JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year ANALOG COMMUNICATIONS Previous Question Paper (March - 2022 - R16)

A square law modulator generates an Amplitude Modulated (AM) signal by passing the sum of a message signal and a carrier signal through a non-linear device that exhibits a square-law characteristic. The device is typically a diode biased in its non-linear region, where the output current i(t) is related to the input voltage v(t) by the expression:

The input voltage is the sum of the message signal m(t) and the carrier signal c(t) = A_ccos(2πf_ct), given by:

Substituting v(t) into the non-linear equation:

Using the trigonometric identity cos²(θ) = (1 + cos(2θ))/2, the last term becomes (a₂A_c²/2) + (a₂A_c²/2)cos(4πf_ct). The expression for i(t) becomes:

To extract the AM signal, the output i(t) is passed through a band-pass filter tuned to the carrier frequency f_c with a bandwidth 2W (where W is the message bandwidth). The filter removes the low-frequency components a₁m(t), a₂m(t)², and the DC term, as well as the high-frequency harmonic 2f_c. The resulting output is:

Defining the modulation index μ = (2a₂/a₁)A_m (where A_m is the peak amplitude of m(t)), this matches the standard form of an AM wave: s(t) = A_c[1 + k_am(t)]cos(2πf_ct), where k_a = 2a₂/a₁.

Double Sideband Suppressed Carrier (DSB-SC) modulation is a process where the carrier signal is suppressed, and only the upper and lower sidebands are transmitted, resulting in higher power efficiency compared to standard AM. The generation of DSB-SC is primarily achieved using a balanced modulator.

A balanced modulator consists of two AM modulators arranged in a balanced configuration such that the carrier signal is cancelled out at the output. If the message signal is m(t) and the carrier signal is c(t) = A_c cos(ω_ct), the output of the balanced modulator is the product of these two signals:

Common methods for generating DSB-SC include:

• Balanced Modulator using Diodes: This circuit uses a bridge or ring configuration of diodes. The message signal is applied in series with the carrier, and the diodes act as switches that reverse the polarity of the carrier at the zero crossings of the message signal, effectively multiplying them.
• Ring Modulator: This is the most common diode-based DSB-SC generator. It uses four diodes in a ring configuration with two transformers. The carrier signal acts as a switching signal that turns the diodes on and off, while the message signal is applied to the input transformer. The output is the product of the message and a square wave carrier, which can be filtered to obtain the desired DSB-SC signal.
• Product Modulator: This uses an analog multiplier IC (like the MC1496) which performs the direct multiplication of the message signal and the high-frequency carrier signal.

The mathematical representation of the DSB-SC signal in the frequency domain is given by the convolution of the Fourier transforms of the message and carrier:

This shows that the spectrum of the message signal is shifted to the carrier frequency f_c, with the carrier component at f_c being suppressed.