Tabla de Transformada Z

1a

x[n]:

δ[n]

X[z]:

1

ROC

Toda z

1b

x[n]:

δ[n-m]

X[z]:

z^-m

ROC

Toda z excepto
0 (si m>0) ó
∞ (si m<0)

2a

x[n]:

μ[n]

X[z]:

$$\frac{z}{z-1}$$

ROC

$$\frac{1}{1-z^{-1}}$$

|z|>1

2b

x[n]:

-μ[-n-1]

X[z]:

$$\frac{z}{z-1}$$

ROC

$$\frac{1}{1-z^{-1}}$$

|z|<1

3

x[n]:

n μ[n]

X[z]:

$$\frac{z}{(z-1)^2}$$

ROC

$$\frac{z^{-1}}{(1- z^{-1})^2}$$

|z|>1

4

x[n]:

n² μ[n]

X[z]:

$$\frac{z(z+1)}{(z-1)^3}$$

5

x[n]:

n³ μ[n]

X[z]:

$$\frac{z(z^2 + 4z + 1)}{(z-1)^4}$$

6a

x[n]:

γⁿ μ[n]

X[z]:

$$\frac{z}{z-\gamma}$$

ROC

$$\frac{1}{1-\gamma z^{-1}}$$

|z|>|γ|

6b

x[n]:

-γⁿ μ[-n-1]

X[z]:

$$\frac{z}{z-\gamma}$$

ROC

$$\frac{1}{1-\gamma z^{-1}}$$

|z|<|γ|

7

x[n]:

γ^n-1 μ[n-1]

X[z]:

$$\frac{1}{z-\gamma}$$

8a

x[n]:

n γⁿ μ[n]

X[z]:

$$\frac{\gamma z}{(z-\gamma)^2}$$

ROC

$$\frac{\gamma z^{-1}}{(1- \gamma z^{-1})^2}$$

|z|>|γ|

8b

x[n]:

-n γⁿ μ[-n-1]

X[z]:

$$\frac{\gamma z}{(z-\gamma)^2}$$

ROC

$$\frac{\gamma z^{-1}}{(1- \gamma z^{-1})^2}$$

|z|<|γ|

8c

x[n]:

(n+1) γⁿ μ[n]

X[z]:

$$\Big[ \frac{z}{z-\gamma}\Big]^2$$

ROC

$$\frac{1}{(1- \gamma z^{-1})^2}$$

|z|>|γ|

9

x[n]:

n² γⁿ μ[n]

X[z]:

$$\frac{\gamma z (z + \gamma)}{(z - \gamma)^3 }$$

10

x[n]:

$$\frac{n(n-1)(n-2) \text{...} (n-m+1)}{\gamma^m m!}\gamma^n \mu[n]$$

X[z]:

$$\frac{ z}{(z-\gamma)^{m+1}}$$

11a

x[n]:

|γ|ⁿ cos(βn) μ[n]

X[z]:

$$\frac{ z \big(z-|\gamma | \cos (\beta ) \big)}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2}$$

11b

x[n]:

|γ|ⁿ sin(βn) μ[n]

X[z]:

$$\frac{ z |\gamma | \sin (\beta )}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2}$$

12a

x[n]:

r|γ|ⁿ cos(βn+θ) μ[n]

X[z]:

$$\frac{ rz[z \cos (\theta) - |\gamma | \cos (\beta -\theta)]}{z^2-(2|\gamma | \cos (\beta ))z +|\gamma |^2}$$

12b

x[n]:

r|γ|ⁿ cos(γn+θ) μ[n]
γ = |γ| e^jβ

X[z]:

$$\frac{\big(0.5r e^{j \theta} \big)z}{z - \gamma} + \frac{\big(0.5r e^{-j \theta} \big)z}{z - \gamma^{*}}$$

12c

x[n]:

r|γ|ⁿ cos(γn+θ) μ[n]

$$r = \sqrt{\frac{A^2|\gamma |^2 + B^2 - 2AaB}{|\gamma |^2 - a^2}}$$

X[z]:

$$\frac{z(Az +B)}{z^2 + 2az + |\gamma|2}$$

$$\beta = \cos ^{-1} \frac{-a}{|\gamma |}$$

$$$\theta = tan^{-1} \frac{ Aa - B}{A \sqrt{|\gamma |^2 - a^2}}$$

13

x[n]:

{aⁿ ; 0≤ n ≤ N-1
{0 ; otro caso

ROC

$$\frac{1-a^N z^{-n}}{1-az^{-1}}$$

|z|>0