Referencia: Lathi Tabla 2.1 p176
Respuesta a estado cero y convolución con Sympy-Python
| No | x1(t) | x2(t) | x1(t)⊗x2(t) = x2(t)⊗x1(t) |
|---|---|---|---|
| 1 | x(t) | δ(t-T) | x(t-T) |
| 2 | eλt μ(t) | μ(t) | \frac{1-e^{\lambda t}}{-\lambda} \mu (t) |
| 3 | μ(t) | μ(t) | t μ(t) |
| 4 | eλ1t μ(t) | eλ2t μ(t) | \frac{e^{\lambda _1 t}-e^{\lambda _2 t}}{\lambda _1 - \lambda _2} \mu(t) |
| \lambda _1 \neq \lambda _2 | |||
| 5 | eλt μ(t) | eλt μ(t) | t eλt μ(t) |
| 6 | t eλt μ(t) | eλt μ(t) | \frac{1}{2} t^2 e^{\lambda t} \mu(t) |
| 7 | t N μ(t) | eλt μ(t) | \frac{N! e^{\lambda t}}{\lambda^{N+1}} \mu(t) - \sum_{k=0}^{N}\frac{N! t^{N-k}}{\lambda^{N+1} (N-k)!} \mu(t) |
| 8 | t M μ(t) | t N μ(t) | \frac{M! N!}{(M+N+1)!} t^{M+N+1} \mu (t) |
| 9 | t eλ1t μ(t) | eλ2t μ(t) | \frac{e^{\lambda_2 t}- e^{\lambda_1 t} + (\lambda_1-\lambda_2)te^{\lambda_1 t}}{(\lambda_1-\lambda_2)^2} \mu (t) |
| 10 | t M eλt μ(t) | t N eλt μ(t) | \frac{M! N!}{(M+N+1)!} t^{M+N+1} e^{\lambda t}\mu (t) |
| 11 | t M eλ1t μ(t) | t^N e^{\lambda_2t} \mu (t) | \sum_{k=0}^{M}\frac{(-1)^k M! (N+k)! t^{M-k} e^{\lambda_1t}}{k!(M-k)!(\lambda_1 - \lambda_2)^{N+k+1}} \mu (t) |
| λ1 ≠λ2 | + \sum_{k=0}^{N}\frac{(-1)^k N! (M+k)! t^{N-k} e^{\lambda_2t}}{k!(N-k)!(\lambda_2 - \lambda_1)^{M+k+1}} \mu (t) | ||
| 12 | e^{\alpha t} \cos (\beta t + \theta) \mu (t) | eλt μ(t) | \frac{\cos(\theta - \phi)e^{\lambda t}-e^{-\alpha t} \cos(\beta t + \theta - \phi)}{\sqrt{(\alpha + \lambda)^2 + \beta^2}} \mu (t) |
| \phi =tan^{-1} \Big[\frac{-\beta}{(\alpha + \lambda})\Big] | |||
| 13 | eλ1t μ(t) | eλ2t μ(-t) | \frac{e^{\lambda_1 t} \mu (t) + e^{\lambda_2 t} \mu (-t)}{\lambda_2 -\lambda_1} |
| \text{Re} \lambda_2 > \text{Re} \lambda_1 | |||
| 14 | eλ1t μ(-t) | eλ2t μ(-t) | \frac{e^{\lambda_1 t} -e^{\lambda_2 t}}{\lambda_2 -\lambda_1} \mu (-t) |