New Laguerre Filter Approximators to the Grünwald-Letnikov Fractional Difference

<table>Optimal Laguerre poles for FFLD model as a function of <svg height="7.9499998" id="M307" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7749996 7.9499998" width="9.7749996" 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="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
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</svg>, with <svg height="17.887501" id="M308" style="vertical-align:-1.97925pt" version="1.1" viewbox="0 0 43.637501 17.887501" width="43.637501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(1.25,0,0,-1.25,90,90)">
<path d="m -71.95,70.93 6.21,0" style="fill:none;stroke:#000000;stroke-width:0.68000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
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<g transform="matrix(.017,-0,0,-.017,.062,15.362)"><path d="M434 650l-5 -28q-60 -8 -75 -20.5t-25 -69.5l-71 -395q-20 -107 -57.5 -168t-106.5 -94q-40 -18 -61 -20l-10 32q77 23 113 108q21 50 41 160l68 377q10 57 -1.5 69.5t-76.5 20.5l6 28h261z" id="x1D43D"></path></g><g transform="matrix(.017,-0,0,-.017,12.54,15.362)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,27.244,15.362)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,35.403,15.362)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
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</svg> and various values of <svg height="11.175" id="M309" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 17.475 11.175" width="17.475" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" id="x1D440"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: New Laguerre Filter Approximators to the Grünwald-Letnikov Fractional Difference 