Question

Consider a modulating signal \(m(t)=2\sin(2\pi 10^3 t)\) is used to modulate a carrier of frequency \(10^6\) Hz. Find the bandwidth for phase modulation. Use phase modulation index \(=10\) units.

• A. \(14000\) Hz
• B. \(24000\) Hz
• C. \(22000\) Hz
• D. \(44000\) Hz

Hint:

Carson’s rule: \[ BW=2(\beta+1)f_m \] used for estimating bandwidth of angle modulation systems.

Answer 1

The Correct Option is C

Concept: Bandwidth of angle modulation can be estimated using Carson’s rule: :contentReference[oaicite:3]{index=3} where:
• \(\beta\) = modulation index
• \(f_m\) = modulating frequency

Step 1: Identify the modulating frequency. Given: \[ m(t)=2\sin(2\pi 10^3 t) \] Comparing with: \[ A\sin(2\pi f_mt) \] we get: \[ f_m=10^3\ \text{Hz} \] \[ f_m=1000\ \text{Hz} \]

Step 2: Write modulation index. Given: \[ \beta=10 \]

Step 3: Apply Carson’s rule. \[ BW=2(\beta+1)f_m \] Substitute values: \[ BW=2(10+1)(1000) \] \[ BW=2(11)(1000) \] \[ BW=22000\ \text{Hz} \]

Step 4: Write final answer. Therefore the bandwidth is: \[ \boxed{22000\ \text{Hz}} \] Hence correct option is: \[ \boxed{(C)} \]