All TV standards use non-linear signals, pre-corrected for the non-linear transfer characteristic of the display CRT. It is here that the most confusion exists, and so this is a VERY important section to understand.
A typical CRT has a non-linear voltage-to-light transfer function with a power law usually denoted by gamma. The value of gamma is theoretically 2.5, but is specified as 2.2 in NTSC systems, 2.8 in PAL systems, and is actually nearer to 2.35 for real CRTs. Any signal destined for display on a CRT must be distorted by an inverse law. In practice, that is impossible because a pure power law has infinite slope (gain) at zero (black). TV systems limit the gain near black to a value between 4 and 5 by offsetting the power law. This has the side advantage of increasing saturation in a way that compensates for the display having a dark surround. For example the ITU-BT.709 specification is:-
Volts = (1 + a) * Light ^ (law) - a for Light > b Volts = slope * Light for Light <= bwhere a=0.099, law=0.45, b=0.018.
and the gain at zero is 4.5. This law is similar to the formula used for L* (see above).
So for accurate colour calculations, this law (or whichever law was actually applied) must be undone to return to linear signals before doing conversions. The law should be reapplied to the results to get the drive signals for the actual display.
A signal that has been gamma-corrected is shown primed (Y', R', G', B' etc). In general, undoing the gamma law will return to linear signals, but that is not always true, especially with the Y' signal, which is not directly related to the CIE Y value. It is a shame that the TV industry used Y' for the luminance channel, because it created a great deal of confusion, most of which still exists. But careful reading of the following section shows the way to performing totally accurate colour calculations using any colour system.
European Y'U'V' (EBU)
ITU.BT-709 HDTV studio production in Y'CbCr
Kodak PhotoYCC colour space for PhotoCD images.