-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathChap_10.html
1690 lines (1323 loc) · 72.7 KB
/
Chap_10.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!doctype html>
<html lang="en" class="no-js">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="author" content="I.T. Young & R. Ligteringen">
<link rel="shortcut icon" href="images/favicon.png">
<meta name="generator" content="mkdocs-1.1.2, mkdocs-material-6.0.2">
<title>10. The Wiener filter - Introduction to Stochastic Signal Processing</title>
<link rel="stylesheet" href="assets/stylesheets/main.38780c08.min.css">
<link rel="stylesheet" href="assets/stylesheets/palette.3f72e892.min.css">
<link href="https://fonts.gstatic.com" rel="preconnect" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Georgia:300,400,400i,700%7CCourier&display=fallback">
<style>body,input{font-family:"Georgia",-apple-system,BlinkMacSystemFont,Helvetica,Arial,sans-serif}code,kbd,pre{font-family:"Courier",SFMono-Regular,Consolas,Menlo,monospace}</style>
<link rel="stylesheet" href="css/extra.css">
<link rel="stylesheet" href="css/imgtxt.css">
<link rel="stylesheet" href="css/tables.css">
<!-- Global site tag (gtag.js) - Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=G-8LE8Z88MMP"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'G-8LE8Z88MMP');
</script>
</head>
<body dir="ltr" data-md-color-scheme="" data-md-color-primary="none" data-md-color-accent="none">
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
<label class="md-overlay" for="__drawer"></label>
<div data-md-component="skip">
<a href="#the-wiener-filter" class="md-skip">
Skip to content
</a>
</div>
<div data-md-component="announce">
</div>
<!-- Application header -->
<header class="md-header" data-md-component="header">
<!-- Top-level navigation -->
<nav class="md-header-nav md-grid" aria-label="Header">
<!-- Link to home -->
<a
href=""
title="Introduction to Stochastic Signal Processing"
class="md-header-nav__button md-logo"
aria-label="Introduction to Stochastic Signal Processing"
>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 003-3 3 3 0 00-3-3 3 3 0 00-3 3 3 3 0 003 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54z"/></svg>
<!--
Insert an <img> here as an alternative for the standard logo if desired.
Then comment out the above two lines
<img src="images/favicon.png" style="margin-top:5px; border: 0px solid lime; border-radius:5px; width:120%; height: auto;" />
-->
</a>
<!-- Button to open drawer -->
<label class="md-header-nav__button md-icon" for="__drawer">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2z"/></svg>
</label>
<!-- Header title -->
<div class="md-header-nav__title" data-md-component="header-title">
<a href="#jumpToBottom">
<span class="md-header-nav__topic">
Introduction to Stochastic Signal Processing
</span>
<span class="md-header-nav__topic">
10. The Wiener filter
</span>
</a>
</div>
<!-- Button to open search dialogue -->
<label class="md-header-nav__button md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0116 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 019.5 16 6.5 6.5 0 013 9.5 6.5 6.5 0 019.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5z"/></svg>
</label>
<!-- Search interface -->
<div class="md-search" data-md-component="search" role="dialog">
<label class="md-search__overlay" for="__search"></label>
<div class="md-search__inner" role="search">
<form class="md-search__form" name="search">
<input type="text" class="md-search__input" name="query" aria-label="Search" placeholder="Search" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" data-md-state="active">
<label class="md-search__icon md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0116 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 019.5 16 6.5 6.5 0 013 9.5 6.5 6.5 0 019.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12z"/></svg>
</label>
<button type="reset" class="md-search__icon md-icon" aria-label="Clear" data-md-component="search-reset" tabindex="-1">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41L17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12 19 6.41z"/></svg>
</button>
</form>
<div class="md-search__output">
<div class="md-search__scrollwrap" data-md-scrollfix>
<div class="md-search-result" data-md-component="search-result">
<div class="md-search-result__meta">
Initializing search
</div>
<ol class="md-search-result__list"></ol>
</div>
</div>
</div>
</div>
</div>
<!-- Repository containing source -->
</nav>
</header>
<div class="md-container" data-md-component="container">
<main class="md-main" data-md-component="main">
<div class="md-main__inner md-grid">
<div class="md-sidebar md-sidebar--primary" data-md-component="navigation">
<div class="md-sidebar__scrollwrap">
<div class="md-sidebar__inner">
<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
<label class="md-nav__title" for="__drawer">
<a href="" title="Introduction to Stochastic Signal Processing" class="md-nav__button md-logo" aria-label="Introduction to Stochastic Signal Processing">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 003-3 3 3 0 00-3-3 3 3 0 00-3 3 3 3 0 003 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54z"/></svg>
<!--
Insert an <img> here as an alternative for the standard logo if desired.
