Properties of the Z-Transform#
|
Property |
Discrete Time Domain |
\(\displaystyle{\mathcal{Z}}\) Transform |
|---|---|---|---|
1 |
Linearity |
\(\displaystyle{af_1[n]+bf_2[n]+\cdots}\) |
\(\displaystyle{aF_1(z)+bF_2(z)+\cdots}\) |
2 |
Shift of \(\displaystyle{x[n]u_0[n]}\) |
\(\displaystyle{f[n-m]u_0[n-m]}\) |
\(\displaystyle{z^{-m}F(z)}\) |
3 |
Left shift |
\(\displaystyle{f[n-m]}\) |
\(\displaystyle{z^{-m}F(z)+\sum_{n=0}^{m-1}f[n-m]z^{-n}}\) |
4 |
Right shift |
\(\displaystyle{f[n+m]}\) |
\(\displaystyle{z^{m}F(z)+\sum_{n=-m}^{-1}f[n+m]z^{-n}}\) |
5 |
Multiplication by \(\displaystyle{a^n}\) |
\(\displaystyle{a^nf[n]}\) |
\(\displaystyle{F\left(\frac{z}{a}\right)}\) |
6 |
Multiplication by \(\displaystyle{e^{-nsT_s}}\) |
\(\displaystyle{e^{-nsT_s}f[n]}\) |
\(\displaystyle{F\left(e^{sT_s}z\right)}\) |
7 |
Multiplication by \(\displaystyle{n}\) |
\(\displaystyle{nf[n]}\) |
\(\displaystyle{-z\frac{d}{dz}F(z)}\) |
8 |
Multiplication by \(\displaystyle{n^2}\) |
\(\displaystyle{n^2f[n]}\) |
\(\displaystyle{-z\frac{d}{dz}F(z)+z^2\frac{d^2}{dz^2}F(z)}\) |
9 |
Summation in time |
\(\displaystyle{\sum_{m=0}^{n}f[m]}\) |
\(\displaystyle{\frac{z}{z-1}F(z)}\) |
10 |
Time convolution |
\(\displaystyle{f_1[n]*f_2[n]}\) |
\(\displaystyle{F_1(z)F_2(z)}\) |
11 |
Frequency convolution |
\(\displaystyle{f_1[n]f_2[n]}\) |
\(\displaystyle{\frac{1}{j2\pi }\oint {x{F_1}(v){F_2}\left( {\frac{z}{v}} \right)} {v^{ - 1}}dv}\) |
12 |
Initial value theorem |
\(\displaystyle{f[0]=\lim_{z\to\infty}F(z)}\) |
|
13 |
Final value theorem |
\(\displaystyle{\lim_{n\to\infty}f[n]=\lim_{z\to 1}(z-1)F(z)}\) |
See also: Wikibooks: Engineering_Tables/Z_Transform_Properties and Z-Transform—WolframMathworld for more complete references.