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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
|
<!DOCTYPE html>
<html>
<!-- Created by GNU Texinfo 7.1.1, https://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Signal Processing (GNU Octave (version 10.3.0))</title>
<meta name="description" content="Signal Processing (GNU Octave (version 10.3.0))">
<meta name="keywords" content="Signal Processing (GNU Octave (version 10.3.0))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta name="viewport" content="width=device-width,initial-scale=1">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="index.html" rel="up" title="Top">
<link href="Image-Processing.html" rel="next" title="Image Processing">
<link href="Geometry.html" rel="prev" title="Geometry">
<style type="text/css">
<!--
a.copiable-link {visibility: hidden; text-decoration: none; line-height: 0em}
div.example {margin-left: 3.2em}
span:hover a.copiable-link {visibility: visible}
strong.def-name {font-family: monospace; font-weight: bold; font-size: larger}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<div class="chapter-level-extent" id="Signal-Processing">
<div class="nav-panel">
<p>
Next: <a href="Image-Processing.html" accesskey="n" rel="next">Image Processing</a>, Previous: <a href="Geometry.html" accesskey="p" rel="prev">Geometry</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<h2 class="chapter" id="Signal-Processing-1"><span>31 Signal Processing<a class="copiable-link" href="#Signal-Processing-1"> ¶</a></span></h2>
<p>This chapter describes the signal processing and fast Fourier
transform functions available in Octave. Fast Fourier transforms are
computed with the <small class="sc">FFTW</small> or <small class="sc">FFTPACK</small> libraries depending on how
Octave is built.
</p>
<a class="anchor" id="XREFfft"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fft"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fft</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-fft"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fft-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-fft-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fft-2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">n</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-fft-2"> ¶</a></span></dt>
<dd><p>Compute the discrete Fourier transform of <var class="var">x</var> using
a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The FFT is calculated along the first non-singleton dimension of the
array. Thus if <var class="var">x</var> is a matrix, <code class="code">fft (<var class="var">x</var>)</code> computes the
FFT for each column of <var class="var">x</var>.
</p>
<p>If called with two arguments, <var class="var">n</var> is expected to be an integer
specifying the number of elements of <var class="var">x</var> to use, or an empty
matrix to specify that its value should be ignored. If <var class="var">n</var> is
larger than the dimension along which the FFT is calculated, then
<var class="var">x</var> is resized and padded with zeros. Otherwise, if <var class="var">n</var> is
smaller than the dimension along which the FFT is calculated, then
<var class="var">x</var> is truncated.
</p>
<p>If called with three arguments, <var class="var">dim</var> is an integer specifying the
dimension of the matrix along which the FFT is performed.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFifft">ifft</a>, <a class="ref" href="#XREFfft2">fft2</a>, <a class="ref" href="#XREFfftn">fftn</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<a class="anchor" id="XREFifft"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ifft"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">ifft</strong> <code class="def-code-arguments">(<var class="var">y</var>)</code><a class="copiable-link" href="#index-ifft"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ifft-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">ifft</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-ifft-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ifft-2"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">ifft</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">n</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-ifft-2"> ¶</a></span></dt>
<dd><p>Compute the inverse discrete Fourier transform of <var class="var">y</var>
using a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The inverse FFT is calculated along the first non-singleton dimension
of the array. Thus if <var class="var">y</var> is a matrix, <code class="code">ifft (<var class="var">y</var>)</code> computes
the inverse FFT for each column of <var class="var">y</var>.
</p>
<p>If called with two arguments, <var class="var">n</var> is expected to be an integer
specifying the number of elements of <var class="var">y</var> to use, or an empty
matrix to specify that its value should be ignored. If <var class="var">n</var> is
larger than the dimension along which the inverse FFT is calculated, then
<var class="var">y</var> is resized and padded with zeros. Otherwise, if <var class="var">n</var> is
smaller than the dimension along which the inverse FFT is calculated,
then <var class="var">y</var> is truncated.
</p>
<p>If called with three arguments, <var class="var">dim</var> is an integer specifying the
dimension of the matrix along which the inverse FFT is performed.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfft">fft</a>, <a class="ref" href="#XREFifft2">ifft2</a>, <a class="ref" href="#XREFifftn">ifftn</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<a class="anchor" id="XREFfft2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fft2"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">fft2</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-fft2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fft2-1"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">fft2</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-fft2-1"> ¶</a></span></dt>
<dd><p>Compute the two-dimensional discrete Fourier transform of <var class="var">A</var> using
a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The optional arguments <var class="var">m</var> and <var class="var">n</var> may be used specify the number of
rows and columns of <var class="var">A</var> to use. If either of these is larger than the
size of <var class="var">A</var>, <var class="var">A</var> is resized and padded with zeros.
</p>
<p>If <var class="var">A</var> is a multi-dimensional matrix, each two-dimensional sub-matrix
of <var class="var">A</var> is treated separately.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFifft2">ifft2</a>, <a class="ref" href="#XREFfft">fft</a>, <a class="ref" href="#XREFfftn">fftn</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<a class="anchor" id="XREFifft2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ifft2"><span><code class="def-type"><var class="var">A</var> =</code> <strong class="def-name">ifft2</strong> <code class="def-code-arguments">(<var class="var">B</var>)</code><a class="copiable-link" href="#index-ifft2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ifft2-1"><span><code class="def-type"><var class="var">A</var> =</code> <strong class="def-name">ifft2</strong> <code class="def-code-arguments">(<var class="var">B</var>, <var class="var">m</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-ifft2-1"> ¶</a></span></dt>
<dd><p>Compute the inverse two-dimensional discrete Fourier transform of <var class="var">B</var>
using a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The optional arguments <var class="var">m</var> and <var class="var">n</var> may be used specify the number of
rows and columns of <var class="var">B</var> to use. If either of these is larger than the
size of <var class="var">B</var>, <var class="var">B</var> is resized and padded with zeros.
</p>
<p>If <var class="var">B</var> is a multi-dimensional matrix, each two-dimensional sub-matrix
of <var class="var">B</var> is treated separately.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfft2">fft2</a>, <a class="ref" href="#XREFifft">ifft</a>, <a class="ref" href="#XREFifftn">ifftn</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<a class="anchor" id="XREFfftn"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fftn"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">fftn</strong> <code class="def-code-arguments">(<var class="var">A</var>)</code><a class="copiable-link" href="#index-fftn"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftn-1"><span><code class="def-type"><var class="var">B</var> =</code> <strong class="def-name">fftn</strong> <code class="def-code-arguments">(<var class="var">A</var>, <var class="var">size</var>)</code><a class="copiable-link" href="#index-fftn-1"> ¶</a></span></dt>
<dd><p>Compute the N-dimensional discrete Fourier transform of <var class="var">A</var> using
a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The optional vector argument <var class="var">size</var> may be used specify the dimensions
of the array to be used. If an element of <var class="var">size</var> is smaller than the
corresponding dimension of <var class="var">A</var>, then the dimension of <var class="var">A</var> is
truncated prior to performing the FFT. Otherwise, if an element of
<var class="var">size</var> is larger than the corresponding dimension then <var class="var">A</var> is
resized and padded with zeros.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFifftn">ifftn</a>, <a class="ref" href="#XREFfft">fft</a>, <a class="ref" href="#XREFfft2">fft2</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<a class="anchor" id="XREFifftn"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ifftn"><span><code class="def-type"><var class="var">A</var> =</code> <strong class="def-name">ifftn</strong> <code class="def-code-arguments">(<var class="var">B</var>)</code><a class="copiable-link" href="#index-ifftn"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ifftn-1"><span><code class="def-type"><var class="var">A</var> =</code> <strong class="def-name">ifftn</strong> <code class="def-code-arguments">(<var class="var">B</var>, <var class="var">size</var>)</code><a class="copiable-link" href="#index-ifftn-1"> ¶</a></span></dt>
<dd><p>Compute the inverse N-dimensional discrete Fourier transform of <var class="var">B</var>
using a Fast Fourier Transform (FFT) algorithm.
