Differentiation Theorem Dual

Theorem: Let $x(n)$ denote a signal with DTFT $X(\ejo )$ , and let

$$X^\prime(\ejo ) \isdefs \frac{d}{d\omega} X(\ejo )$$ (3.40)

denote the derivative of $X$ with respect to $\omega$ . Then we have

$$\zbox {-jn x(n) \;\longleftrightarrow\;\frac{d}{d\omega}X(\ejo )}$$

where $X(\ejo )$ denotes the DTFT of $x(n)$ .

Proof: Using integration by parts, we obtain

$$\begin{eqnarray*} \hbox{\sc IDTFT}_{n}(X^\prime) &\isdef & \frac{1}{2\pi}\int_{-\pi}^\pi X^\prime(\ejo ) e^{j\omega n} d\omega\\ &=& \left. \frac{1}{2\pi}X(\ejo )e^{j\omega t}\right\vert _{-\pi}^{\pi} - \frac{1}{2\pi}\int_{-\pi}^\pi X(\ejo ) (jn)e^{j\omega n} d\omega\\ &=& -jn x(n). \end{eqnarray*}$$

An alternate method of proof is given in §B.3.

Corollary: Perhaps a cleaner statement is as follows:

$$\zbox {- n x(n) \;\longleftrightarrow\;\frac{d}{d(j\omega)}X(\ejo )}$$

This completes our coverage of selected DTFT theorems. The next section adds some especially useful FT theorems having no precise counterpart in the DTFT (discrete-time) case.