Asymptotic Performance Analysis of Large-Scale Active IRS-Aided Wireless Network

In this paper, the dominant factor affecting the performance of active IRS-aided wireless communication networks in Rayleigh fading channel, namely the average signal-to-noise ratio (SNR) gamma0 at IRS, is defined and proposed. Making use of the weak law of large numbers, its simple asymptotic expression is derived as the number $N$ of IRS elements goes to medium-scale and large-scale. When N tends to large-scale, the asymptotic received SNR at user is proved to be a linear increasing function of a product of gamma0 and N. Subsequently, when the BS transmit power is fixed, there exists an optimal limited reflect power at IRS. At this point, more IRS reflect power will degrade the SNR performance. Additionally, under the total power sum of the BS transmit power and the power reflected by the IRS is constrained, an optimal power allocation (PA) strategy is derived and shown to achieve 0.83 bit rate gain over equal PA. Finally, an IRS with finite phase shifters is taken into account, generates phase quantization errors, and further leads to a degradation of receive performance. The corresponding closed-form performance loss expressions for user's asymptotic SNR, achievable rate (AR), and bit error rate (BER) for active IRS and passive IRS are derived. Numerical simulations show that a 3-bit and 2-bit discrete phase shifter is required to achieve a trivial performance loss for a large-scale active and passive IRS, respectively.