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-/Limits [(equation.3.13a) (equation.3.17)] +763 0 obj << +/Names [(equation.3.13) 571 0 R (equation.3.13a) 572 0 R (equation.3.13b) 573 0 R (equation.3.14) 581 0 R (equation.3.15) 582 0 R (equation.3.16) 583 0 R] +/Limits [(equation.3.13) (equation.3.16)] >> endobj -688 0 obj << -/Names [(equation.3.18) 541 0 R (equation.3.19) 543 0 R (equation.3.2) 499 0 R (equation.3.20) 544 0 R (equation.3.21) 545 0 R (equation.3.22) 550 0 R] -/Limits [(equation.3.18) (equation.3.22)] +764 0 obj << +/Names [(equation.3.17) 584 0 R (equation.3.18) 585 0 R (equation.3.19) 587 0 R (equation.3.2) 543 0 R (equation.3.20) 588 0 R (equation.3.21) 589 0 R] +/Limits [(equation.3.17) (equation.3.21)] >> endobj -689 0 obj << -/Names [(equation.3.3) 500 0 R (equation.3.3a) 501 0 R (equation.3.3b) 502 0 R (equation.3.4) 503 0 R (equation.3.4a) 504 0 R (equation.3.4b) 505 0 R] -/Limits [(equation.3.3) (equation.3.4b)] +765 0 obj << +/Names [(equation.3.22) 594 0 R (equation.3.3) 544 0 R (equation.3.3a) 545 0 R (equation.3.3b) 546 0 R (equation.3.4) 547 0 R (equation.3.4a) 548 0 R] +/Limits [(equation.3.22) (equation.3.4a)] >> endobj -690 0 obj << -/Names [(equation.3.4c) 506 0 R (equation.3.5) 507 0 R (equation.3.6) 508 0 R (equation.3.7) 515 0 R (equation.3.7a) 516 0 R (equation.3.7b) 517 0 R] -/Limits [(equation.3.4c) (equation.3.7b)] +766 0 obj << +/Names [(equation.3.4b) 549 0 R (equation.3.4c) 550 0 R (equation.3.5) 551 0 R (equation.3.6) 552 0 R (equation.3.7) 559 0 R (equation.3.7a) 560 0 R] +/Limits [(equation.3.4b) (equation.3.7a)] >> endobj -691 0 obj << -/Names [(equation.3.7c) 518 0 R (equation.3.8) 519 0 R (equation.3.9) 520 0 R (equation.4.1) 563 0 R (equation.4.10) 596 0 R (equation.4.11) 604 0 R] -/Limits [(equation.3.7c) (equation.4.11)] +767 0 obj << +/Names [(equation.3.7b) 561 0 R (equation.3.7c) 562 0 R (equation.3.8) 563 0 R (equation.3.9) 564 0 R (equation.4.1) 608 0 R (equation.4.10) 640 0 R] +/Limits [(equation.3.7b) (equation.4.10)] >> endobj -692 0 obj << -/Names [(equation.4.12) 605 0 R (equation.4.13) 606 0 R (equation.4.14) 607 0 R (equation.4.15) 608 0 R (equation.4.16) 613 0 R (equation.4.17) 618 0 R] -/Limits [(equation.4.12) (equation.4.17)] +768 0 obj << +/Names [(equation.4.11) 648 0 R (equation.4.12) 649 0 R (equation.4.13) 650 0 R (equation.4.14) 651 0 R (equation.4.15) 652 0 R (equation.4.16) 657 0 R] +/Limits [(equation.4.11) (equation.4.16)] >> endobj -693 0 obj << -/Names [(equation.4.18) 620 0 R (equation.4.18a) 621 0 R (equation.4.18b) 622 0 R (equation.4.18c) 623 0 R (equation.4.18d) 624 0 R (equation.4.2) 