Signals & Systems — Complete Revision Notes

Energy, Power & Symmetry

• Energy Signal: $E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty \implies P = 0$.
• Power Signal: $P = \lim_{T\to\infty} \frac{1}{T}\int_{-T/2}^{T/2} |x(t)|^2 dt > 0 \implies E = \infty$.
• Power of $A\cos(\omega_0 t + \theta)$ or $A\sin(\omega_0 t + \theta)$ is $\frac{A^2}{2}$; Power of $A e^{j\omega_0 t}$ is $A^2$.
• Even / Odd Decompositions: $x_e(t) = \frac{x(t)+x(-t)}{2}$, $x_o(t) = \frac{x(t)-x(-t)}{2}$. Orthogonality: $\int_{-\infty}^{\infty} x_e(t)x_o(t) dt = 0$.

Continuous & Discrete Convolution

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau$$

• Area Property: $\text{Area}(y) = \text{Area}(x) \times \text{Area}(h)$
• Width Property: $\text{Width}(y) = \text{Width}(x) + \text{Width}(h)$
• Rectangular Convolution: $\text{rect}(t/T_1) * \text{rect}(t/T_2) = \text{Trapezoid}$ ($T_1 \neq T_2$) or $\text{Triangle}$ ($T_1 = T_2$).
• BIBO Stability: Continuous: $\int_{-\infty}^{\infty} |h(t)| dt < \infty$; Discrete: $\sum_{n=-\infty}^{\infty} |h[n]| < \infty$.

Region of Convergence (ROC) Golden Rules

• ROC never contains any poles!
• Right-sided signal $\implies \text{Re}(s) > \sigma_{max}$ (Right of rightmost pole).
• Left-sided signal $\implies \text{Re}(s) < \sigma_{min}$ (Left of leftmost pole).
• BIBO Stable $\iff$ The $j\omega$-axis is contained inside the ROC.
• Causal & BIBO Stable $\iff$ All poles lie strictly in the Left Half of $s$-plane (LHP).