Relationship Between the Fourier Series and the Continuous and Discrete Fourier Transforms

• Fourier series: discrete frequencies for periodic continuous signals
• CFT: continuous frequency spectrum for general continuous signals (limit of the series as $\text{period } \to\infty$)
• DFT: discrete frequency samples for finite, sampled signals — effectively the Fourier series of one period of a sampled sequence

Takeaways

• Sampling a continuous-time signal maps the continuous-time transform into a discrete-frequency / periodic structure (aliasing phenomena).
• Computing a DFT on $N$ samples produces $N$ complex frequency bins; applying the inverse DFT with the same bins reconstructs the original $N$ samples (assuming no frequency-domain modifications that break perfect reconstruction).