जेड रूपान्तर

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।

${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}}$

${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.\ }$

z-ट्रांसफॉर्म के गुण

	समय-डोमेन्	Z-डोमेन	सिद्धि	ROC
निरूपण	${\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}$	${\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}$		ROC: ${\displaystyle r_{2}<|z|<r_{1}\ }$
रैखिकता (Linearity)	${\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }$	${\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)\ }$	${\displaystyle {\begin{array}{lcl}X(z)=&\\\sum _{n=-\infty }^{\infty }(a_{1}x_{1}(n)+a_{2}x_{2}(n))z^{-n}\ &\\=a_{1}\sum _{n=-\infty }^{\infty }(x_{1}(n))z^{-n}+&\\a_{2}\sum _{n=-\infty }^{\infty }(x_{2}(n))z^{-n}&\\=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{array}}}$	At least the intersection of ROC1 and ROC2
Time expansion	${\displaystyle x_{(k)}[n]={\begin{cases}x[r],&n=rk\\0,&n\not =rk\end{cases}}}$ ${\displaystyle r}$: integer	${\displaystyle X(z^{k})\ }$	${\displaystyle {\begin{array}{lcl}X_{k}(z)=\sum _{n=-\infty }^{\infty }x_{k}(n)z^{-n}=&\\=\sum _{r=-\infty }^{\infty }x(r)z^{-rk}=&\\=\sum _{r=-\infty }^{\infty }x(r)(z^{k})^{-r}=&\\=X(z^{k})\end{array}}}$	R^{1/k}
Time shifting	${\displaystyle x[n-k]\ }$	${\displaystyle z^{-k}X(z)\ }$	${\displaystyle {\begin{array}{lcl}Z\{x[n-k]\}=&\\\sum _{n=0}^{\infty }x[n-k]z^{-n}\&\\{\text{ , let }}j=n-k&\\=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}&\\=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}&\\=z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}&\\=z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}&\\{\text{ , since }}x[\beta ]=0{\text{ if }}\beta <0&\\=z^{-k}X(z)&\\\end{array}}}$	ROC, except ${\displaystyle z=0\ }$ if ${\displaystyle k>0\,}$ and ${\displaystyle z=\infty }$ if ${\displaystyle k<0\ }$
Scaling in the z-domain	${\displaystyle a^{n}x[n]\ }$	${\displaystyle X(a^{-1}z)\ }$	${\displaystyle {\begin{array}{lcl}Z\{a^{n}x[n]\}=&\\\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}&\\=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}&\\=X(a^{-1}z)&\\\end{array}}}$	${\displaystyle |a|r_{2}<|z|<|a|r_{1}\ }$
Time reversal	${\displaystyle x[-n]\ }$	${\displaystyle X(z^{-1})\ }$	${\displaystyle {\begin{array}{lcl}{\mathcal {Z}}\{x(-n)\}=&\\\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\ &\\=\sum _{m=-\infty }^{\infty }x(m)z^{m}\ &\\=\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\ &\\=X(z^{-1})&\\\end{array}}}$	${\displaystyle {\frac {1}{r_{1}}}<|z|<{\frac {1}{r_{2}}}\ }$
Complex conjugation	${\displaystyle x^{*}[n]\ }$	${\displaystyle X^{*}(z^{*})\ }$	${\displaystyle {\begin{array}{lcl}Z\{x^{*}(n)\}=&\\\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\ &\\=\sum _{n=-\infty }^{\infty }[x(n)(z^{*})^{-n}]^{*}\ &\\=[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\ ]^{*}&\\=X^{*}(z^{*})&\\\end{array}}}$	ROC
Real part	${\displaystyle \operatorname {Re} \{x[n]\}\ }$	${\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}$		ROC
Imaginary part	${\displaystyle \operatorname {Im} \{x[n]\}\ }$	${\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}$		ROC
Differentiation	${\displaystyle nx[n]\ }$	${\displaystyle -z{\frac {dX(z)}{dz}}}$	${\displaystyle {\begin{array}{lcl}Z\{nx(n)\}=&\\\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\ &\\=z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\ &\\=-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\ &\\=-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\ &\\=-z{\frac {dX(z)}{dz}}&\\\end{array}}}$	ROC
Convolution	${\displaystyle x_{1}[n]*x_{2}[n]\ }$	${\displaystyle X_{1}(z)X_{2}(z)\ }$	${\displaystyle {\begin{array}{lcl}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}=&\\{\mathcal {Z}}\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\}\ &\\=\sum _{n=-\infty }^{\infty }[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)]z^{-n}\ &\\=\sum _{l=-\infty }^{\infty }x_{1}(l)\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}]\ &\\=[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}][\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}]\ &\\=X_{1}(z)X_{2}(z)&\\\end{array}}}$	At least the intersection of ROC1 and ROC2
Cross-correlation	${\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]\ }$	${\displaystyle R_{x_{1},x_{2}}(z)=X_{1}^{*}(1/z^{*})X_{2}(z)\ }$		At least the intersection of ROC of ${\displaystyle X_{1}(1/z^{*})}$ and ${\displaystyle X_{2}(z)}$
First difference	${\displaystyle x[n]-x[n-1]\ }$	${\displaystyle (1-z^{-1})X(z)\ }$		At least the intersection of ROC of X1(z) and ${\displaystyle |z|>0}$
Accumulation	${\displaystyle \sum _{k=-\infty }^{n}x[k]\ }$	${\displaystyle {\frac {1}{1-z^{-1}}}X(z)}$	${\displaystyle {\begin{array}{lcl}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]\cdot z^{-n}\\=\sum _{n=-\infty }^{\infty }(x[n]+x[n-1]+\\x[n-2]\cdots x[-\infty ])z^{-n}\\=X[z](1+z^{-1}+z^{-2}+z^{-3}\cdots )\\=X[z]\sum _{j=0}^{\infty }z^{-j}\\=X[z]{\frac {1}{1-z^{-1}}}\end{array}}}$
Multiplication	${\displaystyle x_{1}[n]x_{2}[n]\ }$	${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\frac {z}{v}})v^{-1}\mathrm {d} v\ }$		-
Parseval's relation	${\displaystyle \sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\ }$	${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\frac {1}{v^{*}}})v^{-1}\mathrm {d} v\ }$

