AM Demodulation

 

 The block diagram illustrates the process of synchronous (coherent) demodulation, a method used to recover the original message signal from an Amplitude Modulated (AM) waveform. 

 

1. Modulated AM Signal Input:
   The input signal is given by:
   \( s(t) = A_c \left[1 + K_a m(t)\right] \cos(2\pi f_c t) \)
   Where:
   • \( A_c \): Carrier amplitude
   • \( f_c \): Carrier frequency
   • \( m(t) \): Message (modulating) signal
   • \( K_a \): Amplitude sensitivity constant

The Product Modulator (Multiplier)

The core of synchronous demodulation is multiplying the incoming AM signal \( s(t) \) with a locally generated carrier signal \( c(t) \). For ideal demodulation, this local carrier must be a perfect replica of the original transmitter's carrier in both **frequency and phase**:

\( c(t) = \cos(2\pi f_c t + \phi) \)

Here, for perfect synchronization, the phase offset \( \phi \) must ideally be zero. Any deviation from this (frequency or phase error) will degrade the demodulated signal quality, which is a key challenge for coherent receivers.

The multiplication yields the signal \( v(t) \):

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

Using the trigonometric identity \( \cos A \cos B = \frac{1}{2} [\cos(A-B) + \cos(A+B)] \), we get:

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

Since \( \cos(-\phi) = \cos(\phi) \), this simplifies to:

\( v(t) = \underbrace{\frac{A_c}{2} [1 + K_a m(t)] \cos(\phi)}_{\text{Low-Frequency Component}} + \underbrace{\frac{A_c}{2} [1 + K_a m(t)] \cos(4\pi f_c t + \phi)}_{\text{High-Frequency Component}} \)

This multiplication effectively shifts the original message signal back to the baseband (centered around 0 Hz) and also creates a high-frequency component centered at twice the carrier frequency (\( 2f_c \)). The presence of the \( \cos(\phi) \) term highlights the importance of phase synchronization; if \( \phi = \pm \pi/2 \), the desired low-frequency component vanishes (quadrature null effect).


The Low-Pass Filter (LPF)

The output of the multiplier, \( v(t) \), is then passed through a Low-Pass Filter (LPF). The purpose of the LPF is to eliminate the high-frequency component at \( 2f_c \) while preserving the desired low-frequency (baseband) component.

The filter is designed with a cutoff frequency that is:

• Higher than the highest frequency in the message signal \( m(t) \).
• Lower than the high-frequency component at \( 2f_c \).

After filtering, the output signal, \( v_{LPF}(t) \), is:

\( v_{LPF}(t) = \frac{A_c}{2} [1 + K_a m(t)] \cos(\phi) = \frac{A_c}{2} \cos(\phi) + \frac{A_c K_a}{2} m(t) \cos(\phi) \)

This signal contains two parts:

1. A DC offset: \( \frac{A_c}{2} \cos(\phi) \)
2. A scaled version of the desired message signal: \( \frac{A_c K_a}{2} m(t) \cos(\phi) \)

DC Blocking and Amplification

To isolate the original message signal, the DC offset \( \frac{A_c}{2} \cos(\phi) \) must be removed. This is typically done using a simple DC-blocking capacitor or a level-shifting circuit. After removing the DC component, we are left with a scaled version of the message. An amplifier can then be used to scale the signal back to its original amplitude, taking into account the \( \cos(\phi) \) factor, which should ideally be 1 for maximum output.

 

Further Reading

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