Then comment out the above two lines
<img src="images/favicon.png" style="margin-top:5px; border: 0px solid lime; border-radius:5px; width:120%; height: auto;" />
-->
</a>
Introduction to Stochastic Signal Processing
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="Chap_1.html" class="md-nav__link">
1. How to use this iBook
</a>
</li>
<li class="md-nav__item">
<a href="Chap_2.html" class="md-nav__link">
2. Prologue
</a>
</li>
<li class="md-nav__item">
<a href="Chap_3.html" class="md-nav__link">
3. Introduction
</a>
</li>
<li class="md-nav__item">
<a href="Chap_4.html" class="md-nav__link">
4. Characterization of Random Signals
</a>
</li>
<li class="md-nav__item">
<a href="Chap_5.html" class="md-nav__link">
5. Correlations and Spectra
</a>
</li>
<li class="md-nav__item">
<a href="Chap_6.html" class="md-nav__link">
6. Filtering of Stochastic Signals
</a>
</li>
<li class="md-nav__item">
<a href="Chap_7.html" class="md-nav__link">
7. The Langevin Equation – A Case Study
</a>
</li>
<li class="md-nav__item">
<a href="Chap_8.html" class="md-nav__link">
8. Characterizing Signal-to-Noise Ratios
</a>
</li>
<li class="md-nav__item">
<a href="Chap_9.html" class="md-nav__link">
9. The Matched Filter
</a>
</li>
<li class="md-nav__item md-nav__item--active">
<input class="md-nav__toggle md-toggle" data-md-toggle="toc" type="checkbox" id="__toc">
<label class="md-nav__link md-nav__link--active" for="__toc">
10. The Wiener filter
<span class="md-nav__icon md-icon"></span>
</label>
<a href="Chap_10.html" class="md-nav__link md-nav__link--active">
10. The Wiener filter
</a>
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
Table of contents
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="#the-restoration-case-noise" class="md-nav__link">
The restoration case: noise
</a>
</li>
<li class="md-nav__item">
<a href="#using-the-least-mean-square-error-criterion" class="md-nav__link">
Using the least mean-square error criterion
</a>
</li>
<li class="md-nav__item">
<a href="#expressing-the-mean-square-error" class="md-nav__link">
Expressing the mean-square error
</a>
</li>
<li class="md-nav__item">
<a href="#correlation-returns" class="md-nav__link">
Correlation returns
</a>
</li>
<li class="md-nav__item">
<a href="#the-wiener-hopf-equation" class="md-nav__link">
The Wiener-Hopf equation
</a>
</li>
<li class="md-nav__item">
<a href="#as-seen-from-the-fourier-domain" class="md-nav__link">
As seen from the Fourier domain
</a>
</li>
<li class="md-nav__item">
<a href="#what-is-that-least-mean-square-error" class="md-nav__link">
What is that least mean-square error?
