage leakage measurement

The number of independent power vectors (row of the measurement matrix)
may increase by increasing the number of power measurements, M. However,
circuit topology dictates an upper bound on the maximum number of independent
power vectors. But as discussed in Section 2.2, supply voltage and the leakage
current are not linearly dependent. Hence, measuring static power for different
supply voltages results in independent power vectors. We use this fact to increase
the number of the independent power vectors in the measurement matrix.

28



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