此筆記由本人基於我的老師的幻燈片撰寫。
∀ n ∈ Z δ ( n ) := { 1 n = 0 0 n ≠ 0 , ∀ m ∈ Z , n ∈ Z δ ( n − m ) = { 1 n = m 0 n ≠ m {\displaystyle {\underset {n\in \mathbb {Z} }{\forall }}\delta (n):={\begin{cases}1&n=0\\0&n\neq 0\end{cases}},\qquad {\underset {{\color {RoyalBlue}m\in \mathbb {Z} },\,n\in \mathbb {Z} }{\forall }}\delta (n{\color {RoyalBlue}{}-m})={\begin{cases}1&n=\color {RoyalBlue}m\\0&n\neq \color {RoyalBlue}m\end{cases}}}
∀ n ∈ Z u ( n ) := ∑ k = 0 ∞ δ ( n − k ) = { 1 n ⩾ 0 0 n < 0 , ∀ m ∈ Z , n ∈ Z u ( n − m ) = ∑ k = 0 ∞ δ ( n − m − k ) = ∑ k = m ∞ δ ( n − k ) = { 1 n ⩾ m 0 n < m {\displaystyle {\underset {n\in \mathbb {Z} }{\forall }}u(n):={\color {RoyalBlue}\sum _{k=0}^{\infty }}\delta (n-{\color {RoyalBlue}k})={\begin{cases}1&n\geqslant 0\\0&n<0\end{cases}},\quad {\underset {{\color {RoyalBlue}m\in \mathbb {Z} },\,n\in \mathbb {Z} }{\forall }}u(n{\color {RoyalBlue}{}-m})=\sum _{k=0}^{\infty }\delta (n{\color {RoyalBlue}{}-m}-k)=\sum _{k=\color {RoyalBlue}m}^{\infty }\delta (n-k)={\begin{cases}1&n\geqslant \color {RoyalBlue}m\\0&n<\color {RoyalBlue}m\end{cases}}}
∀ n ∈ Z δ ( n ) = ∑ k = 0 0 δ ( n − k ) = u ( n ) − u ( n − 1 ) , ∀ m ∈ Z , n ∈ Z δ ( n − m ) = ∑ k = m m δ ( n − k ) = u ( n − m ) − u ( n − m − 1 ) {\displaystyle {\underset {n\in \mathbb {Z} }{\forall }}\delta (n)=\sum _{k=0}^{\color {RoyalBlue}0}\delta (n-k)=u(n)-u(n-{\color {RoyalBlue}1}),\quad {\underset {{\color {RoyalBlue}m\in \mathbb {Z} },\,n\in \mathbb {Z} }{\forall }}\delta (n{\color {RoyalBlue}{}-m})=\sum _{k=m}^{\color {RoyalBlue}m}\delta (n-k)=u(n-{\color {RoyalBlue}m})-u(n{\color {RoyalBlue}{}-m-1})}
∀ N ∈ Z + , n ∈ Z R N ( n ) := ∑ k = 0 N − 1 δ ( n − k ) = u ( n ) − u ( n − N ) = { 1 0 ⩽ n ⩽ N − 1 0 n < 0 ∨ n > N − 1 {\displaystyle {\underset {{\color {RoyalBlue}N\in \mathbb {Z} ^{+}},\,n\in \mathbb {Z} }{\forall }}R_{N}(n):=\sum _{k=0}^{\color {RoyalBlue}N-1}\delta (n-k)=u(n)-u(n-{\color {RoyalBlue}N})={\begin{cases}1&0\leqslant n\leqslant N-1\\0&n<0\,\lor \,n>N-1\end{cases}}}
∀ N ∈ Z + , m ∈ Z , n ∈ Z R N ( n − m ) = ∑ k = 0 N − 1 δ ( n − m − k ) = ∑ k = m m + N − 1 δ ( n − k ) = u ( n − m ) − u ( n − m − N ) = { 1 m ⩽ n ⩽ m + N − 1 0 n < m ∨ n > m + N − 1 {\displaystyle {\begin{aligned}{\underset {N\in \mathbb {Z} ^{+},\,{\color {RoyalBlue}m\in \mathbb {Z} },\,n\in \mathbb {Z} }{\forall }}R_{N}(n{\color {RoyalBlue}{}-m})&=\sum _{k=0}^{N-1}\delta (n{\color {RoyalBlue}{}-m}-k)=\sum _{k=\color {RoyalBlue}m}^{{\color {RoyalBlue}m+}N-1}\delta (n-k)=u(n-{\color {RoyalBlue}m})-u(n{\color {RoyalBlue}{}-m-N})\\&={\begin{cases}1&{\color {RoyalBlue}m}\leqslant n\leqslant {\color {RoyalBlue}m+{}}N-1\\0&n<{\color {RoyalBlue}m}\,\lor \,n>{\color {RoyalBlue}m+{}}N-1\end{cases}}\end{aligned}}}