</p>
<p>The optional vector argument <var class="var">size</var> may be used specify the dimensions
of the array to be used. If an element of <var class="var">size</var> is smaller than the
corresponding dimension of <var class="var">B</var>, then the dimension of <var class="var">B</var> is
truncated prior to performing the inverse FFT. Otherwise, if an element of
<var class="var">size</var> is larger than the corresponding dimension then <var class="var">B</var> is
resized and padded with zeros.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfftn">fftn</a>, <a class="ref" href="#XREFifft">ifft</a>, <a class="ref" href="#XREFifft2">ifft2</a>, <a class="ref" href="#XREFfftw">fftw</a>.
</p></dd></dl>
<p>Octave uses the <small class="sc">FFTW</small> libraries to perform FFT computations. When Octave
starts up and initializes the <small class="sc">FFTW</small> libraries, they read a system wide
file (on a Unix system, it is typically <samp class="file">/etc/fftw/wisdom</samp>) that
contains information useful to speed up FFT computations. This
information is called the <em class="emph">wisdom</em>. The system-wide file allows
wisdom to be shared between all applications using the <small class="sc">FFTW</small> libraries.
</p>
<p>Use the <code class="code">fftw</code> function to generate and save wisdom. Using the
utilities provided together with the <small class="sc">FFTW</small> libraries
(<code class="command">fftw-wisdom</code> on Unix systems), you can even add wisdom
generated by Octave to the system-wide wisdom file.
</p>
<a class="anchor" id="XREFfftw"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fftw"><span><code class="def-type"><var class="var">method</var> =</code> <strong class="def-name">fftw</strong> <code class="def-code-arguments">("planner")</code><a class="copiable-link" href="#index-fftw"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftw-1"><span><strong class="def-name">fftw</strong> <code class="def-code-arguments">("planner", <var class="var">method</var>)</code><a class="copiable-link" href="#index-fftw-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftw-2"><span><code class="def-type"><var class="var">wisdom</var> =</code> <strong class="def-name">fftw</strong> <code class="def-code-arguments">("dwisdom")</code><a class="copiable-link" href="#index-fftw-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftw-3"><span><strong class="def-name">fftw</strong> <code class="def-code-arguments">("dwisdom", <var class="var">wisdom</var>)</code><a class="copiable-link" href="#index-fftw-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftw-4"><span><code class="def-type"><var class="var">nthreads</var> =</code> <strong class="def-name">fftw</strong> <code class="def-code-arguments">("threads")</code><a class="copiable-link" href="#index-fftw-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftw-5"><span><strong class="def-name">fftw</strong> <code class="def-code-arguments">("threads", <var class="var">nthreads</var>)</code><a class="copiable-link" href="#index-fftw-5"> ¶</a></span></dt>
<dd>
<p>Manage <small class="sc">FFTW</small> wisdom data.
</p>
<p>Wisdom data can be used to significantly accelerate the calculation of the
FFTs, but implies an initial cost in its calculation. When the <small class="sc">FFTW</small>
libraries are initialized, they read a system wide wisdom file (typically in
<samp class="file">/etc/fftw/wisdom</samp>), allowing wisdom to be shared between applications
other than Octave. Alternatively, the <code class="code">fftw</code> function can be used to
import wisdom. For example,
</p>
<div class="example">
<pre class="example-preformatted"><var class="var">wisdom</var> = fftw ("dwisdom")
</pre></div>
<p>will save the existing wisdom used by Octave to the string <var class="var">wisdom</var>.
This string can then be saved to a file and restored using the <code class="code">save</code>
and <code class="code">load</code> commands respectively. This existing wisdom can be
re-imported as follows
</p>
<div class="example">
<pre class="example-preformatted">fftw ("dwisdom", <var class="var">wisdom</var>)
</pre></div>
<p>If <var class="var">wisdom</var> is an empty string, then the wisdom used is cleared.
</p>
<p>During the calculation of Fourier transforms further wisdom is generated.
The fashion in which this wisdom is generated is also controlled by
the <code class="code">fftw</code> function. There are five different manners in which the
wisdom can be treated:
</p>
<dl class="table">
<dt><code class="code">"estimate"</code></dt>
<dd><p>Specifies that no run-time measurement of the optimal means of
calculating a particular is performed, and a simple heuristic is used
to pick a (probably sub-optimal) plan. The advantage of this method is
that there is little or no overhead in the generation of the plan, which
is appropriate for a Fourier transform that will be calculated once.
</p>
</dd>
<dt><code class="code">"measure"</code></dt>
<dd><p>In this case a range of algorithms to perform the transform is considered
and the best is selected based on their execution time.
</p>
</dd>
<dt><code class="code">"patient"</code></dt>
<dd><p>Similar to <code class="code">"measure"</code>, but a wider range of algorithms is
considered.
</p>
</dd>
<dt><code class="code">"exhaustive"</code></dt>
<dd><p>Like <code class="code">"measure"</code>, but all possible algorithms that may be used to
treat the transform are considered.
</p>
</dd>
<dt><code class="code">"hybrid"</code></dt>
<dd><p>As run-time measurement of the algorithm can be expensive, this is a
compromise where <code class="code">"measure"</code> is used for transforms up to the size
of 8192 and beyond that the <code class="code">"estimate"</code> method is used.
</p></dd>
</dl>
<p>The default method is <code class="code">"estimate"</code>. The current method can
be queried with
</p>
<div class="example">
<pre class="example-preformatted"><var class="var">method</var> = fftw ("planner")
</pre></div>
<p>or set by using
</p>
<div class="example">
<pre class="example-preformatted">fftw ("planner", <var class="var">method</var>)
</pre></div>
<p>Note that calculated wisdom will be lost when restarting Octave. However,
the wisdom data can be reloaded if it is saved to a file as described
above. Saved wisdom files should not be used on different platforms since
they will not be efficient and the point of calculating the wisdom is lost.
</p>
<p>The number of threads used for computing the plans and executing the
transforms can be set with
</p>
<div class="example">
<pre class="example-preformatted">fftw ("threads", <var class="var">NTHREADS</var>)
</pre></div>
<p>Note that Octave must be compiled with multi-threaded <small class="sc">FFTW</small> support for
this feature. By default, the number of (logical) processors available to the
current process or <var class="var">3</var> is used (whichever is smaller).
