A Design Method of Sparse Array with High Degrees of Freedom Based on Fourth-Order Cumulants

<div>CRLB for five kinds of sparse arrays based on FOC. (a) CRLB versus input SNR, where <span class="nowrap"><svg height="9.16851pt" id="M308" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.96212 48.3886 9.16851" width="48.3886pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.917,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.18,0)"><path d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.422,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.662,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,41.902,0)"><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"></path></g></svg>.</span> (b) CRLB versus the number of snapshots, where <span class="nowrap"><svg height="9.5404pt" id="M309" style="vertical-align:-0.2324991pt" version="1.1" viewbox="-0.0498162 -9.3079 59.6758 9.5404" width="59.6758pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.005,0)"><path d="M719 650H483V622C553 619 576 606 581 560C585 532 588 487 588 394V148H585L167 650H20V622C67 618 88 610 109 585C127 562 129 557 129 484V264C129 173 126 124 123 93C118 44 92 31 35 28V0H272V28C204 32 181 44 176 95C173 124 170 172 170 264V515H172L598 -9H629V394C629 487 631 532 635 563C639 606 663 619 719 622V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.664,0)"><path d="M631 18C609 24 585 35 559 65C534 91 514 117 478 169C448 214 406 281 389 313C462 346 516 399 516 485C516 545 490 590 449 616C412 641 363 650 290 650H42V622C120 615 128 612 128 527V125C128 40 120 34 38 28V0H300V28C221 34 212 40 212 125V284H244C295 284 312 272 329 244C359 195 395 133 430 84C475 19 516 -3 592 -7C603 -8 615 -8 627 -8L631 18ZM212 316V563C212 591 215 602 223 607C231 613 248 617 277 617C352 617 423 577 423 469C423 415 407 375 368 345C343 324 310 316 260 316H212Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.459,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.722,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,44.962,0)"><path d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.826,0)"><path d="M372 352C427 368 507 407 507 502C507 548 486 592 451 616C420 638 376 650 301 650H44V622C124 616 128 610 128 527V123C128 40 122 34 36 28V0H257C336 0 402 12 453 40C512 72 548 124 548 191C548 291 471 335 372 350V352ZM213 362V558C213 591 216 602 226 608C235 614 258 618 282 618C377 618 415 558 415 490C415 406 369 362 260 362H213ZM213 328H258C380 328 450 283 450 181C450 76 377 35 296 34C231 34 213 53 213 125V328Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

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Figure 8

Figure 8: A Design Method of Sparse Array with High Degrees of Freedom Based on Fourth-Order Cumulants 