Analytical signal is sometime referred as real passband because here you generate Q(t) part from real baseband m(t). The analytic signal is useful when you want more control over the spectrum, especially when generating SSB or working with complex I/Q signals.
Multiplying a real baseband signal by a cosine generally creates both upper and lower sidebands around the carrier.
For example, if the baseband contains a frequency \(f_m\), then
\[\cos(2\pi f_mt)\cos(2\pi f_ct)\]using the product-to-sum identity gives
\[\frac12\cos(2\pi(f_c+f_m)t) +\frac12\cos(2\pi(f_c-f_m)t)\]So you get frequencies at \(f_c+f_m\) and \(f_c-f_m\).
That's double-sideband modulation.
2. What does the analytic signal add?
Suppose you start with a real message \(m(t)\). The Hilbert transform generates a quadrature version:
\[m_a(t)=m(t)+j\hat m(t)\]Here, \(\hat m(t)\) is the Hilbert transform of \(m(t)\).
You can then modulate it using
\[s(t)=\Re\{m_a(t)e^{j2\pi f_ct}\}\]Expanding,
\[s(t)=m(t)\cos(2\pi f_ct) -\hat m(t)\sin(2\pi f_ct)\]The extra sine branch is not just decoration. It allows the two sidebands to cancel each other when the appropriate analytic-signal relationship is used.
Ordinary AM / DSB
Both sidebands are present. The required bandwidth is generally about twice the highest baseband frequency for a real low-pass message.
Hilbert-based SSB
One sideband can be suppressed, reducing the required bandwidth approximately by half.