Simulation
Realizable Systems (Chap 22.4)
Causality and spectral factorization (Chap 22.4)
Simulating a w.s.s. Random Sequence (Chap. 15.4)
- ∫12−12|lnSXf|df<∞
- G(f)=expa02+∑∞n=1anz−n, where
an=∫12−12lnSXfei2πfndf, z=ei2πf
- Causal iff GZ(z) analytic on and outside unit circle.
- SX(f) is real, then SX(z) zeros/poles are in conjugate reciprocal pairs;
SX(f) is nonnegative, then SX(z) zeros/poles are on unit circle and have even order;
RX(0)<∞, then there are no poles on unit circle
- on unit circle, z−1=z∗


Simulating a w.s.s. Random Waveform

- SX(f) is real, then zeros/poles are in conjugate pairs;
SX(f) is nonnegative, then SX(z) zeros/poles have even order;
RX(0)<∞, then there are no real poles.
- Causal iff H(f) analytic on and below real line.
- X(u,t) being real implies poles/zeros of SX(f) are symmetric about the origin.

🏷 Category=Probability