1、 Reference number ISO 18431-2:2004(E) ISO 2004INTERNATIONAL STANDARD ISO 18431-2 First edition 2004-10-15 Mechanical vibration and shock Signal processing Part 2: Time domain windows for Fourier Transform analysis Vibrations et chocs mcaniques Traitement du signal Partie 2: Fentres des domaines temp
2、orels pour analyse par transformation de Fourier ISO 18431-2:2004(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobes licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and install
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6、g from either ISO at the address below or ISOs member body in the country of the requester. ISO copyright office Case postale 56 CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii ISO 2004 All rights reservedISO 18431-2:
7、2004(E) ISO 2004 All rights reserved iiiContents Page Foreword iv Introduction v 1 Scope 1 2 Normative references. 1 3 Terms and definitions. 1 4 Symbols. 2 5 Common time domain windows 2 5.1 General. 2 5.2 Hanning window 3 5.3 Flat-top window. 3 5.4 Rectangular window. 5 6 Examples 6 6.1 Common win
8、dows applied to a truncated sinusoidal signal . 6 6.2 Common windows applied to a non-truncated sinusoidal signal 8 Bibliography . 10 ISO 18431-2:2004(E) iv ISO 2004 All rights reservedForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards b
9、odies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizati
10、ons, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the
11、 ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of t
12、he member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO 18431-2 was prepared by Technical Committee ISO/TC 108, Mechanic
13、al vibration and shock. ISO 18431 consists of the following parts, under the general title Mechanical vibration and shock Signal processing: Part 1: General introduction Part 2: Time domain windows for Fourier Transform analysis The following parts are under preparation: Part 3: Bilinear methods for
14、 joint time-frequency analysis Part 4: Shock response spectrum analysis Part 5: Methods for time-scale analysis ISO 18431-2:2004(E) ISO 2004 All rights reserved vIntroduction Vibration and shock data can consist of displacement, velocity or acceleration measurements, which can be either stationary o
15、r non-stationary with respect to time. For both classes of signals, spectral decomposition with Fourier Transformation is one of the analysis tools. In digital signal processing, there are N uniformly spaced (in time) samples of the observed signal. The application of the Discrete Fourier Transform
16、to these N samples produces a series of simple periodic functions of sines and cosines, whose amplitudes and harmonic balance are determined by the time domain window applied to the N samples. This part of ISO 18431 specifies the three most common windows used. INTERNATIONAL STANDARD ISO 18431-2:200
17、4(E) ISO 2004 All rights reserved 1Mechanical vibration and shock Signal processing Part 2: Time domain windows for Fourier Transform analysis 1 Scope This part of ISO 18431 specifies the algebraic functions which describe a selected set of time domain windows used for pre-processing digitally sampl
18、ed vibration and shock data as a precursor to Discrete Fourier Transform spectral analysis. This selected set consists of Hanning, flat-top and rectangular time windows. This part of ISO 18431 is one of a series of documents that details the tools available for time domain, frequency domain, and joi
19、nt time and frequency domain signal processing. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any
20、 amendments) applies. ISO 2041:1990, Vibration and shock Vocabulary 3 Terms and definitions For the purposes of this document, the terms and definitions given in ISO 2041 and the following apply. 3.1 Discrete Fourier Transform DFT discrete transform in time and frequency, based on the Fourier integr
21、al transform, used to obtain a spectral estimation of N uniformly time-spaced samples of a signal observed over a finite duration 1 2/ s 0 1 () () e N in mN n Xm xn f = = where the symbols are as defined in Clause 4 3.2 Fast Fourier Transform FFT algorithm for computing the Discrete Fourier Transfor
22、m (DFT) with optimized computational efficiency NOTE This algorithm is typically either the Cooley-Tukey (see Reference 1) or Sande-Tukey algorithm. ISO 18431-2:2004(E) 2 ISO 2004 All rights reserved3.3 time windows weighting function applied to an ensemble of sampled data to reduce the amount of en
23、ergy which flows into adjacent frequencies (spectral leakage) caused by sampling a signal that is not periodic within the finite time record of the observation interval, i.e. that has truncated sinusoidal components 4 Symbols a(i) constants for flat-top window B eequivalent noise bandwidth f ssampli
24、ng frequency i index for flat-top window constants m frequency sample n time sample N block size of sampled data; the number of sampled points that are transformed w(n) window function in the time domain W(m) window function in the frequency domain x(n) sampled physical quantity in the time domain X
