GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.
Spring 2015: Circuits & systems II
[TOC]
- continuous time :
$x(t)=\frac{1}{n}\quad\R t$ - discrete time :
$x[n]=\cos{2\pi n}\quad\N n$ - continuous amplitude :
$\R{x(t)}$ - discrete amplitude :
$\N{x(t)}$ - periodic with fundamental period
$T$ (smallest possible) :$x(t)=x(t+T)\quad\forall t$ - energy signal (if finite) :
$\mathcal E=\i{-\infty}{\infty}\abs{x(t)}^2\dt=\s{-\infty}{\infty}\abs{x[n]}^2$ - power signal (if finite) :
$\mathcal P=\l{T\to\infty}\frac{1}{2T}\i{-T}{T}\abs{x(t)}^2\dt=\l{N\to\infty}\frac{1}{2N+1}\s{-T}{T}\abs{x[n]}^2$ - delay :
$x(t-t_0)$
$\begin{CD}@>x(t)\quad x[n]>\text{input}>\boxed{\quad H\quad}@>y(t)\quad y[n]>\text{output}>\end{CD}\qquad y(t)=\sy{x(t)}\quad y[n]=\sy{x[n]}$
- linearity :
$\sy{a_1x_1(t)+a_2x_2(t)}=a_1\sy{x_1(t)}+a_2\sy{x_2(t)}$ - time invariance :
$\sy{x(t)}=y(t)\iff\sy{x(t-T)}=y(t-T)$ - memoryless : output independent of any other inuput
- causality : output independ of future inputs
- stability : output magnitude/amplitude is bouned when input are bounded
- Kronecker-delta :
$\delta_k[n]=\cases{1&n=k\ 0&n\not = k}$ - impulse response :
$h[n]=\sy{\delta_0[n]}=\sy{\delta[n]}$ and$h[n-k]=\sy{\delta_k[n]}=\sy{\delta[n-k]}$ - linear decomposition :
$x[n]=x_1\delta_1[n]+\cdots+x_m\delta_m[n]$ and$y[n]=x_1h[n-1]+\cdots+x_mh[n-m]$ - convolution sum :
$(x*k)[n]=y[n]=\s{k=-\infty}{\infty}x[k]h[n-k]$
- Kronecker-delta :
$\delta_\ep(t)=\cases{\frac{1}{2\ep}&-\ep<t<\ep\ 0&\text{otherwise}}$ - linear decomposition :
$x_ \ep(t)=\i{\infty}{-\infty}x(T)\delta_ \ep(t-T)\d T$ and$y_ \ep(t)=\i{\infty}{-\infty}x(T)h_\ep(t-T)\d T$ - convolution integral :
$(x*h)(t)=y(t)=\l{\ep\to 0}y_\ep(t)=\i{\infty}{-\infty}x(T)h(t-T)\d T$
- commutative :
$(x*h)(t)=(h *x)(t)$ - distributive :
$(x*(h_1+h_2))[n]=(x *h_1)[n]+(x *h_2)[n]$ - associativity :
$((x*h_1) *h_2)(t)=(x *(h_1 *h_2))(t)$ - memoryless : iff
$h[n]=a\delta[n]$ - causal : iff
$h[k]=0;\forall k < 0$ - stable : iff
$\int_{-\infty}^\infty\abs{h(t)}\d t<\infty$ - composition
- parallel :
$y[n]=((h_1+h_2)*x)[n]$ - serie :
$y[n]=(x* h_1 *h_2)[n]$
- parallel :
- homogenous solution :
$y^{(h)}[n]+\alpha y^{(h)}[n_1]=0$ - particular solution :
$y^{(p)}[n]+\alpha y^{(p)}[n_1]=x[n]+\beta x[n-1]$ - "Ansatz" :
$y^{(h)}[n]=cr^n$ and$y^{(p)}[n]=bs^n$ - full solution with initial condition :
$y^{(h)}+y^{(p)}$
- input signal pattern :
$x[n]=e^{j\omega n}$ - frequency response :
$H(j\omega)=\sum_{k=-\infty}^\infty h[k]e^{-j\omega k}$ - LTI :
$y[n]=(x*h)[n]=e^{j\omega n}H(j\omega)$ - existence : iff
