-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathChap_5.html
1743 lines (1394 loc) · 74.1 KB
/
Chap_5.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>5. Correlations and Spectra - 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="#correlations-and-spectra" 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">
5. Correlations and Spectra
</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 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">
5. Correlations and Spectra
<span class="md-nav__icon md-icon"></span>
</label>
<a href="Chap_5.html" class="md-nav__link md-nav__link--active">
5. Correlations and Spectra
</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="#correlations-simple-and-complex" class="md-nav__link">
Correlations: simple and complex
</a>
<nav class="md-nav" aria-label="Correlations: simple and complex">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-delayed-effect" class="md-nav__link">
Example: Delayed effect
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#correlations-and-memory" class="md-nav__link">
Correlations and memory
</a>
</li>
<li class="md-nav__item">
<a href="#the-mechanics-of-correlations-redux" class="md-nav__link">
The mechanics of correlations - redux
</a>
</li>
<li class="md-nav__item">
<a href="#fourier-description-of-correlation-functions" class="md-nav__link">
Fourier description of correlation functions
</a>
</li>
<li class="md-nav__item">
<a href="#a-digression" class="md-nav__link">
A digression
</a>
</li>
<li class="md-nav__item">
<a href="#the-power-density-spectrum-and-its-properties" class="md-nav__link">
The power density spectrum and its properties
</a>
</li>
<li class="md-nav__item">
<a href="#examples-of-power-spectra" class="md-nav__link">
Examples of power spectra
</a>
<nav class="md-nav" aria-label="Examples of power spectra">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-white-noise" class="md-nav__link">
Example: White noise
</a>
</li>
<li class="md-nav__item">
<a href="#example-pink-noise" class="md-nav__link">
Example: Pink noise
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#predicting-the-natural-climate-a-case-study" class="md-nav__link">
Predicting the natural climate - a case study
</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-51" class="md-nav__link">
Problem 5.1
</a>
</li>
<li class="md-nav__item">
<a href="#problem-52" class="md-nav__link">
Problem 5.2
</a>
</li>
<li class="md-nav__item">
<a href="#problem-53" class="md-nav__link">
Problem 5.3
</a>
</li>
<li class="md-nav__item">
<a href="#problem-54" class="md-nav__link">
Problem 5.4
</a>
</li>
<li class="md-nav__item">
<a href="#problem-55" class="md-nav__link">
Problem 5.5
</a>
</li>
<li class="md-nav__item">
<a href="#problem-56" class="md-nav__link">
Problem 5.6
</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-51" class="md-nav__link">
Laboratory Exercise 5.1
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</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">
<a href="Chap_10.html" class="md-nav__link">
10. The Wiener filter
</a>
</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="#correlations-simple-and-complex" class="md-nav__link">
Correlations: simple and complex
</a>
<nav class="md-nav" aria-label="Correlations: simple and complex">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-delayed-effect" class="md-nav__link">
Example: Delayed effect
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#correlations-and-memory" class="md-nav__link">
Correlations and memory
</a>
</li>
<li class="md-nav__item">
<a href="#the-mechanics-of-correlations-redux" class="md-nav__link">
The mechanics of correlations - redux
</a>
</li>
<li class="md-nav__item">
<a href="#fourier-description-of-correlation-functions" class="md-nav__link">
Fourier description of correlation functions
</a>
</li>
<li class="md-nav__item">
<a href="#a-digression" class="md-nav__link">
A digression
</a>
</li>
<li class="md-nav__item">
<a href="#the-power-density-spectrum-and-its-properties" class="md-nav__link">
The power density spectrum and its properties
</a>
</li>
<li class="md-nav__item">
<a href="#examples-of-power-spectra" class="md-nav__link">
Examples of power spectra
</a>
<nav class="md-nav" aria-label="Examples of power spectra">
<ul class="md-nav__list">
<li class="md-nav__item">
<a href="#example-white-noise" class="md-nav__link">
Example: White noise
</a>
</li>
<li class="md-nav__item">
<a href="#example-pink-noise" class="md-nav__link">
Example: Pink noise
</a>
</li>
</ul>
</nav>
</li>
<li class="md-nav__item">
<a href="#predicting-the-natural-climate-a-case-study" class="md-nav__link">
