Amplitude Modulation
&
Demodulation

Amplitude Modulation (AM) is a technique used to modulate a high-frequency carrier wave with a low-frequency message signal. The modulated signal contains the information from the message signal, which can be demodulated at the receiver to recover the original message signal.

The process of deriving Amplitude Modulation (AM) involves multiplying a message signal with a high frequency carrier signal to generate a modulated signal. The mathematical expression for AM modulation can be derived as follows:

Let m(t) be the message signal, c(t) be the carrier signal, and s(t) be the modulated signal.

The carrier signal can be represented as:

c(t) = A_c cos(2πft)

where Ac is the amplitude of the carrier signal and fc is the frequency of the carrier signal.

The message signal can be represented as:

m(t) = A_m cos(2πf_mt)

where Am is the amplitude of the message signal and fm is the frequency of the message signal.

To generate the modulated signal s(t), we multiply the message signal with the carrier signal:

s(t) = (A_c + A_m cos(2πf_mt)) cos(2πf_ct)

Expanding this expression using the trigonometric identity for the product of two cosines, we get:

s(t) = A_c cos(2πf_ct) + 0.5 A_m cos(2π(f_c + f_m)t) + 0.5 A_mcos(2π(f_c - f_m)t)

Thus, the modulated signal s(t) contains three components: the carrier signal, and two sidebands that are spaced apart from the carrier frequency by the frequency of the message signal

In the frequency domain, the AM signal has a spectrum that consists of a carrier signal and two sidebands, which are symmetrically placed around the carrier frequency.

One of the methods used to demodulate AM signals is square-law demodulation, which is based on the non-linear characteristic of a diode. The square-law demodulator circuit consists of a diode, a load resistor, and a capacitor.

The mathematical derivaticn of AM demodulation using square-law detection is as follows:

Let the received AM signal be:

s(t) = (A_c+ A_m m(t)) cos(2πf_ct)

where A_c is the amplitude of the carrier wave, A_m is the amplitude of the message signal, m(t) is the message signal, and f_c is the frequency of the carrier wave.

To demodulate the AM signal using a square law detector, we first pass the received signal through a diode to rectify it. The output of the diode is a half-wave rectified signal, which contains both the positive and negative half cycles of the modulated signal.

The rectified signal is then passed through a low-pass filter (LPF) to remove the high-frequency components and recover the original message signal. The LPF can be implemented using a capacitor and a resistor as shown in the figure below:

The output voltage of the LPF can be derived as follows:

V_out(t) = V_c *(1/RC) * ∫[V_diode(t) - V_c/2] dt

where V_c is the voltage across the capacitor, RC is the time constant of the LPF, and Vdiode(t) is the half-wave rectified output of the diode.

Using the trigonometric identity Cos²(x) = (1/2) * (1 + cos(2x)), we can express the output voltage of the diode as follows:

V_diode(t) = (A_c/2) * (1 + μ*m(t))²

where μ = A_m/A_c is the modulation index.

Substituting Vdiode(t) into the above equation and simplifying, we get:

Vout(t) = (A_mA_c/4) * m(t) + (A_c/2) * (1/RC) * ∫m²(t) dt

The first term in the above equation represents the demodulated message signal, while the second term represents a DC offset. Since the LPF removes the DC component, the output of the LPF is the demodulated message signal.

Therefore, the equation for the demodulated message signal can be written as:

m_demod(t) = (2/π) * Vout(t) / (Ac * μ * Am)

where Vout(t) is the output voltage of the LPF.

In summary, the demodulation of an AM signal using a square law detector involves rectifying the signal, passing it through a LPF to recover the message signal, and then scaling the output to obtain the demodulated message signal