Shift Theorem

The Shift operator is defined as $\hbox{\sc Shift}_{l,n}(y) \mathrel{\stackrel{\Delta}{=}}y(n-l)$ . Since indexing is defined modulo $N$ , $\hbox{\sc Shift}_l(y)$ is a circular right-shift by $l$ samples.

$$\zbox{\hbox{\sc Shift}_l(y) \leftrightarrow e^{-j(\cdot)l}Y}$$

or, more loosely,

$$\zbox{y(n-l) \leftrightarrow e^{-j\omega l}Y(\omega)}$$

i.e.,

$$\hbox{\sc DFT}_k[\hbox{\sc Shift}_l(y)] = \left( e^{-j\omega_k l} \right) Y(\omega_k)$$

$$e^{-j\omega_k l} = \hbox{\emph{Linear Phase Term, slope} = $-l$}$$

• $\angle Y(\omega_k) \,\hbox{\texttt{+=}}\, -\omega_kl$
• Multiplying a spectrum $Y$ by a linear phase term $e^{-j\omega_k l}$ with phase slope $-l$ corresponds to a circular right-shift in the time domain by $l$ samples:
• negative slope $\Rightarrow$ time delay
• positive slope $\Rightarrow$ time advance