Principles of Amplitude Modulation (AM) and Coherent Demodulation
Theoretical Framework:
Amplitude Modulation (AM) is a linear modulation technique wherein the instantaneous amplitude of a high-frequency carrier wave, defined as \( c(t) = \cos(2\pi f_c t) \), is varied in a linear relationship with the baseband message signal \( m(t) \). The analytical expression for the modulated waveform is given by:
\( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(2\pi f_c t) \)
By expansion, the signal consists of the carrier component and the sideband components:
\( s(t) = A_c \cos(2\pi f_c t) + A_c K_a m(t) \cos(2\pi f_c t) \)
Where the parameters are defined as:
- \( A_c \): Unmodulated carrier amplitude
- \( f_c \): Carrier frequency (where \( f_c \gg \) message bandwidth)
- \( m(t) \): Information-bearing (modulating) signal
- \( K_a \): Amplitude sensitivity constant of the modulator
- \( s(t) \): Resultant amplitude-modulated carrier
System Architecture for AM Generation
Fig 1: Schematic Representation of the Amplitude Modulation Process
Modulation Index (\( \mu \)) and Depth
The modulation index (\( \mu \)) is a dimensionless parameter characterizing the degree of carrier excursion relative to the unmodulated level. It is formally defined as:
\( \mu = A_m / A_c \) or \( \mu = K_a A_m \), where:
- \( A_m \): Peak amplitude of the modulating signal \( m(t) \)
- \( A_c \): Carrier peak amplitude
- \( K_a \): Modulator sensitivity
The value of \( \mu \) dictates the envelope integrity and power efficiency:
- \( \mu < 1 \): Under-modulation (Envelope remains strictly positive)
- \( \mu = 1 \): Critical modulation (Maximum efficiency without phase reversal)
- \( \mu > 1 \): Over-modulation (Induces envelope distortion and phase discontinuities)
Empirical Calculation: In practical analysis, \( \mu \) is derived from the envelope extrema:
\( \mu = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}} \)
- \( A_{\text{max}} \): Maximum envelope excursion
- \( A_{\text{min}} \): Minimum envelope excursion
Interactive AM Lab
Message Freq ($f_m$ Hz)
Carrier Freq ($f_c$ Hz)
Msg Amplitude ($A_m$)
Carrier Amplitude ($A_c$)
Demodulation Analysis
LPF Cut-off (Hz):
Frequency Domain Analysis
Spectral Analysis in the Frequency Domain
The spectral density of an AM signal is obtained via the Fourier Transform of the time-domain signal \( s(t) \). Given: \( s(t) = A_c \left[ 1 + K_a m(t) \right] \cos(\omega_c t) \)
Applying the frequency-translation property of the Fourier Transform:
\( S(j\omega) = \pi A_c [ \delta(\omega - \omega_c) + \delta(\omega + \omega_c) ] + \frac{1}{2} K_a A_c [ M(j(\omega - \omega_c)) + M(j(\omega + \omega_c)) ] \)
The resulting spectrum comprises Dirac impulses at the carrier frequencies ±\( \omega_c \), alongside symmetrically shifted replicas of the baseband message spectrum, scaled by the factor \( \frac{1}{2} K_a A_c \).
Fig 2: Dual-domain visualization: Time-domain envelope (left) and Frequency-domain sidebands (right).
Transmission Bandwidth
The required transmission bandwidth (\( B_T \)) for an AM signal is inherently twice the highest frequency component present in the modulating signal \( m(t) \):
BW = 2 \cdot f_{max}
where \( f_{max} \) denotes the baseband bandwidth of the message.
Coherent Demodulation of AM Signals
Fig 3: Synchronous Detection Architecture
Synchronous (coherent) detection is employed to reconstruct the original message signal by multiplying the received signal with a locally generated carrier phase-locked to the transmitter.
Multiplier-Based Detection (Product Modulator)
The received signal \( s(t) \) is mixed with a local oscillator \( c(t) = \cos(2\pi f_c t) \):
\( v(t) = s(t) \cdot c(t) = A_c [1 + K_a m(t)] \cos^2(2\pi f_c t) \)
Applying the trigonometric identity \( \cos^2\theta = \frac{1}{2}(1 + \cos2\theta) \):
\( v(t) = \frac{A_c}{2} [1 + K_a m(t)] + \frac{A_c}{2} [1 + K_a m(t)] \cos(4\pi f_c t) \)
This product yields a baseband term and a high-frequency component centered at \( 2f_c \).
Low-Pass Filtering (LPF) and Baseband Recovery
A low-pass filter stage isolates the baseband signal by suppressing the high-frequency image. The resulting filter output is:
\( v_{LPF}(t) = \frac{A_c}{2} + \frac{A_c K_a}{2} m(t) \)
Following the removal of the DC offset (carrier residue) and subsequent signal conditioning, the original information signal \( m(t) \) is successfully recovered.