∀ x : Z → R , y : Z → R , n ∈ Z ( x ∗ y ) ( n ) := ∑ m = − ∞ ∞ x ( m ) y ( n − m ) , ∀ x : Z → R , n ∈ Z x ( n ) = ∑ m = − ∞ ∞ x ( m ) δ ( n − m ) = ( x ∗ δ ) ( n ) {\displaystyle {\underset {x:\mathbb {Z} \to \mathbb {R} ,\,y:\mathbb {Z} \to \mathbb {R} ,\,n\in \mathbb {Z} }{\forall }}(x*y)(n):={\color {RoyalBlue}\sum _{m=-\infty }^{\infty }}x({\color {RoyalBlue}m})\,y(n{\color {RoyalBlue}{}-m}),\quad {\underset {x:\mathbb {Z} \to \mathbb {R} ,\,n\in \mathbb {Z} }{\forall }}x(n)={\color {RoyalBlue}\sum _{m=-\infty }^{\infty }}x({\color {RoyalBlue}m})\,\delta (n-{\color {RoyalBlue}m})=(x*\delta )(n)}
（以上 R {\displaystyle \mathbb {R} } 可換成 C {\displaystyle \mathbb {C} } ，從此往後不再重複說明）
#離散時間系統#
T [ ⋅ ] {\displaystyle T[\cdot ]} 是線性系統 :⇔ T [ ⋅ ] {\displaystyle :\Leftrightarrow T[\cdot ]} 是離散時間系統 ∧ ∀ a 1 ∈ R , a 2 ∈ R , x 1 ∈ R Z , x 2 ∈ R Z ∀ n ∈ Z T [ a 1 x 1 + a 2 x 2 ] ( n ) = a 1 T [ x 1 ] ( n ) + a 2 T [ x 2 ] ( n ) {\displaystyle \land \,{\underset {a_{1}\in \mathbb {R} ,\,a_{2}\in \mathbb {R} ,\,x_{1}\in \mathbb {R} ^{\mathbb {Z} },\,x_{2}\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[a_{1}x_{1}+a_{2}x_{2}](n)=a_{1}T[x_{1}](n)+a_{2}T[x_{2}](n)}
⇔ T [ ⋅ ] {\displaystyle \Leftrightarrow T[\cdot ]} 是離散時間系統 ∧ ∀ x 1 ∈ R Z , x 2 ∈ R Z ∀ n ∈ Z T [ x 1 + x 2 ] ( n ) = T [ x 1 ] ( n ) + T [ x 2 ] ( n ) ∧ ∀ a ∈ R , x ∈ R Z ∀ n ∈ Z T [ a x ] ( n ) = a T [ x ] ( n ) {\displaystyle \land \,{\underset {x_{1}\in \mathbb {R} ^{\mathbb {Z} },\,x_{2}\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x_{1}+x_{2}](n)=T[x_{1}](n)+T[x_{2}](n)\land {\underset {a\in \mathbb {R} ,\,x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[ax](n)=aT[x](n)}
∀ x ∈ R Z ∀ n ∈ Z T [ x ] ( n ) := x ( n ) sin ( 2 π 9 n + π 7 ) {\displaystyle {\underset {x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x](n):=x(n)\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)}
t r u e ⇔ {\displaystyle true\Leftrightarrow }
∀ a 1 ∈ R , a 2 ∈ R , x 1 ∈ R Z , x 2 ∈ R Z ∀ n ∈ Z T [ a 1 x 1 + a 2 x 2 ] ( n ) = ( a 1 x 1 ( n ) + a 2 x 2 ( n ) ) sin ( 2 π 9 n + π 7 ) = a 1 T [ x 1 ] ( n ) + a 2 T [ x 2 ] ( n ) {\displaystyle {\underset {a_{1}\in \mathbb {R} ,\,a_{2}\in \mathbb {R} ,\,x_{1}\in \mathbb {R} ^{\mathbb {Z} },\,x_{2}\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[a_{1}x_{1}+a_{2}x_{2}](n)=(a_{1}x_{1}(n)+a_{2}x_{2}(n))\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)=a_{1}T[x_{1}](n)+a_{2}T[x_{2}](n)}
⇒ T [ ⋅ ] {\displaystyle \Rightarrow T[\cdot ]} 是線性系統
∀ x ∈ R Z ∀ n ∈ Z T [ x ] ( n ) := a x ( n ) + b ( a ∈ R , b ∈ R ) {\displaystyle {\underset {x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x](n):=ax(n)+b\,(a\in \mathbb {R} ,\,b\in \mathbb {R} )}
