Thực đơn
Biến đổi Z
| Miền thời gian | Biến đổi Z | Ví dụ | ROC | |
|---|---|---|---|---|
| Chú giải | x [ n ] = Z − 1 { X ( z ) } {\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}} | X ( z ) = Z { x [ n ] } {\displaystyle X(z)={\mathcal {Z}}\{x[n]\}} | ||
| Tuyến tính | a 1 x 1 [ n ] + a 2 x 2 [ n ] {\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]} | a 1 X 1 ( z ) + a 2 X 2 ( z ) {\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)} | X ( z ) = ∑ n = − ∞ ∞ ( a 1 x 1 ( n ) + a 2 x 2 ( n ) ) z − n = a 1 ∑ n = − ∞ ∞ x 1 ( n ) z − n + a 2 ∑ n = − ∞ ∞ x 2 ( n ) z − n = a 1 X 1 ( z ) + a 2 X 2 ( z ) {\displaystyle {\begin{aligned}X(z)&=\sum _{n=-\infty }^{\infty }(a_{1}x_{1}(n)+a_{2}x_{2}(n))z^{-n}\\&=a_{1}\sum _{n=-\infty }^{\infty }x_{1}(n)z^{-n}+a_{2}\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\\&=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}}} | Contains ROC1 ∩ ROC2 |
| Thời gian mở rộng | x K [ n ] = { x [ r ] , n = r K 0 , n ≠ r K {\displaystyle x_{K}[n]={\begin{cases}x[r],&n=rK\\0,&n\not =rK\end{cases}}} r: integer | X ( z K ) {\displaystyle X(z^{K})} | X K ( z ) = ∑ n = − ∞ ∞ x K ( n ) z − n = ∑ r = − ∞ ∞ x ( r ) z − r K = ∑ r = − ∞ ∞ x ( r ) ( z K ) − r = X ( z K ) {\displaystyle {\begin{aligned}X_{K}(z)&=\sum _{n=-\infty }^{\infty }x_{K}(n)z^{-n}\\&=\sum _{r=-\infty }^{\infty }x(r)z^{-rK}\\&=\sum _{r=-\infty }^{\infty }x(r)(z^{K})^{-r}\\&=X(z^{K})\end{aligned}}} | R 1 K {\displaystyle R^{\frac {1}{K}}} |
| Decimation | x [ n K ] {\displaystyle x[nK]} | 1 K ∑ p = 0 K − 1 X ( z 1 K ⋅ e − i 2 π K p ) {\displaystyle {\frac {1}{K}}\sum _{p=0}^{K-1}X\left(z^{\tfrac {1}{K}}\cdot e^{-i{\tfrac {2\pi }{K}}p}\right)} | ohio-state.edu or ee.ic.ac.uk | |
| Chuyển dịch thời gian | x [ n − k ] {\displaystyle x[n-k]} | z − k X ( z ) {\displaystyle z^{-k}X(z)} | Z { x [ n − k ] } = ∑ n = 0 ∞ x [ n − k ] z − n = ∑ j = − k ∞ x [ j ] z − ( j + k ) j = n − k = ∑ j = − k ∞ x [ j ] z − j z − k = z − k ∑ j = − k ∞ x [ j ] z − j = z − k ∑ j = 0 ∞ x [ j ] z − j x [ β ] = 0 , β < 0 = z − k X ( z ) {\displaystyle {\begin{aligned}Z\{x[n-k]\}&=\sum _{n=0}^{\infty }x[n-k]z^{-n}\\&=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&&j=n-k\\&=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\&=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\&=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&&x[\beta ]=0,\beta <0\\&=z^{-k}X(z)\end{aligned}}} | ROC, except z = 0 if k > 0 and z = ∞ if k < 0 |
| Scaling in the z-domain | a n x [ n ] {\displaystyle a^{n}x[n]} | X ( a − 1 z ) {\displaystyle X(a^{-1}z)} | Z { a n x [ n ] } = ∑ n = − ∞ ∞ a n x ( n ) z − n = ∑ n = − ∞ ∞ x ( n ) ( a − 1 z ) − n = X ( a − 1 z ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\left\{a^{n}x[n]\right\}&=\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}\\&=X(a^{-1}z)\end{aligned}}} | | a | r 2 < | z | < | a | r 1 {\displaystyle |a|r_{2}<|z|<|a|r_{1}} |
| Đảo thời gian | x [ − n ] {\displaystyle x[-n]} | X ( z − 1 ) {\displaystyle X(z^{-1})} | Z { x ( − n ) } = ∑ n = − ∞ ∞ x ( − n ) z − n = ∑ m = − ∞ ∞ x ( m ) z m = ∑ m = − ∞ ∞ x ( m ) ( z − 1 ) − m = X ( z − 1 ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x(-n)\}&=\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\\&=\sum _{m=-\infty }^{\infty }x(m)z^{m}\\&=\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\\&=X(z^{-1})\\\end{aligned}}} | 1 r 1 < | z | < 1 r 2 {\displaystyle {\tfrac {1}{r_{1}}}<|z|<{\tfrac {1}{r_{2}}}} |
