Mathematical Foundations of Continuous-Time Signals

Trace the historical development of Fourier analysis from Fourier's heat equation work to modern signal processing applications.

Derive the Fourier series from the principle of orthogonality for complex exponentials.

Analyze the convergence of Fourier series using the Dirichlet conditions.

Prove and apply key Fourier transform properties including linearity, time-shifting, frequency-shifting, scaling, and duality.

Derive and interpret the convolution theorem in both time and frequency domains.

Apply Parseval's theorem to relate the energy of a signal in the time and frequency domains.

Derive the Fourier transforms of common signals, including rectangular pulses, sinc functions, and Gaussian functions.