</a>
</li>
<li class="md-nav__item">
<a href="#classic-example-classic-result" class="md-nav__link">
Classic example, classic result
</a>
</li>
<li class="md-nav__item">
<a href="#the-more-general-restoration-case-noise-distortion" class="md-nav__link">
The more general restoration case: noise & distortion
</a>
</li>
<li class="md-nav__item">
<a href="#why-we-avoid-the-inverse-filter" class="md-nav__link">
Why we avoid the inverse filter
</a>
<nav class="md-nav" aria-label="Why we avoid the inverse filter">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-sound-of-distorted-music" class="md-nav__link">
Example: Sound of (distorted) music
</a>
</li>
<li class="md-nav__item">
<a href="#example-cleaning-up-our-act" class="md-nav__link">
Example: Cleaning up our act
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#determining-the-wiener-filter" class="md-nav__link">
Determining the Wiener filter
</a>
</li>
<li class="md-nav__item">
<a href="#problems" class="md-nav__link">
Problems
</a>
<nav class="md-nav" aria-label="Problems">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#problem-101" class="md-nav__link">
Problem 10.1
</a>
</li>
<li class="md-nav__item">
<a href="#problem-102" class="md-nav__link">
Problem 10.2
</a>
</li>
<li class="md-nav__item">
<a href="#problem-103" class="md-nav__link">
Problem 10.3
</a>
</li>
<li class="md-nav__item">
<a href="#problem-104" class="md-nav__link">
Problem 10.4
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercises" class="md-nav__link">
Laboratory Exercises
</a>
<nav class="md-nav" aria-label="Laboratory Exercises">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#laboratory-exercise-101" class="md-nav__link">
Laboratory Exercise 10.1
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-102" class="md-nav__link">
Laboratory Exercise 10.2
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-103" class="md-nav__link">
Laboratory Exercise 10.3
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-104" class="md-nav__link">
Laboratory Exercise 10.4
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-105" class="md-nav__link">
Laboratory Exercise 10.5
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="Chap_11.html" class="md-nav__link">
11. Aspects of Estimation
</a>
</li>
<li class="md-nav__item">
<a href="Chap_12.html" class="md-nav__link">
12. Spectral Estimation
</a>
</li>
<li class="md-nav__item">
<a href="Chap_13.html" class="md-nav__link">
Appendices
</a>
</li>
<li class="md-nav__item">
<a href="info.html" class="md-nav__link">
Information
</a>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div class="md-sidebar md-sidebar--secondary" data-md-component="toc">
<div class="md-sidebar__scrollwrap">
<div class="md-sidebar__inner">
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
Table of contents
</label>
<ul class="md-nav__list" data-md-scrollfix>
<li class="md-nav__item">
<a href="#the-restoration-case-noise" class="md-nav__link">
The restoration case: noise
</a>
</li>
<li class="md-nav__item">
<a href="#using-the-least-mean-square-error-criterion" class="md-nav__link">
Using the least mean-square error criterion
</a>
</li>
<li class="md-nav__item">
<a href="#expressing-the-mean-square-error" class="md-nav__link">
Expressing the mean-square error
</a>
</li>
<li class="md-nav__item">
<a href="#correlation-returns" class="md-nav__link">
Correlation returns
</a>
</li>
<li class="md-nav__item">
<a href="#the-wiener-hopf-equation" class="md-nav__link">
The Wiener-Hopf equation
</a>
</li>
<li class="md-nav__item">
<a href="#as-seen-from-the-fourier-domain" class="md-nav__link">
As seen from the Fourier domain
</a>
</li>
<li class="md-nav__item">
<a href="#what-is-that-least-mean-square-error" class="md-nav__link">
What is that least mean-square error?
