mathematics-physics-wiki/docs/en/physics/mathematical-physics/signal-analysis/fourier-series.md

Fourier series

Theorem: the "Fourier" inner product of two functions g, f: \mathbb{C} \to \mathbb{C} is defined as

\langle f, g \rangle = \int_a^b f(t) \overline g(t) dt

with f, g members of the square integrable functions L^2[a,b] with a,b \in \mathbb{R}.

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The overline generally implies the complex conjugate.

Corollary: the "Fourier" norm of a square integrable function f: \mathbb{C} \to \mathbb{C} is defined as

|f| = \sqrt{\langle f, f \rangle}.

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Proposition: let f: \mathbb{R} \to \mathbb{R} be a periodic function with period T_0 \in \mathbb{R} then the autocorrelation of f will create peaks for t = zT_0 for all t \in \mathbb{R} and z \in \mathbb{Z}.

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Definition: two functions f, g: \mathbb{C} \to \mathbb{C} are orthogonal if and only if

\langle f, g \rangle = 0

Approximating functions

Lemma: a function f: \mathbb{R} \to \mathbb{C} can be approximated with a linear combination of orthogonal functions b_k: \mathbb{R} \to \mathbb{C} given by

\phi_n(t) = \sum_{k=0}^n c_k b_k(t),

for all t \in \mathbb{R} with n \in \mathbb{N} the order. The coefficients c_k \in \mathbb{C} that minimise \|f - \phi_n\| may be determined by

c_k = \frac{\langle f, b_k \rangle}{\langle b_k, b_k \rangle}.

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The orthogonal functions b_k: \mathbb{R} \to \mathbb{C} have not yet been specified. There are many possible choices (Legendre polynomials, Bessel functions, spherical harmonics etc.) for these functions, for the Fourier series specifically we make use trigonometric or more generally imaginary exponential functions.

Lemma: in the special case that b_k: \mathbb{R} \to \mathbb{C} is given by

b_k(t) = \exp(i k \omega_0 t),

for all t \in \mathbb{R} with k \in \mathbb{Z} and \omega_0 \in \mathbb{R} the angular frequency. A periodic function f: \mathbb{R} \to \mathbb{C} with period T_0 = \frac{2\pi}{\omega_0} may be approximated by

\phi_n(t) = \sum_{k = 0}^n c_k e^{i k \omega_0 t},

for all t \in \mathbb{R}. With the coefficients c_k \in \mathbb{C} given by

c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.

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Lemma: For a periodic function f: \mathbb{R} \to \mathbb{C} and its approximation \phi_n given in the above lemma we have

\lim_{n \to \infty} |f - \phi_n | = 0,

implies that the resulting series approximation converges to f. Similarly the series approximation converges also pointwise

\lim_{n \to \infty} |f(t) - \phi_n(t)| = 0,

for all t \in D with D \subseteq \mathbb{R} the interval where f is continuous.

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The Fourier series

With the above lemmas we may state the following theorems.

Theorem: the classical Fourier series of a periodic function f: \mathbb{R} \to \mathbb{C} with period T_0 = \frac{2\pi}{\omega_0} may be given by

f(t) = \sum_{k = -\infty}^\infty c_k e^{i k \omega_0 t},

for all t \in \mathbb{R}. With the coefficients c_k \in \mathbb{C} given by

c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.

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Expanding the Fourier series such that it can also approximate aperiodic functions obtains.

Theorem: the Fourier series of an aperiodic function f: \mathbb{R} \to \mathbb{C} may be given

f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,

for all t \in \mathbb{R}. The expansion coefficient F: \mathbb{R} \to \mathbb{C} is given by

F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt

for all \omega \in \mathbb{R}. Is called the Fourier transform of f and represents the continuous frequency spectrum of f.

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