Low-Complexity User Selection for Rate Maximization in  MIMO Broadcast Channels with Downlink Beamforming

<table>(a) The orthogonal component of vector <svg height="16.9625" id="M130" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.299999 16.9625" width="17.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,12.112)"><path d="M534 24v-24h-234v24q25 3 35.5 16.5t10.5 44.5v224q0 54 -11 75.5t-40 21.5q-36 0 -70 -34q-17 -17 -17 -24v-263q0 -31 10 -44t38 -17v-24h-240v24q37 8 44 16q9 9 9 49v502q0 35 -9 44q-8 9 -45 17v24h193v-280q65 77 147 77q59 0 94.5 -37.5t35.5 -98.5v-250
q0 -34 11.5 -48.5t37.5 -14.5z" id="x1D421"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.075,16.188)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" id="x1D458"></path></g>
</svg> to <svg height="17.4" id="M131" style="vertical-align:-3.27599pt" version="1.1" viewbox="0 0 67.349998 17.4" width="67.349998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,12.687)"><path d="M409 504l-29 -5q-16 60 -44.5 96t-86.5 36q-54 0 -82.5 -32t-28.5 -77q0 -51 30.5 -82.5t96.5 -65.5l35.5 -18t33 -19.5t33.5 -23.5l27 -25.5t24.5 -32t13.5 -36.5t6 -45q0 -80 -62 -134.5t-160 -54.5q-47 0 -98 15q-22 7 -50 21q-8 23 -27 155l30 7q7 -27 18 -52
t30 -51.5t49 -42.5t67 -16q56 0 88 32.5t32 87.5q0 51 -31.5 82t-98.5 67q-80 44 -110 73q-55 53 -55 124q0 75 56 126.5t150 51.5q53 0 126 -23z" id="x53"></path></g><g transform="matrix(.017,-0,0,-.017,8.444,12.687)"><path d="M169 380l92 53q26 16 49 16q80 0 128.5 -56t48.5 -133q0 -108 -72.5 -180.5t-176.5 -91.5q-38 1 -69 18v-131q0 -57 13 -70t71 -18v-27h-236v27q49 5 61 17t12 61v452q0 48 -9.5 58.5t-56.5 16.5v24q66 11 145 43v-79zM169 346v-270q37 -39 94 -39q63 0 101.5 50
t38.5 134q0 78 -38.5 123t-95.5 45q-45 0 -100 -43z" id="x70"></path></g><g transform="matrix(.017,-0,0,-.017,17.352,12.687)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,23.233,12.687)"><path d="M759 25v-25h-339v25q54 6 71.5 20.5t17.5 52.5v228h-241v-228q0 -40 17 -54q16 -13 72 -19v-25h-336v25q54 8 69 20q16 13 16 53v479q0 41 -16.5 55t-68.5 19v25h336v-25q-54 -4 -71.5 -18.5t-17.5 -55.5v-204h241v204q0 41 -17.5 55.5t-71.5 18.5v25h339v-25
q-51 -4 -69.5 -19t-18.5 -55v-479q0 -40 16 -53q15 -12 72 -20z" id="x1D407"></path></g><g transform="matrix(.017,-0,0,-.017,36.458,12.687)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,42.34,12.687)"><path d="M397 143l-20 -4q-20 68 -52 92.5t-84 24.5q-41 0 -76 -34q-36 -33 -36 -85t41 -87q43 -35 102 -35q75 0 134 47t59 129q0 29 -16 56.5t-39.5 51l-46.5 48t-39 57t-16 68.5q0 66 42 116q77 92 211 92q55 0 98 -29q42 -29 42 -87q0 -54 -45 -91q-44 -38 -97 -38v21
q49 5 79 33.5t30 73.5q0 40 -30 60.5t-78 20.5q-64 0 -119 -45q-56 -44 -56 -108q0 -30 16.5 -58.5t40.5 -52.5l48 -49t40.5 -58t16.5 -69q0 -99 -82 -159q-83 -60 -201 -60q-80 0 -139 39q-58 39 -58 118q0 70 50 114q50 43 117 43q63 0 108.5 -40.5t54.5 -115.5z" id="x1D4AE"></path></g><g transform="matrix(.017,-0,0,-.017,54.97,12.687)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g><g transform="matrix(.017,-0,0,-.017,60.852,12.687)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg>. (b) Physical components of the interaction of two selected users <svg height="12.5875" id="M132" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 5.8874998 12.5875" width="5.8874998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,11.762)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
</svg> and <svg height="16.8125" id="M133" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 8.4375 16.8125" width="8.4375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,11.762)"><path d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" id="x1D457"></path></g>
</svg> with third unselected user <svg height="13.55" id="M134" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.8000002 13.55" width="9.8000002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,12.725)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
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</svg>.</table>

The Scientific World Journal

fig1

Figure 1

Figure 1: Low-Complexity User Selection for Rate Maximization in  MIMO Broadcast Channels with Downlink Beamforming 