Praise for Fundamentals of WiMAX

170 Chapter 5 • Multiple-Antenna Techniques5.4.1 DOA-Based BeamformingThe incoming signals to a receiver may consist of desired energy and interference energy—forexample, from other users or from multipath reflections. The various signals can be characterizedin terms of the DOA or the angle of arrival (AOA) of each received signal. Each DOA canbe estimated by using signal-processing techniques, such as the MUSIC, ESPRIT, and MLEalgorithms (see [27, 38] and the references therein). From these acquired DOAs, a beamformerextracts a weighting vector for the antenna elements and uses it to transmit or receive the desiredsignal of a specific user while suppressing the undesired interference signals.When the plane wave arrives at the d-spaced uniform linear array (ULA) with AOA θ, thewave at the first antenna element travels an additional distance of d sin θ to arrive at the secondelement. This difference in propagation distance between the adjacent antenna elements can beformulated as an arrival-time delay, τ = dcsin / θ. As a result, the signal arriving at the secondantenna can be expressed in terms of signal at the first antenna element asy2()= t y1() t exp( − j2 πfcτ),d sin θ= y1( t) exp( − j2 π ).λN r(5.43)For an antenna array with elements all spaced by d , the resulting received signal vectorcan therefore be expressed asT()=[ t y1() t y2() t … yN ()] trd sin θd sin θ T= y1( t)[1 exp( − j2 π ) … exp( −j2 π( Nr−1) )] ,λλa( θ)(5.44)where a( θ)is the array response vector.In the following, we show an example to demonstrate the principle of DOA-based beamforming.Consider a three-element ULA with d = λ/2spacing between the antenna elements.Assume that the desired user’s signal is received with an AOA of θ 1=0—that is, the signal iscoming from the broadside of the ULA—and two interfering signals are received with AOAs ofθ 2 = π/3 and θ 3 = –π/6, respectively. The array response vectors are then given by3πT⎡ − j π2 3a θa θπ⎤− j⎡ j2( 1) =111 [ ] , ( 2)= ⎢1 e e ⎥ , and a( θ3) = ⎢1e e⎣⎢⎦⎥⎣TjπT⎤⎥ .⎦(5.45)The beamforming weight vector w =[ w should increase the antenna gain in the1w2 w ]T3direction of the desired user while simultaneously minimizing the gain in the directions of interferers.Thus, the weight vector w should satisfy the following criterion:( ) ( ) ( ) ⎦ [ ] T*w ⎡⎣a θ1 a θ2 a θ ⎤3=100 ,(5.46)