Group “N” bits together to form N –bits symbols.

These symbols will extend over a period of N $T_b$ where $T_b$ is the duration of one bit.

Due to the grouping of N bits per symbol, we can have 2N =M possible symbols.

These M symbols are represented by sinusoidal signals of duration $T_s =N T_b$ which differ from one another by a phase 2π /M radians. Thus the M-ary PSK waveform can be mathematically represented as,

1. In equation (1) , $∅_m$ represents the symbol phase angle, which is given by,

The M-ary PSK waveforms corresponding to equation (1) are represented by dots in the above fig. This is a signal space diagram in which the co-ordinate axes are formed by the orthonormal waveforms $u_1$ (t) and $u_2$ (t) given by the following equations:

$u_1 (t) = \sqrt{\frac{2}{T_s }} cosω_C t$

The distance of each M-ary PSK waveform from the origin is,