Q.A message signal of frequency 10 kHz and peak voltage of 10 volts is used to modulate a carrier of frequency 1 MHz and peak voltage of 20 volts. Determine

Amplitude Modulation: From Intuition to Precision

Imagine you need to send a voice signal — say, your friend's speech — across a city. The voice itself is a low-frequency wave, maybe a few hundred hertz. If you just broadcast that directly, it won't travel far; low frequencies get absorbed quickly and need enormous antennas. So you need a "vehicle" that can carry that voice over long distances. That vehicle is a high-frequency carrier wave — think of it as a clean, steady sine wave at, say, 1 MHz.

Now, how do you make the carrier carry your voice? You can't just add the two waves together — that would give a messy sum, not a proper transmission. Instead, you let the voice signal control the carrier's amplitude. When your voice is loud, the carrier's amplitude grows; when your voice is soft, the carrier's amplitude shrinks. The carrier's frequency stays fixed — only its envelope (the outer shape of the wave) follows the message.

That is the core intuition: Amplitude Modulation (AM) is the process of imprinting a low-frequency message onto a high-frequency carrier by varying the carrier's amplitude in step with the message.

The Precise Statement

Let the carrier wave be a pure sinusoid:

where $A_c$ is the carrier amplitude and $f_c$ is the carrier frequency (much higher than the message frequency).

Let the message signal be $m(t)$. In standard AM, the modulated signal is:

The amplitude of the carrier is no longer constant — it is $A_c + m(t)$. That sum is the instantaneous amplitude of the carrier, and it varies exactly as $m(t)$ varies.

The modulation index $\mu$ controls how much the carrier amplitude swings. If $\mu > 1$, the envelope distorts — the message is lost. If $\mu = 0$, there is no modulation at all — just the bare carrier.

What You Actually See

If you plot $s(t)$ against time, you see a high-frequency wave whose peaks and troughs trace out the shape of $m(t)$. The upper envelope is $A_c + m(t)$, the lower envelope is $-(A_c + m(t))$. A simple AM receiver (like a diode detector) extracts that envelope and recovers the original message. …