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Undid revision 1230071245 by D.Lazard (talk) Conjugation is what the bar over f(-x) means. Asterisk would also be fine with me. |
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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.
{{Cite book |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |___location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |page=55
}}</ref><ref name=ProakisManolakis/>
'''
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''conjugate symmetric'' if
:<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
A complex valued function is conjugate symmetric is and only if and only if its [[real part]] is an even function and its [[imaginary part]] is an odd function.
A typical example of a conjugate symmetric function is the [[cis function]]
:<math>x \to e^{ix}=\cos x + i\sin x</math>
'''Conjugate antisymmetry:'''
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''conjugate antisymmetric'' if:
:<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
A complex valued function is conjugate antisymmetric is and only if and only if its [[real part]] is an odd function and its [[imaginary part]] is an even function.
===Finite length sequences===
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