564 0 R] -/Limits [(equation.4.18) (equation.4.2)] +769 0 obj << +/Names [(equation.4.17) 662 0 R (equation.4.18) 664 0 R (equation.4.18a) 665 0 R (equation.4.18b) 666 0 R (equation.4.18c) 667 0 R (equation.4.18d) 668 0 R] +/Limits [(equation.4.17) (equation.4.18d)] >> endobj -694 0 obj << -/Names [(equation.4.3) 566 0 R (equation.4.3a) 568 0 R (equation.4.3b) 569 0 R (equation.4.3c) 570 0 R (equation.4.4) 571 0 R (equation.4.4a) 572 0 R] -/Limits [(equation.4.3) (equation.4.4a)] +770 0 obj << +/Names [(equation.4.2) 609 0 R (equation.4.3) 611 0 R (equation.4.3a) 613 0 R (equation.4.3b) 614 0 R (equation.4.3c) 615 0 R (equation.4.4) 616 0 R] +/Limits [(equation.4.2) (equation.4.4)] >> endobj -695 0 obj << -/Names [(equation.4.4b) 573 0 R (equation.4.5) 574 0 R (equation.4.6) 584 0 R (equation.4.6a) 585 0 R (equation.4.6b) 586 0 R (equation.4.6c) 588 0 R] -/Limits [(equation.4.4b) (equation.4.6c)] +771 0 obj << +/Names [(equation.4.4a) 617 0 R (equation.4.4b) 618 0 R (equation.4.5) 619 0 R (equation.4.6) 628 0 R (equation.4.6a) 629 0 R (equation.4.6b) 630 0 R] +/Limits [(equation.4.4a) (equation.4.6b)] >> endobj -696 0 obj << -/Names [(equation.4.6d) 589 0 R (equation.4.6e) 590 0 R (equation.4.7) 591 0 R (equation.4.8) 592 0 R (equation.4.8a) 593 0 R (equation.4.8b) 594 0 R] -/Limits [(equation.4.6d) (equation.4.8b)] +772 0 obj << +/Names [(equation.4.6c) 632 0 R (equation.4.6d) 633 0 R (equation.4.6e) 634 0 R (equation.4.7) 635 0 R (equation.4.8) 636 0 R (equation.4.8a) 637 0 R] +/Limits [(equation.4.6c) (equation.4.8a)] >> endobj -697 0 obj << -/Names [(equation.4.9) 595 0 R (equation.5.1) 630 0 R (equation.5.1a) 631 0 R (equation.5.1b) 632 0 R (equation.5.2) 633 0 R (page.0) 312 0 R] -/Limits [(equation.4.9) (page.0)] +773 0 obj << +/Names [(equation.4.8b) 638 0 R (equation.4.9) 639 0 R (equation.5.1) 681 0 R (equation.5.2) 682 0 R (equation.5.3) 683 0 R (equation.5.4) 684 0 R] +/Limits [(equation.4.8b) (equation.5.4)] >> endobj -698 0 obj << -/Names [(page.1) 368 0 R (page.10) 536 0 R (page.11) 549 0 R (page.12) 562 0 R (page.13) 582 0 R (page.14) 603 0 R] -/Limits [(page.1) (page.14)] +774 0 obj << +/Names [(equation.5.4a) 685 0 R (equation.5.4b) 686 0 R (equation.5.4c) 687 0 R (equation.5.5) 692 0 R (equation.5.5a) 693 0 R (equation.5.5b) 694 0 R] +/Limits [(equation.5.4a) (equation.5.5b)] >> endobj -699 0 obj << -/Names [(page.15) 617 0 R (page.16) 629 0 R (page.2) 411 0 R (page.3) 418 0 R (page.4) 437 0 R (page.5) 454 0 R] -/Limits [(page.15) (page.5)] +775 0 obj << +/Names [(equation.5.5c) 695 0 R (equation.5.6) 696 0 R (equation.5.6a) 697 0 R (equation.5.6b) 