Fourier Analysis is a fundamental mathematical tool used extensively in electrical engineering, particularly in the field of signal processing. It allows engineers to break down complex signals into simpler components, making it easier to analyze, manipulate, and process them. This tutorial will provide a concise overview of Fourier Analysis and its applications in signal processing.
Fourier Series: The foundation of Fourier Analysis lies in the Fourier Series. It states that any periodic function can be represented as an infinite sum of sinusoidal functions with different frequencies, amplitudes, and phases. The Fourier Series for a periodic function $f(t)$ with period $T$ is given by:
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where $a_0$, $a_n$, and $b_n$ are the Fourier coefficients, which can be calculated using the following formulas:
$a_0 = \frac{2}{T} \int_{-T/2}^{T/2} f(t) dt$ $a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \cos \left(\frac{2\pi nt}{T}\right) dt$ $b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(t) \sin \left(\frac{2\pi nt}{T}\right) dt$
The Fourier Transform is an extension of the Fourier Series for non-periodic functions. It converts a time-domain signal into its frequency-domain representation. The Fourier Transform of a continuous-time signal $x(t)$ is given by:
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where $\omega$ is the angular frequency and $j$ is the imaginary unit. The inverse Fourier Transform allows for the reconstruction of the time-domain signal from its frequency-domain representation:
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In digital signal processing, the Discrete Fourier Transform (DFT) is used to analyze discrete-time signals. The DFT of a sequence $x[n]$ of length $N$ is given by:
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where $k = 0, 1, ..., N-1$. The inverse DFT (IDFT) is used to reconstruct the time-domain signal from its frequency-domain representation:
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The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT. It reduces the computational complexity from $O(N^2)$ to $O(N \log N)$, making it practical for real-time signal processing applications. The most common FFT algorithms are the Cooley-Tukey algorithm and the Prime-Factor algorithm.
MATLAB Code Snippets: