जेड रूपान्तर - विकिपीडिया
गणित एवं संकेत प्रसंस्करण में जेड रूपान्तर (Z-transform) किसी डिस्क्रीट टाइम-डोमेन संकेत को समिश्र (कम्प्लेक्स) आवृत्ति-डोमेन में बदलता है। डिस्क्रीट टाइम-डोमेन संकेत से तात्पर्य ऐसे संकेत से है जो केवल कुछ निश्चित समयों पर अशून्य मान रखता है, शेष समय वह शून्य रहता है।
जेड-रूपान्तर को लाप्लास रूपान्तर का विविक्त-समय अनुरूप (discrete-time equivalent) के रूप में समझा जा सकता है। इसका उपयोग आंकिक संकेत प्रसंस्करण (डीएसपी) एवं आंकिक नियंत्रण (डिजिटल कन्ट्रोल) में किया जाता है।
1.1 द्विपक्षीय Z-रूपान्तर
1.2 एकपक्षीय Z-रूपान्तर
3 प्रमुख Z-रूपान्तर युग्म
द्विपक्षीय Z-रूपान्तर[संपादित करें]
एकपक्षीय Z-रूपान्तर[संपादित करें]
X ( z ) = Z { x [ n ] } = ∑ n = 0 ∞ x [ n ] z − n . {\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.\ }
z-ट्रांसफॉर्म के गुण
समय-डोमेन्
Z-डोमेन
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]\}} ROC: r 2 < | z | < r 1 {\displaystyle r_{2}<|z|<r_{1}\ }
रैखिकता (Linearity)
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{array}{lcl}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{array}}} At least the intersection of ROC1 and ROC2
r {\displaystyle 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{array}{lcl}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{array}}} R^{1/k}
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 & , let j = n − k = ∑ j = − k ∞ x [ j ] z − ( j + 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 , since x [ β ] = 0 if β < 0 = z − k X ( z ) {\displaystyle {\begin{array}{lcl}Z\{x[n-k]\}=&\\\sum _{n=0}^{\infty }x[n-k]z^{-n}\&\\{\text{ , let }}j=n-k&\\=\sum _{j=-k}^{\infty }x[j]z^{-(j+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}&\\{\text{ , since }}x[\beta ]=0{\text{ if }}\beta <0&\\=z^{-k}X(z)&\\\end{array}}} ROC, except z = 0 {\displaystyle z=0\ } if k > 0 {\displaystyle k>0\,} and z = ∞ {\displaystyle z=\infty } if k < 0 {\displaystyle k<0\ }
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{array}{lcl}Z\{a^{n}x[n]\}=&\\\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{array}}} | a | r 2 < | z | < | a | r 1 {\displaystyle |a|r_{2}<|z|<|a|r_{1}\ }
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{array}{lcl}{\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{array}}} 1 r 1 < | z | < 1 r 2 {\displaystyle {\frac {1}{r_{1}}}<|z|<{\frac {1}{r_{2}}}\ }
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{array}{lcl}Z\{x^{*}(n)\}=&\\\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\ &\\=\sum _{n=-\infty }^{\infty }[x(n)(z^{*})^{-n}]^{*}\ &\\=[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\ ]^{*}&\\=X^{*}(z^{*})&\\\end{array}}} ROC
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{array}{lcl}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{array}}} ROC
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{array}{lcl}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}=&\\{\mathcal {Z}}\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\}\ &\\=\sum _{n=-\infty }^{\infty }[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)]z^{-n}\ &\\=\sum _{l=-\infty }^{\infty }x_{1}(l)\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}]\ &\\=[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}][\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}]\ &\\=X_{1}(z)X_{2}(z)&\\\end{array}}} At least the intersection of ROC1 and ROC2
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}^{*}(1/z^{*})X_{2}(z)\ } At least the intersection of ROC of X 1 ( 1 / z ∗ ) {\displaystyle X_{1}(1/z^{*})} and X 2 ( z ) {\displaystyle X_{2}(z)}
x [ n ] − x [ n − 1 ] {\displaystyle x[n]-x[n-1]\ } ( 1 − z − 1 ) X ( z ) {\displaystyle (1-z^{-1})X(z)\ } At least the intersection of ROC of X1(z) and | z | > 0 {\displaystyle |z|>0}
∑ k = − ∞ n x [ k ] {\displaystyle \sum _{k=-\infty }^{n}x[k]\ } 1 1 − z − 1 X ( z ) {\displaystyle {\frac {1}{1-z^{-1}}}X(z)} ∑ n = − ∞ ∞ ∑ k = − ∞ n x [ k ] ⋅ z − n = ∑ n = − ∞ ∞ ( x [ n ] + x [ n − 1 ] + x [ n − 2 ] ⋯ x [ − ∞ ] ) z − n = X [ z ] ( 1 + z − 1 + z − 2 + z − 3 ⋯ ) = X [ z ] ∑ j = 0 ∞ z − j = X [ z ] 1 1 − z − 1 {\displaystyle {\begin{array}{lcl}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]\cdot z^{-n}\\=\sum _{n=-\infty }^{\infty }(x[n]+x[n-1]+\\x[n-2]\cdots x[-\infty ])z^{-n}\\=X[z](1+z^{-1}+z^{-2}+z^{-3}\cdots )\\=X[z]\sum _{j=0}^{\infty }z^{-j}\\=X[z]{\frac {1}{1-z^{-1}}}\end{array}}}
