  As a new technology to reconfigure wireless communication environment by
signal reflection controlled by software, intelligent reflecting surface (IRS)
has attracted lots of attention in recent years. Compared with conventional
relay system, the relay system aided by IRS can effectively save the cost and
energy consumption, and significantly enhance the system performance. However,
the phase quantization error generated by IRS with discrete phase shifter may
degrade the receiving performance of the receiver. To analyze the performance
loss arising from IRS phase quantization error, in accordance with the law of
large numbers and Rayleigh distribution, the closed-form expressions for the
signal-to-noise ratio (SNR) performance loss and achievable rate of the double
IRS-aided amplify-and-forward (AF) relay network, which are associated with the
number of phase shifter quantization bits, are derived in the Rayleigh
channels. In addition, their approximate performance loss closed-form
expressions are also derived based on the Taylor series expansion. Simulation
results show that the performance losses of SNR and achievable rate decrease
with the number of quantization bits increases gradually, and increase with the
number $k$ of IRS phase shift elements. The SNR and achievable rate performance
losses of the system are smaller than 0.06dB and 0.03bits/s/Hz when $k$ is
equal to 4 and 3, respectively.
