Discrete-Time Fourier Series

The DTFS is the discrete-time counterpart to continuous-time Fourier series. While continuous-time Fourier series use integrals to calculate expansion coefficients and consist of infinite terms, DTFS uses summations and produces a finite number of terms. For a discrete-time periodic signal with period N0, DTFS yields exactly N0 coefficients representing frequency components.

DTFS coefficients X[k] represent a discrete-time periodic signal in the frequency domain, capturing the amplitude and phase of each frequency component. These coefficients are calculated using summations rather than integrals due to the discrete nature of the signal. They enable transformation of time-domain signals into frequency-domain representations for analysis and processing.

To determine the LTI system response, first compute the DTFS of the input signal to obtain coefficients X[k]. Next, apply the system's frequency response H(ejΩ) to each DTFS term to calculate output coefficients Y[k]. Finally, sum all the responses to obtain the total output signal in the time domain.

The DTFS expansion is finite because discrete-time periodic signals have a fundamental period N0, yielding exactly N0 unique frequency components. Continuous-time periodic signals, by contrast, require an infinite series to represent all frequency components. This finite nature of DTFS makes it computationally efficient for digital signal processing applications.

DTFS is instrumental in analyzing periodic samples from sampled data, identifying specific frequencies within audio signals, and filtering unwanted noise. By transforming signals into the frequency domain, DTFS enables efficient signal analysis and manipulation. It supports applications in telecommunications, audio engineering, and control systems where frequency-domain processing improves performance.

Continuous-time signals define periodicity with respect to a period T and corresponding circular and angular frequencies. Discrete-time signals associate periodicity with a fundamental angular frequency Ωk determined by the period N0. Despite these differences, both use complex-exponential representations to capture periodic behavior and frequency content.

DTFS coefficients are calculated using summations over one period of the discrete-time signal, unlike continuous-time Fourier series which use integrals. This summation-based approach reflects the discrete nature of the signal samples. The resulting coefficients X[k] for k=0,1,2,…,N0−1 represent the frequency components of the periodic signal.