Topic: Digital Signal Processing
What is Discrete Time Fourier Transform (DTFT) of following signal?
$x(n)=a^{n}$ where $n\ge 0$, $-1<a<1$
- A. $X(\omega)=\frac{1}{(1-ae^{-j\omega})}$
- B. $X(\omega)=\frac{1}{(1+ae^{-j\omega})}$
- C. $X(\omega)=\frac{a}{(1-ae^{-j\omega})}$
- D. $X(\omega)=\frac{a}{(1+ae^{-j\omega})}$
Answer: A. $X(\omega)=\frac{1}{(1-ae^{-j\omega})}$
Explanation: $x(n) = a^{n}u(n)$; $-1<a<1$
$X(\omega) = \sum_{n=0}^{\infty}a^{n}e^{-j\omega n} = \sum_{n=0}^{\infty}(ae^{-j\omega})^{n} = \frac{1}{1-ae^{-j\omega}}$