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 baseband signal. This modulation technique can be mathematically represented 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:

• \( K_a \) is the amplitude sensitivity of the modulator
• \( s(t) \) is the modulated signal
• \( A_c \) is the amplitude of the carrier signal
• \( m(t) \) is the modulating (baseband) signal

Block Diagram for Amplitude Modulation

Fig: Amplitude Modulation

Modulation Index (μ) in Amplitude Modulation

The modulation index \( \mu \) quantifies the extent to which the carrier amplitude varies in response to the message signal.

It is commonly defined as:
\( \mu = K_a A_m \), where:

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

The value of \( \mu \) determines the modulation quality:

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

Alternative definition:
Even if \( K_a \) and \( A_m \) are unknown, the modulation index can be calculated directly from the modulated waveform:

\( \mu = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}} \)
where:

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

Frequency Domain Description :

To understand the spectral characteristics of a AM wave, we analyze it in the frequency domain by applying the Fourier transform to its time-domain representation.

The AM signal is defined as:

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

Taking the Fourier transform of both sides, and applying frequency-shifting and delta function properties:

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

This spectrum consists of two impulse functions weighted by the factor πA_c, located at ±ω_c, along with two replicas of the message spectrum that are shifted in frequency by ±ω_c and scaled in amplitude by (1/2)k_aA_c.

Fig: Time-domain (on the left) and frequency-domain (on the right) characteristics of AM produced by a sinusoidal modulating wave. (a) Modulating wave. (b) Carrier wave. (c) AM wave.

Bandwidth

While the provided lab text displays the spectrum of an AM signal, it does not explicitly state the formula for calculating its required bandwidth. The bandwidth is defined as the difference between the highest and lowest frequencies present in the modulated signal's spectrum.

• For a standard AM signal, the bandwidth is precisely twice the highest frequency component found in the modulating signal (also known as the message signal). The formula is:
  BW = 2 * f_m
  Where fm is the maximum frequency of the message signal.

Block Diagram for Amplitude Demodulation

Fig: Amplitude Demodulation