Time Reversal

Definition:

$$\hbox{\sc Flip}_n( x ) \mathrel{\stackrel{\Delta}{=}}x(-n) \mathrel{\stackrel{\Delta}{=}}x(N-n)$$

Note: $x(n) \mathrel{\stackrel{\Delta}{=}}x(n \,$mod$\, N )$ for signals in $\mathbb{C}^N$ (DFT case).

When computing a sampled DTFT using the DFT, we interpret time indices $n=1,2,\ldots,N/2-1$ as positive time indices, and $n=N-1,N-2,\ldots,N/2$ as the negative time indices $n=-1,-2,\ldots,-N/2$ . Under this interpretation, the $\hbox{\sc Flip}$ operator simply reverses a signal in time.

Fourier theorems:

$$\zbox{\hbox{\sc Flip}(x) \leftrightarrow \hbox{\sc Flip}(X)}$$

for $x \in \mathbb{C}^N$ . In the typical special case of real signals ( $x\in\mathbb{R}^N$ ), we have $\hbox{\sc Flip}(X)=\overline{X}$ so that

$$\zbox{\hbox{\sc Flip}(x) \leftrightarrow \overline{X}}$$

Time-reversing a real signal conjugates its spectrum