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfft">fft</a>, <a class="ref" href="#XREFifft">ifft</a>, <a class="ref" href="#XREFfft2">fft2</a>, <a class="ref" href="#XREFifft2">ifft2</a>, <a class="ref" href="#XREFfftn">fftn</a>, <a class="ref" href="#XREFifftn">ifftn</a>.
</p></dd></dl>
<a class="anchor" id="XREFfftconv"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fftconv"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">fftconv</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-fftconv"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftconv-1"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">fftconv</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-fftconv-1"> ¶</a></span></dt>
<dd><p>Convolve two vectors using the FFT for computation.
</p>
<p><code class="code">c = fftconv (<var class="var">x</var>, <var class="var">y</var>)</code> returns a vector of length equal to
<code class="code">length (<var class="var">x</var>) + length (<var class="var">y</var>) - 1</code>. If <var class="var">x</var> and <var class="var">y</var>
are the coefficient vectors of two polynomials, the returned value is the
coefficient vector of the product polynomial.
</p>
<p>The computation uses the FFT by calling the function <code class="code">fftfilt</code>. If
the optional argument <var class="var">n</var> is specified, an N-point FFT is used.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Products-of-Polynomials.html#XREFdeconv">deconv</a>, <a class="ref" href="Products-of-Polynomials.html#XREFconv">conv</a>, <a class="ref" href="Products-of-Polynomials.html#XREFconv2">conv2</a>.
</p></dd></dl>
<a class="anchor" id="XREFfftfilt"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fftfilt"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fftfilt</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">x</var>)</code><a class="copiable-link" href="#index-fftfilt"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftfilt-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fftfilt</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">x</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-fftfilt-1"> ¶</a></span></dt>
<dd><p>Filter <var class="var">x</var> with the FIR filter <var class="var">b</var> using the FFT.
</p>
<p>If <var class="var">x</var> is a matrix, filter each column of the matrix.
</p>
<p>Given the optional third argument, <var class="var">n</var>, <code class="code">fftfilt</code> uses the
overlap-add method to filter <var class="var">x</var> with <var class="var">b</var> using an N-point FFT.
The FFT size must be an even power of 2 and must be greater than or equal to
the length of <var class="var">b</var>. If the specified <var class="var">n</var> does not meet these
criteria, it is automatically adjusted to the nearest value that does.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfilter">filter</a>, <a class="ref" href="#XREFfilter2">filter2</a>.
</p></dd></dl>
<a class="anchor" id="XREFfilter"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-filter"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">filter</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">x</var>)</code><a class="copiable-link" href="#index-filter"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-filter-1"><span><code class="def-type">[<var class="var">y</var>, <var class="var">sf</var>] =</code> <strong class="def-name">filter</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">x</var>, <var class="var">si</var>)</code><a class="copiable-link" href="#index-filter-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-filter-2"><span><code class="def-type">[<var class="var">y</var>, <var class="var">sf</var>] =</code> <strong class="def-name">filter</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">x</var>, [], <var class="var">dim</var>)</code><a class="copiable-link" href="#index-filter-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-filter-3"><span><code class="def-type">[<var class="var">y</var>, <var class="var">sf</var>] =</code> <strong class="def-name">filter</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">x</var>, <var class="var">si</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-filter-3"> ¶</a></span></dt>
<dd><p>Apply a 1-D digital filter to the data <var class="var">x</var>.
</p>
<p><code class="code">filter</code> returns the solution to the following linear, time-invariant
difference equation:
</p>
<div class="example smallexample">
<div class="group"><pre class="example-preformatted"> N M
SUM a(k+1) y(n-k) = SUM b(k+1) x(n-k) for 1<=n<=length(x)
k=0 k=0
</pre></div></div>
<p>where
N=length(a)-1 and M=length(b)-1.
The result is calculated over the first non-singleton dimension of <var class="var">x</var>
or over <var class="var">dim</var> if supplied.
</p>
<p>An equivalent form of the equation is:
</p>
<div class="example smallexample">
<div class="group"><pre class="example-preformatted"> N M
y(n) = - SUM c(k+1) y(n-k) + SUM d(k+1) x(n-k) for 1<=n<=length(x)
k=1 k=0
</pre></div></div>
<p>where
c = a/a(1) and d = b/a(1).
</p>
<p>If the fourth argument <var class="var">si</var> is provided, it is taken as the
initial state of the system and the final state is returned as
<var class="var">sf</var>. The state vector is a column vector whose length is
equal to the length of the longest coefficient vector minus one.
If <var class="var">si</var> is not supplied, the initial state vector is set to all
zeros.
</p>
<p>In terms of the Z Transform, <var class="var">y</var> is the result of passing the
discrete-time signal <var class="var">x</var> through a system characterized by the following
rational system function:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> M
SUM d(k+1) z^(-k)
k=0
H(z) = ---------------------
N
1 + SUM c(k+1) z^(-k)
k=1
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfilter2">filter2</a>, <a class="ref" href="#XREFfftfilt">fftfilt</a>, <a class="ref" href="#XREFfreqz">freqz</a>.
</p></dd></dl>
<a class="anchor" id="XREFfilter2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-filter2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">filter2</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">x</var>)</code><a class="copiable-link" href="#index-filter2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-filter2-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">filter2</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">x</var>, <var class="var">shape</var>)</code><a class="copiable-link" href="#index-filter2-1"> ¶</a></span></dt>
<dd><p>Apply the 2-D FIR filter <var class="var">b</var> to <var class="var">x</var>.
</p>
<p>If the argument <var class="var">shape</var> is specified, return an array of the desired
shape. Possible values are:
</p>
<dl class="table">
<dt><code class="code">"full"</code></dt>
<dd><p>pad <var class="var">x</var> with zeros on all sides before filtering.
</p>
</dd>
<dt><code class="code">"same"</code></dt>
<dd><p>unpadded <var class="var">x</var> (default)
</p>
</dd>
<dt><code class="code">"valid"</code></dt>
<dd><p>trim <var class="var">x</var> after filtering so edge effects are no included.
</p></dd>
</dl>
<p>Note this is just a variation on convolution, with the parameters reversed
and <var class="var">b</var> rotated 180 degrees.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Products-of-Polynomials.html#XREFconv2">conv2</a>.
</p></dd></dl>
<a class="anchor" id="XREFfreqz"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-freqz"><span><code class="def-type">[<var class="var">h</var>, <var class="var">w</var>] =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">n</var>, "whole")</code><a class="copiable-link" href="#index-freqz"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-1"><span><code class="def-type">[<var class="var">h</var>, <var class="var">w</var>] =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(<var class="var">b</var>)</code><a class="copiable-link" href="#index-freqz-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-2"><span><code class="def-type">[<var class="var">h</var>, <var class="var">w</var>] =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>)</code><a class="copiable-link" href="#index-freqz-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-3"><span><code class="def-type">[<var class="var">h</var>, <var class="var">w</var>] =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-freqz-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-4"><span><code class="def-type"><var class="var">h</var> =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(<var class="var">b</var>, <var class="var">a</var>, <var class="var">w</var>)</code><a class="copiable-link" href="#index-freqz-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-5"><span><code class="def-type">[<var class="var">h</var>, <var class="var">w</var>] =</code> <strong class="def-name">freqz</strong> <code class="def-code-arguments">(…, <var class="var">Fs</var>)</code><a class="copiable-link" href="#index-freqz-5"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz-6"><span><strong class="def-name">freqz</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-freqz-6"> ¶</a></span></dt>
<dd>
<p>Return the complex frequency response <var class="var">h</var> of the rational IIR filter
whose numerator and denominator coefficients are <var class="var">b</var> and <var class="var">a</var>,
respectively.