b ≠ 0 ⇒ ∀ x 1 ∈ R Z , x 2 ∈ R Z ∀ n ∈ Z T [ x 1 + x 2 ] ( n ) − T [ x 1 ] ( n ) − T [ x 2 ] ( n ) = − b ≠ 0 ⇒ T [ ⋅ ] {\displaystyle b\neq 0\Rightarrow {\underset {x_{1}\in \mathbb {R} ^{\mathbb {Z} },\,x_{2}\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x_{1}+x_{2}](n)-T[x_{1}](n)-T[x_{2}](n)=-b\neq 0\Rightarrow T[\cdot ]} 是非線性系統
b = 0 ⇒ ∀ x 1 ∈ R Z , x 2 ∈ R Z ∀ n ∈ Z T [ x 1 + x 2 ] ( n ) − T [ x 1 ] ( n ) − T [ x 2 ] ( n ) = − b = 0 ⇒ T [ ⋅ ] {\displaystyle b=0\Rightarrow {\underset {x_{1}\in \mathbb {R} ^{\mathbb {Z} },\,x_{2}\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x_{1}+x_{2}](n)-T[x_{1}](n)-T[x_{2}](n)=-b=0\Rightarrow T[\cdot ]} 是線性系統
T [ ⋅ ] {\displaystyle T[\cdot ]} 是時不變系統（移不變系統） :⇔ T [ ⋅ ] {\displaystyle :\Leftrightarrow T[\cdot ]} 是離散時間系統 ∧ ∀ x ∈ R Z ∀ m ∈ Z ∀ n ∈ Z T [ ⟨ x ( k − m ) ⟩ k ∈ Z ] ( n ) = T [ x ] ( n − m ) {\displaystyle \land \,{\underset {x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {m\in \mathbb {Z} }{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[\langle x(k-m)\rangle _{k\in \mathbb {Z} }](n)=T[x](n-m)}
∀ x ∈ R Z ∀ m ∈ Z ∀ n ∈ Z T [ ⟨ x ( k − m ) ⟩ k ∈ Z ] ( n ) − T [ x ] ( n − m ) = x ( n − m ) ( sin ( 2 π 9 n + π 7 ) − sin ( 2 π 9 ( n − m ) + π 7 ) ) {\displaystyle {\underset {x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {m\in \mathbb {Z} }{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[\langle x(k-m)\rangle _{k\in \mathbb {Z} }](n)-T[x](n-m)=x(n-m)\left(\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)-\sin \left({\frac {2\pi }{9}}(n-m)+{\frac {\pi }{7}}\right)\right)}
h := T [ δ ] {\displaystyle h:=T[\delta ]}
T [ ⋅ ] {\displaystyle T[\cdot ]} 是 LSI 系統 ⇒ {\displaystyle \Rightarrow }
∀ x ∈ R Z ∀ n ∈ Z T [ x ] ( n ) = T [ ⟨ ∑ m = − ∞ ∞ x ( m ) δ ( k − m ) ⟩ k ∈ Z ] ( n ) = ∑ m = − ∞ ∞ x ( m ) T [ ⟨ δ ( k − m ) ⟩ k ∈ Z ] ( n ) {\displaystyle {\underset {x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[x](n)=T\left[\left\langle \sum _{m=-\infty }^{\infty }x(m)\delta (k-m)\right\rangle _{k\in \mathbb {Z} }\right](n)=\sum _{m=-\infty }^{\infty }x(m)T[\langle \delta (k-m)\rangle _{k\in \mathbb {Z} }](n)}
= ∑ m = − ∞ ∞ x ( m ) T [ δ ] ( n − m ) = ( x ∗ T [ δ ] ) ( n ) = ( x ∗ h ) ( n ) {\displaystyle =\sum _{m=-\infty }^{\infty }x(m)T[\delta ](n-m)=(x*T[\delta ])(n)=(x*h)(n)}
∀ x ∈ R Z , y ∈ R Z ∀ n ∈ Z ( x ∗ y ) ( n ) = ∑ m = − ∞ ∞ x ( m ) y ( n − m ) = ∑ m = − ∞ ∞ x ( n − ( n − m ) ) y ( n − m ) = ∑ k = − ∞ ∞ y ( k ) x ( n − k ) = ( y ∗ x ) ( n ) {\displaystyle {\underset {x\in \mathbb {R} ^{\mathbb {Z} },\,y\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}(x*y)(n)=\sum _{m=-\infty }^{\infty }x(m)y(n-m)=\sum _{m=-\infty }^{\infty }x(n-(n-m))y(n-m)=\sum _{k=-\infty }^{\infty }y(k)x(n-k)=(y*x)(n)}
