Analyzes discrete-time signals \(x[n]\) and linear time-invariant (LTI) systems. Discrete Fourier Transform (DFT): \(X[k] = \sum_{n=0}^{N-1} x[n] e^{-i 2\pi k n / N}\). Convolution Theorem: \((x * h)[n] \iff X[k] \cdot H[k]\). Butterworth Digital Filter magnitude response: \(|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega/\omega_c)^{2N}}}\) providing maximally flat passband without ripple. Nyquist-Shannon Sampling Theorem guarantees exact reconstruction if sampling rate \(f_s \ge 2 f_{\max}\).