Multi Dimensional Constellation and Shaping Gain

Hi
How binary symbols are mapped to multidimensional constellation points? Like in APSK symbols are mapped to constellation points of different Phase and Amplitude.
And what is the "shaping gain" ?
Thanks

Multidimensional constellation is a general expression for orthogonal constellation. Because if two signals are othogonal to each other, you should define at least two dimensions to identify these 2 signals. With that approach, if you arbitrarily define a modulation scheme that assigns your signals to some points and if you cannot represent all these functions with a single basis function, then one or more signals should be orthogonal. i.e. their vectorel multiplication is zero.

To grasp the underlying mechanism, you can think QPSK. As you remember, you use x and y coordinates to make a constellation for QPSK signaling. And we know x and y axis are othogonal each other, so, signals are orthogonal to each other. Hence, QPSK has simply a two-dimensional signaling. In QPSK, you just change phases, but changes are special like each pair signals orthogonal to each other. So, 4 signals can be identified by using 2 dimensions.

Similarly, ASK, BPSK are one-dimensional, 8PSK, QAM, 16QAM are two-dimensional modulations.

With using FSK (Frequency Shift Keying), one can provide multidimensional constellation. With spacing frequency such a manner that every signal will be orthogonal to each other, that corresponds to multidimensional signaling. However, it is obviously not valid to draw a constellation diagram, because we cannot draw higher than 3 dimensions.

Hope that helps. You may search for Gram-Schmidt procedure in order to go in detail.