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।
परिभाषा
द्विपक्षीय Z-रूपान्तर
- <math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} </math>
एकपक्षीय Z-रूपान्तर
- <math>X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n}. \ </math>
गुण
| समय-डोमेन् | Z-डोमेन | सिद्धि | ROC | |
|---|---|---|---|---|
| निरूपण | <math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math> | <math>X(z)=\mathcal{Z}\{x[n]\}</math> | z|<r_1 \ </math> | |
| रैखिकता (Linearity) | <math>a_1 x_1[n] + a_2 x_2[n]\ </math> | <math>a_1 X_1(z) + a_2 X_2(z) \ </math> | <math>\begin{array} {lcl} X(z) = & \\
\sum_{n=-\infty}^{\infty} (a_1x_1(n)+a_2x_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_1X_1(z) + a_2X_2(z)\end{array} </math>
|
At least the intersection of ROC1 and ROC2 |
| Time expansion | <math>x_{(k)}[n] = \begin{cases} x[r], & n = rk \\ 0, & n \not= rk \end{cases}</math>
<math>r</math>: integer |
<math>X(z^k) \ </math> | <math>\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}</math> |
R^{1/k} |
| Time shifting | <math>x[n-k]\ </math> | <math>z^{-k}X(z) \ </math> | <math> \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} </math> |
ROC, except <math>z=0\ </math> if <math>k>0\,</math> and <math>z=\infty</math> if <math>k<0\ </math> |
| Scaling in
the z-domain |
<math>a^n x[n]\ </math> | <math>X(a^{-1}z) \ </math> | <math>\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} </math> |
a|r_2<|z|<|a|r_1 \ </math> |
| Time reversal | <math>x[-n]\ </math> | <math>X(z^{-1}) \ </math> | <math>\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} </math> |
z|<\frac{1}{r_2} \ </math> |
| Complex conjugation | <math>x^*[n]\ </math> | <math>X^*(z^*) \ </math> | <math>\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} </math> |
ROC |
| Real part | <math>\operatorname{Re}\{x[n]\}\ </math> | <math>\frac{1}{2}\left[X(z)+X^*(z^*) \right]</math> | ROC | |
| Imaginary part | <math>\operatorname{Im}\{x[n]\}\ </math> | <math>\frac{1}{2j}\left[X(z)-X^*(z^*) \right]</math> | ROC | |
| Differentiation | <math>nx[n]\ </math> | <math> -z \frac{dX(z)}{dz}</math> | <math>\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} </math> |
ROC |
| Convolution | <math>x_1[n] * x_2[n]\ </math> | <math>X_1(z)X_2(z) \ </math> | <math>\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} </math> |
At least the intersection of ROC1 and ROC2 |
| Cross-correlation | <math>r_{x_1,x_2}=x_1^*[-n] * x_2[n]\ </math> | <math>R_{x_1,x_2}(z)=X_1^*(1/z^*)X_2(z)\ </math> | At least the intersection of ROC of <math>X_1(1/z^*)</math> and <math>X_2(z)</math> | |
| First difference | <math>x[n] - x[n-1] \ </math> | <math> (1-z^{-1})X(z) \ </math> | z|>0</math> | |
| Accumulation | <math>\sum_{k=-\infty}^{n} x[k]\ </math> | <math> \frac{1}{1-z^{-1} }X(z)</math> | <math>\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}</math>
|
|
| Multiplication | <math>x_1[n]x_2[n]\ </math> | <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v \ </math> | z|<r_{1u}r_{2u} \ </math> |- | |
| Parseval's relation | <math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n]\ </math> | <math>\frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\frac{1}{v^*})v^{-1}\mathrm{d}v \ </math> |
- <math>x[0]=\lim_{z\rightarrow \infty}X(z) \ </math>, If <math>x[n]\,</math> causal
- <math>x[\infty]=\lim_{z\rightarrow 1}(z-1)X(z) \ </math>, Only if poles of <math>(z-1)X(z) \ </math> are inside the unit circle
प्रमुख Z-रूपान्तर युग्म
यहाँ:
- <math>u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}</math>
- <math>\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}</math>
| Signal, <math>x[n]</math> | Z-transform, <math>X(z)</math> | ROC | |
|---|---|---|---|
| 1 | <math>\delta[n] \, </math> | <math>1\, </math> | <math> \mbox{all }z\, </math> |
| 2 | <math>\delta[n-n_0] \,</math> | <math> z^{-n_0} \, </math> | <math> z \neq 0\,</math> |
| 3 | <math>u[n] \,</math> | <math> \frac{1}{1-z^{-1} }</math> | z| > 1\,</math> |
| 4 | <math>\, e^{-\alpha n} u[n] </math> | <math> 1 \over 1-e^{-\alpha }z^{-1}</math> | z| > |e^{-\alpha}| \,</math> |
| 5 | <math> -u[-n-1] \,</math> | <math> \frac{1}{1 - z^{-1}}</math> | z| < 1\,</math> |
| 6 | <math> n u[n] \,</math> | <math> \frac{z^{-1}}{(1-z^{-1})^2}</math> | z| > 1\,</math> |
| 7 | <math> - n u[-n-1] \,</math> | <math> \frac{z^{-1} }{ (1 - z^{-1})^2 }</math> | z| < 1 \,</math> |
| 8 | <math>n^2 u[n] \,</math> | <math> \frac{ z^{-1} (1 + z^{-1})}{(1 - z^{-1})^3} </math> | z| > 1\,</math> |
| 9 | <math> - n^2 u[-n - 1] \,</math> | <math> \frac{ z^{-1} (1 + z^{-1})}{(1 - z^{-1})^3} </math> | z| < 1\,</math> |
| 10 | <math>n^3 u[n] \,</math> | <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2})}{(1-z^{-1})^4} </math> | z| > 1\,</math> |
| 11 | <math>- n^3 u[-n -1] \,</math> | <math> \frac{z^{-1} (1 + 4 z^{-1} + z^{-2})}{(1-z^{-1})^4} </math> | z| < 1\,</math> |
| 12 | <math>a^n u[n] \,</math> | <math> \frac{1}{1-a z^{-1}}</math> | z| > |a|\,</math> |
| 13 | <math>-a^n u[-n-1] \,</math> | <math> \frac{1}{1-a z^{-1}}</math> | z| < |a|\,</math> |
| 14 | <math>n a^n u[n] \,</math> | <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> | z| > |a|\,</math> |
| 15 | <math>-n a^n u[-n-1] \,</math> | <math> \frac{az^{-1} }{ (1-a z^{-1})^2 }</math> | z| < |a|\,</math> |
| 16 | <math>n^2 a^n u[n] \,</math> | <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> | z| > |a|\,</math> |
| 17 | <math>- n^2 a^n u[-n -1] \,</math> | <math> \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} </math> | z| < |a|\,</math> |
| 18 | <math>\cos(\omega_0 n) u[n] \,</math> | <math> \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> | z| >1\,</math> |
| 19 | <math>\sin(\omega_0 n) u[n] \,</math> | <math> \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }</math> | z| >1\,</math> |
| 20 | <math>a^n \cos(\omega_0 n) u[n] \,</math> | <math> \frac{ 1-a z^{-1} \cos(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> | z| > |a|\,</math> |
| 21 | <math>a^n \sin(\omega_0 n) u[n] \,</math> | <math> \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }</math> | z| > |a|\,</math> |