Mathematical Formulation
Single-sideband has the mathematical form of quadrature amplitude modulation in the special case where one of the baseband waveforms is derived from the other, instead of being independent messages:
where is the message, is its Hilbert transform, and is the radio carrier frequency.
is real-valued; therefore its Fourier transform, is Hermitian symmetrical about the axis. Double sideband modulation of to frequency moves the axis of symmetry to and the two sides of each axis are called sidebands. Single-sideband modulation eliminates one sideband, while preserving To understand this, a useful mathematical concept is the analytic representation of :
- where
The Fourier transform of equals for but it has no negative-frequency components. And yet is fully recoverable as the real part of The product of with function shifts the one-sided Fourier transform by amount No negative-frequency components are created, so the result is an analytic representation of the single sideband signal:
Therefore, using Euler's formula to expand
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