Distributed Collaborative Beamforming in the Presence of Angular Scattering

In this paper, a collaborative beamformer (CB) is considered to achieve a dual-hop communication from a source to a receiver, through a wireless network comprised of K independent terminals. Whereas previous works neglect the scattering effect to assume a plane-wave single-ray propagation channel termed here as monochromatic (with reference to its angular distribution), a multi-ray channel termed as polychromatic due to the presence of scattering is considered, thereby broadening the range of applications in real-world environments. Taking into account the scattering effects, the weights of the so-called polychromatic CB (P-CB) are designed so as to minimize the received noise power while maintaining the desired power equal to unity. Unfortunately, their derivation in closed-form is analytically intractable due to the complex nature of polychromatic channels. However, when the angular spread (AS) is relatively small to moderate, it is proven that a polychromatic channel may be properly approximated by two rays and hence considered as bichromatic. Exploiting this fact, we introduce a new bichromatic CB (B-CB) whose weights can be derived in closed-form and, further, accurately approximate the P-CB´s weights. Yet these weights, which turn out to be locally uncomputable at every terminal, are unsuitable for a distributed implementation. In order to circumvent this shortcoming, we exploit the asymptotic expression at large K of the B-CB whose weights could be locally computed at every terminal and, further, well-approximate their original counterparts. The performances of the so-obtained bichromatic distributed CB (B-DCB) and its advantages against the monochromatic DCB (M-DCB), which is designed without accounting for scattering, are analytically proved and further verified by simulations at practical values of K.