Why Single sideband generated digitally normally uses the phasing method: the audio is split into two paths that are identical in amplitude but 90 degrees apart in phase across the whole audio band, then each drives a mixer fed by carriers also 90 degrees apart, so one sideband adds and the other cancels. The DSP block that produces that broadband 90 degree phase shift is the Hilbert transform filter, usually implemented as an FIR filter with antisymmetric coefficients. Without it you would need a very sharp analog filter to remove the unwanted sideband.
Watch out An elliptical filter is a real analog-style filter response with equal ripple in passband and stopband and very steep skirts, useful for filter-method SSB or anti-aliasing, but it shifts phase only incidentally; a notch filter removes a single tone, and an adaptive filter changes its coefficients on the fly for noise or interference reduction.
Hilbert = 90 degrees across all audio frequencies, the phasing-method key to SSB.