16.7:
Discrete-Time Fourier Series
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.
16.7:
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:
- Compute the DTFS of the Input Signal: Calculate the DTFS coefficients X[k] for the input signal x[n].
- Calculate the Output Response for Each DTFS Term: Use the system's frequency response H(ejΩ) to determine the output for each frequency component. The output DTFS coefficients Y[k] are given by
where Ω =
- 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 Ωk = 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.