可換成 
，從此往後不再重複說明）
![{\displaystyle T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e23d4d9416b995643ca974f95de039d0fc89782d)
是線性系統 ![{\displaystyle :\Leftrightarrow T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f58be8b26dcf82436bc91711b0780c73c4bb3fd7)
是離散時間系統 =a_{1}T[x_{1}](n)+a_{2}T[x_{2}](n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/339a8fcdddf7a62abe71119be763ac42600c16ac)
![{\displaystyle \Leftrightarrow T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc75c5eab20aeba35ec5c209cc979b14ee327a21)
是離散時間系統 =T[x_{1}](n)+T[x_{2}](n)\land {\underset {a\in \mathbb {R} ,\,x\in \mathbb {R} ^{\mathbb {Z} }}{\forall }}\ {\underset {n\in \mathbb {Z} }{\forall }}T[ax](n)=aT[x](n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2efdc3d2e81fcf9cfc90ae0c236f9e937cd66465)
:=x(n)\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a69477d0fcf7085999353b05c8d4959b416fd13)
=(a_{1}x_{1}(n)+a_{2}x_{2}(n))\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)=a_{1}T[x_{1}](n)+a_{2}T[x_{2}](n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7795fe9ec182ab1df601d6c396b8acda57d10a7)
![{\displaystyle \Rightarrow T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0708b5c7c0efc182af56f94b28420eec146f441)
是線性系統
:=ax(n)+b\,(a\in \mathbb {R} ,\,b\in \mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3179a0607063c3c9d6061de3474a46294d5468)
-T[x_{1}](n)-T[x_{2}](n)=-b\neq 0\Rightarrow T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a768276167546a32e5b8c70a15ca56f8f0d2d9)
是非線性系統
-T[x_{1}](n)-T[x_{2}](n)=-b=0\Rightarrow T[\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26f52c607ac55f0bf0c256493a726dcb6abd8188)
是線性系統
是時不變系統（移不變系統） 是離散時間系統 =T[x](n-m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e143ecbb2075eba2b0eafedbce5a6ab450117f92)
-T[x](n-m)=x(n-m)\left(\sin \left({\frac {2\pi }{9}}n+{\frac {\pi }{7}}\right)-\sin \left({\frac {2\pi }{9}}(n-m)+{\frac {\pi }{7}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6e2a50db7ed23c9715cec289e96e60a9f832c8)
![{\displaystyle h:=T[\delta ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f09960b7144ba885686228f243d6f252eef6c7a)
是 LSI 系統 
=T\left[\left\langle \sum _{m=-\infty }^{\infty }x(m)\delta (k-m)\right\rangle _{k\in \mathbb {Z} }\right](n)=\sum _{m=-\infty }^{\infty }x(m)T[\langle \delta (k-m)\rangle _{k\in \mathbb {Z} }](n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70959eff299eec545b9a45a77ff4913e56a69f4d)
=(x*T[\delta ])(n)=(x*h)(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858cc926196f7af6aa5312b9443208613be5181a)