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}({\frac {z}{v}})v^{-1}\mathrm {d} v\ } -
x [ 0 ] = lim z → ∞ X ( z ) {\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ } , If x [ n ] {\displaystyle x[n]\,} causal
x [ ∞ ] = lim z → 1 ( z − 1 ) X ( z ) {\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ } , Only if poles of ( z − 1 ) X ( z ) {\displaystyle (z-1)X(z)\ } are inside the unit circle
प्रमुख Z-रूपान्तर युग्म[संपादित करें]
u [ n ] = { 1 , n ≥ 0 0 , n < 0 {\displaystyle u[n]={\begin{cases}1,&n\geq 0\\0,&n<0\end{cases}}}
δ [ n ] = { 1 , n = 0 0 , n ≠ 0 {\displaystyle \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}}
4 e − α n u [ n ] {\displaystyle \,e^{-\alpha n}u[n]} 1 1 − e − α z − 1 {\displaystyle 1 \over 1-e^{-\alpha }z^{-1}} | z | > | e − α | {\displaystyle |z|>|e^{-\alpha }|\,}
5 − u [ − n − 1 ] {\displaystyle -u[-n-1]\,} 1 1 − z − 1 {\displaystyle {\frac {1}{1-z^{-1}}}} | z | < 1 {\displaystyle |z|<1\,}
6 n u [ n ] {\displaystyle nu[n]\,} z − 1 ( 1 − z − 1 ) 2 {\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}} | z | > 1 {\displaystyle |z|>1\,}
7 − n u [ − n − 1 ] {\displaystyle -nu[-n-1]\,} z − 1 ( 1 − z − 1 ) 2 {\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}} | z | < 1 {\displaystyle |z|<1\,}
8 n 2 u [ n ] {\displaystyle n^{2}u[n]\,} z − 1 ( 1 + z − 1 ) ( 1 − z − 1 ) 3 {\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}} | z | > 1 {\displaystyle |z|>1\,}
10 n 3 u [ n ] {\displaystyle n^{3}u[n]\,} z − 1 ( 1 + 4 z − 1 + z − 2 ) ( 1 − z − 1 ) 4 {\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}} | z | > 1 {\displaystyle |z|>1\,}
11 − n 3 u [ − n − 1 ] {\displaystyle -n^{3}u[-n-1]\,} z − 1 ( 1 + 4 z − 1 + z − 2 ) ( 1 − z − 1 ) 4 {\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}} | z | < 1 {\displaystyle |z|<1\,}
12 a n u [ n ] {\displaystyle a^{n}u[n]\,} 1 1 − a z − 1 {\displaystyle {\frac {1}{1-az^{-1}}}} | z | > | a | {\displaystyle |z|>|a|\,}
13 − a n u [ − n − 1 ] {\displaystyle -a^{n}u[-n-1]\,} 1 1 − a z − 1 {\displaystyle {\frac {1}{1-az^{-1}}}} | z | < | a | {\displaystyle |z|<|a|\,}
14 n a n u [ n ] {\displaystyle na^{n}u[n]\,} a z − 1 ( 1 − a z − 1 ) 2 {\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}} | z | > | a | {\displaystyle |z|>|a|\,}
15 − n a n u [ − n − 1 ] {\displaystyle -na^{n}u[-n-1]\,} a z − 1 ( 1 − a z − 1 ) 2 {\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}} | z | < | a | {\displaystyle |z|<|a|\,}
16 n 2 a n u [ n ] {\displaystyle n^{2}a^{n}u[n]\,} a z − 1 ( 1 + a z − 1 ) ( 1 − a z − 1 ) 3 {\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}} | z | > | a | {\displaystyle |z|>|a|\,}
17 − n 2 a n u [ − n − 1 ] {\displaystyle -n^{2}a^{n}u[-n-1]\,} a z − 1 ( 1 + a z − 1 ) ( 1 − a z − 1 ) 3 {\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}} | z | < | a | {\displaystyle |z|<|a|\,}
18 cos ⁡ ( ω 0 n ) u [ n ] {\displaystyle \cos(\omega _{0}n)u[n]\,} 1 − z − 1 cos ⁡ ( ω 0 ) 1 − 2 z − 1 cos ⁡ ( ω 0 ) + z − 2 {\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}} | z | > 1 {\displaystyle |z|>1\,}
19 sin ⁡ ( ω 0 n ) u [ n ] {\displaystyle \sin(\omega _{0}n)u[n]\,} z − 1 sin ⁡ ( ω 0 ) 1 − 2 z − 1 cos ⁡ ( ω 0 ) + z − 2 {\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}} | z | > 1 {\displaystyle |z|>1\,}
20 a n cos ⁡ ( ω 0 n ) u [ n ] {\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,} 1 − a z − 1 cos ⁡ ( ω 0 ) 1 − 2 a z − 1 cos ⁡ ( ω 0 ) + a 2 z − 2 {\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}} | z | > | a | {\displaystyle |z|>|a|\,}
21 a n sin ⁡ ( ω 0 n ) u [ n ] {\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,} a z − 1 sin ⁡ ( ω 0 ) 1 − 2 a z − 1 cos ⁡ ( ω 0 ) + a 2 z − 2 {\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}} | z | > | a | {\displaystyle |z|>|a|\,}
फुरिये रूपान्तर
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