MaA-2025F-9.@2025-06-16T13:00D90

$ a_0u(t)+a_1u'(t)+\cdots=f(t)

$ \implies\mathcal F(f)(\omega)=G(\omega)\mathcal F(u)(\omega)

$ \omega:=\frac{2\pi}{T}

$ \mathcal F_{T,1}(f)(\omega)\sim\frac1{N\varDelta t}\sum_{0\le n<N}f(n\varDelta t)e^{-i\omega n\varDelta t}\varDelta t

$ = \frac1N\sum_{0\le n<N}f(n\varDelta t){e^{i\omega\varDelta t}}^{-n}

$ :=\mathcal Z_N(f(n\varDelta t))(e^{i\omega\varDelta t})

$ u_n:=u(n\varDelta t),f_n:=f(n\varDelta t)

$ \mathcal Z_N(u_\bullet)(z)=\frac1N\sum_{0\le n<N}\left(-\sum_{1\le j\le M}a_ju_{n-j}+\sum_{0\le j\le K}b_jf_{n-j}\right)z^{-n}

$ = -\sum_{1\le j\le M}a_j\frac1N\sum_{0\le n<N}u_{n-j}z^{-n}+\sum_{0\le j\le K}b_j\frac1N\sum_{0\le n<N}f_{n-j}z^{-n}

$ = -\sum_{1\le j\le M}a_jz^{-j}\frac1N\sum_{0\le n<N}u_{n-j}z^{-(n-j)}+\sum_{0\le j\le K}b_jz^{-j}\frac1N\sum_{0\le n<N}f_{n-j}z^{-(n-j)}

$ = -\sum_{1\le j\le M}a_jz^{-j}\mathcal Z_N(u_\bullet)(z)+\sum_{0\le j\le K}b_jz^{-j}\mathcal Z_N(f_\bullet)(z)

$ \because\frac1N\sum_{0\le n<N}u_{n-j}z^{-(n-j)}=\frac1N\sum_{-j\le n<N-j}u_nz^{-n}

$ = \frac1N\sum_{-j\le n<0}u_nz^{-n}+\frac1N\sum_{0\le n<N-j}u_nz^{-n}

$ = \frac1N\sum_{-j\le n<0}u_{n+N}z^{N-n}+\frac1N\sum_{0\le n<N-j}u_nz^{-n}

$ \because z^N=e^{i\omega N\varDelta t}=1

$ \because u_{n+N}=u(n\varDelta t+T)=u(n\varDelta t)=u_n

$ = \frac1N\sum_{N-j\le n<N}u_{n}z^{-n}+\frac1N\sum_{0\le n<N-j}u_nz^{-n}

$ =\frac1N\sum_{0\le n<N}u_nz^{-n}

$ =\mathcal Z_N(u_\bullet)(z)

$ = -M\mathcal Z_M(a_\bullet)(z)\mathcal Z_N(u_\bullet)(z)+K\mathcal Z_K(b_\bullet)(z)\mathcal Z_N(f_\bullet)(z)

$ \iff(1+M\mathcal Z_M(a_\bullet)(z))\mathcal Z_N(u_\bullet)(z)=K\mathcal Z_K(b_\bullet)(z)\mathcal Z_N(f_\bullet)(z)

$ \therefore G(\omega)=\frac{\mathcal F_{T,1}(u)(\omega)}{\mathcal F_{T,1}(f)(\omega)}\sim\frac{\mathcal Z_N(u(n\varDelta t))(e^{i\omega\varDelta t})}{\mathcal Z_N(f(n\varDelta t))(e^{i\omega\varDelta t})}=\frac{K\mathcal Z_K(b_\bullet)(e^{i\omega\varDelta t})}{1+M\mathcal Z_M(a_\bullet)(e^{i\omega\varDelta t})}

$ \mathcal F_{d,n}(u_\bullet)=\frac1N\sum_{0\le k<N}u_ne^{-2\pi i\frac{nk}{N}}

どうにかスッキリ整理できないかな