Formulario

Trigonometría

Desfases de \(\tfrac{\pi}{2}\) y de \(\pi\), y cambio de signo del ángulo:

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

Con \(n\) entero:

\[ \cos(\pi n) = (-1)^n \qquad\qquad \sin(\pi n) = 0 \]

Suma y diferencia de ángulos:

\[ \begin{aligned} \cos(A \pm B) &= \cos A\cos B \mp \sin A\sin B \\ \sin(A \pm B) &= \sin A\cos B \pm \cos A\sin B \end{aligned} \]

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:

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

Juntar seno-más-coseno en un solo coseno con fase (\(a > 0\)):

\[ \begin{aligned} a\cos(\omega t) + b\sin(\omega t) &= \sqrt{a^2+b^2}\,\cos\!\left(\omega t - \arctan\frac{b}{a}\right) \\ a\cos(\Omega n) + b\sin(\Omega n) &= \sqrt{a^2+b^2}\,\cos\!\left(\Omega n - \arctan\frac{b}{a}\right) \end{aligned} \]

Integrales

Exponencial (\(a \neq 0\)):

\[ \int_0^t e^{a(t-\theta)}\,d\theta = \int_0^t e^{a\theta}\,d\theta = \frac{e^{at} - 1}{a} \]

Exponencial por coseno:

\[ \begin{aligned} \int e^{a\theta}\cos(b\theta)\,d\theta &= \frac{e^{a\theta}}{\sqrt{a^2+b^2}}\cos\!\left(b\theta - \arctan\frac{b}{a}\right) \qquad (a > 0) \\[6pt] &= \frac{e^{a\theta}}{a^2+b^2}\left(a\cos b\theta + b\sin b\theta\right) \end{aligned} \]

Exponencial por seno:

\[ \begin{aligned} \int e^{a\theta}\sin(b\theta)\,d\theta &= \frac{e^{a\theta}}{\sqrt{a^2+b^2}}\sin\!\left(b\theta - \arctan\frac{b}{a}\right) \qquad (a > 0) \\[6pt] &= \frac{e^{a\theta}}{a^2+b^2}\left(a\sin b\theta - b\cos b\theta\right) \end{aligned} \]

Series

Geométrica finita (\(a \neq 1\), \(n_1 \leq n_2\)):

\[ \begin{aligned} \sum_{k=n_1}^{n_2} a^k &= \frac{a^{n_1} - a^{n_2+1}}{1 - a} \\[6pt] \sum_{k=0}^{n} a^{n-k} = \sum_{k=0}^{n} a^k &= \frac{1 - a^{n+1}}{1 - a} \qquad (n_1 = 0,\ n_2 = n) \end{aligned} \]

Geométrica infinita (\(|a| < 1\)):

\[ \begin{aligned} \sum_{k=n_1}^{\infty} a^k &= \frac{a^{n_1}}{1 - a} \\[6pt] \sum_{k=0}^{\infty} a^k &= \frac{1}{1 - a} \qquad (n_1 = 0) \end{aligned} \]

Exponencial por coseno (\(|a| < 1\)):

\[ \sum_{k=0}^{\infty} a^k\cos(\Omega k) = \frac{1 - a\cos\Omega}{1 - 2a\cos\Omega + a^2} \]

Exponencial por seno (\(|a| < 1\)):

\[ \sum_{k=0}^{\infty} a^k\sin(\Omega k) = \frac{a\sin\Omega}{1 - 2a\cos\Omega + a^2} \]

Exponencial por coseno, suma finita (cualquier \(a\) que no anule el denominador):

\[ \sum_{k=0}^{n} a^{n-k}\cos(\Omega k) = \frac{\cos(\Omega n) - a\cos\big(\Omega(n+1)\big) - a^{n+1}\left(\cos\Omega - a\right)}{1 - 2a\cos\Omega + a^2} \]

Exponencial por seno, suma finita (cualquier \(a\) que no anule el denominador):

\[ \sum_{k=0}^{n} a^{n-k}\sin(\Omega k) = \frac{\sin(\Omega n) - a\sin\big(\Omega(n+1)\big) + a^{n+1}\sin\Omega}{1 - 2a\cos\Omega + a^2} \]

Delta

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

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	\((x_1 * x_2)(t) = \displaystyle\int_{-\infty}^{\infty} x_1(\theta)\,x_2(t-\theta)\,d\theta\)	\((x_1 * x_2)[n] = \displaystyle\sum_{k=-\infty}^{\infty} x_1[k]\,x_2[n-k]\)