the ways in which we can manip- ulate them, we now turn to their use in analyzing various aspects of LPTV system behavior. In what follows, it is shown how HTMs can be used to perform multi-tone (section 3.5.1), stability (section 3.5.2) and noise (section 3.5.3) analysis of LPTV sys- items. 3.5.1 Multi-tone analysis 3.5.1 Multi-tone analysis Let us be given an LPTV system characterized by the HTM H(s). To this system, we apply an input signal u(t) with equivalent baseband components (envelopes) that equal (3.100) where Um E C. This corresponds to injecting extra (small-signal) tones into the system. These tones are located at Aw + mwo. The Laplace transforms of the envelopes in (3.100) equal Um(s)======== Ur,m (3.101) They all contain a pole on the imaginary axis with Ur,m as the associated residue. With an LPTV system's input-output behavior captured by (3.43), the output signal's equivalent baseband components are obtained from (3.102) where Ur = Just as it is the case for the input signal components, the Laplace transforms of the output signal components also contain a pole on the imaginary axis at S = jw. The associated vector of residues equals (3.103) with Y = Yr,-1 Y1;0 In steady state, the output signal's equiva- lent baseband components hence become yn(t) )=Yrnejson (3.104) Equation (3.103) relates the tones in the output spectrum to the tones in the input spec- trum and allows efficient evaluation of the LPTV system's steady-state response to multi-tone input signals. 3.5.2 Stability analysis The similarity between the input-output relation (3.76), characterizing the closed-loop behavior of a time-varying feedback system, and its counterpart in traditional LTI feed- back analysis raises the question whether powerful techniques for frequency-domain stability analysis, like Nyquist and Bode [Bode45], can be extended to the time-varying case. The answer turns out to be affirmative. Results that follow are similar to the ones in [Were91a, Were91b]. They are slightly simplified by exploiting the repetitive 