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TitleNon-stationary Signal Analysis using Wavelet Transform,Short-time Fourier Transform and Wigner-Vi
LanguageEnglish
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Total Pages34
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Page 1

Technical
Review
Non-stationary Signal Analysis using Wavelet T ransform,
Short-time Fourier T ransform and Wigner-V ille Distribution

No. 2 – 1996

ISSN 007–2621

BV 0049–11

HEADQUARTERS: DK-2850 Nærum · Denmark · Telephone: +45 4580 0500
Fax: +45 4580 1405 · www.bksv.com · [email protected]

Australia (+61) 2 9889-8888 · Austria (+43) 1 865 74 00 · Brazil (+55) 11 5188-8161 · Canada (+1) 514 695-8225
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Germany (+49) 421 17 87 0 · Hong Kong (+852) 2548 7486 · Hungary (+36) 1 215 83 05 · Ireland (+353) 1 807 4083
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Page 2

Previously issued numbers of
Brüel & Kjær Technical Review
1 – 1996 Calibration Uncertainties & Distortion of Microphones.

Wide Band Intensity Probe. Accelerometer Mounted Resonance Test
2 – 1995 Order Tracking Analysis
1 – 1995 Use of Spatial Transformation of Sound Fields (STSF) Techniques in the

Automative Industry
2 – 1994 The use of Impulse Response Function for Modal Parameter Estimation

Complex Modulus and Damping Measurements using Resonant and
Non-resonant Methods (Damping Part II)

1 – 1994 Digital Filter Techniques vs. FFT Techniques for Damping
Measurements (Damping Part I)

2 – 1990 Optical Filters and their Use with the Type 1302 & Type 1306
Photoacoustic Gas Monitors

1 – 1990 The Brüel & Kjær Photoacoustic Transducer System and its Physical
Properties

2 – 1989 STSF — Practical instrumentation and application
Digital Filter Analysis: Real-time and Non Real-time Performance

1 – 1989 STSF — A Unique Technique for scan based Near-Field Acoustic
Holography without restrictions on coherence

2 – 1988 Quantifying Draught Risk
1 – 1988 Using Experimental Modal Analysis to Simulate Structural Dynamic

Modifications
Use of Operational Deflection Shapes for Noise Control of Discrete
Tones

4 – 1987 Windows to FFT Analysis (Part II)
Acoustic Calibrator for Intensity Measurement Systems

3 – 1987 Windows to FFT Analysis (Part I)
2 – 1987 Recent Developments in Accelerometer Design

Trends in Accelerometer Calibration
1 – 1987 Vibration Monitoring of Machines
4 – 1986 Field Measurements of Sound Insulation with a Battery-Operated

Intensity Analyzer
Pressure Microphones for Intensity Measurements with Significantly
Improved Phase Properties
Measurement of Acoustical Distance between Intensity Probe
Microphones
Wind and Turbulence Noise of Turbulence Screen, Nose Cone and
Sound Intensity Probe with Wind Screen

3 – 1986 A Method of Determining the Modal Frequencies of Structures with
Coupled Modes
Improvement to Monoreference Modal Data by Adding an Oblique
Degree of Freedom for the Reference
(Continued on cover page 3)

Page 17

A much more fundamental version of the uncertainty relationship uses the
second order moments of the weighting function, g (t) around a suitable point
as the time duration (also called RMS duration), and the second order moment
of the corresponding frequency filter shape around a suitable point as the
bandwidth (also called the RMS bandwidth). See Eq. (4) and Fig. 6.

(4)

The normalisation factor, Eg is the energy of the window function. It can be
shown that this product can never be smaller than 1/4π. Ref. [5]. Note that in
all examples (except for WVD, where Δ t is the sampling interval), the time
duration is indicated as twice the RMS duration and the bandwidth as twice
the RMS bandwidth, which is more relevant from the user’s point of view,
when dealing with bandpass filtering.

Fig. 10. Heisenberg’s ellipses for WT. Y-axis is both a frequency
and a frequency resolution axis

ΔtRMS
1

Eg
------- t t0–( )

2
g t( )

2
dt

∞–

∞+











1 2⁄

=

ΔfRMS
1

Eg
------- f f0–( )

2

∞–

∞+

∫ G f( )
2
df

⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞

1 2⁄

=

960596e

High

Low
Δt

Δf

Δt

Δt
Δf

Δt
Δf

Δf

Frequency

Time resolution

WT
13

Page 18

The optimum choice of weighting function is a Gaussian shape, both for the
STFT and the WT. In this case we actually achieve an uncertainty product
which equals 1/4π.