• Initial value theorem

${\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }$, If ${\displaystyle x[n]\,}$ causal

• Final value theorem

${\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }$, Only if poles of ${\displaystyle (z-1)X(z)\ }$ are inside the unit circle

यहाँ:

• ${\displaystyle u[n]={\begin{cases}1,&n\geq 0\\0,&n<0\end{cases}}}$
• ${\displaystyle \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}}$

	Signal, ${\displaystyle x[n]}$	Z-transform, ${\displaystyle X(z)}$	ROC
1	${\displaystyle \delta [n]\,}$	${\displaystyle 1\,}$	${\displaystyle {\mbox{all }}z\,}$
2	${\displaystyle \delta [n-n_{0}]\,}$	${\displaystyle z^{-n_{0}}\,}$	${\displaystyle z\neq 0\,}$
3	${\displaystyle u[n]\,}$	${\displaystyle {\frac {1}{1-z^{-1}}}}$	${\displaystyle |z|>1\,}$
4	${\displaystyle \,e^{-\alpha n}u[n]}$	${\displaystyle 1 \over 1-e^{-\alpha }z^{-1}}$	${\displaystyle |z|>|e^{-\alpha }|\,}$
5	${\displaystyle -u[-n-1]\,}$	${\displaystyle {\frac {1}{1-z^{-1}}}}$	${\displaystyle |z|<1\,}$
6	${\displaystyle nu[n]\,}$	${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$	${\displaystyle |z|>1\,}$
7	${\displaystyle -nu[-n-1]\,}$	${\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}$	${\displaystyle |z|<1\,}$
8	${\displaystyle n^{2}u[n]\,}$	${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$	${\displaystyle |z|>1\,}$
9	${\displaystyle -n^{2}u[-n-1]\,}$	${\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}$	${\displaystyle |z|<1\,}$
10	${\displaystyle n^{3}u[n]\,}$	${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$	${\displaystyle |z|>1\,}$
11	${\displaystyle -n^{3}u[-n-1]\,}$	${\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}$	${\displaystyle |z|<1\,}$
12	${\displaystyle a^{n}u[n]\,}$	${\displaystyle {\frac {1}{1-az^{-1}}}}$	${\displaystyle |z|>|a|\,}$
13	${\displaystyle -a^{n}u[-n-1]\,}$	${\displaystyle {\frac {1}{1-az^{-1}}}}$	${\displaystyle |z|<|a|\,}$
14	${\displaystyle na^{n}u[n]\,}$	${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$	${\displaystyle |z|>|a|\,}$
15	${\displaystyle -na^{n}u[-n-1]\,}$	${\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}$	${\displaystyle |z|<|a|\,}$
16	${\displaystyle n^{2}a^{n}u[n]\,}$	${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$	${\displaystyle |z|>|a|\,}$
17	${\displaystyle -n^{2}a^{n}u[-n-1]\,}$	${\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}$	${\displaystyle |z|<|a|\,}$
18	${\displaystyle \cos(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$	${\displaystyle |z|>1\,}$
19	${\displaystyle \sin(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}$	${\displaystyle |z|>1\,}$
20	${\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$	${\displaystyle |z|>|a|\,}$
21	${\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}$	${\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}$	${\displaystyle |z|>|a|\,}$

• लाप्लास रूपान्तर
• फुरिये रूपान्तर

• Z-Transform table of some common Laplace transforms
• Mathworld's entry on the Z-transform
• Z-Transform threads in Comp.DSP
• Z-Transform Module by John H. Mathews