</p>
<p>The response is evaluated at <var class="var">n</var> angular frequencies between 0 and
2*pi.
</p>
<p>The output value <var class="var">w</var> is a vector of the frequencies.
</p>
<p>If <var class="var">a</var> is omitted, the denominator is assumed to be 1 (this
corresponds to a simple FIR filter).
</p>
<p>If <var class="var">n</var> is omitted, a value of 512 is assumed. For fastest computation,
<var class="var">n</var> should factor into a small number of small primes.
</p>
<p>If the fourth argument, <code class="code">"whole"</code>, is omitted the response is
evaluated at frequencies between 0 and
pi.
</p>
<p><code class="code">freqz (<var class="var">b</var>, <var class="var">a</var>, <var class="var">w</var>)</code>
</p>
<p>Evaluate the response at the specific frequencies in the vector <var class="var">w</var>.
The values for <var class="var">w</var> are measured in radians.
</p>
<p><code class="code">[…] = freqz (…, <var class="var">Fs</var>)</code>
</p>
<p>Return frequencies in Hz instead of radians assuming a sampling rate
<var class="var">Fs</var>. If you are evaluating the response at specific frequencies
<var class="var">w</var>, those frequencies should be requested in Hz rather than radians.
</p>
<p><code class="code">freqz (…)</code>
</p>
<p>Plot the magnitude and phase response of <var class="var">h</var> rather than returning them.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfreqz_005fplot">freqz_plot</a>.
</p></dd></dl>
<a class="anchor" id="XREFfreqz_005fplot"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-freqz_005fplot"><span><strong class="def-name">freqz_plot</strong> <code class="def-code-arguments">(<var class="var">w</var>, <var class="var">h</var>)</code><a class="copiable-link" href="#index-freqz_005fplot"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-freqz_005fplot-1"><span><strong class="def-name">freqz_plot</strong> <code class="def-code-arguments">(<var class="var">w</var>, <var class="var">h</var>, <var class="var">freq_norm</var>)</code><a class="copiable-link" href="#index-freqz_005fplot-1"> ¶</a></span></dt>
<dd><p>Plot the magnitude and phase response of <var class="var">h</var>.
</p>
<p>If the optional <var class="var">freq_norm</var> argument is true, the frequency vector
<var class="var">w</var> is in units of normalized radians. If <var class="var">freq_norm</var> is false, or
not given, then <var class="var">w</var> is measured in Hertz.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfreqz">freqz</a>.
</p></dd></dl>
<a class="anchor" id="XREFsinc"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sinc"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">sinc</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-sinc"> ¶</a></span></dt>
<dd><p>Compute the sinc function.
</p>
<p>Return
sin (pi*x) / (pi*x).
</p></dd></dl>
<a class="anchor" id="XREFunwrap"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-unwrap"><span><code class="def-type"><var class="var">b</var> =</code> <strong class="def-name">unwrap</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-unwrap"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-unwrap-1"><span><code class="def-type"><var class="var">b</var> =</code> <strong class="def-name">unwrap</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">tol</var>)</code><a class="copiable-link" href="#index-unwrap-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-unwrap-2"><span><code class="def-type"><var class="var">b</var> =</code> <strong class="def-name">unwrap</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">tol</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-unwrap-2"> ¶</a></span></dt>
<dd><p>Unwrap radian phases by adding or subtracting multiples of 2*pi as
appropriate to remove jumps greater than <var class="var">tol</var>.
</p>
<p><var class="var">tol</var> defaults to pi.
</p>
<p><code class="code">unwrap</code> will work along the dimension <var class="var">dim</var>. If <var class="var">dim</var>
is unspecified it defaults to the first non-singleton dimension.
</p>
<p><code class="code">unwrap</code> ignores all non-finite input values (Inf, NaN, NA).
</p>
</dd></dl>
<a class="anchor" id="XREFarch_005ffit"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-arch_005ffit"><span><code class="def-type">[<var class="var">a</var>, <var class="var">b</var>] =</code> <strong class="def-name">arch_fit</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">x</var>, <var class="var">p</var>, <var class="var">iter</var>, <var class="var">gamma</var>, <var class="var">a0</var>, <var class="var">b0</var>)</code><a class="copiable-link" href="#index-arch_005ffit"> ¶</a></span></dt>
<dd><p>Fit an ARCH regression model to the time series <var class="var">y</var> using the scoring
algorithm in Engle’s original ARCH paper.
</p>
<p>The model is
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2
</pre></div></div>
<p>in which <em class="math">e(t)</em> is <em class="math">N(0, h(t))</em>, given a time-series vector
<var class="var">y</var> up to time <em class="math">t-1</em> and a matrix of (ordinary) regressors <var class="var">x</var>
up to <em class="math">t</em>. The order of the regression of the residual variance is
specified by <var class="var">p</var>.
</p>
<p>If invoked as <code class="code">arch_fit (<var class="var">y</var>, <var class="var">k</var>, <var class="var">p</var>)</code> with a positive
integer <var class="var">k</var>, fit an ARCH(<var class="var">k</var>, <var class="var">p</var>) process, i.e., do the above
with the <em class="math">t</em>-th row of <var class="var">x</var> given by
</p>
<div class="example">
<pre class="example-preformatted">[1, y(t-1), ..., y(t-k)]
</pre></div>
<p>Optionally, one can specify the number of iterations <var class="var">iter</var>, the
updating factor <var class="var">gamma</var>, and initial values <em class="math">a0</em> and <em class="math">b0</em>
for the scoring algorithm.
</p></dd></dl>
<a class="anchor" id="XREFarch_005frnd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-arch_005frnd"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">arch_rnd</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">b</var>, <var class="var">t</var>)</code><a class="copiable-link" href="#index-arch_005frnd"> ¶</a></span></dt>
<dd><p>Simulate an ARCH sequence of length <var class="var">t</var> with AR coefficients <var class="var">b</var> and
CH coefficients <var class="var">a</var>.
</p>
<p>The result <em class="math">y(t)</em> follows the model
</p>
<div class="example smallexample">
<pre class="example-preformatted">y(t) = b(1) + b(2) * y(t-1) + ... + b(lb) * y(t-lb+1) + e(t),
</pre></div>
<p>where <em class="math">e(t)</em>, given <var class="var">y</var> up to time <em class="math">t-1</em>, is
<em class="math">N(0, h(t))</em>, with
</p>
<div class="example smallexample">
<pre class="example-preformatted">h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(la) * e(t-la+1)^2
</pre></div>
</dd></dl>
<a class="anchor" id="XREFarch_005ftest"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-arch_005ftest"><span><code class="def-type">[<var class="var">pval</var>, <var class="var">lm</var>] =</code> <strong class="def-name">arch_test</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">x</var>, <var class="var">p</var>)</code><a class="copiable-link" href="#index-arch_005ftest"> ¶</a></span></dt>
<dd><p>For a linear regression model
</p>
<div class="example">
<pre class="example-preformatted">y = x * b + e
</pre></div>
<p>perform a Lagrange Multiplier (LM) test of the null hypothesis of no
conditional heteroscedascity against the alternative of CH(<var class="var">p</var>).