698 0 R (equation.5.6c) 699 0 R (equation.6.1) 706 0 R] +/Limits [(equation.5.5c) (equation.6.1)] >> endobj -700 0 obj << -/Names [(page.6) 474 0 R (page.7) 486 0 R (page.8) 497 0 R (page.9) 514 0 R (section*.3) 370 0 R (section.1.1) 6 0 R] -/Limits [(page.6) (section.1.1)] +776 0 obj << +/Names [(equation.6.1a) 707 0 R (equation.6.1b) 708 0 R (equation.6.2) 709 0 R (page.0) 344 0 R (page.1) 400 0 R (page.10) 558 0 R] +/Limits [(equation.6.1a) (page.10)] >> endobj -701 0 obj << -/Names [(section.1.2) 18 0 R (section.1.3) 30 0 R (section.1.4) 46 0 R (section.1.5) 50 0 R (section.1.6) 62 0 R (section.1.7) 74 0 R] -/Limits [(section.1.2) (section.1.7)] +777 0 obj << +/Names [(page.11) 580 0 R (page.12) 593 0 R (page.13) 607 0 R (page.14) 626 0 R (page.15) 647 0 R (page.16) 661 0 R] +/Limits [(page.11) (page.16)] >> endobj -702 0 obj << -/Names [(section.2.1) 102 0 R (section.2.2) 114 0 R (section.3.1) 134 0 R (section.3.2) 138 0 R (section.3.3) 150 0 R (section.3.4) 170 0 R] -/Limits [(section.2.1) (section.3.4)] +778 0 obj << +/Names [(page.17) 680 0 R (page.18) 691 0 R (page.19) 705 0 R (page.2) 451 0 R (page.3) 455 0 R (page.4) 462 0 R] +/Limits [(page.17) (page.4)] >> endobj -703 0 obj << -/Names [(section.3.5) 182 0 R (section.3.6) 194 0 R (section.4.1) 226 0 R (section.4.2) 254 0 R (section.4.3) 262 0 R (section.4.4) 278 0 R] -/Limits [(section.3.5) (section.4.4)] +779 0 obj << +/Names [(page.5) 481 0 R (page.6) 498 0 R (page.7) 519 0 R (page.8) 530 0 R (page.9) 541 0 R (section*.3) 402 0 R] +/Limits [(page.5) (section*.3)] >> endobj -704 0 obj << -/Names [(section.5.1) 294 0 R (subsection.1.1.1) 10 0 R (subsection.1.1.2) 14 0 R (subsection.1.2.1) 22 0 R (subsection.1.2.2) 26 0 R (subsection.1.3.1) 34 0 R] -/Limits [(section.5.1) (subsection.1.3.1)] +780 0 obj << +/Names [(section.1.1) 6 0 R (section.1.2) 18 0 R (section.1.3) 30 0 R (section.1.4) 46 0 R (section.1.5) 50 0 R (section.1.6) 62 0 R] +/Limits [(section.1.1) (section.1.6)] >> endobj -705 0 obj << -/Names [(subsection.1.3.2) 38 0 R (subsection.1.3.3) 42 0 R (subsection.1.5.1) 54 0 R (subsection.1.5.2) 58 0 R (subsection.1.6.1) 66 0 R (subsection.1.6.2) 70 0 R] -/Limits [(subsection.1.3.2) (subsection.1.6.2)] +781 0 obj << +/Names [(section.1.7) 74 0 R (section.2.1) 102 0 R (section.2.2) 114 0 R (section.3.1) 134 0 R (section.3.2) 138 0 R (section.3.3) 150 0 R] +/Limits [(section.1.7) (section.3.3)] >> endobj -706 0 obj << -/Names [(subsection.1.7.1) 78 0 R (subsection.1.7.2) 82 0 R (subsection.1.7.3) 86 0 R (subsection.1.7.4) 90 0 R (subsection.1.7.5) 94 0 R (subsection.2.1.1) 106 0 R] -/Limits [(subsection.1.7.1) (subsection.2.1.1)] +782 0 obj << +/Names [(section.3.4) 170 0 R (section.3.5) 182 0 R (section.3.6) 194 0 R (section.4.1) 226 0 R (section.4.2) 254 0 R (section.4.3) 262 0 R] +/Limits [(section.3.4) (section.4.3)] >> endobj -707 0 obj << -/Names [(subsection.2.1.2) 110 0 R (subsection.2.2.1) 118 0 R (subsection.2.2.2) 122 0 R (subsection.2.2.3) 126 0 R (subsection.3.2.1) 142 0 R (subsection.3.2.2) 146 0 R] -/Limits [(subsection.2.1.2) (subsection.3.2.2)] +783 0 obj << +/Names [(section.4.4) 278 0 R (section.5.1) 294 0 R (section.6.1) 326 0 R (subsection.1.1.1) 10 0 R (subsection.1.1.2) 14 0 R (subsection.1.2.1) 22 0 R] +/Limits [(section.4.4) (subsection.1.2.1)] >> endobj -708 0 obj << -/Names [(subsection.3.3.1) 154 0 R (subsection.3.3.2) 158 0 R (subsection.3.3.3) 162 0 R (subsection.3.3.4) 166 0 R (subsection.3.4.1) 174 0 R (subsection.3.4.2) 178 0 R] -/Limits [(subsection.3.3.1) (subsection.3.4.2)] +784 0 obj << +/Names [(subsection.1.2.2) 26 0 R (subsection.1.3.1) 34 0 R (subsection.1.3.2) 38 0 R (subsection.1.3.3) 42 0 R (subsection.1.5.1) 54 0 R (subsection.1.5.2) 58 0 R] +/Limits [(subsection.1.2.2) (subsection.1.5.2)] >> endobj -709 0 obj << -/Names [(subsection.3.5.1) 186 0 R (subsection.3.5.2) 190 0 R (subsection.3.6.1) 198 0 R (subsection.3.6.2) 202 0 R (subsection.3.6.3) 206 0 R (subsection.3.6.4) 210 0 R] -/Limits [(subsection.3.5.1) (subsection.3.6.4)] +785 0 obj << +/Names [(subsection.1.6.1) 66 0 R (subsection.1.6.2) 70 0 R (subsection.1.7.1) 78 0 R (subsection.1.7.2) 82 0 R (subsection.1.7.3) 86 0 R (subsection.1.7.4) 90 0 R] +/Limits [(subsection.1.6.1) (subsection.1.7.4)] >> endobj -710 0 obj << -/Names [(subsection.3.6.5) 214 0 R (subsection.3.6.6) 218 0 R (subsection.4.1.1) 230 0 R (subsection.4.1.2) 234 0 R (subsection.4.1.3) 238 0 R (subsection.4.1.4) 242 0 R] -/Limits [(subsection.3.6.5) (subsection.4.1.4)] +786 0 obj << +/Names [(subsection.1.7.5) 94 0 R (subsection.2.1.1) 106 0 R (subsection.2.1.2) 110 0 R (subsection.2.2.1) 118 0 R (subsection.2.2.2) 122 0 R (subsection.2.2.3) 126 0 R] +/Limits [(subsection.1.7.5) (subsection.2.2.3)] >> endobj -711 0 obj << -/Names [(subsection.4.1.5) 246 0 R (subsection.4.1.6) 250 0 R (subsection.4.2.1) 258 0 R (subsection.4.3.1) 266 0 R (subsection.4.3.2) 270 0 R (subsection.4.3.3) 274 0 R] -/Limits [(subsection.4.1.5) (subsection.4.3.3)] +787 0 obj << +/Names [(subsection.3.2.1) 142 0 R (subsection.3.2.2) 146 0 R (subsection.3.3.1) 154 0 R (subsection.3.3.2) 158 0 R (subsection.3.3.3) 162 0 R (subsection.3.3.4) 166 0 R] +/Limits [(subsection.3.2.1) (subsection.3.3.4)] >> endobj -712 0 obj << -/Names [(subsection.4.4.1) 282 0 R (subsection.4.4.2) 286 0 R (subsection.5.1.1) 298 0 R (subsection.5.1.2) 302 0 R (subsubsection*.10) 619 0 R (subsubsection*.4) 455 0 R] -/Limits [(subsection.4.4.1) (subsubsection*.4)] +788 0 obj << +/Names [(subsection.3.4.1) 174 0 R (subsection.3.4.2) 178 0 R (subsection.3.5.1) 186 0 R (subsection.3.5.2) 190 0 R (subsection.3.6.1) 198 0 R (subsection.3.6.2) 202 0 R] +/Limits [(subsection.3.4.1) (subsection.3.6.2)] >> endobj -713 0 obj << -/Names [(subsubsection*.5) 457 0 R (subsubsection*.6) 542 0 R (subsubsection*.7) 565 0 R (subsubsection*.8) 583 0 R (subsubsection*.9) 609 0 R] -/Limits [(subsubsection*.5) (subsubsection*.9)] +789 0 obj << +/Names [(subsection.3.6.3) 206 0 R (subsection.3.6.4) 210 0 R (subsection.3.6.5) 214 0 R (subsection.3.6.6) 218 0 R (subsection.4.1.1) 230 0 R (subsection.4.1.2) 234 0 R] +/Limits [(subsection.3.6.3) (subsection.4.1.2)] >> endobj -714 0 obj << -/Kids [677 0 R 678 0 R 679 0 R 680 0 R 681 0 R 682 0 R] -/Limits [(Doc-Start) (equation.1.5b)] +790 0 obj << +/Names [(subsection.4.1.3) 238 0 R (subsection.4.1.4) 242 0 R (subsection.4.1.5) 246 0 R (subsection.4.1.6) 250 0 R (subsection.4.2.1) 258 0 R (subsection.4.3.1) 266 0 R] +/Limits [(subsection.4.1.3) (subsection.4.3.1)] >> endobj -715 0 obj << -/Kids [683 0 R 684 0 R 685 0 R 686 0 R 687 0 R 688 0 R] -/Limits [(equation.1.5c) (equation.3.22)] +791 0 obj << +/Names [(subsection.4.3.2) 270 0 R (subsection.4.3.3) 274 0 R (subsection.4.4.1) 282 0 R (subsection.4.4.2) 286 0 R (subsection.5.1.1) 298 0 R (subsection.5.1.2) 302 0 R] +/Limits [(subsection.4.3.2) (subsection.5.1.2)] >> endobj -716 0 obj << -/Kids [689 0 R 690 0 R 691 0 R 692 0 R 693 0 R 694 0 R] -/Limits [(equation.3.3) (equation.4.4a)] +792 0 obj << +/Names [(subsection.5.1.3) 306 0 R (subsection.5.1.4) 310 0 R (subsection.5.1.5) 314 0 R (subsection.5.1.6) 318 0 R (subsection.6.1.1) 330 0 R (subsection.6.1.2) 334 0 R] +/Limits [(subsection.5.1.3) (subsection.6.1.2)] >> endobj -717 0 obj << -/Kids [695 0 R 696 0 R 697 0 R 698 0 R 699 0 R 700 0 R] -/Limits [(equation.4.4b) (section.1.1)] +793 0 obj << +/Names [(subsubsection*.10) 663 0 R (subsubsection*.4) 499 0 R (subsubsection*.5) 501 0 R (subsubsection*.6) 586 0 R (subsubsection*.7) 610 0 R (subsubsection*.8) 627 0 R] +/Limits [(subsubsection*.10) (subsubsection*.8)] >> endobj -718 0 obj << -/Kids [701 0 R 702 0 R 703 0 R 704 0 R 705 0 R 706 0 R] -/Limits [(section.1.2) (subsection.2.1.1)] +794 0 obj << +/Names [(subsubsection*.9) 653 0 R] +/Limits [(subsubsection*.9) (subsubsection*.9)] >> endobj -719 0 obj << -/Kids [707 0 R 708 0 R 