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


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


The magnitude of the complex envelope is then


and its phase is


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, ...

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