Shift Theorem

Theorem: For any $x\in\mathbb{C}^N$ and any integer $\Delta$ ,

$$\zbox {\hbox{\sc DFT}_k[\hbox{\sc Shift}_\Delta(x)] = e^{-j\omega_k\Delta} X(k).}$$

Proof:

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

The shift theorem is often expressed in shorthand as

$$\zbox {x(n-\Delta) \longleftrightarrow e^{-j\omega_k\Delta}X(\omega_k).}$$

The shift theorem says that a delay in the time domain corresponds to a linear phase term in the frequency domain. More specifically, a delay of $\Delta$ samples in the time waveform corresponds to the linear phase term $e^{-j \omega_k \Delta}$ multiplying the spectrum, where $\omega_k\isdeftext 2\pi k/N$ .^7.14Note that spectral magnitude is unaffected by a linear phase term. That is, $\left\vert e^{-j \omega_k \Delta}X(k)\right\vert = \left\vert X(k)\right\vert$ .