</p>
<p>I.e., the model is
</p>
<div class="example">
<pre class="example-preformatted">y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
</pre></div>
<p>given <var class="var">y</var> up to <em class="math">t-1</em> and <var class="var">x</var> up to <em class="math">t</em>,
<em class="math">e</em>(t) is <em class="math">N(0, h(t))</em> with
</p>
<div class="example">
<pre class="example-preformatted">h(t) = v + a(1) * e(t-1)^2 + ... + a(p) * e(t-p)^2,
</pre></div>
<p>and the null is <em class="math">a(1)</em> == … == <em class="math">a(p)</em> == 0.
</p>
<p>If the second argument is a scalar integer, <em class="math">k</em>, perform the same
test in a linear autoregression model of order <em class="math">k</em>, i.e., with
</p>
<div class="example">
<pre class="example-preformatted">[1, y(t-1), ..., y(t-<var class="var">k</var>)]
</pre></div>
<p>as the <em class="math">t</em>-th row of <var class="var">x</var>.
</p>
<p>Under the null, LM approximately has a chisquare distribution with
<var class="var">p</var> degrees of freedom and <var class="var">pval</var> is the <em class="math">p</em>-value (1
minus the CDF of this distribution at LM) of the test.
</p>
<p>If no output argument is given, the <em class="math">p</em>-value is displayed.
</p></dd></dl>
<a class="anchor" id="XREFarma_005frnd"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-arma_005frnd"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">arma_rnd</strong> <code class="def-code-arguments">(<var class="var">a</var>, <var class="var">b</var>, <var class="var">v</var>, <var class="var">t</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-arma_005frnd"> ¶</a></span></dt>
<dd><p>Return a simulation of the ARMA model.
</p>
<p>The ARMA model is defined by
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">x(n) = a(1) * x(n-1) + ... + a(k) * x(n-k)
+ e(n) + b(1) * e(n-1) + ... + b(l) * e(n-l)
</pre></div></div>
<p>in which <var class="var">k</var> is the length of vector <var class="var">a</var>, <var class="var">l</var> is the length of
vector <var class="var">b</var> and <var class="var">e</var> is Gaussian white noise with variance <var class="var">v</var>.
The function returns a vector of length <var class="var">t</var>.
</p>
<p>The optional parameter <var class="var">n</var> gives the number of dummy <var class="var">x</var>(<var class="var">i</var>)
used for initialization, i.e., a sequence of length <var class="var">t</var>+<var class="var">n</var> is
generated and <var class="var">x</var>(<var class="var">n</var>+1:<var class="var">t</var>+<var class="var">n</var>) is returned. If <var class="var">n</var>
is omitted, <var class="var">n</var> = 100 is used.
</p></dd></dl>
<a class="anchor" id="XREFautoreg_005fmatrix"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-autoreg_005fmatrix"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">autoreg_matrix</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">k</var>)</code><a class="copiable-link" href="#index-autoreg_005fmatrix"> ¶</a></span></dt>
<dd><p>Given a time series (vector) <var class="var">y</var>, return a matrix with ones in the first
column and the first <var class="var">k</var> lagged values of <var class="var">y</var> in the other columns.
</p>
<p>In other words, for <var class="var">t</var> > <var class="var">k</var>,
<code class="code">[1, <var class="var">y</var>(<var class="var">t</var>-1), …, <var class="var">y</var>(<var class="var">t</var>-<var class="var">k</var>)]</code> is the
t-th row of the result.
</p>
<p>The resulting matrix may be used as a regressor matrix in autoregressions.
</p></dd></dl>
<a class="anchor" id="XREFbartlett"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-bartlett"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">bartlett</strong> <code class="def-code-arguments">(<var class="var">m</var>)</code><a class="copiable-link" href="#index-bartlett"> ¶</a></span></dt>
<dd><p>Return the filter coefficients of a Bartlett (triangular) window of length
<var class="var">m</var>.
</p>
<p>For a definition of the Bartlett window see, e.g.,
A.V. Oppenheim & R. W. Schafer,
<cite class="cite">Discrete-Time Signal Processing</cite>.
</p></dd></dl>
<a class="anchor" id="XREFblackman"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-blackman"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">blackman</strong> <code class="def-code-arguments">(<var class="var">m</var>)</code><a class="copiable-link" href="#index-blackman"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-blackman-1"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">blackman</strong> <code class="def-code-arguments">(<var class="var">m</var>, "periodic")</code><a class="copiable-link" href="#index-blackman-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-blackman-2"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">blackman</strong> <code class="def-code-arguments">(<var class="var">m</var>, "symmetric")</code><a class="copiable-link" href="#index-blackman-2"> ¶</a></span></dt>
<dd><p>Return the filter coefficients of a Blackman window of length
<var class="var">m</var>.
</p>
<p>If the optional argument <code class="code">"periodic"</code> is given, the periodic form
of the window is returned. This is equivalent to the window of length
<var class="var">m</var>+1 with the last coefficient removed. The optional argument
<code class="code">"symmetric"</code> is equivalent to not specifying a second argument.
</p>
<p>For a definition of the Blackman window, see, e.g.,
A.V. Oppenheim & R. W. Schafer,
<cite class="cite">Discrete-Time Signal Processing</cite>.
</p></dd></dl>
<a class="anchor" id="XREFdetrend"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-detrend"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">detrend</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">p</var>)</code><a class="copiable-link" href="#index-detrend"> ¶</a></span></dt>
<dd><p>If <var class="var">x</var> is a vector, <code class="code">detrend (<var class="var">x</var>, <var class="var">p</var>)</code> removes the
best fit of a polynomial of order <var class="var">p</var> from the data <var class="var">x</var>.
</p>
<p>If <var class="var">x</var> is a matrix, <code class="code">detrend (<var class="var">x</var>, <var class="var">p</var>)</code> does the same
for each column in <var class="var">x</var>.
</p>
<p>The second argument <var class="var">p</var> is optional. If it is not specified, a value of
1 is assumed. This corresponds to removing a linear trend.
</p>
<p>The order of the polynomial can also be given as a string, in which case
<var class="var">p</var> must be either <code class="code">"constant"</code> (corresponds to <code class="code"><var class="var">p</var>=0</code>)
or <code class="code">"linear"</code> (corresponds to <code class="code"><var class="var">p</var>=1</code>).
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Polynomial-Interpolation.html#XREFpolyfit">polyfit</a>.
</p></dd></dl>
<a class="anchor" id="XREFdiffpara"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-diffpara"><span><code class="def-type">[<var class="var">d</var>, <var class="var">dd</var>] =</code> <strong class="def-name">diffpara</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">a</var>, <var class="var">b</var>)</code><a class="copiable-link" href="#index-diffpara"> ¶</a></span></dt>
<dd><p>Return the estimator <var class="var">d</var> for the differencing parameter of an
integrated time series.
