1、BRITISH STANDARD BS ISO 18431-2:2004 Incorporating corrigendum June 2008 Mechanical vibration and shock Signal processing Part 2: Time domain windows for Fourier Transform analysis ICS 17.160 BS ISO 18431-2:2004 This British Standard was published under the authority of the Standards Policy and Stra
2、tegy Committee on 18 February 2005 BSI 2009 ISBN 978 0 580 64384 2 National foreword This British Standard is the UK implementation of ISO 18431-2:2004, incorporating corrigendum June 2008. The start and finish of text introduced or altered by corrigendum is indicated in the text by tags. Text alter
3、ed by ISO corrigendum June 2008 is indicated in the text by . The UK participation in its preparation was entrusted by Technical Committee GME/21, Mechanical vibration and shock, to Subcommittee GME/21/2, Vibration and shock measuring instruments and testing equipment. A list of organizations repres
4、ented on this subcommittee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. Compliance with a British Standard cannot confer immunity from legal obligations. Amendm
5、ents/corrigenda issued since publication Date Comments 28 February 2009 Implementation of ISO corrigendum June 2008 (corrected version of corrigendum of October 2004) Reference number ISO 18431-2:2004(E) OSI 4002INTERNATIONAL STANDARD ISO 18431-2 First edition 2004-10-15 Mechanical vibration and sho
6、ck Signal processing Part 2: Time domain windows for Fourier Transform analysis Vibrations et chocs mcaniques Traitement du signal Partie 2: Fentres des domaines temporels pour analyse par transformation de Fourier BSISO184312:2004 ii ii I SO 4002 All irthgs ersedevr iiiContents Page Foreword iv Int
7、roduction 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 windows applied to a truncated sinusoidal signal . 6 6.2 Common windows
8、applied to a non-truncated sinusoidal signal 8 Bibliography . 10 BS ISO 18431-2:2004iv I SO 4002 All irthgs ersedevrForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standa
9、rds 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 organizations, governmental and non-governmental, in liaison with ISO, also tak
10、e 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 ISO/IEC Directives, Part 2. The main task of technical committees is
11、 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 the member bodies casting a vote. Attention is drawn to the possibilit
12、y 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, Mechanical vibration and shock. ISO 18431 consists of the following parts, un
13、der 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 joint time-frequency analysis Part 4: Shock response spectrum analys
14、is Part 5: Methods for time-scale analysis BS ISO 18431-2:2004 I SO 4002 All irthgs ersedevr vIntroduction Vibration and shock data can consist of displacement, velocity or acceleration measurements, which can be either stationary or non-stationary with respect to time. For both classes of signals,
15、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 to these N samples produces a series of simple periodic functions of
16、 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. BS ISO 18431-2:2004 blankI SO 4002 All irthgs ersedevr 1Mechanical vibration and shock Signal processing P
17、art 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 sampled vibration and shock data as a precursor to Discrete Fourier Transform spectral ana
18、lysis. 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 joint time and frequency domain signal processing. 2 Normative references The following
19、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 amendments) applies. ISO 2041:1990, Vibration and shock Vocabulary 3 Terms and defin
20、itions 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 integral transform, used to obtain a spectral estimation of N uniformly time-spaced samples
21、 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 Transform (DFT) with optimized computational efficiency NOTE This algorithm is typically eith
22、er the Cooley-Tukey (see Reference 1) or Sande-Tukey algorithm. BS ISO 18431-2:20042 3.3 time windows weighting function applied to an ensemble of sampled data to reduce the amount of energy which flows into adjacent frequencies (spectral leakage) caused by sampling a signal that is not periodic wit
23、hin 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 ssampling frequency i index for flat-top window constants m frequency sample n time sample N block size of sampled data;
24、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(n) Digital Fourier Tranform of x(n t) 5 Common time domain windows 5.1 General There are three common time domain
25、 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. Table 1 Window properties Highest sidelobe Sidelobe rolloff Noise bandwidth Maximum amplitude error Window type
26、dB dB/decade No. of lines adB Hanning 31,5 60 1,50 1,4 Flat-top 93,0 0 3,77 0,01 Rectangular 13,3 20 1,00 3,9 aRelative to line spacing. BS ISO 18431-2:2004 3The noise bandwidth and maximum amplitude error imply that, where knowledge of amplitude is paramount (e.g. during calibration), either the fl
27、at top or Hanning window is appropriate and, where frequency resolution is paramount (e.g. for identifying sidebands), either the rectangular or Hanning window is 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 us
28、e 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 (2) 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
29、 per second. 5.3 Flat-top window For the purposes of this part of ISO 18431, the flat top window is defined as (3) where n = 0, 1, N 1 a 1= 1,932 617 19 a 2= +1,286 132 81 a 3= 0,387 695 31 a 4= +0,032 226 56 N is the number of samples in the time record. Figure 2 shows an example of a 1 024 point f
30、lat-top window sampled (f s ) at 1 024 samples per second. w(n) = 1 cos(2 n/N) where n = 0, 1, N 1 w(n) = 1 + a 1 cos(2 n/N) + a 2 cos(4 n/N) + a 3 cos(6 n/N) + a 4 cos(8 n/N) BS ISO 18431-2:2004 4 I SO 4002 All irthgs ersedevra) Key X sample Y amplitude, w(n) b) Key X frequency, in hertz Y amplitud
31、e, W(m) Figure 1 Hanning window for 1 024 samples BS ISO 18431-2:2004 I SO 4002 All irthgs ersedevr 5a) Key X sample Y amplitude, w(n) b) Key X frequency, in hertz Y amplitude, W(m) Figure 2 Flat-top window for 1 024 samples 5.4 Rectangular window For the purposes of this part of ISO 18431, the rect
32、angular 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 samples per second. BS ISO 18431-2:20046 I SO 4002 All irthgs ersedevra) Key X sample Y amplitude, w(n) b
33、) 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 of a sine wave of 4 1/2 cycles sampled at 1 024 samples per second (f s ); the results are independe
34、nt of a phase shift. It shows the extent of the noise bandwidth and amplitude error. BS ISO 18431-2:2004 I SO 4002 All irthgs ersedevr 7a) 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 4 Example of the common
35、windows applied to a truncated sinusoid of 4 1/2 cycles BS ISO 18431-2:20048 I SO 4002 All irthgs ersedevrTable 2 Common windows applied to a truncated sinusoidal signal Frequency Hanning Flat-top Rectangular 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
36、 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-truncated sinusoidal signal Figure 5 and Table 3 show an example of a sine wave of 4 cycles sampled a
37、t 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 sinusoidal signal Frequency Hanning Flat-top Rectangular 0 0,0000 0,0000 0,0000 1 0,0000 0,1940 0,0
38、000 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 0,0000 0,0000 BS ISO 18431-2:2004 I SO 4002 All irthgs ersedevr 9a) Sine wave Key X sample Y signal, x
39、(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 BS ISO 18431-2:200410 I SO 4002 All irthgs ersedevrBibliography 1 COOLEY, J.W. and TUKEY, J.W. An Algorithm for the M
40、achine 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. Proceedings of the IEEE, 66, January 1978, pp. 51-83 3 RANDALL, R.B. Frequency Analysis. 3 rdEd.,
41、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
42、and Kjaer, Technical Review, No. 4, 1987 (ISSN 007-2621) BS ISO 18431-2:2004 blankBS ISO 18431-2:2004 BSI Group Headquarters 389 Chiswick High Road, London W4 4AL, UK Tel +44 (0)20 8996 9001 Fax +44 (0)20 8996 7001 British Standards Institution (BSI) BSI is the independent national body responsible
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