====== Foaia de formule — partial PS ======

[[ps:start|← inapoi la cuprins]] · [[ps:simulare-partial|Simulare partial]]

Tot ce trebuie sa ai in cap, pe o pagina. Citeste-o in dimineata partialului.

===== Cap 0 — Baze =====

  * **liniar:** $S[a_1x_1 + a_2x_2] = a_1S[x_1] + a_2S[x_2]$; test rapid $S[0] = 0$ · **invariant:** $x(t-\tau) \to y(t-\tau)$
  * $Gx(t)$, $x(t-\tau)$ → LTI · $x^2(t)$ → neliniar · $x(t)\cos(2\pi f_c t)$ → liniar, **variant** · $x(t) + 1$ → neliniar
  * $z = a + jb = re^{j\theta}$, $r = \sqrt{a^2+b^2}$, $\theta = \tan^{-1}\frac{b}{a}$; polar: modulele se inmultesc, unghiurile se aduna
  * $e^{j\theta} = \cos\theta + j\sin\theta$ · $\cos(2\pi ft) = \frac{e^{j2\pi ft} + e^{-j2\pi ft}}{2}$ · $\sin(2\pi ft) = \frac{e^{j2\pi ft} - e^{-j2\pi ft}}{2j}$
  * $A\cos(2\pi f_0 t)$ → linii **$A/2$** la **$\pm f_0$**; constanta $C$ → linie $C$ la 0; sin si cos: acelasi $|X|$
  * semnal real → $|X(f)|$ simetric

===== Cap 1 — Functia de transfer, filtre =====

  * $Z_R = R$, $Z_C = \frac{1}{j2\pi fC}$ · $H(f) = \frac{V_{out}}{V_{in}}$ (depinde de $f$ si de componente)
  * RC trece-jos: $H = \frac{1}{1 + j2\pi fRC}$, $|H| = \frac{1}{\sqrt{1 + (f/f_c)^2}}$, $\angle H = -\arctan(f/f_c)$
  * RC trece-sus: $H = \frac{j2\pi fRC}{1 + j2\pi fRC}$ · ambele: $f_c = \frac{1}{2\pi RC}$
  * $|H(f_c)| = \frac{\max|H|}{\sqrt 2}$ ⇔ putere la jumatate ⇔ −3 dB · la $f_c$ (RC): faza $\mp 45^\circ$
  * LTI: $A\cos(2\pi f_0t + \varphi) \to |H(f_0)|A\cos(2\pi f_0 t + \varphi + \angle H(f_0))$ — **aceeasi frecventa**
  * FTJ / FTS / FTB / FOB: te uiti unde $|H|$ e mare

===== Cap 2 — Seria Fourier =====

  * $s(t) = \sum c_k e^{j2\pi kt/T}$, $c_k = \frac1T\int_0^T s(t)e^{-j2\pi kt/T}dt$; armonica $k$ la $k/T$; $c_0$ = media
  * unda patrata: $c_k = \frac{2A}{j\pi k}$, doar $k$ impar
  * puls: $c_k = A\frac{\Delta}{T}e^{-j\pi k\Delta/T}\mathrm{sinc}(\frac{\pi k\Delta}{T})$ · $c_k = 0$ la $k = m\frac{T}{\Delta}$ ($m \ne 0$) · $c_0 = A\Delta/T$
  * $\mathrm{sinc}(x) = \frac{\sin x}{x}$, $\mathrm{sinc}(0) = 1$ · puls ingust ↔ spectru lat
  * real: $c_{-k} = c_k^*$ · par: $c_k$ real · impar: $c_k$ imaginar · $s(t-\tau) \leftrightarrow c_ke^{-j2\pi k\tau/T}$ (doar faza)
  * Parseval: $\frac1T\int_0^T s^2 = \sum|c_k|^2$ · cosinus: $P = A^2/2$ · THD $= \frac{P - c_0^2 - 2|c_1|^2}{P - c_0^2}$
  * filtrare: $c'_k = H(k/T)c_k$
  * comunicare: "00" → 0, "01" → $f_1$, "10" → $f_2$, "11" → ambele; $f_i$ multipli de $1/T$

===== Cap 3 — Transformata Fourier, AM =====

  * $S(f) = \int s(t)e^{-j2\pi ft}dt$, $s(t) = \int S(f)e^{j2\pi ft}df$ · aperiodic → spectru continuu
  * puls $\Delta$: $|P(f)| = \Delta|\mathrm{sinc}(\pi f\Delta)|$, zerouri la $f = m/\Delta$ · $S(0)$ = DC
  * **$s(t)e^{j2\pi f_0t} \leftrightarrow S(f - f_0)$** · $s(t)\cos(2\pi f_ct) \leftrightarrow \frac12S(f - f_c) + \frac12S(f + f_c)$
  * $\cos A\cos B = \frac12\cos(A - B) + \frac12\cos(A + B)$
  * **AM** $(1 + s(t))\cos(2\pi f_ct)$: sageti $\frac12$ la $\pm f_c$ + copii $\frac12S$ in jurul $\pm f_c$; banda = $2\times$ frecventa max a mesajului
  * demodulare: inmultesti cu $\cos(2\pi f_c t)$ + FTJ

===== Cap 4 — Esantionare =====

  * replici la fiecare $f_s$: $X(f) = \sum c_kS(f - kf_s)$ · **Nyquist:** $f_s \geq 2f_{max}$ · informatia: $[-f_s/2, f_s/2]$
  * lant A/D: FTJ analog anti-aliasing → esantionare → cuantizare · recuperare: FTJ la $W$
  * aliasing: $f$ apare la $f - mf_s \in [-f_s/2, f_s/2]$; nu se repara dupa ADC
  * **bandpass:** FTB analog, $f_s \geq 2B$, $r = f_c \bmod f_s$: banda $[r - \frac B2, r + \frac B2]$ trebuie in $(0, \frac{f_s}{2})$ (ok) sau $(\frac{f_s}{2}, f_s)$ (inversat); ideal $r = f_s/4$, $f_s = \frac{4f_c}{2m+1}$
  * inversare → $\times(-1)^n$ (muta cu $f_s/2$)
  * **IQ:** $s\,e^{-j2\pi f_ct} = i + jq$, $i = s\cos$, $q = -s\sin$; la $r = f_s/4$: $e^{-j\pi n/2} = [1, -j, -1, j]$ (cos $[1,0,-1,0]$, $-\sin$ $[0,-1,0,1]$)
  * complex: $f_s' \geq B$ · real: $f_s' \geq 2B$ · decimare: $s_M(n) = s(nM)$, $f_s' = f_s/M$, FTJ inainte

===== Cap 5 — Cuantizare, SNR, dB =====

  * $2^B$ niveluri, $\Delta = \frac{2A}{2^B}$ · eroare $\pm\frac\Delta2$, RMS $\frac{\Delta}{\sqrt{12}}$, putere $\frac{\Delta^2}{12}$; ireversibila
  * $\mathrm{SNR} = \frac{P_s}{P_z}$ · $\mathrm{SNR}_{ADC} = \frac32 2^{2B} \approx 6B + 2$ dB (8 biti → 50 dB, 16 biti → 98 dB)
  * $\mathrm{dB} = 10\log_{10}\frac{P_2}{P_1}$ · **$10X$ dB = $10^X$ ori** · 3 dB ≈ ×2 · −3 dB ≈ ½ · dB-urile se aduna
  * dBm: fata de 1 mW (0 dBm = 1 mW, 30 dBm = 1 W) · amplitudini: $20\log_{10}$