</a>
</li>
<li class="md-nav__item">
<a href="#classic-example-classic-result" class="md-nav__link">
Classic example, classic result
</a>
</li>
<li class="md-nav__item">
<a href="#the-more-general-restoration-case-noise-distortion" class="md-nav__link">
The more general restoration case: noise & distortion
</a>
</li>
<li class="md-nav__item">
<a href="#why-we-avoid-the-inverse-filter" class="md-nav__link">
Why we avoid the inverse filter
</a>
<nav class="md-nav" aria-label="Why we avoid the inverse filter">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-sound-of-distorted-music" class="md-nav__link">
Example: Sound of (distorted) music
</a>
</li>
<li class="md-nav__item">
<a href="#example-cleaning-up-our-act" class="md-nav__link">
Example: Cleaning up our act
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#determining-the-wiener-filter" class="md-nav__link">
Determining the Wiener filter
</a>
</li>
<li class="md-nav__item">
<a href="#problems" class="md-nav__link">
Problems
</a>
<nav class="md-nav" aria-label="Problems">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#problem-101" class="md-nav__link">
Problem 10.1
</a>
</li>
<li class="md-nav__item">
<a href="#problem-102" class="md-nav__link">
Problem 10.2
</a>
</li>
<li class="md-nav__item">
<a href="#problem-103" class="md-nav__link">
Problem 10.3
</a>
</li>
<li class="md-nav__item">
<a href="#problem-104" class="md-nav__link">
Problem 10.4
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercises" class="md-nav__link">
Laboratory Exercises
</a>
<nav class="md-nav" aria-label="Laboratory Exercises">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#laboratory-exercise-101" class="md-nav__link">
Laboratory Exercise 10.1
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-102" class="md-nav__link">
Laboratory Exercise 10.2
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-103" class="md-nav__link">
Laboratory Exercise 10.3
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-104" class="md-nav__link">
Laboratory Exercise 10.4
</a>
</li>
<li class="md-nav__item">
<a href="#laboratory-exercise-105" class="md-nav__link">
Laboratory Exercise 10.5
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div class="md-content">
<article class="md-content__inner md-typeset">
<h1 id="the-wiener-filter">The Wiener filter<a class="headerlink" href="#the-wiener-filter" title="Permanent link">¶</a></h1>
<p>In the previous chapter we showed that a desired effect, a maximized SNR, could be achieved by the suitable choice of a linear filter, a matched filter. We will now address a more difficult problem: the use of linear filters to estimate a <em>stochastic</em> signal in the presence of noise. We let <span class="arithmatex">\(x[n]\)</span> be a stochastic, ergodic signal from a process with known
statistics.</p>
<h2 id="the-restoration-case-noise">The restoration case: noise<a class="headerlink" href="#the-restoration-case-noise" title="Permanent link">¶</a></h2>
<p>In the presence of noise and using a linear filter, we wish to produce an <em>estimate</em> of <span class="arithmatex">\(x[n]\)</span> which we call <span class="arithmatex">\({x_e}[n].\)</span> This is frequently termed signal <em>restoration</em> because we are attempting to restore a signal <span class="arithmatex">\(x[n]\)</span> that has become damaged, corrupted, and/or noisy. We want the <em>best</em> estimate where the definition of “best” will be explained. We assume that <span class="arithmatex">\(x[n]\)</span> is real as is the noise process <span class="arithmatex">\(N[n].\)</span> The total signal <span class="arithmatex">\(r[n]\)</span> composed of <span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(N[n]\)</span> is to be processed to produce the estimate <span class="arithmatex">\({x_e}[n]\)</span> of the original <span class="arithmatex">\(x[n].\)</span> The model is shown in <a href="#fig:fig_Wiener1">Figure 10.1</a>.</p>
