Properties of the DTFT We cover a few here and you can read about the rest in the textbook.

LINEARITY

$$ax_1[n] + b x_2[n] \longleftrightarrow a X_1 (\Omega) + b X_2 (\Omega)$$

TIME SHIFT

MODULATION - Frequency Shift

CONVOLUTION IN TIME

As usual,

$$x_1[n] \ast x_2[n] \longleftrightarrow X_1(\Omega) X_2 (\Omega)$$

Ex. Given $h[n]=a^n u[n], \vert a\vert<1$. Find its inverse system $h_i[n]$.

$x[n] = (.9)^{\vert n\vert}$. Find its DTFT.

MULTIPLICATION OF SIGNALS

$$x_1[n]x_2[n] \longleftrightarrow {1 \over 2 \pi} X_1 (\Omega ) \bigotimes X_2 (\Omega)$$

where $\bigotimes$ denotes CIRCULAR CONVOLUTION:

$$X_1 (\Omega) \bigotimes X_2 (\Omega) = \int_{2 \pi} X_1 ( \theta) X_2 (\Omega-\theta))d\theta$$

$$y[n]=x_1[n] x_2[n] \longleftrightarrow {\rm Take\ its\ DTFT:}$$

$$Y(\Omega) = \sum_n y[n] e^{-j \Omega n} = \sum_n x_1[n] x_2[n] e^{ -j \Omega n}$$

$$x_1 [n] = {1 \over 2 \pi} \int_{2 \pi} X_1(a) e^{jan} da$$

$$x_2 [n] = {1 \over 2 \pi} \int_{2 \pi} X_2(b) e^{jbn} db$$

$$Y(\Omega) = \sum_n \left[ {1 \over 2 \pi} \int_{da} X_1(a) e^... ...er 2\pi} \int_{db} X_2 (b) e^{jbn} db \right] e^{-j\Omega n}$$

$$= \left( {1\over 2 \pi} \right)^2 \int_{da} \int_{db} X_1(a) X_2(b) \sum_n e^{j (a+b)n} e^{-j \Omega n} da db$$

Now,

$$\sum_n e^{j(a+b)n} e^{-j \Omega n}$$

is just the DTFT of

$$e^{j(a+b)n}$$

that is,

$$e^{j(a+b)n} \leftrightarrow 2 \pi \delta(\Omega -a -b )$$

So,

$$\begin{eqnarray*} Y(\Omega) &=& { 1\over 2\pi} \int_{da \over 2 \pi} \int_{db ... ... \over 2 \pi} X_1 (\Omega ) \bigotimes X_2 (\Omega) \nonumber \end{eqnarray*}$$