Why DSP transmitters build SSB with the phasing method: the audio is split into two paths whose components differ by 90 degrees (a Hilbert transform provides the wideband phase shift), and those in-phase and quadrature signals modulate carriers that are also 90 degrees apart. When the two products are summed, one sideband adds in phase and the other cancels, leaving a single sideband with no crystal filter required. This quadrature I/Q approach is easy in software because exact 90 degree shifts and perfectly matched amplitudes are just arithmetic.
Watch out The idea of converting mixing products to voltages and subtracting them in adder circuits sounds like analog cancellation but it ignores the essential phase relationship, and a frequency synthesizer only sets the operating frequency, it cannot strip a sideband. Emulating crystal characteristics describes the filter method, not what a quadrature modulator does.
SSB in software = I and Q, 90 degrees apart. Quadrature cancels one sideband.