Method of Generating Continuous Spatial Correlations (16-Nov-2009)
Disclosed is a method of generating an always continuous spatial correlation (i.e., not a block-to-block discrete spatial correlation) of a process/device/circuit parameter for a desired spatial correlation. We use only (I + J + 2M) random variables to model a two-dimensional region which has been divided into in IxJ sub-regions. The spread of correlation at a fixed distance can be reduced by increasing the number of random variables (I + J + 2M).
Method of Generating Continuous Spatial Correlations
Consider spatial correlation among a set of semiconductor device parameters. Examples include FET channel length, FET channel width, OP resistor's resistance value, MIMCAP's capacitance value, VNCAP (BEOL metal wires formed capacitor) capacitance values, ring oscillator's
period/speed, other logic circuits (
NAND,
NO
R
, etc)
period. In each of the above examples, each
instance has the same mean value and the same standard deviation, but the correlation between any two instances vary (say, decrease) with the separation/distance between them.
Prelude. We consider a rectangular region, 0 ≤
x
≤ bJ , for a part of chip region or for the whole chip region. For a stochastic/random variable
P
≤ aI , 0 ≤
y
(r) located at position r = (
x
,
y
) in a
chip/die, we denote its mean as
μ, and denote its standard deviation as σ, namely,
1
/
2
1
/
2
P ≤
≤
(
r
)
= σ
μ
σ
μ r
r
r
r
,
(
)
≡( ) ( ) .
0
P
(
)
−
P
(
)
2 bJ
y
aI
x
P
=
(
)
−
2
=
0
≤
≤
,
(0.1)
We further denote the correlation of two instances
P
(r1) and
P
(r2) of the parameter as
r P
P
≡ r
P
(
r
1
)
−
P
(
r
1
)
P
(
r
2
)
−
P
(
r
2
)
(
(
r
)
μ −
)
(
(
μ
( 2
2
1
,
r
( )( ) ( )
( )
=
1
−
)
C
) .
2
)
P
(
r
1
)
−
P
(
r
)
2
1
P
(
r
)
−
P
(
σ
2
r
2
)
2
(0.2)
It is easy to see that
.
r
r C
C = (0.3)
The absolute value of correlation coefficient is upper-ward bounded,
.
1
)
,
( 2
1 ≤
( 1
2
2
1 r
,
)
(
r
,
)
r
r
C (0.4)
Without loss of generality, one can introduce a mean-zero and normalized random variable
p
(r)
such that
P
( r
r p
μ
+
σ
(
)
.
)= (0.5) In terms of
p
( 2 bJ
y
aI
x
p
p ≤
≤
(r), Eq. (0.1) becomes
( ) ,
0
,
0
,
1
)
(
= r
r (0.6) and Eq. (0.2) is reduced to
.
)
(
)
(
)
,
( 2
1
2
1 r
)
0
,
=
≤
≤
r p
p
C = (0.7)
Drawbacks of prior art.
In a typical block-based (or grid-based) spatial correlation, there is discontinuity at a block boundary. Moreover, if a spatial correlation function decays to 0 when its distance is increased to M or (M + 1) grid/block point apart, then the jump of a correlation value at each grid boundary
point is about 1/
M (Fig. 1). Figure 1 also illustrates that the spread of correlation value at a fixed distance is also about 1/M .
r
r
1
Fig. 1. Prior-art: Discontinuity in a block-based spatial correlation .
I. One-dimensional continuous spatial correlations: Let the whole one-dimensional region be 0 ≤
x
≤ aI (Fig. 2). The requirements (0.6) then
simplifies to
( 2 aI
x
x
p
x
p ≤
≤
)
=
0
,
(
)
=
1
,
0
.
(1.1)
Also let each of
g
0,
g
1,
g
2, … be an independent stochastic/random variable of mean zero and
standard deviation one,
,
0 2 L
=
,
=
1
,
,
0
,
1
,
2
= i
g
g i
i (1.2a)
.
,
,
2
,
1
,
0
,
,
0 j
i
j
i
g
g j
i ≠
=
= L (1.2b)
2
Fig. 2. One-dimensio...
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