VSB modulation, motivation and use in TV

What is VSB?

Vestigial sideband (VSB) modulation transmits one sideband almost completely plus a small part (a "vestige") of the other sideband. It is obtained by passing a DSB signal through a VSB filter whose response has odd symmetry about $f_c$:

$$H(f-f_c) + H(f+f_c) = \text{constant for } |f| \le W$$

Its bandwidth is $B_T = W + f_v$, where $f_v$ is the vestige width ($W < B_T < 2W$).

 |H(f)|
  1 |      ________________
    |     /
 0.5|    /  <- odd symmetry
    |   /      about fc
  0 +--/-----------------------> f
     fc-fv fc fc+fv     fc+W

Motivation for VSB

• DSB wastes bandwidth ($2W$), which is costly for wide-band signals.
• SSB saves bandwidth but needs a filter with a very sharp cut-off at $f_c$. This is impossible when the message has large low-frequency and DC content, because there is no gap around $f_c$ to place the filter transition.
• VSB is a compromise: bandwidth close to SSB, but the filter can have a gradual roll-off, so it is easy to build. Low frequencies, including DC, are kept without distortion.
• With a carrier added, VSB can be detected with a simple envelope detector (with small distortion).

Why VSB suits television

1. A video signal has a large bandwidth (about 4.2 MHz in NTSC, 5–5.5 MHz in PAL). DSB would need 8–11 MHz per channel; VSB needs only about 5.75 MHz (PAL video) and allows a 7–8 MHz channel including sound.
2. Video has important content at very low frequencies and DC (background brightness). SSB filtering would damage it; VSB preserves it.
3. The VSB filter's gradual slope is practical and gives little phase distortion, which matters for picture quality.
4. A carrier is sent with the VSB signal (VSB + C), so cheap TV receivers can use an envelope detector.
5. In PAL, the full upper sideband (5 MHz) and a 0.75–1.25 MHz vestige of the lower sideband are sent.