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Thus, the and u(t), whose Fourier transforms are u˜ (− j ω), H( j ω), and u( Fourier transform on both sides of Eq. 108) Finally, from Eq. 109) Necessity We show that if the n-port is passive, then the general hybrid matrix H(s) must be positive real. To this end, we apply a particular excitation of the form u(t) = [u 1 (t), u 2 (t), . . 111b) with k = 1, 2, . . , n; where s0 = σ0 + j ω0 is an arbitrary point in the open RHS, and ck , φk , and φ0 are arbitrary real constants. Note that each excitation signal has the same associated σ0 , ω0 , and φ0 , but different ck and φk .

103) the total energy stored in the n-port. On the real-frequency axis, Eq. 104) Consider y˜ ( j w), u( ˜ j w), and H( j w) as the Fourier transforms of y(t), u(t), and h(t), respectively. 106) which is recognized as the convolution of the functions of the forms u (−t), h(t), ˜ j ω). Thus, the and u(t), whose Fourier transforms are u˜ (− j ω), H( j ω), and u( Fourier transform on both sides of Eq. 108) Finally, from Eq. 109) Necessity We show that if the n-port is passive, then the general hybrid matrix H(s) must be positive real.

16 A causal and nonanticipative oneport network. to be linear if the superposition principle holds, that is, if ya (t) and yb (t) are the responses of the excitations ua (t) and ub (t), respectively, of an n-port, then for any choice of real scalars α and β, the vector αya (t) + βyb (t) represents the response of the excitation αua (t) + βub (t). When we speak of an n-port that can support n linearly independent excitations, we mean that if {uk (t), yk (t); k = 1, 2, . } denotes the set of all excitation-response pairs that can be supported by the n-port, there exist n linearly independent excitation vectors uk (t)’s in the set.

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