<figure class="figaltcap fullsize" id="fig:fig_Wiener1"><img src="images/Fig_10_1.gif" /><figcaption><strong>Figure 10.1:</strong> LTI filter <span class="arithmatex">\(h[n\rbrack\)</span> to estimate a stochastic signal <span class="arithmatex">\(x[n\rbrack\)</span> in the presence of additive noise.</figcaption>
</figure>
<p>We start with:</p>
<div class="" id="eq:Weq1">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.1)</td>
<td class="eqTableEq">
<div>$${x_e}[n] = \sum\limits_{m = - \infty }^{ + \infty } {r[n - m]} h[m]$$</div>
</td>
</tr>
</table>
</div>
<p>and we define “best” in terms of a measure of the difference (error) between the “true” stochastic signal <span class="arithmatex">\(x[n]\)</span> and the estimate <span class="arithmatex">\({x_e}[n].\)</span></p>
<h2 id="using-the-least-mean-square-error-criterion">Using the least mean-square error criterion<a class="headerlink" href="#using-the-least-mean-square-error-criterion" title="Permanent link">¶</a></h2>
<p>Specifically we look at the mean (expected) square error:</p>
<div class="" id="eq:Weq2">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.2)</td>
<td class="eqTableEq">
<div>$$e = mean\text{-}squared\;error = E\left\{ {{{\left| {{x_e}[n] - x[n]} \right|}^2}} \right\}$$</div>
</td>
</tr>
</table>
</div>
<p>and we propose to choose <span class="arithmatex">\(h[n]\)</span> in order to minimize this error. We will need a basic result from least mean-square estimation theory and this can be found in <a href="Chap_13.html#appendix-i-mean-square-error-minimization">Appendix I</a>. </p>
<p>To determine a minimum, we look at:</p>
<div class="" id="eq:Weq3">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.3)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{\frac{{\partial e}}{{\partial h}}}&{ = \frac{\partial }{{\partial h}}E\left\{ {{{\left| {{x_e}[n] - x[n]} \right|}^2}} \right\}}\\
{\,\,\,}&{ = \frac{\partial }{{\partial h}}E\left\{ {{{\left( {\sum\limits_{m = - \infty }^{ + \infty } {r[n - m]} h[m] - x[n]} \right)}^2}} \right\} = 0}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>That is, we vary the filter to choose the one that gives the minimum mean-square error. Note that the restriction to real signals and systems ensures that we can replace the non-differentiable absolute value operation <span class="arithmatex">\({\left| \bullet \right|^2}\)</span> with the differentiable operation <span class="arithmatex">\({\left( \bullet \right)^2}\)</span> in <a href="Chap_10.html#eq:Weq3">Equation 10.3</a>. </p>
<p>The equation above, of course, yields an extremum but it can be shown that this is a minimum; see <a href="#problem-101">Problem 10.1</a>. The development of this approach follows those in the references Castleman<sup id="fnref:castleman1996"><a class="footnote-ref" href="#fn:castleman1996">1</a></sup>, Lee<sup id="fnref:lee1960"><a class="footnote-ref" href="#fn:lee1960">2</a></sup>, and Papoulis<sup id="fnref:papoulis1977"><a class="footnote-ref" href="#fn:papoulis1977">3</a></sup>.</p>
<h2 id="expressing-the-mean-square-error">Expressing the mean-square error<a class="headerlink" href="#expressing-the-mean-square-error" title="Permanent link">¶</a></h2>
<p>Based upon <a href="Chap_13.html#appendix-i-mean-square-error-minimization">Appendix I</a>, and starting from <a href="Chap_10.html#eq:Weq1">Equation 10.1</a> and <a href="Chap_10.html#eq:Weq2">Equation 10.2</a>, we have:</p>
<div class="" id="eq:Weq4">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.4)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = E\left\{ {{{\left( {{x_e} - x} \right)}^2}} \right\} = E\left\{ {{{\left( {{x_e}[n] - x[n]} \right)}^2}} \right\}}\\
{\,\,\,}&{ = E\left\{ {{{\left( {r[n] \otimes h[n] - x[n]} \right)}^2}} \right\}}\\
