Notes on discrete-time Fourier series and transform
6 hours ago
- Discrete-time Fourier series (DTFS) represents N-periodic discrete signals as a finite sum of discrete complex exponentials, with no convergence issues due to the finite sum.
- The DTFS coefficients are obtained by orthogonality of complex exponentials, yielding a closed-form expression for the coefficients.
- The DTFT extends the Fourier representation to finite-duration discrete signals, producing a continuous and periodic frequency-domain function.
- The DTFT and its inverse form an integral pair, analogous to the continuous Fourier transform, allowing analysis of non-periodic discrete signals.
- DTFS coefficients are equally spaced samples of the DTFT, linking the two transforms in a manner similar to the continuous case.
- The DTFT inherits valuable properties such as linearity, time-shifting, and convolution, though not detailed in the notes.
- Appendices cover properties of discrete complex exponentials, sum of exponentials, and sine series decomposition, providing mathematical groundwork.