  Compared to passive intelligent reflecting surface (IRS), active IRS is
viewed as a more efficient promising technique to combat the double-fading
impact in IRS-aided wireless network. In this paper, in order to boost the
achievable rate of user in such a wireless network, three enhanced-rate
iterative beamforming methods are proposed by designing the amplifying factors
and the corresponding phases at active IRS. The first method, maximizing the
simplified signal-to-noise ratio (Max-SSNR) is designed by omitting the
cross-term in the definition of rate. Using the Rayleigh-Ritz (RR) theorem,
Max-SSNR-RR is proposed to iteratively optimize the norm of beamforming vector
and its associated normalized vector. In addition, generalized maximum ratio
reflection (GMRR) is presented with a closed-form expression, which is
motivated by the maximum ratio combining. To further improve rate, maximizing
SNR (Max-SNR) is designed by fractional programming (FP), which is called
Max-SNR-FP. Simulation results show that the proposed three methods make an
obvious rate enhancement over Max-reflecting signal-to-noise ratio (Max-RSNR),
maximum ratio reflection (MRR), selective ratio reflecting (SRR), equal gain
reflection (EGR) and passive IRS, and are in increasing order of rate
performance as follows: Max-SSNR-RR, GMRR, and Max-SNR-FP.