Predicting the natural climate - a case study
</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-51" class="md-nav__link">
Problem 5.1
</a>
</li>
<li class="md-nav__item">
<a href="#problem-52" class="md-nav__link">
Problem 5.2
</a>
</li>
<li class="md-nav__item">
<a href="#problem-53" class="md-nav__link">
Problem 5.3
</a>
</li>
<li class="md-nav__item">
<a href="#problem-54" class="md-nav__link">
Problem 5.4
</a>
</li>
<li class="md-nav__item">
<a href="#problem-55" class="md-nav__link">
Problem 5.5
</a>
</li>
<li class="md-nav__item">
<a href="#problem-56" class="md-nav__link">
Problem 5.6
</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-51" class="md-nav__link">
Laboratory Exercise 5.1
</a>
</li>
</ul>
</nav>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div class="md-content">
<article class="md-content__inner md-typeset">
<h1 id="correlations-and-spectra">Correlations and Spectra<a class="headerlink" href="#correlations-and-spectra" title="Permanent link">¶</a></h1>
<p>Let <span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(y[n]\)</span> be two ergodic random processes. As stated in <a href="Chap_4.html#the-ergodic-process">Chapter 4</a>, this means that both processes are stationary.</p>
<h2 id="correlations-simple-and-complex">Correlations: simple and complex<a class="headerlink" href="#correlations-simple-and-complex" title="Permanent link">¶</a></h2>
<p>Based upon <a href="Chap_4.html#autocorrelation">Equation 4.19</a> and <a href="Chap_4.html#cross-correlation">Equation 4.21</a>, we have that the autocorrelation function of the random process <span class="arithmatex">\(x\)</span> and the cross-correlation function between processes <span class="arithmatex">\(x\)</span> and <span class="arithmatex">\(y\)</span> are:</p>
<div class="mainresult" id="eq:correlations">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.1)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\varphi _{xx}}[k] = E\left\{ {x[n]{x^*}[n + k]} \right\}\\
{\varphi _{xy}}[k] = E\left\{ {x[n]{y^*}[n + k]} \right\}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>While <span class="arithmatex">\(x\)</span> and <span class="arithmatex">\(y\)</span> are random processes, the auto- and cross-correlation
functions are just ordinary, deterministic (not random) functions of the
variable <span class="arithmatex">\(k.\)</span> Why is that?</p>
<p>Because, like the mean <span class="arithmatex">\(\left( \mu \right)\)</span> and standard deviation <span class="arithmatex">\(\left( \sigma \right)\)</span> of familiar probability distributions, the correlations and covariances describe
parameters—characteristics—of the probability distributions that
generate random variables but are, themselves, not random.</p>
<p>The auto- and cross-covariance functions are given by:</p>
<div class="" id="eq:covariances">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.2)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\gamma _{xx}}[k] = {\varphi _{xx}}[k] - {\left| {{m_x}} \right|^2}\\
{\gamma _{xy}}[k] = {\varphi _{xy}}[k] - {m_x}m_y^* \end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>By setting <span class="arithmatex">\(k = 0,\)</span> we see that</p>
<div class="" id="eq:meansqpos">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.3)</td>
<td class="eqTableEq">
<div>$${\varphi _{xx}}[0] = E\left\{ {{{\left| {x[n]} \right|}^2}} \right\} = mean{\text -}square\;of\;process\; \ge 0$$</div>
</td>
</tr>
</table>
</div>
<p>and </p>
<div class="" id="eq:varpos">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.4)</td>
<td class="eqTableEq">
<div>$${\gamma _{xx}}[0] = \sigma _x^2 \ge 0$$</div>
</td>
</tr>
</table>
</div>
<p>Further, we have:</p>
<div class="" id="eq:autoeven">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.5)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\varphi _{xx}}[k] = \varphi _{xx}^*[ - k]\\
{\gamma _{xx}}[k] = \gamma _{xx}^*[ - k]
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<div class="" id="eq:crossHermitian">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.6)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\varphi _{xy}}[k] = \varphi _{yx}^*[ - k]\\
{\gamma _{xy}}[k] = \gamma _{yx}^*[ - k]
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>It is straightforward and, therefore, left to you to show in <a href="#problem-51">Problem 5.1</a> that <a href="Chap_5.html#eq:autoeven">Equation 5.5</a> and the second part of <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a> are correct. The proof
of the first part of <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a> is given below.</p>
