Civil Engineering Reference
In-Depth Information
This leads to the relations
While matching the second and third central moments would certainly yield
values of and in practice‚ it was found that a time-shifted gamma function
provided more accurate results. Essentially‚ this involves matching the first
moment to a time shift‚
since the first moment of
is
this results
in setting
While closed forms for the median do not exist‚ it is relatively simple to
store them in two-dimensional look-up tables that are functions of and
If the input is not a step‚ but can be modeled as a saturated ramp (which
can be written as the difference between two time-shifted ramp functions)‚ the
response can be captured by two time-shifted gamma functions‚ separated by
the rise time of the saturated ramp. For details‚ the reader is referred to [KP98].
The h-gamma metric.
This technique was found to have limited accuracy
in cases such as those when interconnect resistance plays a large part‚ and
a modification‚ called the h-gamma method‚ was proposed in [LAP98]. The
method is predicated on the observation that the Laplace transform of the
step response can be divided into two components: a forced response and a
homogeneous response‚ as shown below‚ and illustrated in Figure 3.16:
where
is the forced response (
is the unit step function) and
is the homogeneous response‚
and
are the step response in the time
and frequency domains‚ respectively‚ and
is the transfer function in the
Laplace domain.
For an RC mesh‚ the homogeneous response increases monotonically from
0 to 1 under a step excitation‚ and the h-gamma method treats as a
probability density function that is approximated by the gamma distribution.
Specifically‚ in the frequency domain‚ the homogeneous response is represented
by
