Editing Fourier analysis (section)

==Symmetry properties==
When the real and imaginary parts of a complex function are decomposed into their [[Even and odd functions#Even–odd decomposition|even and odd parts]], there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform''':'''<ref name=Proakis/>

:<math>
\begin{array}{rccccccccc}
\text{Time domain} & s & = & s_{_{\text{RE}}} & + & s_{_{\text{RO}}} & + & i s_{_{\text{IE}}} & + & \underbrace{i\ s_{_{\text{IO}}}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\text{Frequency domain} & S & = & S_\text{RE} & + & \overbrace{\,i\ S_\text{IO}\,} & + & i S_\text{IE} & + & S_\text{RO}
\end{array}
</math>

From this, various relationships are apparent, for example''':'''
*The transform of a real-valued function <math>(s_{_{RE}}+s_{_{RO}})</math> is the [[Even and odd functions#Complex-valued functions|''conjugate symmetric'']] function <math>S_{RE}+i\ S_{IO}.</math>  Conversely, a ''conjugate symmetric'' transform implies a real-valued time-domain.
*The transform of an imaginary-valued function <math>(i\ s_{_{IE}}+i\ s_{_{IO}})</math> is the [[Even and odd functions#Complex-valued functions|''conjugate antisymmetric'']] function <math>S_{RO}+i\ S_{IE},</math> and the converse is true.
*The transform of a [[Even and odd functions#Complex-valued functions|''conjugate symmetric'']] function <math>(s_{_{RE}}+i\ s_{_{IO}})</math> is the real-valued function <math>S_{RE}+S_{RO},</math> and the converse is true.
*The transform of a [[Even and odd functions#Complex-valued functions|''conjugate antisymmetric'']] function <math>(s_{_{RO}}+i\ s_{_{IE}})</math> is the imaginary-valued function <math>i\ S_{IE}+i\ S_{IO},</math> and the converse is true.