Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.

For a discrete-time periodic signal x[n] with period N₀, the DTFS coefficients X[k] are calculated using the formula:

Where k=0,1,2,…,N₀−1. These coefficients X[k] represent the signal in the frequency domain, capturing the amplitude and phase of each frequency component.

To determine the response of a Linear Time-Invariant (LTI) system to a discrete-time periodic signal, a systematic approach is followed:

1. Compute the DTFS of the Input Signal: Calculate the DTFS coefficients  X[k] for the input signal x[n].
2. Calculate the Output Response for Each DTFS Term: Use the system's frequency response H(e^jΩ) to determine the output for each frequency component. The output DTFS coefficients Y[k] are given by
   
   where Ω =
3. Sum the Responses: Finally, sum the contributions of all DTFS terms to obtain the total output signal in the time domain.

In continuous-time signals, periodicity is defined with respect to a period T, corresponding to circular and angular frequencies. For a discrete-time signal, periodicity is associated with a fundamental angular frequency Ω_k = EQUATION, where N₀ is the period of the discrete signal. The DTFS expansion is finite, consisting of N terms, contrasting with the infinite series in continuous-time Fourier series.

DTFS plays a crucial role in digital signal processing (DSP), particularly in analyzing and manipulating periodic signals derived from sampled data. It is instrumental in tasks such as identifying specific frequencies within audio signals, enhancing or suppressing certain frequency components, and filtering out unwanted noise. By transforming signals into the frequency domain, DTFS facilitates efficient signal analysis and processing, enabling improved performance in various applications such as telecommunications, audio engineering, and control systems.