  This paper proposes new accurate approximations for average error probability
(AEP) of a communication system employing either $M$-phase-shift keying (PSK)
or differential quaternary PSK with Gray coding (GC-DQPSK) modulation schemes
over $\kappa-\mu$ shadowed fading channel. Firstly, new accurate approximations
of error probability (EP) of both modulation schemes are derived over additive
white Gaussian noise (AWGN) channel. Leveraging the trapezoidal integral
method, a tight approximate expression of symbol error probability for $M$-PSK
modulation is presented, while new upper and lower bounds for Marcum
$Q$-function of the first order (MQF), and subsequently those for bit error
probability (BER) under DQPSK scheme, are proposed. Next, these bounds are
linearly combined to propose a highly refined and accurate BER's approximation.
The key idea manifested in the decrease property of modified Bessel function
$I_{v}$, strongly related to MQF, with its argument $v$. Finally, theses
approximations are used to tackle AEP's approximation under $\kappa-\mu$
shadowed fading. Numerical results show the accuracy of the presented
approximations compared to the exact ones.
