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Chapter 20: Problem 2746

A \(1000 \mathrm{KHz}\) carrier is modulated with \(800 \mathrm{~Hz}\) audio signals. What are the frequencies of first pair side band? (A) \(1000 \mathrm{KHz}\) and \(800 \mathrm{~Hz}\) (B) \(800 \mathrm{~Hz}\) and \(999.2 \mathrm{KHz}\) (C) \(1000.8 \mathrm{KHz}\) and \(999.2 \mathrm{KHz}\) (D) \(1000.8 \mathrm{KHz}\) and \(1000 \mathrm{KHz}\)

Short Answer

Expert verified
The short answer is: \(1000.8 \mathrm{KHz}\) and \(999.2 \mathrm{KHz}\).

Step by step solution

01

Identify the carrier frequency and modulating frequency

We are given a carrier signal with a frequency of 1000 KHz and a modulating audio signal with a frequency of 800 Hz. Let's represent these frequencies as 𝑓𝑐 (carrier frequency) and 𝑓𝑚 (modulating frequency).
02

Convert frequencies to the same unit

Before performing any calculations, let's convert the modulating frequency to the same unit as the carrier frequency: 800 Hz = 0.8 KHz Now, 𝑓𝑐 = 1000 KHz and 𝑓𝑚 = 0.8 KHz.
03

Calculate the sideband frequencies

Using the formulas for the upper sideband (USB) and the lower sideband (LSB) frequencies resulting from the modulation process, we can now calculate the frequencies of the first pair of sidebands: Upper Sideband Frequency (USB) = 𝑓𝑐 + 𝑓𝑚 Lower Sideband Frequency (LSB) = 𝑓𝑐 - 𝑓𝑚
04

Plug in the values and compute the sideband frequencies

Let's plug in the values for 𝑓𝑐 and 𝑓𝑚 and compute the USB and LSB frequencies: USB = 1000 KHz + 0.8 KHz = 1000.8 KHz LSB = 1000 KHz - 0.8 KHz = 999.2 KHz
05

Determine the correct answer

Now that we have calculated the frequencies of the first pair of sidebands, we can determine the correct option: (A) 1000 KHz and 800 Hz (B) 800 Hz and 999.2 KHz (C) 1000.8 KHz and 999.2 KHz (D) 1000.8 KHz and 1000 KHz The correct answer is (C) 1000.8 KHz and 999.2 KHz.

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