Signal processing relies on the Nyquist sampling theorem, which states that to accurately reconstruct a signal, it must be sampled at least twice as fast as its highest frequency component. Sampling below this rate causes aliasing, a form of distortion in which high-frequency components masquerade as lower frequencies. To analyze signals in the frequency domain, engineers use the Fourier transform, which decomposes any signal into its constituent sinusoids. The discrete version, the DFT, can be computed efficiently via the FFT algorithm, reducing computational complexity from \(O(N^2)\) to \(O(N \log N)\).
Filters selectively pass or attenuate signals based on frequency. A low-pass filter passes frequencies below a cutoff and attenuates higher ones, while a high-pass filter does the opposite. A simple RC low-pass filter has a cutoff frequency \(f_c = 1/(2\pi RC)\), at which the output power drops to half its input power, corresponding to the -3 dB point, where voltage is approximately 70.7% of the maximum. A band-pass filter combines these properties to pass only a specific range of frequencies, and the quality factor \(Q = f_0/\Delta f\) describes how sharply the filter selects around its center frequency, with higher \(Q\) indicating greater selectivity.
Underlying all of electrical engineering are the laws of electromagnetics. Coulomb's law describes the force between point charges as \(F = k q_1 q_2 / r^2\). Gauss's law relates the electric flux through a closed surface to the enclosed charge, \(\oint \mathbf{E} \cdot d\mathbf{A} = Q_{enc}/\epsilon_0\). Faraday's law states that a changing magnetic flux through a loop induces an EMF, \(\mathcal{E} = -d\Phi_B/dt\), where the negative sign embodies Lenz's law, the principle that induced currents oppose the change that created them. Ampère's law, extended by Maxwell to include changing electric fields, is \(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I_{enc} + \epsilon_0 \, d\Phi_E/dt)\). Maxwell's four equations, namely Gauss's law for electricity, Gauss's law for magnetism (which forbids magnetic monopoles), Faraday's law, and the Ampère-Maxwell law, together predict the existence of electromagnetic waves traveling in vacuum at speed \(c = 1/\sqrt{\mu_0 \epsilon_0} \approx 3 \times 10^8\) m/s.
A suite of powerful theorems simplifies the analysis of linear circuits. Impedance \(Z = R + jX\) generalizes Ohm's law to AC, combining resistance and reactance. The Thévenin equivalent reduces any linear circuit to a single voltage source \(V_{th}\) in series with a resistance \(R_{th}\), while the Norton equivalent reduces it to a current source \(I_N\) in parallel with a resistance \(R_N\), with \(V_{th} = I_N \times R_N\) and \(R_{th} = R_N\). Superposition states that the response to multiple sources is the algebraic sum of the responses to each source acting alone. The maximum power transfer theorem says that a load receives maximum power when its resistance matches the source resistance, or in AC circuits, when its impedance is the complex conjugate of the source impedance. Finally, the Wheatstone bridge, a network of four resistors balanced when \(R_1/R_2 = R_3/R_4\), provides a classical means of measuring an unknown resistance with high precision.