709 0 R 710 0 R 711 0 R 712 0 R] -/Limits [(subsection.2.1.2) (subsubsection*.4)] +795 0 obj << +/Kids [753 0 R 754 0 R 755 0 R 756 0 R 757 0 R 758 0 R] +/Limits [(Doc-Start) (equation.1.5a)] >> endobj -720 0 obj << -/Kids [713 0 R] -/Limits [(subsubsection*.5) (subsubsection*.9)] +796 0 obj << +/Kids [759 0 R 760 0 R 761 0 R 762 0 R 763 0 R 764 0 R] +/Limits [(equation.1.5b) (equation.3.21)] >> endobj -721 0 obj << -/Kids [714 0 R 715 0 R 716 0 R 717 0 R 718 0 R 719 0 R] -/Limits [(Doc-Start) (subsubsection*.4)] +797 0 obj << +/Kids [765 0 R 766 0 R 767 0 R 768 0 R 769 0 R 770 0 R] +/Limits [(equation.3.22) (equation.4.4)] >> endobj -722 0 obj << -/Kids [720 0 R] -/Limits [(subsubsection*.5) (subsubsection*.9)] +798 0 obj << +/Kids [771 0 R 772 0 R 773 0 R 774 0 R 775 0 R 776 0 R] +/Limits [(equation.4.4a) (page.10)] >> endobj -723 0 obj << -/Kids [721 0 R 722 0 R] +799 0 obj << +/Kids [777 0 R 778 0 R 779 0 R 780 0 R 781 0 R 782 0 R] +/Limits [(page.11) (section.4.3)] +>> endobj +800 0 obj << +/Kids [783 0 R 784 0 R 785 0 R 786 0 R 787 0 R 788 0 R] +/Limits [(section.4.4) (subsection.3.6.2)] +>> endobj +801 0 obj << +/Kids [789 0 R 790 0 R 791 0 R 792 0 R 793 0 R 794 0 R] +/Limits [(subsection.3.6.3) (subsubsection*.9)] +>> endobj +802 0 obj << +/Kids [795 0 R 796 0 R 797 0 R 798 0 R 799 0 R 800 0 R] +/Limits [(Doc-Start) (subsection.3.6.2)] +>> endobj +803 0 obj << +/Kids [801 0 R] +/Limits [(subsection.3.6.3) (subsubsection*.9)] +>> endobj +804 0 obj << +/Kids [802 0 R 803 0 R] /Limits [(Doc-Start) (subsubsection*.9)] >> endobj -724 0 obj << -/Dests 723 0 R +805 0 obj << +/Dests 804 0 R >> endobj -725 0 obj << +806 0 obj << /Type /Catalog -/Pages 675 0 R -/Outlines 676 0 R -/Names 724 0 R +/Pages 751 0 R +/Outlines 752 0 R +/Names 805 0 R /PageMode/UseOutlines/PageLabels<>1<>]>> -/OpenAction 305 0 R +/OpenAction 337 0 R >> endobj -726 0 obj << +807 0 obj << /Author()/Title()/Subject()/Creator(LaTeX with hyperref package)/Producer(Latex TUD-CD \(C. v. 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<851A7ABB4E12DF8788024423186EF1BB>] >> +<< /Size 808 +/Root 806 0 R +/Info 807 0 R +/ID [<72A7D8E4093942D3197CF2AE2A9CE912> <72A7D8E4093942D3197CF2AE2A9CE912>] >> startxref -229007 +242533 %%EOF diff --git a/document.tex b/document.tex index 2f5b2cf..730bc74 100644 --- a/document.tex +++ b/document.tex @@ -493,17 +493,65 @@ Das Spektrum eines Realen Zufallsprozesses ist komplett im Intervall $[0,\pi]$ b \end{subequations} +\chapter{Filter} +\section{Lineare Filter} +Wenn $X(n)$ und $Y(n)$ \emph{stationr} (\ref{stationarystrict}) sind, $h(n)$ eine Impulsantwort eines