Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis D. Ponteggia 1 M. Di Cola 2 1 Audiomatica, Firenze, ITALY 2 Audio Labs

Time-Frequency Characterization of Time-Frequency Characterization of Loudspeaker Responses Using Loudspeaker Responses Using

Wavelet AnalysisWavelet Analysis

D. Ponteggia1 M. Di Cola2

1Audiomatica, Firenze, ITALY2Audio Labs Systems, Milano, ITALY

123rd AES Convention, 2007 October 5-8 New York, NY

Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola

2

OutlineOutline

• Introduction

• Loudspeaker Characterization

• The Continuous Wavelet Transform

• Practical Examples

• Conclusions


3

MotivationMotivation

• This work is a direct spin-off of a previous work presented at AES 121th in San Francisco last year:M. Di Cola, M. T. Hadelich, D. Ponteggia, D. Saronni, “Linear Phase Crossover Filters Advantages in Concert Sound Reinforcement Systems: a practical approach”

• While trying to show the temporal effects of different crossover strategies, we found out that the available analysis tool were not easy to manage.

• Phase-time relationship is well documented in literature but still not well understood by loudspeaker system designers.


4

MotivationMotivation

• We need simpler tools to visualize the loudspeaker system response.

• This led us to research new tools to investigate the joint time-frequency characterization of loudspeaker systems.

• After a brief literature research, we turned our attention to the Wavelet theory.

• Even though Wavelet is a relatively recent topic, we found out that was yet used for loudspeaker impulse response analysis.


5

Loudspeaker As Linear SystemLoudspeaker As Linear System

• A loudspeaker (at least its linear model) can be fully described by means of its Impulse Response IR.

• The IR is usually collected using computer based measuring instruments. Thanks to the fact that the IR is stored in a computer, post-processing is easily feasible.

m easurem ent environm ent

PC

DUT

m ic


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Fourier Transform PairFourier Transform Pair

• By means of the Fourier transform pair (in its radial form) is it possible to switch back and forth from time domain to frequency domain:


7

Dual DomainDual Domain

Impulse response

Complex FrequencyResponse

From D.Davis, “Sound System Engineering”


8

The Impulse Response (IR)The Impulse Response (IR)

• Impulse Response of a two way loudspeaker system:

5.7 7.6 9.5 11 13 15 17 19 21 22 24ms

0.100

0.060

0.020

-0.020

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Pa

CLIO

LogChirp - Impulse Response 21-09-2006 16.22.03

CH B dBSPL Unsmoothed 48kHz 16K Rectangular Start 0.00ms Stop 341.31ms FreqLO 2.93Hz Length 341.31ms


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Complex Frequency ResponseComplex Frequency Response

• Complex Frequency Response of a two way loudspeaker system:

20 50 100 200 500 1k 2k 5k 10k 20k20 Hz

110.0 360.0

Deg

100.0 216.0

90.0 72.0

80.0 -72.0

70.0 -216.0

60.0 -360.0

CLIO

LogChirp - Frequency Response

CH B Unsmoothed 48kHz 16K Rectangular Start 8.02ms Stop 27.15ms FreqLO 52.29Hz Length 19.12ms


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IR vs Complex Freq. ResponseIR vs Complex Freq. Response

• Impulse Response:– display very little information on the frequency domain– post-processing, as the ETC, can help to get more

informations

• Complex Frequency Response:– The phase part of the response is useful to understand the

temporal behavior of the system (example crossover alignment)

– unfortunately phase is buried into the propagation term

– phase/time relationship is not simple as may appear


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Time ViewsTime Views

• We have already showed that from the IR is not easy to infer the frequency components involved into the time distortion

• Another time views has been developed to better understand the temporal behaviour of the system, but without gaining much more info on the spectral aspect.

• Between them we have:– Step Response– ETC


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Step ResponseStep Response

5.7 7.6 9.5 11 13 15 17 19 21 22 24ms

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CLIO

LogChirp - Step Response 21-09-2006 16.22.03



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ETCETC

5.7 7.6 9.5 11 13 15 17 19 21 22 24ms

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CLIO

LogChirp - ETC Plot 21-09-2006 16.22.03



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Spectral ViewsSpectral Views

• The complex frequency response can be showed as magnitude and phase response.

• It is common practice to check the time alignment of a loudspeaker system by looking at its phase response.

