CTFT convolution; DTFT of 2ⁿu[n]

Convolution property of CTFT

Statement: If $x(t) \leftrightarrow X(j\omega)$ and $h(t) \leftrightarrow H(j\omega)$, then

$$y(t) = x(t) * h(t) \;\longleftrightarrow\; Y(j\omega) = X(j\omega)\,H(j\omega)$$

Proof: The convolution is $y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau$. Its FT is

$$\begin{aligned} Y(j\omega) &= \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau\right]e^{-j\omega t}dt \\ &= \int_{-\infty}^{\infty} x(\tau)\left[\int_{-\infty}^{\infty} h(t-\tau)e^{-j\omega t}dt\right]d\tau \end{aligned}$$

The inner integral is the FT of $h(t)$ shifted by $\tau$, which by the time-shifting property is $e^{-j\omega\tau}H(j\omega)$:

$$\begin{aligned} Y(j\omega) &= \int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega\tau}H(j\omega)\,d\tau \\ &= H(j\omega)\int_{-\infty}^{\infty} x(\tau)e^{-j\omega\tau}d\tau = X(j\omega)H(j\omega) \end{aligned}$$

Hence convolution in time becomes multiplication in frequency. This is why the output of an LTI system is found simply as $Y = XH$, where $H(j\omega)$ is the frequency response.

DTFT of $x[n] = 2^n u[n]$

$$X(e^{j\omega}) = \sum_{n=0}^{\infty} 2^n e^{-j\omega n} = \sum_{n=0}^{\infty}\left(2e^{-j\omega}\right)^n$$

This geometric series converges only if $|2e^{-j\omega}| < 1$, i.e. $2 < 1$, which is false. Also $\sum|x[n]| = \sum 2^n = \infty$, so the sequence is not absolutely summable. The DTFT of $2^n u[n]$ does not exist (its z-transform $\frac{1}{1-2z^{-1}}$, ROC $|z|>2$, does not include the unit circle).

The question is usually intended as $x[n] = \left(\frac{1}{2}\right)^n u[n]$, which is solved below.

$$X(e^{j\omega}) = \sum_{n=0}^{\infty}\left(\tfrac12 e^{-j\omega}\right)^n = \frac{1}{1 - 0.5\,e^{-j\omega}}$$ $$|X(e^{j\omega})| = \frac{1}{\sqrt{1.25 - \cos\omega}}, \qquad \angle X(e^{j\omega}) = -\tan^{-1}\frac{0.5\sin\omega}{1 - 0.5\cos\omega}$$

 |X|
     2 |                        *
       |                     *     *
       |
       |                  *           *
       |               *                 *
       |         *  *                       *  *
  0.67 |*  *  *                                   *  *  *
       +-------------------------------------------------> w
       -pi                      0                      pi

 phase (degrees)
   +30 |            *  *  *
       |         *           *
       |      *
       |   *
     0 |*                       *                       *
       |                                             *
       |                                          *
       |                           *           *
   -30 |                              *  *  *
       +-------------------------------------------------> w
       -pi                      0                      pi

The magnitude is even (low-pass, peak 2 at $\omega = 0$), the phase is odd (extremes $\pm30^\circ$ at $\omega = \mp\pi/3$), and both repeat every $2\pi$.