Simulation
Realizable Systems (Chap 22.4)
Causality and spectral factorization (Chap 22.4)
Simulating a w.s.s. Random Sequence (Chap. 15.4)
- \( \int_{-\frac{1}{2}}^{\frac{1}{2}} |\ln S_X{f}| \mathrm{d} f < \infty \)
- \( G(f) = \exp{\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n z^{-n} } \), where \( a_n = \int_{-\frac{1}{2}}^{\frac{1}{2}} \ln S_X{f} e^{i2\pi fn} \mathrm{d} f \), \( z=e^{i2\pi f} \)
- Causal iff \( G_Z(z) \) analytic on and outside unit circle.
- \( S_X(f) \) is real, then \( S_X(z) \) zeros/poles are in conjugate reciprocal pairs; \( S_X(f) \) is nonnegative, then \( S_X(z) \) zeros/poles are on unit circle and have even order; \( R_X(0) <\infty \), then there are no poles on unit circle
- on unit circle, \( z^{-1} = z^{*} \)
Simulating a w.s.s. Random Waveform
- \( S_X(f) \) is real, then zeros/poles are in conjugate pairs; \( S_X(f) \) is nonnegative, then \( S_X(z) \) zeros/poles have even order; \( R_X(0) <\infty \), 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 \( S_X(f) \) are symmetric about the origin.
