Application of Nonredundant Sampling Representations of Electromagnetic Fields to NF-FF Transformation Techniques

<table>Amplitude of the NF <svg height="12.175" id="M145" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 9.9624996 12.175" width="9.9624996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.675)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g>
</svg>-component on the line <svg height="12.175" id="M146" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 9.9624996 12.175" width="9.9624996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.675)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g>
</svg> = 0, <svg height="8.4250002" id="M147" style="vertical-align:-0.273pt" version="1.1" viewbox="0 0 8.5874996 8.4250002" width="8.5874996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,8.013)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
t-47.5 -67l-25 13q15 39 38 78q40 66 83 66q57 0 120 -12q52 -10 80 -10q29 0 67 42z" id="x1D467"></path></g>
</svg> = 12<i>λ</i> as function of <svg height="7.9499998" id="M148" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.625 7.9499998" width="9.625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.675)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
</svg>. The samples distribution (crosses) when adopting a uniform sampling in <svg height="15.9125" id="M149" style="vertical-align:-2.57985pt" version="1.1" viewbox="0 0 7.7375002 15.9125" width="7.7375002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,12.638)"><path d="M425 682q0 -40 -49 -59q-36 -13 -110 -13q-31 -18 -52.5 -49.5t-21.5 -67.5q0 -34 17.5 -57.5t45.5 -28.5q63 15 97 15q18 0 18 -16q0 -13 -22 -36t-34 -23q-50 0 -93 8q-47 -20 -80.5 -66.5t-33.5 -96.5q0 -76 84 -103q33 -11 75 -21q101 -26 101 -93q0 -51 -39 -95.5
t-76 -68.5l-22 20q60 69 60 111q0 21 -14 34t-55 22q-21 4 -63 16q-68 19 -101.5 52.5t-33.5 96.5q0 125 157 206q-27 10 -46 36t-19 58q0 84 103 152q-43 3 -71.5 21t-28.5 45q0 37 65 59l17 -6q-9 -14 -9 -33q0 -21 17.5 -36t51.5 -19q82 53 147 53q18 0 18 -17z" id="x1D709"></path></g>
</svg> is also shown.</table>

International Journal of Antennas and Propagation

fig4

Figure 4

Figure 4: Application of Nonredundant Sampling Representations of Electromagnetic Fields to NF-FF Transformation Techniques 