Conjugation and Reversal

Theorem: For any $x\in{\bf C}^N$ ,

$\displaystyle \zbox {\overline{x} \;\longleftrightarrow\;\hbox{\sc Flip}(\overline{X}).}$

Proof:

$\begin{eqnarray*} \hbox{\sc DFT}_k(\overline{x}) &\isdef & \sum_{n=0}^{N-1}\ov... ...n) e^{-j 2\pi n(-k)/N}} \isdef \hbox{\sc Flip}_k(\overline{X}) \end{eqnarray*}$

Theorem: For any $x\in{\bf C}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(\overline{x}) \;\longleftrightarrow\;\overline{X}.}$

Proof: Making the change of summation variable $m\isdeftext N-n$ , we get

$\begin{eqnarray*} \hbox{\sc DFT}_k(\hbox{\sc Flip}(\overline{x})) &\isdef & \s... ...sum_{m=0}^{N-1}x(m) e^{-j 2\pi m k/N}} \isdef \overline{X(k)}. \end{eqnarray*}$

Theorem: For any $x\in{\bf C}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\hbox{\sc Flip}(X).}$

Proof:

$\begin{eqnarray*} \hbox{\sc DFT}_k[\hbox{\sc Flip}(x)] &\isdef & \sum_{n=0}^{N-1... ...-1}x(m) e^{j 2\pi mk/N} \isdef X(-k) \isdef \hbox{\sc Flip}_k(X) \end{eqnarray*}$

Corollary: For any $x\in{\bf R}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\overline{X}}$ $\displaystyle \mbox{($x$\ real).}$

Proof: Picking up the previous proof at the third formula, remembering that is real,

$\displaystyle \sum_{m=0}^{N-1}x(m) e^{j 2\pi mk/N} = \overline{\sum_{m=0}^{N-1}... .../N}} = \overline{\sum_{m=0}^{N-1}x(m) e^{-j 2\pi mk/N}} \isdef \overline{X(k)}$

when

is real.

Thus, conjugation in the frequency domain corresponds to reversal in the time domain. Another way to say it is that negating spectral phase flips the signal around backwards in time.

Corollary: For any $x\in{\bf R}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(X) = \overline{X}}$ $\displaystyle \mbox{($x$\ real).}$

Proof: This follows from the previous two cases.

Definition: The property $X(-k)=\overline{X(k)}$ is called Hermitian symmetry or ``conjugate symmetry.'' If $X(-k)=-\overline{X(k)}$ , it may be called skew-Hermitian.