{\,\,\,}&{ = E\left\{ {{{\left( {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]h[k]} - x[n]} \right)}^2}} \right\}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>We might have expected that the left-hand side of <a href="Chap_10.html#eq:Weq4">Equation 10.4</a> would be <span class="arithmatex">\(e[n]\)</span> instead of <span class="arithmatex">\(e.\)</span> The expectation operation, <span class="arithmatex">\(E\left\{ \bullet \right\},\)</span> combined with the assumption of ergodicity assures that the result <span class="arithmatex">\(e\)</span> is independent of <span class="arithmatex">\(n.\)</span></p>
<p>We now apply the key result—orthogonality—from <a href="Chap_13.html#eq:Ap1eq3">Equation 13.3</a> and rewrite this as:</p>
<div class="" id="eq:Weq5">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.5)</td>
<td class="eqTableEq">
<div>$$0 = \frac{{de}}{{d{h_i}}} = \,2E\left\{ {\left( {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]h[k]} - x[n]} \right)r[n - i]} \right\}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="correlation-returns">Correlation returns<a class="headerlink" href="#correlation-returns" title="Permanent link">¶</a></h2>
<p>Rearranging this expression gives:</p>
<div class="" id="eq:Weq6">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.6)</td>
<td class="eqTableEq">
<div>$$E\left\{ {x[n]r[n - i]} \right\} = E\left\{ {r[n - i]\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]} h[k]} \right\}$$</div>
</td>
</tr>
</table>
</div>
<p>The term on the left side of the equation is the cross-correlation function between <span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(r[n].\)</span> Because both signals are real, we have from <a href="Chap_5.html#eq:autoeven">Equation 5.5</a> and <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a>:</p>
<div class="" id="eq:Weq7">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.7)</td>
<td class="eqTableEq">
<div>$$E\left\{ {x[n]r[n - i]} \right\} = {\varphi _{xr}}[ - i] = \varphi _{rx}^*[i] = {\varphi _{rx}}[i]$$</div>
</td>
</tr>
</table>
</div>
<p>Because the signal <span class="arithmatex">\(r[n - i]\)</span> is not a function of <span class="arithmatex">\(k\)</span> and the order of the operations of summation and expectation can—with all the usual caveats—be reversed, the term on the right side of the equation can be rewritten as:</p>
<div class="" id="eq:Weq8">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.8)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
E\left\{ {\sum\limits_{k = - \infty }^{ + \infty } {r[n - k]r[n - i]} h[k]} \right\} = \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{k = - \infty }^{ + \infty } {E\left\{ {r[n - k]r[n - i]} \right\}} h[k]
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Again we have made use of <a href="Chap_4.html#eq:additive">Equation 4.13</a> which says that expectation as an operator distributes over sums.</p>
<h2 id="the-wiener-hopf-equation">The Wiener-Hopf equation<a class="headerlink" href="#the-wiener-hopf-equation" title="Permanent link">¶</a></h2>
<p>The result is an implicit expression for the optimum filter <span class="arithmatex">\(h[n]\)</span>:</p>
<div class="" id="eq:Weq9">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.9)</td>
<td class="eqTableEq">
<div>$${\varphi _{rx}}[k] = \sum\limits_{m = - \infty }^{ + \infty } {{\varphi _{rr}}[k - m]} h[m] = {\varphi _{rr}}[k] \otimes h[k]$$</div>
</td>
</tr>
</table>
</div>
<p>Note the way the variable names “<span class="arithmatex">\(m\)</span>” and “<span class="arithmatex">\(k\)</span>” are used in order to be
consistent with earlier notation, for example, <a href="Chap_5.html#eq:autoeven">Equation 5.5</a> and <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a>.</p>
<p>We distinguish between two cases of this famous equation, the <a href="https://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf_method">Wiener-Hopf
