# Dragon Notes

UNDER CONSTRUCTION
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# $$\bn{z}$$-Transform Properties

 Property $$x[n]$$ $$\bn{X}(\bn{z})$$ $$[1]\ \t{Linearity}$$ $$\ds C_1 x_1[n]+C_2 x_2[n]$$ ⬌ $$\ds C_1\bn{X}_1(\bn{z})+C_2\bn{X}_2(\bn{z})$$ $$[2]\ \t{Time delay by }1$$ $$\ds x[n-1]\ u[n]$$ ⬌ $$\ds \frac{1}{\z}\Xz +x[-1]$$ $$[3]\ \t{Time delay by }m$$ $$\ds x[n-m]\ u[n]$$ ⬌ $$\ds \frac{1}{\z^m}\Xz +\frac{1}{\z^m}\sum_{i=1}^{m}x[-i]\z^i$$ $$[4]\ \t{Right shift by }m$$ $$\ds x[n-m]\ u[n-m]$$ ⬌ $$\ds \frac{1}{\z^m}\Xz$$ $$[5]\ \t{Time advance by }1$$ $$\ds x[n+1]\ u[n]$$ ⬌ $$\ds \z\Xz -\z\ x[0]$$ $$[6]\ \t{Time advance by }m$$ $$\ds x[n+m]\ u[n]$$ ⬌ $$\ds \z^m \Xz + \z^m \sum_{i=0}^{m-1}x[i]\z^{-i}$$ $$[7]\ \t{Multiplication by }a^n$$ $$\ds \bn{a}^n x[n]\ u[n]$$ ⬌ $$\ds \X\Frac{\z}{\bn{a}}$$ $$[8]\ \t{Multiplication by }n$$ $$\ds n x[n]\ u[n]$$ ⬌ $$\ds -\z\frac{d\Xz}{d\z}$$ $$[9]\ \t{Time scaling}$$ $$\ds x[n/k]\ u[n]$$ ⬌ $$\ds \X (\z^k),\ \ k\t{ = pos. integer},\ n\t{ = multiple of }k$$ $$[10]\ \t{Time reversal}$$ $$\ds x[-n]\ u[n]$$ ⬌ $$\ds \X (1/\z)$$ $$[11]\ \t{Summation}$$ $$\ds \s{}\sum_{k=0}^{n}\ns{}x[k]\ u[n]$$ ⬌ $$\ds \frac{\z}{\z -1}\Xz$$ $$[12]\ \t{Convolution}$$ $$\ds x_1[n]*x_2[n]\ u[n]$$ ⬌ $$\ds \X_1 (\z) \X_2 (\z)$$ $$[13]\ \t{Initial value}$$ $$\ds x[0]$$ $$\vphantom{\lmt{z}{\infty}}\bf{=}\hspace{1px}$$ $$\ds \ilim{\z}\Xz$$ $$[14]\ \t{Final value}$$ $$\ds \ilim{n}\ x[n]$$ $$\vphantom{\lmt{z}{\infty}}\bf{=}\hspace{1px}$$ $$\ds \lim_{\z\ra 1}[(\z -1)\Xz],\ \$$asm $$x[\infty]\t{ exists}$$