Modulation of a Carrier Signal

- Modulation is a process where a high-frequency carrier signal is modified by a low-frequency information signal.
- Amplitude modulation (AM) is a type of modulation where the amplitude of the carrier signal is varied in proportion to the instantaneous amplitude of the modulating signal.

Symmetrical Square Wave Modulation

- A symmetrical square wave has equal positive and negative pulse widths and a 50% duty cycle.
- The period of the square wave is 100 sec, which means the frequency of the modulating signal is 1/100 Hz or 0.01 Hz.
- The modulating signal has a frequency much lower than the carrier signal, which is 1 MHz or 1,000,000 Hz.

Frequency Components of the Modulated Signal

- When a carrier signal is modulated by a symmetrical square wave, the frequency components of the resulting modulated signal can be calculated using Fourier analysis.
- The carrier signal is given by cos(2πfct), where fc is the carrier frequency of 1 MHz.
- The modulating signal is a symmetrical square wave with a frequency of 0.01 Hz.
- The resulting modulated signal can be expressed as:

cos(2πfct) [1 + mcos(2πfmt)]

where m is the modulation index and fM is the frequency of the modulating signal.

- The modulation index can be calculated as the ratio of the amplitude of the modulating signal to the amplitude of the carrier signal.

m = Am / Ac

where Am is the amplitude of the modulating signal and Ac is the amplitude of the carrier signal.

- In this case, the modulation index is not given, but it can be assumed to be less than 1, which means the modulation is a low-level AM.

- The frequency components of the modulated signal can be calculated using Fourier analysis:

cos(2πfct) [1 + mcos(2πfmt)]

= cos(2πfct) + m/2 cos[2π(fc-fM)t] + m/2 cos[2π(fc+fM)t]

- The first term is the carrier signal, which has a frequency of 1 MHz.
- The second term has a frequency of (fc - fM) = (1 MHz - 0.01 Hz) = 999,999.99 Hz, which is close to 1 MHz.
- The third term has a frequency of (fc + fM) = (1 MHz + 0.01 Hz) = 1,000,000.01 Hz, which is also close to 1 MHz.

Answer Explanation

- Option A: 990 KHz is close to the carrier frequency of 1 MHz, so it is likely to be present in the modulated signal.
- Option B: 1010 KHz is also close to the carrier frequency of 1 MHz, so it is likely to be present in the modulated signal.
- Option C: 1020 KHz is not close to the carrier frequency of 1 MHz or any of the frequency components calculated using Fourier analysis, so it is not likely to be present in the modulated signal.
- Option D: 1030 KHz is close to the third frequency component calculated using Fourier analysis, which is 1,000,000.01 Hz. Therefore, it is likely to be present in the modulated