Convolución de Sumas - Tabla y propiedades

Tabla de suma de convolución

Referencia: Lathi Tabla 3.1 p285

\gamma_1 \neq \gamma_2

No	x₁[n]	x₂[n]	x₁[n]⊗x₂[n] = x₂[n]⊗x₁[n]
1	δ[n-k]	x[n]	x[n-k]
2	$\gamma^{n} \mu[n]$	μ[n]	$\frac{1-\gamma^{n+1}}{1-\gamma} \mu[n]$
3	μ[n]	μ[n]	(n+1) μ[n]
4	$\gamma_1^{n} \mu[n]$	$\gamma_2^{n} \mu[n]$	$\frac{\gamma_1^{n+1} - \gamma_2^{n+1}}{\gamma_1 - \gamma_2} \mu[n]$
5	$\gamma_1^{n} \mu[n]$	$\gamma_2^{n} \mu[-(n+1) ]$	$\frac{\gamma_1}{\gamma_2 - \gamma_1} \gamma_1^{n} \mu[n] +$ $+ \frac{\gamma_2}{\gamma_2 - \gamma_1} \gamma_2^{n} \mu[-(n+1)]$
			$\|\gamma_2\| > \|\gamma_1\|$
6	$n\gamma_1^{n} \mu[n]$	$\gamma_2^{n} \mu[n]$	$\frac{\gamma_1 \gamma_2}{(\gamma_1 - \gamma_2)^2} \Big[ \gamma_2^{n} - \gamma_1^{n} + \frac{\gamma_1 - \gamma_2}{\gamma_2}n \gamma_1^n \Big] \mu [n]$
			$\gamma_1\neq \gamma_2$
7	n μ[n]	n μ[n]	$\frac{1}{6} n (n-1) (n+1) \mu [n]$
8	$\gamma^{n} \mu[n]$	$\gamma^{n} \mu[n]$	$(n+1) \gamma^{n} \mu[n]$
9	$\gamma^{n} \mu[n]$	$n \mu[n]$	$\Big[ \frac{\gamma(\gamma^{n}-1)+n(1-\gamma)}{(1-\gamma)^2} \Big] \mu[n]$
10	$\|\gamma_1\|^{n} \cos (\beta n + \theta) \mu [k]$	$\|\gamma_2\|^{n}\mu [n]$	$\frac{1}{R} \Big[ \|\gamma_1\|^{n+1} \cos [\beta (n+1) +\theta -\phi] -\gamma_2 ^{n+1} \cos (\theta - \phi) \Big] \mu[n]$
			$R=\Big[\|\gamma_1\|^2 + \gamma_2^2 -2\|\gamma_1\|\gamma_2 \cos(\beta) \Big]^{\frac{1}{2}}$
			$\phi = \tan ^{-1} \Big[ \frac{\|\gamma_1\| \sin(\beta)}{\|\gamma_1\| \cos (\beta) -\gamma_2} \Big]$
11	$\mu [n]$	$n\mu [n]$	$\frac{n(n+1)}{2}\mu [n]$

x_1[n] \circledast x_2[n] = x_2[n] \circledast x_1[n]

x_1[n] \circledast (x_2[n]+x_3[n])= (x_1[n] \circledast x_2[n]) + (x_1[n] \circledast x_3[n])

x_1[n] \circledast (x_2[n] \circledast x_3[n]) = (x_1[n] \circledast x_2[n]) \circledast x_3[n]

x_1[n] \circledast x_2[n] = c[n]

entonces

x_1[n-m] \circledast x_2[n-p] = c[n-m-p]

x[n] \circledast \delta[n] = x[n]

Si x1[n] y x2[n] tienen anchos finitos W1 y W2, el ancho de la convolución entre ellos es W1+W2.