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Category:Statistics templates

Category:Sidebar templates Category:Statistics templates Category:Sidebar templates by topic

Symmetry properties



 * If $$f$$ is a real function, then $$c_{n} = c_{-n}^*$$ (Hermitian symmetric) which implies:
 * $$\Re{(c_{n})} = \Re{(c_{-n})}$$ (real part is even symmetric)
 * $$\Im{(c_{n})} = -\Im{(c_{-n})}$$ (imaginary part is odd symmetric)
 * $$|c_{n}| = |c_{-n}|$$ (absolut value is even symmetric)
 * $$\arg{(c_{n})} = \arg{(-c_{-n})}$$ (argument is odd symmetric)
 * If $$f$$ is a real and even function ($$f(x)=f(-x) \forall x \in \mathbb{R}$$), then all coefficients $$c_n$$ are real and $$c_{n} = c_{-n}$$ (even symmetric) which implies:
 * $$b_n = 0$$ for all $$n \ge 1$$
 * If $$f$$ is a real and odd function ($$f(x)=-f(-x) \forall x \in \mathbb{R}$$), then all coefficients $$c_n$$ are purely imaginary and $$c_{n} = -c_{-n}$$ (odd symmetric) which implies:
 * $$a_n = 0$$ for all $$n \ge 0$$
 * If $$f$$ is a purely imaginary function, then $$c_{n} = -c_{-n}^*$$ which implies:
 * $$\Re{(c_{n})} = -\Re{(c_{-n})}$$ (real part is odd symmetric)
 * $$\Im{(c_{n})} = \Im{(c_{-n})}$$ (imaginary part is even symmetric)
 * $$|c_{n}| = |c_{-n}|$$ (absolut value is even symmetric)
 * $$\arg{(c_{n})} = \arg{(-c_{-n})}$$ (argument is odd symmetric)
 * If $$f$$ is a purely imaginary and even function ($$f(x)=f(-x) \forall x \in \mathbb{R}$$), then all coefficients $$c_n$$ are purely imaginary and $$c_{n} = c_{-n}$$ (even symmetric).
 * If $$f$$ is a purely imaginary and odd function ($$f(x)=-f(-x) \forall x \in \mathbb{R}$$), then all coefficients $$c_n$$ are real and $$c_{n} = -c_{-n}$$ (odd symmetric).

Table of Fourier Series coefficients
Some common pairsof periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies:


 * $$f(x)$$ designates a periodic function defined on $$0 \le x < T $$.
 * $$a_k, b_k, a_0$$ designates a ...
 * $$c_k$$ designates a ...

Properties
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
 * $$x[n]^{*}$$ is the complex conjugate of $$x[n]$$.
 * $$f(x),g(x)$$ designate a $$T$$-periodic functions defined on $$0 < x \le T $$.
 * $$\hat{f}[n], \hat{g}[n]$$ designates the Fourier series coefficients (exponential form) of $$f$$ and $$g$$ as defined in equation TODO!!!