Q.Define the modulation index of an amplitude-modulated wave and state the condition its value must satisfy for the modulated wave to remain undistorted.

Modulation Index: The Depth of the Message

Imagine you are holding a microphone, speaking into it. Your voice is a weak electrical signal — the modulating signal. To send it across a city, you need to "ride" it on a much stronger, high-frequency wave — the carrier wave. Amplitude modulation (AM) is the process of making the carrier's strength (its amplitude) change in step with your voice.

The modulation index is simply a number that tells you how much the carrier's amplitude is being varied by the message. It answers the question: "How deep is the imprint of the message on the carrier?"

The Intuition: A Handshake

Think of the carrier as a tall, steady pillar of height $A_c$. Your voice signal is a smaller, wavy force of maximum height $A_m$ that pushes and pulls on the top of that pillar. The modulation index $m$ is the ratio:

If $A_m$ is very small compared to $A_c$, the pillar barely wobbles — the modulation is shallow ($m$ is small). If $A_m$ is almost as large as $A_c$, the pillar wobbles a lot — the modulation is deep ($m$ is close to 1). If $A_m$ exceeds $A_c$, the pillar would be pushed so hard that it gets "over-modulated" and the message gets distorted.

The Precise Statement

For a sinusoidal modulating signal $v_m(t) = A_m \cos(2\pi f_m t)$ and a carrier $v_c(t) = A_c \cos(2\pi f_c t)$, the amplitude-modulated wave is:

where the modulation index $m$ is defined as:

This is a dimensionless number, always between 0 and 1 for distortion-free AM.

What Happens at Different Values?