equation</a>. The variable <span class="arithmatex">\(k\)</span> represents the interval over which the process is observed. In the first case, <span class="arithmatex">\(k > 0\)</span> and this represents the case of <em>causal</em> observations. That is, in principle, it is only possible to estimate <span class="arithmatex">\(x[n]\)</span> from data that have already been gathered. The estimate <span class="arithmatex">\({x_e}[n]\)</span> can only have a causal dependence on <span class="arithmatex">\(r[n]\)</span> and the (causal) filter choice <span class="arithmatex">\(h[n].\)</span> With this condition the solution of the Wiener-Hopf equation is extremely
difficult, far beyond what we introduce here.</p>
<p>We will concentrate, instead, on the case <span class="arithmatex">\(- \infty \le k \le + \infty\)</span> which admits a direct solution through application of Fourier techniques. While this case is somewhat less realistic for temporal signals (but not for spatial signals), it will enable us to develop some insights into the character of filters developed as solutions to the Wiener-Hopf equation. Such filters, independent of the conditions on <span class="arithmatex">\(k,\)</span> are known
as <em>Wiener filters</em> after a mathematical giant of the 20<sup>th</sup> century <a href="https://en.wikipedia.org/wiki/Norbert_Wiener">Prof. Norbert Wiener</a> (1894-1964).</p>
<h2 id="as-seen-from-the-fourier-domain">As seen from the Fourier domain<a class="headerlink" href="#as-seen-from-the-fourier-domain" title="Permanent link">¶</a></h2>
<p>We start by taking the Fourier transform of both sides of <a href="Chap_10.html#eq:Weq9">Equation 10.9</a> to produce:</p>
<div class="" id="eq:Weq10">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.10)</td>
<td class="eqTableEq">
<div>$${S_{rx}}(\Omega ) = {S_{rr}}(\Omega )H(\Omega )$$</div>
</td>
</tr>
</table>
</div>
<p>The desired filter is, of course, <span class="arithmatex">\(H(\Omega )\)</span> and this is given by:</p>
<div class="" id="eq:Weq11">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.11)</td>
<td class="eqTableEq">
<div>$$H(\Omega ) = \frac{{{S_{rx}}(\Omega )}}{{{S_{rr}}(\Omega )}}$$</div>
</td>
</tr>
</table>
</div>
<h2 id="what-is-that-least-mean-square-error">What is that least mean-square error?<a class="headerlink" href="#what-is-that-least-mean-square-error" title="Permanent link">¶</a></h2>
<p>Through a series of manipulations (following section 10.3 in Papoulis<sup id="fnref2:papoulis1977"><a class="footnote-ref" href="#fn:papoulis1977">3</a></sup>),
we show that the total minimum error <span class="arithmatex">\(e\)</span> is:</p>
<div class="mainresult" id="eq:Weq12">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.12)</td>
<td class="eqTableEq">
<div>$$e = \frac{1}{{2\pi }}\int\limits_{ - \pi }^{ + \pi } {\left( {{S_{xx}}(\Omega ) - {S_{rx}}(\Omega )H(\Omega )} \right)} d\Omega$$</div>
</td>
</tr>
</table>
</div>
<p>We start with <a href="Chap_10.html#eq:Weq4">Equation 10.4</a> and using some algebra and the definition of the autocorrelation function we find:</p>
<div class="" id="eq:Weq13">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.13)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = E\left\{ {{{\left( {{x_e}[n] - x[n]} \right)}^2}} \right\} }\\
{\,\,\,}&{ = E\left\{ {{x^2}[n]} \right\} + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}\\
{\,\,\,}&{ = {\varphi _{xx}}[0] + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Continuing with the use of <a href="Chap_10.html#eq:Weq1">Equation 10.1</a> this becomes a lengthy—somewhat inelegant— expression:</p>
<div class="" id="eq:Weq14">
<table class="eqTable">
<tr>
<td class="eqTableTag">(10.14)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
e&{ = {\varphi _{xx}}[0] + E\left\{ {x_e^2[n]} \right\} - 2E\left\{ {{x_e}[n]x[n]} \right\}}\\
{}&{ = {\varphi _{xx}}[0]\,\, + }\\