Amplitude Modulation (AM) and Demodulation

Theory :

Amplitude modulation (AM) is the process in which the amplitude of a carrier wave, denoted as \( c(t) = \cos(2\pi f_c t) \), is varied in proportion to the message signal \( m(t) \). Mathematically, this is expressed as:

\( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(2\pi f_c t) \)

or equivalently:

\( s(t) = A_c \cos(2\pi f_c t) + A_c K_a m(t) \cos(2\pi f_c t) \)

where:

• \( A_c \) = carrier amplitude
• \( f_c \) = carrier frequency
• \( m(t) \) = message (modulating) signal
• \( K_a \) = amplitude sensitivity of the modulator
• \( s(t) \) = amplitude-modulated signal

Block Diagram for Amplitude Modulation

Fig 1: Amplitude Modulation

Modulation Index (\( \mu \))

The modulation index \( \mu \) quantifies how much the carrier amplitude varies in response to the message signal. It is defined as:

\( \mu = A_m / A_c \) or \( \mu = K_a A_m \), where:

• \( A_m \) = peak amplitude of the message signal \( m(t) \)
• \( A_c \) = carrier amplitude
• \( K_a \) = amplitude sensitivity

The modulation index determines modulation quality:

• \( \mu < 1 \): Under-modulation
• \( \mu = 1 \): 100% modulation (ideal)
• \( \mu > 1 \): Over-modulation (causes distortion)

Alternative definition: If \( K_a \) and \( A_m \) are unknown, \( \mu \) can be calculated from the modulated waveform:
\( \mu = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}} \)

• \( A_{\text{max}} \) = maximum amplitude of the modulated signal
• \( A_{\text{min}} \) = minimum amplitude of the modulated signal

Frequency Domain Description

The frequency-domain representation of an AM signal is obtained by applying the Fourier transform to \( s(t) \):

\( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(\omega_c t) \)

Applying Fourier transform and frequency-shifting properties:

\( S(j\omega) = \pi A_c [ \delta(\omega - \omega_c) + \delta(\omega + \omega_c) ] + \frac{1}{2} K_a A_c [ M(j(\omega - \omega_c)) + M(j(\omega + \omega_c)) ] \)

This spectrum consists of two impulses at ±\( \omega_c \) with amplitude \( \pi A_c \), plus two replicas of the message spectrum shifted to ±\( \omega_c \) and scaled by \( \frac{1}{2} K_a A_c \).

Fig 2: Time-domain (left) and frequency-domain (right) representation of AM with a sinusoidal message.

Bandwidth

The bandwidth of an AM signal is twice the highest frequency component of the message signal \( m(t) \):

BW = 2 * f_m

where \( f_m \) is the maximum frequency in the message signal.

Amplitude Demodulation

Fig 3: Amplitude Demodulation

Synchronous (coherent) demodulation recovers the original message signal. The AM input is:

\( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(2\pi f_c t) \)

• \( A_c \): carrier amplitude
• \( f_c \): carrier frequency
• \( m(t) \): message signal
• \( K_a \): amplitude sensitivity

Product Modulator (Multiplier)

Multiply \( s(t) \) with a locally generated carrier \( c(t) = \cos(2\pi f_c t) \):

\( v(t) = s(t) \cdot c(t) = A_c [1 + K_a m(t)] \cos^2(2\pi f_c t) \)

Using \( \cos^2\theta = \frac{1}{2}(1 + \cos2\theta) \):

\( v(t) = \frac{A_c}{2} [1 + K_a m(t)] + \frac{A_c}{2} [1 + K_a m(t)] \cos(4\pi f_c t) \)

The first term is the desired low-frequency component; the second term is a high-frequency component at \( 2f_c \).

Low-Pass Filter (LPF)

The LPF removes the high-frequency component while preserving the low-frequency message. The output is:

\( v_{LPF}(t) = \frac{A_c}{2} + \frac{A_c K_a}{2} m(t) \)

After DC removal and optional amplification, the original message signal \( m(t) \) is recovered.

Envelope Detector

Coherent demodulation above requires a locally generated carrier that is perfectly synchronized in frequency and phase with the transmitter -- an added cost and complexity. AM's key practical advantage over DSB-SC is that it can instead be demodulated with a much simpler and cheaper envelope detector: a diode rectifier followed by a low-pass RC filter, with no local oscillator needed at all.

This works because a full-carrier AM signal's amplitude envelope directly follows the message, as long as the envelope never crosses zero:

\( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(2\pi f_c t) \)

The diode passes only the positive half-cycles (half-wave rectification), and the RC filter then smooths this rectified signal to trace out its envelope, \( A_c [1 + K_a m(t)] \) -- which is exactly the message signal riding on a DC offset.

This only works correctly when the modulation index \( \mu = K_a \cdot \max|m(t)| \le 1 \). If \( \mu > 1 \), the AM signal is said to be over-modulated: \( 1 + K_a m(t) \) goes negative for part of the cycle, the envelope no longer follows \( m(t) \) faithfully, and the envelope detector output becomes visibly distorted. This is why AM transmitters are designed to keep \( \mu \le 1 \) at all times.