Angular Beamforming Technique for MIMO Beamforming System

Table 4

Element phasing, beam direction, and interelement phasing for the Butler matrix shown in Figure 3 (conceptual).

𝐛𝑡,𝐛𝑟	𝐸1  (𝑙=1)	𝐸2  (𝑙=2)	𝐸3  (𝑙=3)	𝐸4  (𝑙=4)	Beam; direction	Interelement; phasing
Port 1; (𝑚=1)	1√4𝑒^∘−𝑗45	1√4𝑒^∘−𝑗180	1√4𝑒^∘𝑗45	1√4𝑒^∘−𝑗90	138.6°	−135°
Port 2; (𝑚=2)	1√4𝑒^∘𝑗0	1√4𝑒^∘−𝑗45	1√4𝑒^∘−𝑗90	1√4𝑒^∘−𝑗135	104.5°	−45°
Port 3; (𝑚=3)	1√4𝑒^∘−𝑗135	1√4𝑒^∘−𝑗90	1√4𝑒^∘−𝑗45	1√4𝑒^∘−𝑗0	75.5°	45°
Port 4; (𝑚=4)	1√4𝑒^∘−𝑗90	1√4𝑒^∘𝑗45	1√4𝑒^∘−𝑗180	1√4𝑒^∘−𝑗45	41.4°	135°