Publicado en por Edison Del Rosario

Transformada z -Tabla

Referencia: Lathi Tabla 5.1 Transformada z p492. Oppenheim tabla 10.2 p776, Schaum Hsu Tabla 4-1 p170.

Tabla de Transformada z
No. x[n] X[z] ROC
1a δ[n] 1 Toda z
1b δ[n-m] z-m Toda z excepto
0 (si m>0) ó
∞ (si m<0)
2a μ[n] zz−1 \frac{z}{z-1} 11−z−1 \frac{1}{1-z^{-1}} |z|>1
2b -μ[-n-1] zz−1 \frac{z}{z-1} 11−z−1 \frac{1}{1-z^{-1}} |z|<1
3 n μ[n] z(z−1)2 \frac{z}{(z-1)^2} z−1(1−z−1)2 \frac{z^{-1}}{(1- z^{-1})^2} |z|>1
4 n2 μ[n] z(z+1)(z−1)3 \frac{z(z+1)}{(z-1)^3}
5 n3 μ[n] z(z2+4z+1)(z−1)4 \frac{z(z^2 + 4z + 1)}{(z-1)^4}
6a γn μ[n] zz−γ \frac{z}{z-\gamma} 11−γz−1 \frac{1}{1-\gamma z^{-1}} |z|>|γ|
6b -γn μ[-n-1] zz−γ \frac{z}{z-\gamma} 11−γz−1 \frac{1}{1-\gamma z^{-1}} |z|<|γ|
7 γn-1 μ[n-1] 1z−γ \frac{1}{z-\gamma}
8a n γn μ[n] γz(z−γ)2 \frac{\gamma z}{(z-\gamma)^2} γz−1(1−γz−1)2 \frac{\gamma z^{-1}}{(1- \gamma z^{-1})^2} |z|>|γ|
8b -n γn μ[-n-1] γz(z−γ)2 \frac{\gamma z}{(z-\gamma)^2} γz−1(1−γz−1)2 \frac{\gamma z^{-1}}{(1- \gamma z^{-1})^2} |z|<|γ|
8c (n+1) γn μ[n] [zz−γ]2 \Big[ \frac{z}{z-\gamma}\Big]^2 1(1−γz−1)2 \frac{1}{(1- \gamma z^{-1})^2} |z|>|γ|
9 n2 γn μ[n] γz(z+γ)(z−γ)3 \frac{\gamma z (z + \gamma)}{(z - \gamma)^3 }
10 n(n−1)(n−2)...(n−m+1)γmm!γnμ[n]\frac{n(n-1)(n-2) \text{...} (n-m+1)}{\gamma^m m!}\gamma^n \mu[n] z(z−γ)m+1 \frac{ z}{(z-\gamma)^{m+1}}
11a |γ|n cos(βn) μ[n] z(z−∣γ∣cos⁡(β))z2−(2∣γ∣cos⁡(β))z+∣γ∣2 \frac{ z \big(z-|\gamma | \cos (\beta ) \big)}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2} |z|>γ
11b |γ|n sin(βn) μ[n] z∣γ∣sin⁡(β)z2−(2∣γ∣cos⁡(β))z+∣γ∣2 \frac{ z |\gamma | \sin (\beta )}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2} |z|>γ
12a r|γ|n cos(βn+θ) μ[n] rz[zcos⁡(θ)−∣γ∣cos⁡(β−θ)]z2−(2∣γ∣cos⁡(β))z+∣γ∣2 \frac{ rz[z \cos (\theta) - |\gamma | \cos (\beta -\theta)]}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2}
12b r|γ|n cos(βn+θ) μ[n]
γ = |γ| ejβ		 (0.5rejθ)zz−γ+(0.5re−jθ)zz−γ∗ \frac{\big(0.5r e^{j \theta} \big)z}{z - \gamma} + \frac{\big(0.5r e^{-j \theta} \big)z}{z - \gamma^{*}}
12c r|γ|n cos(βn+θ) μ[n] z(Az+B)z2+2az+∣γ∣2\frac{z(Az +B)}{z^2 + 2az + |\gamma|2}
r=A2∣γ∣2+B2−2AaB∣γ∣2−a2r = \sqrt{\frac{A^2|\gamma |^2 + B^2 - 2AaB}{|\gamma |^2 - a^2}} β=cos⁡−1−a∣γ∣\beta = \cos ^{-1} \frac{-a}{|\gamma |}

θ=tan−1Aa−BA∣γ∣2−a2\theta = tan^{-1} \frac{ Aa - B}{A \sqrt{|\gamma |^2 - a^2}}
13 {an ; 0≤ n ≤ N-1
{0  ; otro caso		 1−aNz−n1−az−1\frac{1-a^N z^{-n}}{1-az^{-1}} |z|>0

Transformada z – Tabla de propiedades