On the Relation between NARX Clusters and Even/Odd Nonlinearities through Frequency-Domain Analysis

<div>Case (II) including a Gaussian noise with <svg height="12.2911pt" id="M333" style="vertical-align:-0.2129993pt" version="1.1" viewbox="-0.0498162 -12.0781 44.6253 12.2911" width="44.6253pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,7.657,-5.984)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,16.576,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,28.314,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,34.817,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,37.906,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> (× excited, ∘ odd, <svg height="6.36877pt" id="M334" style="vertical-align:-0.04980993pt" version="1.1" viewbox="-0.0498162 -6.31896 8.47661 6.36877" width="8.47661pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M485 147C487 173 478 200 466 219C419 224 389 226 342 249C389 272 419 273 465 279C479 297 489 321 485 351C463 366 435 372 413 371C385 332 369 308 325 278C329 331 342 357 361 401C352 421 336 442 308 453C285 442 265 421 255 401C274 357 287 331 291 278C248 308 232 333 203 370C181 372 155 369 131 351C129 325 138 298 150 279C198 274 227 272 274 249C227 226 197 225 151 219C138 201 127 177 131 147C153 132 181 126 204 127C231 166 248 190 291 220C287 167 274 141 256 97C265 77 280 56 308 45C331 56 351 77 361 97C342 141 329 167 325 220C369 190 385 165 413 128C435 126 461 129 485 147Z" id="g5-43"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="616" vert-adv-y="616"></glyph.data></g></svg> even, — noise level).</div>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: On the Relation between NARX Clusters and Even/Odd Nonlinearities through Frequency-Domain Analysis 