Topic: Signals and Systems
If a discrete-time LTI system has a frequency response $H(e^{j\omega})=1/(1-0.5e^{-j\omega})$, what is its impulse response $h[n]$?
- A. $(0.5)^{n}u[n]$
- B. $(0.5)^{(n-1)}u[n-1]$
- C. $u[n]-0.5u[n-1]$
- D. $\delta[n]-0.5\delta[n-1]$
Answer: A. $(0.5)^{n}u[n]$
Explanation: The given frequency response $H(e^{j\omega})$ represents the Discrete-Time Fourier Transform (DTFT) of the impulse response $h[n]$. The general DTFT pair for a right-sided exponential sequence is $\mathcal{F}\\{a^n u[n]\\} = \frac{1}{1 - a e^{-j\omega}}$, provided $|a| < 1$. Matching this with the given $H(e^{j\omega})$, we have $a = 0.5$. Therefore, the inverse DTFT gives the impulse response $h[n] = (0.5)^n u[n]$.