 into account. Using, for example, the rank-three approximation G(s)~ G[-1,1], ( s) (3.119) 3.5 LPTV SYSTEM ANALYSIS USING HTMS x 10-3 -0.02 -0.04 -0.06 -0.08 Re{?} Re{?} x 10-4 Figure 3.20: Estimating the position of the eigenvalues of G(s) using (a) the Gershgorin theorem and (b) the eigenvalues obtained from a rank-reduced approximation. with G[-1,1],- (s) containing the rows - 1, 0 and 1 of G(s), we find stability to be deter- mined by the eigenvalues of H2(s-jwo)Hfb(s) Greduced = H2(s)Hgb(s-jwo) H2(s)Hfb(s+jwo) (3.120) H2(s-jwo)Hfb(s) These eigenvalues are given by (2(2a }. (3.121) Fig. 3.20(b) plots the non-zero eigenvalues in (3.121) for S = jw with W ranging from -wo/2 to wo/2. The eigenvalues turn out to be over an order of magnitude smaller than the worst-case upperbound (3.118) predicted by the Gershgorin theorem. Observing (3.121), it is seen that the eigenvalues can only be large in magnitude when the product H2 (s)Hfb(s) remains large for frequencies up to WO. Only then, the stability of the time-varying - feedback loop becomes an issue. 3.5.3 Noise analysis 3.5.3 Noise analysis The remainder of this discussion on LPTV system analysis using HTMs is devoted noise analysis. Just as LPTV system behavior transfers deterministic signal compo- nents from one frequency band to another, SO does it transfer random (stochastic) signal components. A complex-valued stochastic process n(t) is characterized by a probabil- ity density function (PDF) Pn (s(t)). This PDF specifies the probability for n(t) to be near a particular sample s(t). For a rigorous introduction to stochastic processes, see [Mid87]. Given a particular system and a random input signal's PDF, the task of finding the PDF that characterizes the output signal is often a very difficult one. However, in many instances, one is not interested in the entire stochastic characterization of n(t). Often, first- and second-order stochastic behavior, i.e. the mean and autocorrelation function mm(t) = E{n(t)} (3.122) Rnnt,t)=E{n(t)nt-t)*} (3.123) contain all 