| Liên hợp phức | x ∗ [ n ] {\displaystyle x^{*}[n]} | X ∗ ( z ∗ ) {\displaystyle X^{*}(z^{*})} | Z { x ∗ ( n ) } = ∑ n = − ∞ ∞ x ∗ ( n ) z − n = ∑ n = − ∞ ∞ [ x ( n ) ( z ∗ ) − n ] ∗ = [ ∑ n = − ∞ ∞ x ( n ) ( z ∗ ) − n ] ∗ = X ∗ ( z ∗ ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x^{*}(n)\}&=\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\\&=\sum _{n=-\infty }^{\infty }\left[x(n)(z^{*})^{-n}\right]^{*}\\&=\left[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\right]^{*}\\&=X^{*}(z^{*})\end{aligned}}} | |
| Phần thực | Re { x [ n ] } {\displaystyle \operatorname {Re} \{x[n]\}} | 1 2 [ X ( z ) + X ∗ ( z ∗ ) ] {\displaystyle {\tfrac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]} | ||
| Phần ảo | Im { x [ n ] } {\displaystyle \operatorname {Im} \{x[n]\}} | 1 2 j [ X ( z ) − X ∗ ( z ∗ ) ] {\displaystyle {\tfrac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]} | ||
| Vi phân | n x [ n ] {\displaystyle nx[n]} | − z d X ( z ) d z {\displaystyle -z{\frac {dX(z)}{dz}}} | Z { n x ( n ) } = ∑ n = − ∞ ∞ n x ( n ) z − n = z ∑ n = − ∞ ∞ n x ( n ) z − n − 1 = − z ∑ n = − ∞ ∞ x ( n ) ( − n z − n − 1 ) = − z ∑ n = − ∞ ∞ x ( n ) d d z ( z − n ) = − z d X ( z ) d z {\displaystyle {\begin{aligned}{\mathcal {Z}}\{nx(n)\}&=\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\\&=z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\\&=-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\\&=-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\\&=-z{\frac {dX(z)}{dz}}\end{aligned}}} | |
| Tích chập | x 1 [ n ] ∗ x 2 [ n ] {\displaystyle x_{1}[n]*x_{2}[n]} | X 1 ( z ) X 2 ( z ) {\displaystyle X_{1}(z)X_{2}(z)} | Z { x 1 ( n ) ∗ x 2 ( n ) } = Z { ∑ l = − ∞ ∞ x 1 ( l ) x 2 ( n − l ) } = ∑ n = − ∞ ∞ [ ∑ l = − ∞ ∞ x 1 ( l ) x 2 ( n − l ) ] z − n = ∑ l = − ∞ ∞ x 1 ( l ) [ ∑ n = − ∞ ∞ x 2 ( n − l ) z − n ] = [ ∑ l = − ∞ ∞ x 1 ( l ) z − l ] [ ∑ n = − ∞ ∞ x 2 ( n ) z − n ] = X 1 ( z ) X 2 ( z ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}&={\mathcal {Z}}\left\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right\}\\&=\sum _{n=-\infty }^{\infty }\left[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right]z^{-n}\\&=\sum _{l=-\infty }^{\infty }x_{1}(l)\left[\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}\right]\\&=\left[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}\right]\!\!\left[\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}\right]\\&=X_{1}(z)X_{2}(z)\end{aligned}}} | Contains ROC1 ∩ |
| r x 1 , x 2 = x 1 ∗ [ − n ] ∗ x 2 [ n ] {\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]} | R x 1 , x 2 ( z ) = X 1 ∗ ( 1 z ∗ ) X 2 ( z ) {\displaystyle R_{x_{1},x_{2}}(z)=X_{1}^{*}({\tfrac {1}{z^{*}}})X_{2}(z)} | Contains the intersection of ROC of X 1 ( 1 z ∗ ) {\displaystyle X_{1}({\tfrac {1}{z^{*}}})} and X 2 ( z ) {\displaystyle X_{2}(z)} | ||
| Vi phân bậc một | x [ n ] − x [ n − 1 ] {\displaystyle x[n]-x[n-1]} | ( 1 − z − 1 ) X ( z ) {\displaystyle (1-z^{-1})X(z)} | Contains the intersection of ROC of X1(z) and z ≠ 0 | |
| Tích lũy | ||||
| Tích | x 1 [ n ] x 2 [ n ] {\displaystyle x_{1}[n]x_{2}[n]} | 1 j 2 π ∮ C X 1 ( v ) X 2 ( z v ) v − 1 d v {\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\tfrac {z}{v}})v^{-1}\mathrm {d} v} | At least r 1 l r 2 l < | z | < r 1 u r 2 u {\displaystyle r_{1l}r_{2l}<|z|<r_{1u}r_{2u}} |- |
Định lý Parseval
Định lý giá trị ban đầu: nếu x[n] là quan hệ nhân quả, thì
Định lý giá trị cuối cùng: nếu các cực của (z−1)X(z) ở trong vòng tròn đơn vị, thì
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