Formulario

Trigonometría

Sumas y diferencias a productos:

\[ \begin{aligned} \cos A + \cos B &= 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} \\ \cos A - \cos B &= -2\sin\frac{A+B}{2}\sin\frac{A-B}{2} \\ \sin A + \sin B &= 2\sin\frac{A+B}{2}\cos\frac{A-B}{2} \\ \sin A - \sin B &= 2\cos\frac{A+B}{2}\sin\frac{A-B}{2} \end{aligned} \]

Productos a sumas:

\[ \begin{aligned} \sin A\cos B &= \tfrac{1}{2}\left[\sin(A+B) + \sin(A-B)\right] \\ \cos A\cos B &= \tfrac{1}{2}\left[\cos(A-B) + \cos(A+B)\right] \\ \sin A\sin B &= \tfrac{1}{2}\left[\cos(A-B) - \cos(A+B)\right] \end{aligned} \]

Integrales

Exponencial por coseno:

\[ \int e^{a\theta}\cos(b\theta)\,d\theta = \frac{e^{a\theta}}{\sqrt{a^2+b^2}}\cos\!\left(b\theta - \arctan\frac{b}{a}\right) \]

Delta

Propiedad	Continuo: \(\delta(t)\)	Discreto: \(\delta[n]\)
Producto	\(\displaystyle\int_{-\infty}^{\infty} \delta(t-a)\,\delta(t-b)\,dt = \delta(a-b)\)	\(\displaystyle\sum_{n=-\infty}^{\infty} \delta[n-a]\,\delta[n-b] = \delta[a-b]\)
Desplazamiento	\(\displaystyle\int_{-\infty}^{\infty} x(t)\,\delta(t-a)\,dt = x(a)\)	\(\displaystyle\sum_{n=-\infty}^{\infty} x[n]\,\delta[n-a] = x[a]\)

Correlación y convolución

	Continuo	Discreto
Correlación	\(R_{x_1x_2}(\tau) = \displaystyle\int_{-\infty}^{\infty} x_1(t)\,x_2(t+\tau)\,dt\)	\(R_{x_1x_2}[l] = \displaystyle\sum_{n=-\infty}^{\infty} x_1[n]\,x_2[n+l]\)
Autocorrelación	\(R_{xx}(\tau) = \displaystyle\int_{-\infty}^{\infty} x(t)\,x(t+\tau)\,dt\)	\(R_{xx}[l] = \displaystyle\sum_{n=-\infty}^{\infty} x[n]\,x[n+l]\)
Convolución	\((x_1 * x_2)(t) = \displaystyle\int_{-\infty}^{\infty} x_1(\theta)\,x_2(t-\theta)\,d\theta\)	\((x_1 * x_2)[n] = \displaystyle\sum_{k=-\infty}^{\infty} x_1[k]\,x_2[n-k]\)