Coherent detection is one of the method to demodulate the signal modulated by PSK.
Coherent detection multiplies the signal by the carrier wave.
The signal modulated by PSK can be written as:
\[s(t) = A\cos{(2\pi ft+\phi)}\]
where \(A\) is amplitutde, \(f\) is frequency, and \(\phi\) is the phase difference.
Multiplying this signal by the carrier wave:
\[c(t) = \cos{(2\pi ft)}\]
The following equation can be obtained.
\[\begin{eqnarray}
s(t)\cdot c(t) & = & A\cos{(2\pi ft+\phi)}\cos{(2\pi ft)} \\
& = & \frac{A}{2}(\cos{(4\pi ft+\phi)}+\cos{(\phi)})
\end{eqnarray}\]
By applying to LPF, high frequency components are removed,
\[s(t)\cdot c(t) = \frac{A}{2}\cos{(\phi)}\]
Thus, the original signal can be obtained.