The transfer function and the frequency response of a multidimensional continuous-pulse system (Q1375019)

This paper proposes the concepts of parametric transfer function (PTF) and parametric frequency response (PFR) of a multidimensional linear continuous-pulse system. Assume the fractions \(F(s)= p^{-1}_F (s)q_F(s)\) and \(G(s)= p_G^{-1} (s)q_G(s)\) are strictly proper and proper respectively, the matrices \(p_F(s)\) and \(p_G (s)\) are complete (i.e., their inverses are noncancellable). The multidimensional digital filter (MDF) implemented by the digital computer \(C\) is assumed to be a matrix \(w_d(\zeta)= \alpha^{-1} (\zeta)\beta (\zeta)\) where \(\zeta= e^{-sT}\), \(\alpha (\zeta)= \alpha_0 +\alpha_1\zeta +\cdots +\alpha_r \zeta^r\) and \(\beta(\zeta) =\beta_0+\beta_1 \zeta+ \cdots +\beta_r \zeta^r\) \((T\) is the period of quantization). Then, the following results can be gained for the system under consideration: 1) The PTF exists uniquely and is defined by \(w(s,t)= F(s)+ \varphi_{FGH} (T,s,t)w_d(s) [E-\varphi_{FGH} (T,s,0) w_d(s)]^{-1}F(s)\) where \(w_d(s)= w_d(\zeta) |_{\zeta =e^{-sT}}\), \(H(s)\) is the transfer function of the forming element in the digital-to-analog converter, and \(\varphi_{FGH} (T,s,t)= {1\over T} \sum^\infty_{k=-\infty} F(s+ki \omega) G(s+ki \omega)H (s+ki \omega)e^{ki \omega t}\), \(\omega= {2\pi\over T}\). 2) The above PTF admits the representation \(w(s,t)=r(s,t)/ \Delta^*(s)\), where \(r(s,t)\) \(r(s,t+T)\) is an integer function of the argument \(s\) for every \(t\), and the system's characteristic function \(\Delta^*(s) =\Delta (\zeta) |_{\zeta =e^{-sT}} =\text{det}\left | \begin{smallmatrix} a(\zeta) & \zeta b(\zeta) \\ -\beta (\zeta) & \alpha(\zeta) \end{smallmatrix} \right|\). 3) For a nonresonant system, the frequency response is defined on the entire axis \(\nu\), and \(\| w(i\nu,t) \|<R (=\text{const)}\) for \(-\infty<t <\infty\) and \(-\infty <\nu <\infty\), where \(\|\;\|\) is a certain norm of a finite-dimensional number matrix. In addition, \(\| w(i\nu,t) \|\to 0\) for \(|\nu |\to \infty\), and the rate of decrease is the same as the rate of decrease of \(\| F(i\nu) \|\to 0\) for \(|\nu |\to \infty\).