</p>
<p>The frequencies from <em class="math">[2*pi*a/t, 2*pi*b/T]</em> are used for the
estimation. If <var class="var">b</var> is omitted, the interval
<em class="math">[2*pi/T, 2*pi*a/T]</em> is used. If both <var class="var">b</var> and <var class="var">a</var> are omitted
then <em class="math">a = 0.5 * sqrt (T)</em> and <em class="math">b = 1.5 * sqrt (T)</em> is used, where
<em class="math">T</em> is the sample size. If <var class="var">x</var> is a matrix, the differencing
parameter of each column is estimated.
</p>
<p>The estimators for all frequencies in the intervals described above is
returned in <var class="var">dd</var>.
</p>
<p>The value of <var class="var">d</var> is simply the mean of <var class="var">dd</var>.
</p>
<p>Reference: P.J. Brockwell & R.A. Davis. <cite class="cite">Time Series:
Theory and Methods</cite>. Springer 1987.
</p></dd></dl>
<a class="anchor" id="XREFdurbinlevinson"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-durbinlevinson"><span><code class="def-type">[<var class="var">newphi</var>, <var class="var">newv</var>] =</code> <strong class="def-name">durbinlevinson</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">oldphi</var>, <var class="var">oldv</var>)</code><a class="copiable-link" href="#index-durbinlevinson"> ¶</a></span></dt>
<dd><p>Perform one step of the Durbin-Levinson algorithm.
</p>
<p>The vector <var class="var">c</var> specifies the autocovariances
<code class="code">[gamma_0, …, gamma_t]</code> from lag 0 to <var class="var">t</var>, <var class="var">oldphi</var>
specifies the coefficients based on <var class="var">c</var>(<var class="var">t</var>-1) and <var class="var">oldv</var>
specifies the corresponding error.
</p>
<p>If <var class="var">oldphi</var> and <var class="var">oldv</var> are omitted, all steps from 1 to <var class="var">t</var> of
the algorithm are performed.
</p></dd></dl>
<a class="anchor" id="XREFfftshift"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fftshift"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fftshift</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-fftshift"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-fftshift-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">fftshift</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-fftshift-1"> ¶</a></span></dt>
<dd><p>Perform a shift of the vector <var class="var">x</var>, for use with the <code class="code">fft</code> and
<code class="code">ifft</code> functions, in order to move the frequency 0 to the center of
the vector or matrix.
</p>
<p>If <var class="var">x</var> is a vector of <em class="math">N</em> elements corresponding to <em class="math">N</em> time
samples spaced by <em class="math">dt</em>, then
<code class="code">fftshift (fft (<var class="var">x</var>))</code> corresponds to frequencies
</p>
<div class="example">
<pre class="example-preformatted">f = [ -(ceil((N-1)/2):-1:1), 0, (1:floor((N-1)/2)) ] * df
</pre></div>
<p>where <em class="math">df = 1 / (N * dt)</em>.
</p>
<p>If <var class="var">x</var> is a matrix, the same holds for rows and columns. If <var class="var">x</var>
is an array, then the same holds along each dimension.
</p>
<p>The optional <var class="var">dim</var> argument can be used to limit the dimension along
which the permutation occurs.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFifftshift">ifftshift</a>.
</p></dd></dl>
<a class="anchor" id="XREFifftshift"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-ifftshift"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">ifftshift</strong> <code class="def-code-arguments">(<var class="var">y</var>)</code><a class="copiable-link" href="#index-ifftshift"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-ifftshift-1"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">ifftshift</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">dim</var>)</code><a class="copiable-link" href="#index-ifftshift-1"> ¶</a></span></dt>
<dd><p>Undo the action of the <code class="code">fftshift</code> function.
</p>
<p>For even length <var class="var">x</var>, <code class="code">fftshift</code> is its own inverse, but odd lengths
differ slightly.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfftshift">fftshift</a>.
</p></dd></dl>
<a class="anchor" id="XREFfractdiff"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-fractdiff"><span><code class="def-type"><var class="var">fd</var> =</code> <strong class="def-name">fractdiff</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">d</var>)</code><a class="copiable-link" href="#index-fractdiff"> ¶</a></span></dt>
<dd><p>Compute the fractional differences <em class="math">(1-L)^d x</em> where <em class="math">L</em>
denotes the lag-operator and <em class="math">d</em> is greater than -1.
</p></dd></dl>
<a class="anchor" id="XREFhamming"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-hamming"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hamming</strong> <code class="def-code-arguments">(<var class="var">m</var>)</code><a class="copiable-link" href="#index-hamming"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-hamming-1"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hamming</strong> <code class="def-code-arguments">(<var class="var">m</var>, "periodic")</code><a class="copiable-link" href="#index-hamming-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-hamming-2"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hamming</strong> <code class="def-code-arguments">(<var class="var">m</var>, "symmetric")</code><a class="copiable-link" href="#index-hamming-2"> ¶</a></span></dt>
<dd><p>Return the filter coefficients of a Hamming window of length <var class="var">m</var>.
</p>
<p>If the optional argument <code class="code">"periodic"</code> is given, the periodic form
of the window is returned. This is equivalent to the window of length
<var class="var">m</var>+1 with the last coefficient removed. The optional argument
<code class="code">"symmetric"</code> is equivalent to not specifying a second argument.
</p>
<p>For a definition of the Hamming window see, e.g.,
A.V. Oppenheim & R. W. Schafer,
<cite class="cite">Discrete-Time Signal Processing</cite>.
</p></dd></dl>
<a class="anchor" id="XREFhanning"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-hanning"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hanning</strong> <code class="def-code-arguments">(<var class="var">m</var>)</code><a class="copiable-link" href="#index-hanning"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-hanning-1"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hanning</strong> <code class="def-code-arguments">(<var class="var">m</var>, "periodic")</code><a class="copiable-link" href="#index-hanning-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-hanning-2"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">hanning</strong> <code class="def-code-arguments">(<var class="var">m</var>, "symmetric")</code><a class="copiable-link" href="#index-hanning-2"> ¶</a></span></dt>
<dd><p>Return the filter coefficients of a Hanning window of length <var class="var">m</var>.
</p>
<p>If the optional argument <code class="code">"periodic"</code> is given, the periodic form
of the window is returned. This is equivalent to the window of length
<var class="var">m</var>+1 with the last coefficient removed. The optional argument
<code class="code">"symmetric"</code> is equivalent to not specifying a second argument.
</p>
<p>For a definition of the Hanning window see, e.g.,
A.V. Oppenheim & R. W. Schafer,
<cite class="cite">Discrete-Time Signal Processing</cite>.
</p></dd></dl>
<a class="anchor" id="XREFhurst"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-hurst"><span><code class="def-type"><var class="var">H</var> =</code> <strong class="def-name">hurst</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-hurst"> ¶</a></span></dt>
<dd><p>Estimate the Hurst parameter of sample <var class="var">x</var> via the rescaled range
statistic.
</p>
<p>If <var class="var">x</var> is a matrix, the parameter is estimated for every column.
</p></dd></dl>
<a class="anchor" id="XREFpchip"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-pchip"><span><code class="def-type"><var class="var">pp</var> =</code> <strong class="def-name">pchip</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-pchip"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pchip-1"><span><code class="def-type"><var class="var">yi</var> =</code> <strong class="def-name">pchip</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>, <var class="var">xi</var>)</code><a class="copiable-link" href="#index-pchip-1"> ¶</a></span></dt>
<dd><p>Return the Piecewise Cubic Hermite Interpolating Polynomial (pchip) of
points <var class="var">x</var> and <var class="var">y</var>.
