Environmental Engineering Reference
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derived. The behavior of linear periodically time-varying systems in the
presence of noise inputs has been studied using autocorrelation function
and the network functions derived in Chapter 3. We have shown that al-
though the power spectral density of the output of periodically switched
linear circuits with stationary noise sources is periodically time-varying,
its average value is time-invariant and is determined from the aliasing
transfer functions from the noise sources to the output of the circuits.
An adjoint network-based noise analysis algorithm has been developed.
The effectiveness and efficiency of the method have been assessed using
practical examples with measurement results.
Finally, statistical analysis of periodically switched linear circuits in
the frequency domain using the first-order second-moment has been de-
veloped. We have shown that the method is computationally efficient
and gives accurate results when the coefficient of variance of circuit ele-
ments is low.
APPENDIX 11.A: Solution of Adjoint Network of
Periodically Switched Linear Circuits
In this appendix, we show that the adjoint network of a given periodically switched
linear circuit can be solved efficiently by utilizing the intrinsic relationship of these
two circuits.
When taking into account the time reversal characteristic of the adjoint network,
it can be shown that
and
of
N
relate to those of
by
where and are the conductance and capacitance matrices of
The superscript
T
denotes matrix transpose. Since
where
A
is a non-singular square matrix, also because
we
have
Therefore, the LU-factorization of can be obtained directly from that of
without additional computation. Also, because
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