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change even symmetric to conjugate symmetric and odd symmetric to conjugate antisymmetric IAW Oppenheim & Schaefer Tag: Reverted |
Reverted 2 edits by Bob K (talk): "conjugate symmetric" implies that some complex conjugation appears is the formula |
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===Complex-valued functions===
The definitions for even and odd symmetry for [[Complex number|complex-valued]] functions of a real argument are similar to the real case but involve [[complex conjugation]].
'''Even symmetry:'''
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''
:<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
'''Odd symmetry:'''
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''
:<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
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'''Even symmetry:'''
A ''N''-point sequence is called ''
:<math>f(n) = f(N-n) \quad \text{for all } n \in \left\{ 1,\ldots,N-1 \right\}.</math>
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'''Odd symmetry:'''
A ''N''-point sequence is called ''
:<math>f(n) = -f(N-n) \quad \text{for all } n \in \left\{1,\ldots,N-1\right\}. </math>
Such a sequence is sometimes called an '''anti-palindromic sequence'''; see also [[Palindromic polynomial|Antipalindromic polynomial]].
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