25、(n) Digital Fourier Tranform of x(n t) 5 Common time domain windows 5.1 General There are three common time domain windows in use with Fourier analysis: Hanning, flat top and rectangular. NOTE The latter is not really an algebraically applied window, but is included in this document for completeness
26、. Table 1 Window properties Window type Highest sidelobe dB Sidelobe rolloff dB/decade Noise bandwidthNo. of lines aMaximum amplitude error dB Hanning 32 60 1,50 1,4 Flat-top 93 0 3,77 0,01 Rectangular 13 20 1,00 3,9 a Relative to line spacing. ISO 18431-2:2004(E) ISO 2004 All rights reserved 3The n
27、oise bandwidth and maximum amplitude error imply that, where knowledge of amplitude is paramount (e.g. during calibration), either the flat top or Hanning window is appropriate and, where frequency resolution is paramount (e.g. for identifying sidebands), either the rectangular or Hanning window is
28、appropriate. The equivalent noise bandwidth is () () 1 2 0s e 2 1 0 1 1 N n N n wn N f B N wn N = = = (1) NOTE More information on the use of time domain windows can be found in References 2, 3 and 4. 5.2 Hanning window For the purposes of this part of ISO 18431, the Hanning window is defined as ()
29、= 1 cos(2/N) (2) where = 0, 1, 1 N is the number of samples in the time record. Figure 1 shows an example of a 1 024-point Hanning window sampled (f s ) at 1 024 samples per second. 5.3 Flat-top window For the purposes of this part of ISO 18431, the flat top window is defined as w(n)= 1 + a(1)cos(2n
30、/N) + a(2)cos(4n/N) + a(3)cos(6n/N) + a(4)cos(8n/N) (3) where n = 0, 1, N 1 a(1)= 1,933 a(2)= +1,286 a(3)= 0,388 a(4)= +0,0322 N is the number of samples in the time record. Figure 2 shows an example of a 1 024 point flat-top window sampled (f s ) at 1 024 samples per second. ISO 18431-2:2004(E) 4 I
31、SO 2004 All rights reserveda) Key X sample Y amplitude, w(n) b) Key X frequency, in hertz Y amplitude, W(m) Figure 1 Hanning window for 1 024 samples ISO 18431-2:2004(E) ISO 2004 All rights reserved 5a) Key X sample Y amplitude, w(n) b) Key X frequency, in hertz Y amplitude, W(m) Figure 2 Flat-top w
32、indow for 1 024 samples 5.4 Rectangular window For the purposes of this part of ISO 18431, the rectangular window is defined as w(n) = 1 (4) where n = 0, 1, N 1 N is the number of samples in the time record. Figure 3 shows an example of a 1 024 point rectangular window sampled (f s ) at 1 024 sample
33、s per second. ISO 18431-2:2004(E) 6 ISO 2004 All rights reserveda) Key X sample Y amplitude, w(n) b) Key X frequency, in hertz Y amplitude, W(m) Figure 3 Rectangular window for 1 024 samples 6 Examples 6.1 Common windows applied to a truncated sinusoidal signal Figure 4 and Table 2 show an example o
34、f a sine wave of 4 1/2 cycles sampled at 1 024 samples per second (f s ); the results are independent of a phase shift. It shows the extent of the noise bandwidth and amplitude error. ISO 18431-2:2004(E) ISO 2004 All rights reserved 7a) Sine wave Key X sample Y signal, x(n) b) Hanning window c) Flat
35、-top window d) Rectangular window Key X frequency, in hertz Y X(m) Figure 4 Example of the common windows applied to a truncated sinusoid of 4 1/2 cycles ISO 18431-2:2004(E) 8 ISO 2004 All rights reservedTable 2 Common windows applied to a truncated sinusoidal signal Frequency Hanning Flat-top Recta
36、ngular 0 0,0073 0,0033 0,1415 1 0,0101 0,0694 0,1488 2 0,0254 0,3988 0,1763 3 0,1705 0,8507 0,2546 4 0,8483 0,9989 0,6741 5 0,8492 0,9990 0,6031 6 0,1695 0,8506 0,1819 7 0,0240 0,3989 0,0997 8 0,0079 0,0693 0,0655 9 0,0035 0,0017 0,0472 10 0,0019 0,0000 0,0359 6.2 Common windows applied to a non-tru
37、ncated sinusoidal signal Figure 5 and Table 3 show an example of a sine wave of 4 cycles sampled at 1 024 samples per second (f s ); the results are independent of a phase shift. It shows the extent of the noise bandwidth with zero amplitude error. Table 3 Common windows applied to a non-truncated s
38、inusoidal signal Frequency Hanning Flat-top Rectangular 0 0,0000 0,0000 0,0000 1 0,0000 0,1940 0,0000 2 0,0000 0,6430 0,0000 3 0,5000 0,9665 0,0000 4 1,0000 1,0000 1,0000 5 0,5000 0,9665 0,0000 6 0,0000 0,6430 0,0000 7 0,0000 0,1940 0,0000 8 0,0000 0,0160 0,0000 9 0,0000 0,0000 0,0000 10 0,0000 0,00
39、00 0,0000 ISO 18431-2:2004(E) ISO 2004 All rights reserved 9a) Sine wave Key X sample Y signal, x(n) b) Hanning window c) Flat-top window d) Rectangular window Key X frequency, in hertz Y X(m) Figure 5 Example of the common windows applied to a non-truncated sinusoid of 4 cycles ISO 18431-2:2004(E)
40、10 ISO 2004 All rights reservedBibliography 1 COOLEY, J.W. and TUKEY, J.W. An Algorithm for the Machine Computation of the Complex Fourier Series. Mathematics of Computation, 19, April 1965, pp. 297-301 2 HARRIS, F.J. On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. P
41、roceedings of the IEEE, 66, January 1978, pp. 51-83 3 RANDALL, R.B. Frequency Analysis. 3 rdEd., Bruel & Kjaer, Glostrup, Denmark, 1987 (ISBN 87 87355 07 8) 4 GADE, S. and HERLUFSEN, H. Use of Weighting Functions in DFT/FFT Analysis: Part I. Bruel and Kjaer, Technical Review, No. 3, 1987 (ISSN 007-2621) 5 GADE, S. and HERLUFSEN, H. Use of Weighting Functions in DFT/FFT Analysis: Part II. Bruel and Kjaer, Technical Review, No. 4, 1987 (ISSN 007-2621) ISO 18431-2:2004(E) ICS 17.160 Price based on 10 pages ISO 2004 All rights reserved
