Periodic Spectra and Discrete Sequences: A Fourier-Series Derivation of the Sampling Theorem

A concise derivation of the sampling theorem is presented starting directly from a periodic spectrum, without invoking the impulse function or impulse train. The central observation is that the Fourier series coefficients of a periodic spectrum are precisely the uniformly spaced samples of the inverse Fourier transform of one spectral period. This establishes a direct duality between periodic spectra and discrete sequences, mirroring the classical Fourier-series duality between periodic signals and discrete spectra. The sampling theorem, sinc reconstruction formula, and Poisson summation formula then follow naturally as consequences of this structure. The derivation requires only elementary Fourier analysis and may provide a conceptually simpler introduction to sampling theory for students encountering the subject for the first time.