## 5 Simulation and Results
Specifically, by reformulating the problem as QCQP and introducing an auxiliary matrix variable, the difficult rank-one constraint is relaxed into a convex form. This makes the problem solvable using convex optimisation tools such as CVX which is mentioned before. Alongside this, two intuitive yet practical beamforming schemes are investigated: _AP-user MRT_ and _AP-IRS MRT_. The _AP-user MRT_ aligns the transmit beamforming vector with the direct AP-to-user channel. This approach is particularly effective when the user is close to the AP and the direct link dominates. However, in environments where the direct path is blocked such as indoor scenarios with thick walls its performance degrades significantly. In contrast, the _AP-IRS MRT_ steers the AP's beamforming vector toward the IRS instead of the user, assuming that the reflected path dominates. This method proves useful when the user is near the IRS and can benefit from a strong reflected signal. Nevertheless, neither of these two MRT strategies jointly optimise the IRS phase shifts and AP beamforming, resulting in suboptimal performance when the channel conditions are more balanced or complex. In addition to these schemes, a _random phase shift_ strategy is also considered as a low-complexity benchmark. In this case, the phase shifts applied by the IRS elements are randomly selected from a uniform distribution over \([0,2\pi)\), and the AP performs MRT based on the combined AP-user and AP-IRS-user channel. Although this method requires no channel state information or optimisation, it can still provide a modest performance gain due to signal reflections. However, the improvement scales only linearly with the number of IRS elements \(N\), whereas optimised designs based on SDR or alternating optimisation achieve a quadratic (\(N^{2}\)) gain. This clearly highlights the advantage of jointly optimising the active and passive beamforming components. Finally, a conventional _Without IRS_ is considered, where the IRS is disabled and the system relies solely on the direct AP-to-user transmission. The AP applies MRT using only the direct channel. This setup serves as a useful baseline performance reference point to demonstrate the limitations in terms of coverage and power efficiency, especially when the direct link is either weak or blocked. Simulation results consistently show that the required transmit power in this case is significantly higher, particularly for users located far from the AP or in NLoS environments. (1) _Transmit Power at AP vs. AP-User Distance:_ In Figure 3, we have evaluated the horizontal distance between the AP and user against the transmit power required for every scheme. Since signal attenuation is higher for the system without the IRS, it can be seen that the user from the AP requires a higher transmit power at the AP. The transmit power requirement can be reduced by installing an IRS. However, the work also demonstrated that in an IRS-enabled wireless network, transmit power efficiency is not always guaranteed at every AP-user distance. When the user is placed further away from the AP and nearer to the IRS, it enables to get a stronger reflected signal from the IRS. So, the user who is closer to either the AP (\(d=22\) m) or IRS (\(d=46\) m), for example, uses less transmitted power than the user who is much far from both distance (\(d=39\) m). This scenario also indicates that the signal coverage may be successfully enhanced by using just a passive IRS rather than installing a second AP or active relay. For instance, with the same transmit power of 12 dBm, the network's coverage is roughly 33 m without the IRS, but by implementing the suggested combined beamforming designs with an IRS, this value is increased to be above 50 m.  
Note that although the proposed alternating optimisation technique outperforms the existing schemes, it is more computationally demanding than the other schemes. For example, the 'SDR' method does not solve the problem iteratively and stops optimisation after the first round.