Real Power ($P$): $P = |\mathbf{V}_{rms}| |\mathbf{I}_{rms}| \cos(\theta_v - \theta_i)$ (Watts). $P = \text{Re}(S)$.

Reactive Power ($Q$): $Q = |\mathbf{V}_{rms}| |\mathbf{I}_{rms}| \sin(\theta_v - \theta_i)$ (VAR). $Q = \text{Im}(S)$.

Power Factor (PF): $\cos(\theta_v - \theta_i) = P/|S|$.

Z-parameters: $V_1 = z_{11}I_1 + z_{12}I_2$, $V_2 = z_{21}I_1 + z_{22}I_2$.
Y-parameters: $I_1 = y_{11}V_1 + y_{12}V_2$, $I_2 = y_{21}V_1 + y_{22}V_2$.
h-parameters (hybrid): $V_1 = h_{11}I_1 + h_{12}V_2$, $I_2 = h_{21}I_1 + h_{22}V_2$.
ABCD-parameters (transmission): $V_1 = AV_2 - BI_2$, $I_1 = CV_2 - DI_2$.

Continuous-time: $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$.
Discrete-time: $y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]$.
Properties: Commutative ($x*h = h*x$), Associative ($(x*h)*g = x*(h*g)$), Distributive ($x*(h_1+h_2) = x*h_1 + x*h_2$).

Continuous-time FT: $X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$.
Inverse CTFT: $x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega$.
Common Pairs:
- $\delta(t) \leftrightarrow 1$
- $e^{-at}u(t) \leftrightarrow \frac{1}{a+j\omega}$
- $rect(t/\tau) \leftrightarrow \tau \text{sinc}(\omega \tau / 2)$
Properties:
- Linearity: $ax(t) + by(t) \leftrightarrow aX(j\omega) + bY(j\omega)$.
- Time Shift: $x(t-t_0) \leftrightarrow e^{-j\omega t_0} X(j\omega)$.
- Frequency Shift (Modulation): $e^{j\omega_0 t} x(t) \leftrightarrow X(j(\omega - \omega_0))$.
- Time Scaling: $x(at) \leftrightarrow \frac{1}{|a|} X(j\omega/a)$.
- Differentiation: $\frac{d}{dt}x(t) \leftrightarrow j\omega X(j\omega)$.
- Integration: $\int_{-\infty}^t x(\tau) d\tau \leftrightarrow \frac{1}{j\omega} X(j\omega) + \pi X(0)\delta(\omega)$.
- Convolution: $x(t) * h(t) \leftrightarrow X(j\omega)H(j\omega)$.

Causality: Impulse response $h(t)=0$ for $t<0$. Output only depends on present and past inputs.
Stability (BIBO): System is stable if $\int_{-\infty}^{\infty} |h(t)| dt < \infty$. Bounded input produces bounded output.
Linearity: $T[ax_1(t)+bx_2(t)] = aT[x_1(t)]+bT[x_2(t)]$.
Time Invariance: If $y(t) = T[x(t)]$, then $y(t-t_0) = T[x(t-t_0)]$.

Nyquist Rate: $f_s \ge 2f_{max}$, where $f_{max}$ is the highest frequency component in the signal.
Aliasing: Occurs if $f_s < 2f_{max}$, high frequencies appear as lower frequencies.
Reconstruction: Low-pass filter used to recover original signal from sampled version.

Unilateral Laplace Transform: $X(s) = \int_{0^-}^{\infty} x(t) e^{-st} dt$.
Inverse Laplace Transform: Use partial fraction expansion and lookup tables.
Region of Convergence (ROC): Range of $s$ for which $X(s)$ converges. Important for causality and stability.
System Function: $H(s) = Y(s)/X(s)$. Poles and zeros of $H(s)$ determine system behavior.
Stability: For causal systems, all poles must be in the left-half of the s-plane ($\text{Re}(s) < 0$).</