e is a LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES more fundamental one: it relates the structure of HTMs to the waveform's non-unique linear periodically modulated signal model. This point of view is much more fruitful in extending LPTV theory to other classes of LTV systems. As a consequence of (3.53), it is sufficient to know H(s) for all s=o+ jw with o E R and WE [-wo/2,wo/2]. Furthermore, if H(s) is analytic in the right-hand complex plane, i.e. the LPTV system is stable and contains no right-hand plane poles, it is even sufficient to know H(jw) for w E [-wo/2,000/2]. Using Cauchy's integral theorem and analytic continuation [Arf85], it is possible to find H(s),Ts E C. Another important consequence of the structural relation (3.53) is the repetitive struc- ture of the poles and eigenvalues of H(s). As will be seen in section 3.5.2, this is important for LPTV stability analysis. If pec is a pole of H(s), i.e. (3.55) with H(s) the HTM's two-norm evaluated at S E C, then (3.53) immediately implies that p+jwo is also a pole of H(s). Furthermore, given a complex number a and a vector U such that for a given S H(s)U=AU, (3.56) we find, again using (3.53), that (3.57) This means that if 2 is an eigenvalue of H(s), it is also an eigenvalue of H(s+jwo). Hence, poles repeat themselves in the direction of the imaginary axis while, in that same direction, H(s) is characterized by the periodicity of its eigenvalues. 3.3.5 On the oo-dimensional nature of HTMs One of the main problems in dealing with HTMs is their oo-dimensional matrix struc- ture. As always, care must be taken in dealing with 8 in a sense that a proper limiting procedure is required: how is infinity being approached? As different paths to 8 may lead to different results, many paradoxes in mathematics can be traced down to careless specification of a proper limiting procedure. In this text, any co-dimensional HTM H(s) should be considered as the limit of a sequence of square K-dimensional submatrices HK(s). Written formally: (3