Welcome back! In this article, we will delve into the fascinating world of the Fourier series and how it allows us to approximate periodic functions using an infinite sum of sines and cosines of increasing frequencies. This concept forms the basis for the fast Fourier transform method, which has found applications in compression, numerical solutions of differential equations, and much more.
Previously, we explored how to represent functions using only cosines and sines. Now, we will take it a step further and examine how complex-valued functions can be represented using a sum of complex coefficients multiplied by the function e to the power of i k x. Here, we extend our representation to two pi periodic functions, ranging from -pi to pi.
To better understand the complex Fourier series, let’s recall Euler’s expansion of e to the power of i k x, which is just the sum of cosine kx and i sine kx. By using this expansion and expressing the series in terms of complex-valued functions, we achieve a more compact and versatile representation of the Fourier series that encompasses both real and complex functions.
To formalize this representation, we define a set of functions, psi k, where each psi k corresponds to e to the power of i k x. These functions form an orthogonal basis in our function space, meaning each psi k is orthogonal to every other psi k, except when j equals k. Mathematically, this orthogonality is expressed through the inner product of psi j and psi k, which evaluates to 0 when j is not equal to k.
By leveraging the orthogonality of psi j and psi k, we can compute the inner product and establish that it equals 2 pi when j is equal to k. This shows that the psi k functions, i.e., e to the power of i k x, provide an orthogonal basis for our function space. Consequently, we can represent any function, f(x), using a projection of f(x) onto each psi k direction, multiplied by the corresponding psi k vector.
In practice, this implies that the complex-valued Fourier series can be expressed as an infinite sum, where each term represents the component of f(x) in the psi k direction, scaled by the inner product of f(x) and psi k. Importantly, to account for the norm of psi k, we normalize the series by dividing it by 1 over 2 pi.
This geometrically intuitive understanding of the Fourier series as a projection onto an orthogonal basis highlights the similarities between this concept and regular vector spaces. Just as we can represent vectors in terms of x and y directions, here we can represent functions using the orthogonal basis formed by psi k functions.
In future articles, we will explore how to approximate functions using the Fourier series and examine the challenges that arise when dealing with functions containing discontinuities. Furthermore, we will derive the Fourier transform integral, which is a powerful tool in the field of signal processing.
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Contents
FAQs
Q: How does the complex Fourier series differ from the Fourier series using only sines and cosines?
A: The complex Fourier series allows us to represent both real and complex-valued functions using a sum of complex coefficients multiplied by the function e to the power of i k x. This representation is more compact and versatile compared to the Fourier series using only sines and cosines.
Q: What is the significance of the orthogonal basis formed by the psi k functions?
A: The orthogonal basis formed by the psi k functions allows us to represent any function in terms of its components projected onto each psi k direction. This representation is akin to how vectors can be represented using x and y components in regular vector spaces.
Q: How is the Fourier transform integral related to the complex Fourier series?
A: The Fourier transform integral is a powerful tool that provides a continuous representation of a function in terms of its frequency components. It allows us to transform a function from the time domain to the frequency domain. The complex Fourier series provides a discrete representation of a periodic function using a sum of complex coefficients, while the Fourier transform integral extends this concept to non-periodic functions.
Conclusion
In this article, we explored the complex Fourier series and its ability to represent both real and complex-valued functions using a sum of complex coefficients multiplied by e to the power of i k x. We discovered that the psi k functions form an orthogonal basis, providing a versatile and compact representation of functions in terms of their components projected onto each psi k direction. This understanding opens the door to approximating functions and deriving the Fourier transform integral. Stay tuned to learn more about these exciting topics in future articles!
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