The Discrete-Time Fourier Series is a counterpart to the Fourier-series expansion of continuous-time periodic signals.

Calculating the expansion coefficients of the DTFS involves summations rather than integrals. The response of a Linear Time-Invariant system to a discrete-time periodic signal can be determined through a step-by-step process.

First, the DTFS of the input signal must be computed. The output response to each DTFS term is calculated using the system's frequency response. Finally, these results are summed up to obtain the total output signal.

A periodic signal in continuous time has a period with circular and angular frequencies, represented by a complex-exponential Fourier series. Similarly, a discrete-time periodic signal has a fundamental angular frequency.

The DTFS expansion consists of finite terms, unlike the Fourier series for continuous time, which consists of an infinite number of terms.

In digital signal processing, DTFS aids in examining periodic samples, pinpointing specific frequencies, and effectively filtering out undesirable noise.