Engineering Handbook/Mathematics/Z Transformation

Z Transformation Properties

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! 	! Time domain ! Z-domain ! ROC |- ! Notation | $$x[n]=\mathcal{Z}^{-1}\{X(z)\}$$ | $$X(z)=\mathcal{Z}\{x[n]\}$$ | ROC: $$r_2<|z|0\,$$ and $$z=\infty$$ if $$k<0\ $$ |- ! Scaling in the z-domain | $$a^n x[n]\ $$ | $$X(a^{-1}z) \ $$ | $$|a|r_2<|z|<|a|r_1 \ $$ |- ! Time reversal | $$x[-n]\ $$ | $$X(z^{-1}) \ $$ | $$\frac{1}{r_2}<|z|<\frac{1}{r_1} \ $$ |- ! Conjugation | $$x^*[n]\ $$ | $$X^*(z^*) \ $$ | ROC |- ! Real part | $$\operatorname{Re}\{x[n]\}\ $$ | $$\frac{1}{2}\left[X(z)+X^*(z^*) \right]$$ | ROC |- ! Imaginary part | $$\operatorname{Im}\{x[n]\}\ $$ | $$\frac{1}{2j}\left[X(z)-X^*(z^*) \right]$$ | ROC |- ! Differentiation | $$nx[n]\ $$ | $$ -z \frac{\mathrm{d}X(z)}{\mathrm{d}z}$$ | ROC |- ! Convolution | $$x_1[n] * x_2[n]\ $$ | $$X_1(z)X_2(z) \ $$ | At least the intersection of ROC1 and ROC2 |- ! Correlation | $$r_{x_1,x_2}(l)=x_1[l] * x_2[-l]\ $$ | $$R_{x_1,x_2}(z)=X_1(z)X_2(z^{-1})\ $$ | At least the intersection of ROC of X1(z) and X2($$z^{-1}$$) |- ! Multiplication | $$x_1[n]x_2[n]\ $$ | $$\frac{1}{j2\pi}\oint_C X_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v \ $$ | At least $$r_{1l}r_{2l}<|z|=0$$, $$u[n]=0$$ for $$n<0$$
 * $$\delta[n] = 1 $$ for $$n=0$$, $$\delta[n] = 0 $$ otherwise


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! !! Signal, $$x[n]$$ !! Z-transform, $$X(z)$$ !! ROC
 * 1 || $$\delta[n] \, $$ || $$1\, $$ ||$$ \mbox{all }z\, $$
 * 2 || $$\delta[n-n_0] \,$$ || $$ z^{-n_0} \, $$ || $$ z \neq 0\,$$
 * 3 || $$u[n] \,$$ || $$ \frac{1}{1-z^{-1} }$$ || $$|z| > 1\,$$
 * 4 || $$- u[-n-1] \,$$ || $$ \frac{1}{1 - z^{-1}}$$ ||$$|z| < 1\,$$
 * 5 || $$n u[n] \,$$ || $$ \frac{z^{-1}}{( 1-z^{-1} )^2}$$ || $$|z| > 1\,$$
 * 6 || $$ - n u[-n-1] \,$$ || $$ \frac{z^{-1} }{ (1 - z^{-1})^2 }$$ ||$$ |z| < 1 \,$$
 * 7 || $$n^2 u[n] \,$$ || $$ \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} $$ || $$|z| > 1\,$$
 * 8 || $$ - n^2 u[-n - 1] \,$$ || $$ \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} $$ || $$|z| < 1\,$$
 * 9 || $$n^3 u[n] \,$$ || $$ \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} $$ || $$|z| > 1\,$$
 * 10 || $$- n^3 u[-n -1] \,$$ || $$ \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} $$ || $$|z| < 1\,$$
 * 11 || $$a^n u[n] \,$$ || $$ \frac{1}{1-a z^{-1}}$$ ||$$ |z| > |a|\,$$
 * 12 || $$-a^n u[-n-1] \,$$ || $$ \frac{1}{1-a z^{-1}}$$ ||$$|z| < |a|\,$$
 * 13 || $$n a^n u[n] \,$$ || $$ \frac{az^{-1} }{ (1-a z^{-1})^2 }$$ || $$|z| > |a|\,$$
 * 14 || $$-n a^n u[-n-1] \,$$ || $$ \frac{az^{-1} }{ (1-a z^{-1})^2 }$$ ||$$ |z| < |a|\,$$
 * 15 || $$n^2 a^n u[n] \,$$ || $$ \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} $$ || $$|z| > |a|\,$$
 * 16 || $$- n^2 a^n u[-n -1] \,$$ || $$ \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} $$ || $$|z| < |a|\,$$
 * 17 || $$\cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 18 || $$\sin(\omega_0 n) u[n] \,$$ || $$ \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 19 || $$a^n \cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * 20 || $$a^n \sin(\omega_0 n) u[n] \,$$ || $$ \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * }
 * 11 || $$a^n u[n] \,$$ || $$ \frac{1}{1-a z^{-1}}$$ ||$$ |z| > |a|\,$$
 * 12 || $$-a^n u[-n-1] \,$$ || $$ \frac{1}{1-a z^{-1}}$$ ||$$|z| < |a|\,$$
 * 13 || $$n a^n u[n] \,$$ || $$ \frac{az^{-1} }{ (1-a z^{-1})^2 }$$ || $$|z| > |a|\,$$
 * 14 || $$-n a^n u[-n-1] \,$$ || $$ \frac{az^{-1} }{ (1-a z^{-1})^2 }$$ ||$$ |z| < |a|\,$$
 * 15 || $$n^2 a^n u[n] \,$$ || $$ \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} $$ || $$|z| > |a|\,$$
 * 16 || $$- n^2 a^n u[-n -1] \,$$ || $$ \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} $$ || $$|z| < |a|\,$$
 * 17 || $$\cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 18 || $$\sin(\omega_0 n) u[n] \,$$ || $$ \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 19 || $$a^n \cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * 20 || $$a^n \sin(\omega_0 n) u[n] \,$$ || $$ \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * }
 * 17 || $$\cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 18 || $$\sin(\omega_0 n) u[n] \,$$ || $$ \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }$$ ||$$ |z| >1\,$$
 * 19 || $$a^n \cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * 20 || $$a^n \sin(\omega_0 n) u[n] \,$$ || $$ \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * }
 * 19 || $$a^n \cos(\omega_0 n) u[n] \,$$ || $$ \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * 20 || $$a^n \sin(\omega_0 n) u[n] \,$$ || $$ \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }$$ ||$$ |z| > |a|\,$$
 * }
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