<p>We start from the second part of <a href="Chap_5.html#eq:correlations">Equation 5.1</a> and replace <span class="arithmatex">\(n + k\)</span> with
<span class="arithmatex">\(m\)</span>:</p>
<div class="arithmatex">\[\begin{array}{*{20}{l}}
{{\varphi _{xy}}[k]}&{ = E\left\{ {x[n]\,{y^*}[n + k]} \right\}\mathop \Rightarrow \limits^{m = n + k} }\\
{}&\begin{array}{l}
= E\left\{ {x[m - k]\,{y^*}[m]} \right\}\\
= {\left( {E\left\{ {y[m]\,{x^*}[m - k]} \right\}} \right)^*}
\end{array}\\
{}&{ = \varphi _{yx}^*[ - k]}
\end{array}\]</div>
<p>which proves the first part of <a href="Chap_5.html#eq:crossHermitian">Equation 5.6</a> .</p>
<h4 id="example-delayed-effect">Example: Delayed effect<a class="headerlink" href="#example-delayed-effect" title="Permanent link">¶</a></h4>
<p>As a simple example we consider the case where <span class="arithmatex">\(y[n] = x[n - {n_o}],\)</span> a
delayed version of <span class="arithmatex">\(x[n].\)</span> Then:</p>
<div class="" id="eq:eq_delayed_effect">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.7)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{{\varphi _{yy}}[k]}&{ = E\left\{ {y[n]\,{y^*}[n + k]} \right\}}\\
{}&{ = E\left\{ {x[n - {n_0}]\,{x^*}[n - {n_0} + k]} \right\}}\\
{}&{ = {\varphi _{xx}}[k]}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p><em>What does this mean</em>? The conclusion associated with this example is
that the autocorrelation of a time signal is independent of any time
shift.</p>
<h2 id="correlations-and-memory">Correlations and memory<a class="headerlink" href="#correlations-and-memory" title="Permanent link">¶</a></h2>
<p>We would now like to consider what may happen to the correlation between
<span class="arithmatex">\(x[n]\)</span> and <span class="arithmatex">\(x[n + k]\)</span> as <span class="arithmatex">\(k\)</span> becomes larger. For many processes, as <span class="arithmatex">\(k \to \infty ,\)</span> the value of the random signal at time <span class="arithmatex">\(n\)</span> becomes less dependent upon the value at time <span class="arithmatex">\(n + k.\)</span> A simple example might be that the weather today is similar to the weather
yesterday but less similar to the weather of four weeks ago. As <span class="arithmatex">\(k \to \infty ,\)</span> we have </p>
<div class="" id="eq:autointhelimit">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.8)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
\mathop {\lim }\limits_{k \to \infty } \left\{ {{\varphi _{xx}}[k] = E\left\{ {x[n]{x^*}[n + k]} \right\}} \right\}\\
\,\,\,\,\,\,\,\,\,\, = E\left\{ {x[n]} \right\}E\left\{ {{x^*}[n + k]} \right\} = {\left| {{m_x}} \right|^2}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>This result frequently holds as well for the cross-correlation giving:</p>
<div class="" id="eq:crossinthelimit">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.9)</td>
<td class="eqTableEq">
<div>$$\mathop {\lim }\limits_{k \to \infty } {\varphi _{xy}}[k] = {m_x}m_y^*$$</div>
</td>
</tr>
</table>
</div>
<p>Finally these two results may be used to show that for many important cases:</p>
<div class="" id="eq:covarinthelimit">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.10)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
\mathop {\lim }\limits_{k \to \infty } {\gamma _{xx}}[k] = 0\\
\mathop {\lim }\limits_{k \to \infty } {\gamma _{xy}}[k] = 0
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>As <span class="arithmatex">\(k \to \infty ,\)</span> correlations and covariances can exhibit a form of memory
loss, of forgetting their relation to earlier values.</p>
<p>These results provide us with tools that can help us estimate the long
term correlation and variation of one or more random processes. We have
specifically stated “many processes” but not “all processes”. See <a href="#problem-52">Problem 5.2</a> to explore this.</p>
<h2 id="the-mechanics-of-correlations-redux">The mechanics of correlations - redux<a class="headerlink" href="#the-mechanics-of-correlations-redux" title="Permanent link">¶</a></h2>
<p>An issue that we have avoided until now is: how do we actually calculate the autocorrelation or cross-correlation of a random signal? According to <a href="Chap_4.html#eq:stat_corr-def2">Equation 4.24</a>, we need to know the joint probability distributions at two points <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(n + m.\)</span> Under the assumption of stationarity this reduces from <span class="arithmatex">\(\varphi [n,n + m]\)</span> to <span class="arithmatex">\(\varphi [m].\)</span> But given an actual random process such as air pressure, internet traffic, or radioactive emissions, we almost never know the joint probability distribution including the values of all of its parameters. Under the assumption of ergodicity, as described in Chapter 4 <a href="Chap_4.html#properties-of-averaging">here</a> and <a href="Chap_4.html#the-ergodic-process">here</a>, an alternative method is available to us.</p>
