Digital Communications: Fundamentals and Applications, 3rd Edition

4.6 Complex Envelope

The description of real-world modulators and demodulators is facilitated by the use of complex notation, which began to be covered in Section 4.2.1 and continues here. Any real bandpass waveform s(t) can be represented using complex notation as

s(t)=Re{g(t)ejω0t}(4.57)

where g(t) is known as the complex envelope, expressed as

g(t)=x(t)+jy(t)=|g(t)|ejθ(t)=R(t)ejθ(t)(4.58)

The magnitude of the complex envelope is then

R⁡(t)=|g⁡(t)|=x2⁢(t)+y2⁢(t)(4.59)

and its phase is

θ(t)=tan−1y(t)x(t)(4.60)

With respect to Equation (4.57), we can call g(t) the baseband message or data in complex form and ejω₀t the carrier wave in complex form. The product of these two represents modulation, and s(t), the real part of this product, ...