</p>
<p>If called with two arguments, return the piecewise polynomial <var class="var">pp</var>
that may be used with <code class="code">ppval</code> to evaluate the polynomial at specific
points.
</p>
<p>When called with a third input argument, <code class="code">pchip</code> evaluates the pchip
polynomial at the points <var class="var">xi</var>. The third calling form is equivalent to
<code class="code">ppval (pchip (<var class="var">x</var>, <var class="var">y</var>), <var class="var">xi</var>)</code>.
</p>
<p>The variable <var class="var">x</var> must be a strictly monotonic vector (either increasing
or decreasing) of length <var class="var">n</var>.
</p>
<p><var class="var">y</var> can be either a vector or array. If <var class="var">y</var> is a vector then it
must be the same length <var class="var">n</var> as <var class="var">x</var>. If <var class="var">y</var> is an array then
the size of <var class="var">y</var> must have the form
<code class="code">[<var class="var">s1</var>, <var class="var">s2</var>, …, <var class="var">sk</var>, <var class="var">n</var>]</code>
The array is reshaped internally to a matrix where the leading dimension is
given by
<code class="code"><var class="var">s1</var> * <var class="var">s2</var> * … * <var class="var">sk</var></code>
and each row of this matrix is then treated separately. Note that this is
exactly opposite to <code class="code">interp1</code> but is done for <small class="sc">MATLAB</small>
compatibility.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="One_002ddimensional-Interpolation.html#XREFspline">spline</a>, <a class="ref" href="Polynomial-Interpolation.html#XREFppval">ppval</a>, <a class="ref" href="Polynomial-Interpolation.html#XREFmkpp">mkpp</a>, <a class="ref" href="Polynomial-Interpolation.html#XREFunmkpp">unmkpp</a>.
</p></dd></dl>
<a class="anchor" id="XREFperiodogram"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-periodogram"><span><code class="def-type">[<var class="var">Pxx</var>, <var class="var">w</var>] =</code> <strong class="def-name">periodogram</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-periodogram"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-periodogram-1"><span><code class="def-type">[<var class="var">Pxx</var>, <var class="var">w</var>] =</code> <strong class="def-name">periodogram</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win</var>)</code><a class="copiable-link" href="#index-periodogram-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-periodogram-2"><span><code class="def-type">[<var class="var">Pxx</var>, <var class="var">w</var>] =</code> <strong class="def-name">periodogram</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win</var>, <var class="var">nfft</var>)</code><a class="copiable-link" href="#index-periodogram-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-periodogram-3"><span><code class="def-type">[<var class="var">Pxx</var>, <var class="var">f</var>] =</code> <strong class="def-name">periodogram</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win</var>, <var class="var">nfft</var>, <var class="var">Fs</var>)</code><a class="copiable-link" href="#index-periodogram-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-periodogram-4"><span><code class="def-type">[<var class="var">Pxx</var>, <var class="var">f</var>] =</code> <strong class="def-name">periodogram</strong> <code class="def-code-arguments">(…, "<var class="var">range</var>")</code><a class="copiable-link" href="#index-periodogram-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-periodogram-5"><span><strong class="def-name">periodogram</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-periodogram-5"> ¶</a></span></dt>
<dd><p>Return the periodogram (Power Spectral Density) of <var class="var">x</var>.
</p>
<p>The possible inputs are:
</p>
<dl class="table">
<dt><var class="var">x</var></dt>
<dd>
<p>data vector. If <var class="var">x</var> is real-valued a one-sided spectrum is estimated.
If <var class="var">x</var> is complex-valued, or <code class="code">"<var class="var">range</var>"</code> specifies
<code class="code">"twosided"</code>, the full spectrum is estimated.
</p>
</dd>
<dt><var class="var">win</var></dt>
<dd><p>window weight data. If window is empty or unspecified a default rectangular
window is used. Otherwise, the window is applied to the signal
(<code class="code"><var class="var">x</var> .* <var class="var">win</var></code>) before computing the periodogram. The window
data must be a vector of the same length as <var class="var">x</var>.
</p>
</dd>
<dt><var class="var">nfft</var></dt>
<dd><p>number of frequency bins. The default is 256 or the next higher power of
2 greater than the length of <var class="var">x</var>
(<code class="code">max (256, 2.^nextpow2 (length (x)))</code>). If <var class="var">nfft</var> is greater
than the length of the input then <var class="var">x</var> will be zero-padded to the length
of <var class="var">nfft</var>.
</p>
</dd>
<dt><var class="var">Fs</var></dt>
<dd><p>sampling rate. The default is 1.
</p>
</dd>
<dt><var class="var">range</var></dt>
<dd><p>range of spectrum. <code class="code">"onesided"</code> computes spectrum from
[0:nfft/2+1]. <code class="code">"twosided"</code> computes spectrum from
[0:nfft-1].
</p></dd>
</dl>
<p>The optional second output <var class="var">w</var> are the normalized angular frequencies.
For a one-sided calculation <var class="var">w</var> is in the range [0, pi] if <var class="var">nfft</var>
is even and [0, pi) if <var class="var">nfft</var> is odd. Similarly, for a two-sided
calculation <var class="var">w</var> is in the range [0, 2*pi] or [0, 2*pi) depending on
<var class="var">nfft</var>.
</p>
<p>If a sampling frequency is specified, <var class="var">Fs</var>, then the output frequencies
<var class="var">f</var> will be in the range [0, <var class="var">Fs</var>/2] or [0, <var class="var">Fs</var>/2) for
one-sided calculations. For two-sided calculations the range will be
[0, <var class="var">Fs</var>).
</p>
<p>When called with no outputs the periodogram is immediately plotted in the
current figure window.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFfft">fft</a>.
</p></dd></dl>
<a class="anchor" id="XREFsinetone"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sinetone"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">sinetone</strong> <code class="def-code-arguments">(<var class="var">freq</var>, <var class="var">rate</var>, <var class="var">sec</var>, <var class="var">ampl</var>)</code><a class="copiable-link" href="#index-sinetone"> ¶</a></span></dt>
<dd><p>Return a sinetone of frequency <var class="var">freq</var> with a length of <var class="var">sec</var>
seconds at sampling rate <var class="var">rate</var> and with amplitude <var class="var">ampl</var>.
</p>
<p>The arguments <var class="var">freq</var> and <var class="var">ampl</var> may be vectors of common size.
</p>
<p>The defaults are <var class="var">rate</var> = 8000, <var class="var">sec</var> = 1, and <var class="var">ampl</var> = 64.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsinewave">sinewave</a>.
</p></dd></dl>
<a class="anchor" id="XREFsinewave"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sinewave"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">sinewave</strong> <code class="def-code-arguments">(<var class="var">m</var>, <var class="var">n</var>, <var class="var">d</var>)</code><a class="copiable-link" href="#index-sinewave"> ¶</a></span></dt>
<dd><p>Return an <var class="var">m</var>-element vector with <var class="var">i</var>-th element given by
<code class="code">sin (2 * pi * (<var class="var">i</var>+<var class="var">d</var>-1) / <var class="var">n</var>)</code>.