<p>When considering the autocorrelation function, for example, instead of
using <a href="Chap_4.html#eq:stat_corr-def2">Equation 4.24</a> we can use <a href="Chap_4.html#eq:time_acorr">Equation 4.30</a>. Both of these equations are repeated below in <a href="Chap_5.html#eq:acorrtimeavg">Equation 5.11</a> and they illustrate that for ergodic processes ensemble averages can be exchanged for time averages.</p>
<div class="" id="eq:acorrtimeavg">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.11)</td>
<td class="eqTableEq">
<div>$$\begin{array}{*{20}{l}}
{{\varphi _{xx}}[k]}&{ = E\left\{ {x[n]\,{x^*}[n + k]} \right\}}\\
{}&{ = \; < x[n]\,{x^*}[n + k] > }\\
{}&{ = \mathop {\lim }\limits_{N \to \infty } \frac{1}{{2N + 1}}\sum\limits_{n = - N}^{ + N} {x[n]\,{x^*}[n + k]} }
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>Even with this reformulation there remains a problem. We never have an
infinite amount of recorded data; <span class="arithmatex">\(N\)</span> never goes to infinity. This
brings us instead to <a href="Chap_4.html#eq:finite_acorr">Equation 4.34</a> where a
finite amount of data is used to estimate a correlation function. This
means that we can use the following formulation.</p>
<div class="" id="eq:acorrtimeavg_finite">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.12)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\varphi _{xx}}[k] = \;\frac{1}{{2N + 1}}\sum\limits_{n = - N}^{ + N} {x[n]\,{x^*}[n + k]} \\
{\varphi _{xy}}[k] = \;\frac{1}{{2N + 1}}\sum\limits_{n = - N}^{ + N} {x[n]\,{y^*}[n + k]}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>As the term <span class="arithmatex">\(1/\left( {2N + 1} \right)\)</span> is only a scale factor—albeit an important one when comparing estimates of correlation functions based upon data sets of different lengths—we further simplify this to:</p>
<div class="" id="eq:acorrtimeavg_scaled">
<table class="eqTable">
<tr>
<td class="eqTableTag">(5.13)</td>
<td class="eqTableEq">
<div>$$\begin{array}{l}
{\varphi _{xx}}[k] = \;\sum\limits_{n = - N}^{ + N} {x[n]\,{x^*}[n + k]} \\
{\varphi _{xy}}[k] = \;\sum\limits_{n = - N}^{ + N} {x[n]\,{y^*}[n + k]}
\end{array}$$</div>
</td>
</tr>
</table>
</div>
<p>The formulas for calculating the autocorrelation and crosscorrelation of
stochastic signals are now similar to the ones for deterministic signals Oppenheim<sup id="fnref:oppenheim1996"><a class="footnote-ref" href="#fn:oppenheim1996">1</a></sup>.</p>
<p>To summarize, if the processes we are studying are ergodic, we use finite
data sets to calculate correlation functions (and means and covariance
functions). Papoulis<sup id="fnref:papoulis1965"><a class="footnote-ref" href="#fn:papoulis1965">2</a></sup> stated this in a most
succinct way:</p>
<p><a class="quote" href="">“x(t) is ergodic in the most general form if (with probability 1) all its statistics can be determined from a single function x(t,ζ) of the process.”</a></p>
<p>This, of course, holds for discrete-time ergodic processes as well.</p>
<h2 id="fourier-description-of-correlation-functions">Fourier description of correlation functions<a class="headerlink" href="#fourier-description-of-correlation-functions" title="Permanent link">¶</a></h2>
<p>We now look at the Fourier transform of the autocorrelation function for
the complex ergodic random process <span class="arithmatex">\(x[n].\)</span> We should remember that
<span class="arithmatex">\({\varphi _{xx}}[k]\)</span> is not a random process. The Fourier
transform of the autocorrelation function—if it exists—is called the
<em>power spectrum</em> (or power density spectrum) of the random process.
This name will be explained later.</p>
<p>Existence of the Fourier transform is not a trivial matter. A random
process <span class="arithmatex">\(x[n]\)</span> might be ergodic but it need not be absolutely
integrable, a condition for the existence of its Fourier spectrum; see
Section 5.1.3 of Oppenheim<sup id="fnref2:oppenheim1996"><a class="footnote-ref" href="#fn:oppenheim1996">1</a></sup>. The same holds for the autocorrelation function associated with <span class="arithmatex">\(x[n].\)</span></p>
<h2 id="a-digression">A digression<a class="headerlink" href="#a-digression" title="Permanent link">¶</a></h2>
<p>Ignoring the issue of convergence can lead to surprising results.
Consider, for example, the <em>deterministic</em> signal given by:</p>