This can be visualised by showing the Heisenberg’s ellipses, where the area
of all ellipses is the same. For the STFT the ellipses have constant shapes,
while for the WT the length of the frequency width is proportional to the fre-
quency at which it is located.

Thus the STFT offers constant time and frequency resolution, while the WT
offers good frequency resolution at low frequency and good time resolution at
high frequencies. See Figs. 9 and 10.

Fig. 11. Speech analysis using STFT. Sentence (French), “Dès que le tambour bat...”,
2Δ f = 99 Hz, 2Δ t = 3.2 ms
14

Page 33

Previously issued numbers of
Brüel & Kjær Technical Review

(Continued from cover page 2)
2 – 1986 Quality in Spectral Match of Photometric Transducers

Guide to Lighting of Urban Areas
1 – 1986 Environmental Noise Measurements
4 – 1985 Validity of Intensity Measurements in Partially Diffuse Sound Field

Influence of Tripods and Microphone Clips on the Frequency Response
of Microphones

3 – 1985 The Modulation Transfer Function in Room Acoustics
RASTI: A Tool for Evaluating Auditoria

2 – 1985 Heat Stress
A New Thermal Anemometer Probe for Indoor Air Velocity
Measurements

1 – 1985 Local Thermal Discomfort
4 – 1984 Methods for the Calculation of Contrast

Proper Use of Weighting Functions for Impact Testing
Computer Data Acquisition from Brüel & Kjær Digital Frequency
Analyzers 2131/2134 Using their Memory as a Buffer

3 – 1984 The Hilbert Transform
Microphone System for Extremely Low Sound Levels
Averaging Times of Level Recorder 2317

2 – 1984 Dual Channel FFT Analysis (Part II)
1 – 1984 Dual Channel FFT Analysis (Part I)
4 – 1983 Sound Level Meters — The Atlantic Divide

Design Principles for Integrating Sound Level Meters

Special technical literature
Brüel & Kjær publishes a variety of technical literature which can be obtained
from your local Brüel & Kjær representative.
The following literature is presently available:

❍ Modal Analysis of Large Structures – Multiple Exciter Systems (English)
❍ Acoustic Noise Measurements (English), 5th. Edition
❍ Noise Control (English, French)
❍ Frequency Analysis (English), 3rd. Edition
❍ Catalogues (several languages)
❍ Product Data Sheets (English, German, French, Russian)

Furthermore, back copies of the Technical Review can be supplied as shown in the
list above. Older issues may be obtained provided they are still in stock.

Page 34

Technical
Review
Non-stationary Signal Analysis using Wavelet Transform,
Short-time Fourier Transform and Wigner-Ville Distribution

No.2–1996

ISSN 007 – 2621
BV 0049 – 11

HEADQUARTERS: DK-2850 Nærum · Denmark · Telephone: +45 4580 0500
Fax: +45 4580 1405 · www.bksv.com · [email protected]

Australia (+61) 2 9889-8888 · Austria (+43) 1 865 74 00 · Brazil (+55) 11 5188-8161 · Canada (+1) 514 695-8225
China (+86)10680 29906 · Czech Republic (+420) 2 6702 1100 · Finland (+358) 9 521300 · France (+33) 1 69 90 71 00
Germany (+49) 421 17 87 0 · Hong Kong (+852) 2548 7486 · Hungary (+36) 1 215 83 05 · Ireland (+353) 1 807 4083
Italy (+39) 0257 68061 · Japan (+81) 3 5715 1612 · Netherlands (+31) 318 55 9290 · Norway (+47) 66 771155
Poland (+48) 22 816 75 56 · Portugal (+351) 21 4169 040 · Republic of Korea (+82) 2 3473 0605 · Singapore (+65) 6377 4512
Slovak Republic (+421) 25 443 0701 · Spain (+34) 91 659 0820 · Sweden (+46) 33 22 56 22 · Switzerland (+41) 44 880 7035
Taiwan (+886) 2 2502 7255 · United Kingdom (+44) 14 38 739 000 · USA (+1) 800 332 2040

Local representatives and service organisations worldwide

BV0049-11.qxp 14-05-2007 09:02 Page 1

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