ECE 45 reference sheet

Screw derivation.

A cleaned up version of my old cheat sheet.

Math things

Remember \omega_0 = \frac{2\pi}{T}. Also, for phasors, 

        A\cos(\omega t) \to \boxed{H(\omega)} \to A|H(\omega)| \cos(\omega t + \measuredangle H(\omega))

Zeger fetishes

            \begin{aligned}
            2 \cos t &= e^{jt} + e^{-jt} \\
            2j \sin t &= e^{jt} - e^{-jt}
            \end{aligned}

Dirac delta things

            \begin{aligned}
            x(t) \, \delta(t - a) &= x(a) \, \delta(t - a) \\
            \int_{-\infty}^{\infty} x(t) \, \delta(t - a) \; dt &= x(a) \\
            x(t) * \delta(t - a) &= x(t - a)
            \end{aligned}

Convolutions

            \begin{aligned}
            x(t) * h(t) &= \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \\
            &= \int_{-\infty}^{\infty} x(t - \tau) h(\tau) d\tau \\
            y(t - a - b) &= f(t - a) * h(t - b) ~ \text{(shift)} \\
            y'(t) &= f'(t) * h(t) ~ \text{(derivative)}
            \end{aligned}

For Desmos

You can paste the code directly into Desmos.

\sinc

            \sinc t = \frac{\sin t}{t}

\rect, r(t) 

r\left(t\right)=\left\{-\frac{1}{2}\le t\le\frac{1}{2}:1,0\right\}

Unit function, u(t)

Wolfram Alpha calls this the step function, step(x).

u\left(t\right)=\left\{t\le0:0,1\right\}

Triangle/ramp, T(t)

T\left(t\right)=\left\{0\le t<1:t,0\right\}

Impulse train

            \begin{aligned}
            s(t) &= \sum_{n = -\infty}^\infty \delta(t - nT) \\
            S(\omega) &= \omega_s \sum_{n = -\infty}^\infty \delta(\omega - n\omega_s) \\
            X_s(\omega) &= \frac{1}{T_s} \sum_{n = -\infty}^\infty X(\omega - n\omega_s) ~ \text{(for $x(t) s(t)$)}
            \end{aligned}

Fourier series stuff

Fourier series

            f(t) = \sum_{n = -\infty}^\infty F_n e^{jn\omega_0 t}

Fourier coefficients

            F_n = \frac{1}{T} \int_T f(t) e^{-jn\omega_0 t} dt

LTI systems

            Y_n = X_n H(n\omega_0)

Properties

            \begin{aligned}
            f(t - a) &\is F_n e^{jn\omega_0 a} ~ \text{(time shift)} \\
            f'(t) &\is jn\omega_0 F_n ~ \text{(derivative)} \\
            f(t) g(t) &\is \sum_{k = -\infty}^\infty F_k G_{n - k} ~ \text{(multiplication)} \\
            \frac{1}{T} \int_T |f(t)|^2 dt &\is \sum_{n = -\infty}^\infty |F_n|^2 ~ \text{(Parseval's)} \\
            f^*(t) &\is F^*_{-n}
            \end{aligned}

Examples

\cos(kt) F_{\pm 1} = \frac{1}{2}, others 0
\sin(kt) F_{-1} = \frac{1}{2j}, F_1 = -\frac{1}{2j}, others 0
|\sin t| F_{-1} = \frac{1}{2j}, F_1 = -\frac{1}{2j}, others 0
triangle wave* F_0 = \frac{1}{2}, others \frac{j}{2\pi n}

*f(t) = t between 0 and 1, repeating

Fourier transforms time

Fourier transform

            F(\omega) = \int_{-\infty}^\infty f(t) e^{-j\omega t} dt

Inverse fourier transform

            f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{j\omega t} d\omega

LTI systems

            \begin{aligned}
            y(t) &= x(t) * h(t) \\
            Y(\omega) &= X(\omega) H(\omega)
            \end{aligned}

Properties

            \begin{aligned}
            f(t - a) &\is F(\omega) e^{-j\omega a} ~ \text{(time shift)} \\
            f(t) e^{jat} &\is F(\omega - a) ~ \text{(frequency shift)} \\
            f(t) * g(t) &\is F(\omega) G(\omega) ~ \text{(convolution)} \\
            f(t) g(t) &\is \frac{1}{2\pi} F(\omega) * G(\omega) ~ \text{(multiplication)} \\
            \frac{df(t)}{dt} &\is j\omega F(\omega) ~ \text{(derivative)} \\
            -jtf(t) &\is \frac{dF}{d\omega} ~ \text{(derivative)} \\
            F(t) &\is 2\pi f(-\omega) ~ \text{(duality/symmetry)} \\
            \sum_{n = -\infty}^\infty F_n e^{jn\omega_0 t} &\is \sum_{n = -\infty}^\infty F_n \cdot 2\pi\delta(\omega - n\omega_0) ~ \text{(Fourier series)}
            \end{aligned}

There are other less-used properties, like Parseval's theorem, in the old cheat sheet.

Examples

            \begin{aligned}
            \sinc(a t) &\is \frac{\pi}{a} \rect\left(\frac{\omega}{2a}\right) \\
            \rect\left(\frac{t}{a}\right) &\is a\sinc\left(\frac{\omega a}{2}\right) \\
            \cos(a t) &\is \pi\delta(\omega - a) + \pi\delta(\omega + a) \\
            \sin(a t) &\is \frac{\pi}{j}\delta(\omega - a) - \frac{\pi}{j}\delta(\omega + a) \\
            \delta(t) &\is 1 \\
            1 &\is 2\pi\delta(\omega) ~ \text{($e^{j\omega_0t}$ can be derived by freq. shift)} \\
            e^{-at} u(t) &\is \frac{1}{a + j\omega} ~ \text{(where $a > 0$)} \\
            \frac{1}{a + jt} &\is 2\pi e^{a\omega} u(-\omega)
            \end{aligned}

Laplace transform

        X(s) = \int_{-\infty}^\infty x(t) e^{-st} dt

Properties

            \begin{aligned}
            x(t - a) &\is e^{-as} X(s) ~ \text{(time shift)} \\
            e^{at} x(t) &\is X(s - a) ~ \text{(frequency shift)} \\
            -t x(t) &\is \frac{dX(s)}{ds} ~ \text{(derivative)} \\
            t^n x(t) &\is (-1)^n X^{(n)}(s) ~ \text{($n$th derivative)}
            \end{aligned}

Examples

            \begin{aligned}
            e^{-at} u(t) &\is \frac{1}{s + a} & \text{ROC:} & \re(s) > \re(-a) \\
            -e^{-at} u(-t) &\is \frac{1}{s + a} & \text{ROC:} & \re(s) < \re(-a) \\
            \delta(t) &\is 1 & \text{ROC:} & \, \text{all of $\mathbb{C}$} \\
            \cos(at) u(t) &\is \frac{s}{s^2 + a^2} & \text{ROC:} & \re(s) > 0 \\
            \sin(at) u(t) &\is \frac{a}{s^2 + a^2} & \text{ROC:} & \re(s) > 0 \\
            e^{-a|t|} &\is \frac{-2a}{s^2 - a^2} & \text{ROC:} & -\re(a) < \re(s) < \re(a)
            \end{aligned}