Signal Processing

Mathematical Principles & Theorems

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 \(f_s \ge 2 f_{\max}\).

Operating Instructions

• Generate synthesized test signals (Multi-tone Sinusoids, Square Wave, Gaussian Noise).
• Configure digital filter type (Low-pass, High-pass, Band-pass), cutoff frequency \(f_c\), and filter order \(N\).
• Click Process to compute time-domain filtering and frequency spectrum via FFT.
• Inspect dual oscilloscope displays comparing raw versus filtered waveforms and harmonic magnitude bars.