LTI-Systems ist + und das Filter \emph{stabil} (\ref{Stability}) ist, existiert mit \emph{Wahrscheinlichkeit 1} (\ref{conv:one}) +\begin{equation} \label{eq:linfil} +Y(n)=\sum\limits_{k=-\infty}^\infty h(k)X(n-k)=\sum\limits_{k=-\infty}^\infty h(n-k)X(k) +\end{equation} +\subsection{Stabilitt} \label{Stability} +Die Stabilitt eines Filters ist gegeben, wenn: +\begin{equation} +\sum|h(n)|<\infty +\end{equation} +\subsection{Eigenschaften eines Linearen Filters} +\begin{itemize} + \item Ist $X(n)$ \emph{stationr} (\ref{stationarystrict}) und $\E[|X(n)|]<\infty$, dann ist $Y(n)$ \emph{stationr} + \item $Y(n)$ wird linearer Prozess genannt (linear process) +\end{itemize} +\subsection{Instabiler linearer Filter} +Ist das Filter nicht \emph{stabil} (\ref{Stability}), aber $\int|H(e^{j\omega}|d\omega<\infty)$ trifft zu +und fr $X(n) \quad \sum|c_{XX}(n)<\infty$, sodann existiert im \emph{Mean-Square-Sense} (\ref{conv:mss}) die Formel (\ref{eq:linfil}) +und $Y(n)$ ist \emph{stationr im weiteren Sinne} (\ref{stationary}) mit +\begin{equation} +\mu_Y=\E[Y(n)]=\sum\limits_{k=-\infty}^\infty h(k)\E[X(n-k)]=\mu_XH(e^{j0}) +\end{equation} +\subsection{Kreukovarianz des Ausgangs des Filters} +\begin{subequations} +\begin{align} +c_{YX}&=\sum\limits_{k=-\infty}^\infty h(k)c_{XX}(\kappa -k) \\ +C_{YX}(e^{j\omega})&=H(e^{j\omega})C_{XX}(e^{j\omega})\\ +c_{YX}&=\int\limits_{-\pi}^\pi H(e^{j\omega})C_{XX}(e^{j\omega})e^{-j\omega\kappa}\frac{d\omega}{2\pi} +\end{align} +\end{subequations} +\subsection{Kreukovarianz des Ausgangs zweier Filter} +\begin{subequations} +\begin{align} +c_{Y_1Y_2}(\kappa)&=\sum\limits_{k=-\infty}^\infty \sum\limits_{l=-\infty}^\infty h_1(k)h_2(l)\cdot c_{X_1X_2}(\kappa -k+l) \\ +c_{Y_1Y_2}(\kappa)&=h_1(\kappa)\star h_2(\kappa)^*\star c_{X_1X_2}(\kappa)\\ +C_{Y_1Y_2}(e^{j\omega})&=H_1(e^{j\omega})H_2(e^{j\omega})^*C_{X_1X_2}(e^{j\omega}) +\end{align} +\end{subequations} - +\subsection{Kaskade linearer Filter} +\begin{subequations} +\begin{align} +H(e^{j\omega})&=\prod\limits_{i=1}^L H_i(e^{j\omega}) \\ +C_{YY}(e^{j\omega})&=C_{XX}(e^{j\omega})\prod\limits_{i=1}^L\left| H_i(e^{j\omega})\right|^2 \\ +C_{YX}(e^{j\omega})&=C_{XX}(e^{j\omega})\prod\limits_{i=1}^L H_i(e^{j\omega}) +\end{align} +\end{subequations} \chapter{Sonstiges} \section{Spezielle Funktionen} -\subsection{Gaussian white noise process} +\subsection{Gaussian white noise process} \label{whitenoise} Gausches weies Rauschen ist immer \emph{stationr} (\ref{stationarystrict}) \begin{subequations} \begin{align} +\E[W(n)]=0 r_{WW}(\kappa)&=\sigma_W^2\delta(\kappa) \\ S_{WW}(e^{j \omega})&=\sigma_W^2 \end{align}