$\sum_{k=-\infty}^\infty\abs{h[k]}<\infty$
- input signal pattern :
$x(t)=e^{j\omega t}$ - frequency response :
$H(j\omega)=\int_{-\infty}^\infty h(\tau)e^{-j\omega\tau}\d\tau$ - LTI :
$y(t)=(x*h)(t)=e^{j\omega t}H(j\omega)$ - existence : iff
$\int_{-\infty}^\infty\abs{h(\tau)}\d\tau<\infty$
- real valued :
$h$ is real$H(-j\omega)=H^*(j\omega)$ - module even symmeric :
$\abs{H(j\omega)}=\abs{H(-j\omega)}$ - argument odd symmetric :
$\arg{H(j\omega)}=-\arg{H(-j\omega)}$
- periodic input :
$x(t)$ with$\omega_k=\frac{2\pi}{T}k$ - fourier series :
$\tilde{x}(t)=\sumi X_k e^{j 2\pi kt/T} $ - series coefficient :
$X_k=\frac{1}{T}\int_0^T x(t)e^{-j 2\pi k t /T}\d t$
- Drichlet :
$x(t)=\tilde{x}(t)$ - linearity : if fundamental period is
$T$ then$Ax(t)+By(t)$ give$AX_k+BY_k$ - time shifting : for
$x(t-t_0)$ , multiplication by a constant$Y_k=e^{-j 2\pi kt_0/T}X_k$ - conjugation :
$x^*(t)$ give$X_{-k}^ *$ (if$x(t)$ real$\iff x(t)=x^ *(t)$ , then$X_k=X _{-k}^ *$ ) - time reversal :
$x(-t)$ give$X_{-k}$ - parseval :
$\frac{1}{T}\int_0^T\abs{x(t)}^2\d t=\sumi \abs{X_k}^2$ - real valued and odd for
$x(t)$ : we have$X_k=X_{-k}^*=X _{-k}$
- fourier transform :
$x(t)=\frac{1}{2\pi}\inti X(j\omega)e^{j\omega t}\d\omega$ - transform coefficient :
$X(j\omega)=\inti x(t)e^{-j\omega t}\d t$
- time scaling :
$x(at)$ give$\frac{1}{a}X(\frac{j\omega}{a})$ if$a\ge0$ or$-\frac{1}{a}X(\frac{j\omega}{a})$ if$a<0$ - box-functions transform is sinc
- input :
$x(t)=e^{st}$ - output :
$y(t)=\inti e^{s(t-\tau)}h(\tau)\d\tau=e^{st}H(s)$ - replace frequency response
- transfer function :
$H(s)=\inti h(\tau)e^{-s\tau}\d\tau$ - radius of convergence (ROC) :
$ROC(H(s))={s : \Re(s)> -a}$ for signal$e^{-a\tau}$ - stable : containing y-axis
- anti-causal : until minus infinity
- causal : until infinity
- pole : cross
- zero : circle
- if
$s=jw$ , Laplace = Fourier - linear
- convolution : multiplication
-
$\frac{\d{}}{\d x}x(t)$ :$sX(s)$ - controller : output + input go through it before system
- p-controller (proportionnal) :
$G(s)=k$ - pi-controller :
$G(s)=k_1+k_2/s$ - pid-controller :
$G(s)=k_1+k_2/s+sk_3$
- p-controller (proportionnal) :
- input :
$x[n]=z_0^n=(ae^{jw})^n$ - output :
$y[n]=\sumi h[k]z_0^{n-k}=z_0^n H(z_0)$ - z-transform :
$H(z_0)=\sumi h[k]z_0^{-k}$ - convergence
- stable : containing unit circle
- anti-causal : until zero
- causal : until infinity
- if
$z=e^{jw}$ , z-transform = discrete Fourier - linear
- wavelets : downsample, then upsample, under which condition is signal reconstructed ?
- downsampling : get rid of odd or substitute twice the variable