</p>
<p>The default value for <var class="var">d</var> is 0 and the default value for <var class="var">n</var> is
<var class="var">m</var>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsinetone">sinetone</a>.
</p></dd></dl>
<a class="anchor" id="XREFspectral_005fadf"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-spectral_005fadf"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_adf</strong> <code class="def-code-arguments">(<var class="var">c</var>)</code><a class="copiable-link" href="#index-spectral_005fadf"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-spectral_005fadf-1"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_adf</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">win</var>)</code><a class="copiable-link" href="#index-spectral_005fadf-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-spectral_005fadf-2"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_adf</strong> <code class="def-code-arguments">(<var class="var">c</var>, <var class="var">win</var>, <var class="var">b</var>)</code><a class="copiable-link" href="#index-spectral_005fadf-2"> ¶</a></span></dt>
<dd><p>Return the spectral density estimator given a vector of autocovariances
<var class="var">c</var>, window name <var class="var">win</var>, and bandwidth, <var class="var">b</var>.
</p>
<p>The window name, e.g., <code class="code">"triangle"</code> or <code class="code">"rectangle"</code> is
used to search for a function called <code class="code"><var class="var">win</var>_lw</code>.
</p>
<p>If <var class="var">win</var> is omitted, the triangle window is used.
</p>
<p>If <var class="var">b</var> is omitted, <code class="code">1 / sqrt (length (<var class="var">x</var>))</code> is used.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFspectral_005fxdf">spectral_xdf</a>.
</p></dd></dl>
<a class="anchor" id="XREFspectral_005fxdf"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-spectral_005fxdf"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_xdf</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-spectral_005fxdf"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-spectral_005fxdf-1"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_xdf</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win</var>)</code><a class="copiable-link" href="#index-spectral_005fxdf-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-spectral_005fxdf-2"><span><code class="def-type"><var class="var">sde</var> =</code> <strong class="def-name">spectral_xdf</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win</var>, <var class="var">b</var>)</code><a class="copiable-link" href="#index-spectral_005fxdf-2"> ¶</a></span></dt>
<dd><p>Return the spectral density estimator given a data vector <var class="var">x</var>, window
name <var class="var">win</var>, and bandwidth, <var class="var">b</var>.
</p>
<p>The window name, e.g., <code class="code">"triangle"</code> or <code class="code">"rectangle"</code> is used to
search for a function called <code class="code"><var class="var">win</var>_sw</code>.
</p>
<p>If <var class="var">win</var> is omitted, the triangle window is used.
</p>
<p>If <var class="var">b</var> is omitted, <code class="code">1 / sqrt (length (<var class="var">x</var>))</code> is used.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFspectral_005fadf">spectral_adf</a>.
</p></dd></dl>
<a class="anchor" id="XREFspencer"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-spencer"><span><code class="def-type"><var class="var">savg</var> =</code> <strong class="def-name">spencer</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-spencer"> ¶</a></span></dt>
<dd><p>Return Spencer’s 15-point moving average of each column of
<var class="var">x</var>.
</p></dd></dl>
<a class="anchor" id="XREFstft"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-stft"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-stft"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-stft-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win_size</var>)</code><a class="copiable-link" href="#index-stft-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-stft-2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win_size</var>, <var class="var">inc</var>)</code><a class="copiable-link" href="#index-stft-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-stft-3"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win_size</var>, <var class="var">inc</var>, <var class="var">num_coef</var>)</code><a class="copiable-link" href="#index-stft-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-stft-4"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">win_size</var>, <var class="var">inc</var>, <var class="var">num_coef</var>, <var class="var">win_type</var>)</code><a class="copiable-link" href="#index-stft-4"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-stft-5"><span><code class="def-type">[<var class="var">y</var>, <var class="var">c</var>] =</code> <strong class="def-name">stft</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-stft-5"> ¶</a></span></dt>
<dd><p>Compute the short-time Fourier transform of the vector <var class="var">x</var> with
<var class="var">num_coef</var> coefficients by applying a window of <var class="var">win_size</var> data
points and an increment of <var class="var">inc</var> points.
</p>
<p>Before computing the Fourier transform, one of the following windows
is applied:
</p>
<dl class="table">
<dt><code class="code">"hanning"</code></dt>
<dd><p>win_type = 1
</p>
</dd>
<dt><code class="code">"hamming"</code></dt>
<dd><p>win_type = 2
</p>
</dd>
<dt><code class="code">"rectangle"</code></dt>
<dd><p>win_type = 3
</p></dd>
</dl>
<p>The window names can be passed as strings or by the <var class="var">win_type</var> number.
</p>
<p>The following defaults are used for unspecified arguments:
<var class="var">win_size</var> = 80, <var class="var">inc</var> = 24, <var class="var">num_coef</var> = 64, and
<var class="var">win_type</var> = 1.
</p>
<p><code class="code"><var class="var">y</var> = stft (<var class="var">x</var>, …)</code> returns the absolute values of the
Fourier coefficients according to the <var class="var">num_coef</var> positive frequencies.
</p>
<p><code class="code">[<var class="var">y</var>, <var class="var">c</var>] = stft (<var class="var">x</var>, …)</code> returns the entire
STFT-matrix <var class="var">y</var> and a 3-element vector <var class="var">c</var> containing the window
size, increment, and window type, which is needed by the <code class="code">synthesis</code>
function.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsynthesis">synthesis</a>.
</p></dd></dl>
<a class="anchor" id="XREFsynthesis"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-synthesis"><span><code class="def-type"><var class="var">x</var> =</code> <strong class="def-name">synthesis</strong> <code class="def-code-arguments">(<var class="var">y</var>, <var class="var">c</var>)</code><a class="copiable-link" href="#index-synthesis"> ¶</a></span></dt>
<dd><p>Compute a signal from its short-time Fourier transform <var class="var">y</var> and a
3-element vector <var class="var">c</var> specifying window size, increment, and window type.
</p>
<p>The values <var class="var">y</var> and <var class="var">c</var> can be derived by
</p>
<div class="example">
<pre class="example-preformatted">[<var class="var">y</var>, <var class="var">c</var>] = stft (<var class="var">x</var> , ...)
</pre></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFstft">stft</a>.
</p></dd></dl>
<a class="anchor" id="XREFyulewalker"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-yulewalker"><span><code class="def-type">[<var class="var">a</var>, <var class="var">v</var>] =</code> <strong class="def-name">yulewalker</strong> <code class="def-code-arguments">(<var class="var">c</var>)</code><a class="copiable-link" href="#index-yulewalker"> ¶</a></span></dt>
<dd><p>Fit an AR (p)-model with Yule-Walker estimates given a vector <var class="var">c</var> of
autocovariances <code class="code">[gamma_0, …, gamma_p]</code>.
</p>
<p>Returns the AR coefficients, <var class="var">a</var>, and the variance of white noise,
<var class="var">v</var>.
</p></dd></dl>
</div>
<hr>
<div class="nav-panel">
<p>
Next: <a href="Image-Processing.html">Image Processing</a>, Previous: <a href="Geometry.html">Geometry</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
