### 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 *ej*ω_{0}*t* 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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