• A direct relationship between phase and time delay is possible only for all-pass LTI systems:


15

A Closer Look To The A Closer Look To The Measurement EnvironmentMeasurement Environment

• A closer look to the measurement environment shows that the measured response is the sum of the loudspeaker system under test plus the sound propagation term:

• The sound propagation can be modeled as a simple delay (in case of short distances). To recover the loudspeaker system phase response we need to remove the propagation delay:


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Phase Frequency ResponsePhase Frequency Response(as measured)(as measured)

20 50 100 200 500 1k 2k 5k 10k 20k20 Hz

100.0 360.0

Deg

80.0 216.0

60.0 72.0

40.0 -72.0

20.0 -216.0

0.0 -360.0

CLIO




17

Delay Removal TechniquesDelay Removal Techniques

• To remove the propagation delay we need to make some a priori assumption on the measurement model.

• In the paper we have analyzed three commonly used techniques:– Impulse Time Maximum– Excess Phase Group Delay– Geometrical

• We do not want to go into the details during this presentation, here we can state that choosing a “correct” value for the propagation delay is not straightforward!


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Phase Frequency Response Phase Frequency Response (delay removed)(delay removed)

20 50 100 200 500 1k 2k 5k 10k 20k20 Hz

100.0 360.0

Deg

80.0 216.0

60.0 72.0

40.0 -72.0

20.0 -216.0

0.0 -360.0

CLIO




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Linear Phase ResponseLinear Phase Response

• An ideal perfect system will exhibit a flat magnitude response and a linear phase response (in a linear frequency axis graph)

• It is engineering practice to look at frequency response graphs with frequency log scale

• In case of complete removal of delay the phase plot must be flat, a deviation from linearity is easily seen and magnified by the log freq axis

• In case of not complete removal of delay, the phase plot is a curve with negative slope, it could be more difficult to check deviations from linearity


20

Linear Phase ResponseLinear Phase Response


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Joint Time-Frequency ViewsJoint Time-Frequency Views

• Since we are not completely satisfied by the two previous views of the system response, there is a need to get some joint time-frequency descriptions:– Cumulative Spectral Decay CSD– Short Time Fourier Transform STFT– Wigner Distribution– Wavelet Analysis

• While the CSD and STFT are well known and accepted, the Wigner and the Wavelet transform have not yet gained popularity.


22

Cumulative Spectral DecayCumulative Spectral Decay

• The CSD is calculated by means of FT of progressively shorter sections of the IR.


23

Cumulative Spectral DecayCumulative Spectral Decay

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37.5 ms

100 1k 10k 20k20 Hz CLIO

Waterfall 26-07-2007 14.41.48

Cumulative Spectral Decay Rise 0.580ms Unsmoothed


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Short Time Fourier TransformShort Time Fourier Transform

• The idea of the STFT is to follow the temporal evolution of the IR and to apply FT to each section:

• The main drawback of the STFT is its fixed resolution over the time-frequency plane. The choice of the FFT size is linked to the section length.

• STFT is of little help to the analysis of wide-band long-duration signals as the IR of a loudspeaker system.


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Short Time Fourier TransformShort Time Fourier Transform

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37.5 ms

100 1k 10k 20k20 Hz CLIO

Waterfall 26-07-2007 14.42.16

Energy Time Frequency Unsmoothed


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Wigner-Ville DistributionWigner-Ville Distribution

• The Wigner was already used for loudspeaker IR analysis, but it exhibits cross-components artifact.


27

Continuous Wavelet TransformContinuous Wavelet Transform

• The Continuous Wavelet Transform is defined as the inner product between the IR and a scaled and translated version of a function called “mother wavelet”:

• The CWT can be wrote as:

The factor 1/sqrt(a) is added to normalize the energy of the scaled wavelets.


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• The Wavelet Transform can be loosely interpreted as a correlation function between the IR and the scaled and translated wavelets.– low scale (high frequency) wavelets are short duration

functions and they are good for the analysis of high frequency-short duration events

– high scale (low frequency) wavelets are long duration functions and they are good for the analysis of low frequency-long duration events

• The Wavelet Analysis can be understood as a constant-Q analysis– it is a good tool to investigate long duration wide-band

signals


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• The uncertainty principle states that the temporal and bandwidth resolutions product:

• It can be shown that the function with minimum product is the Gaussian pulse.

• Therefore a good candidate as a mother wavelet is a modulated Gaussian pulse:


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• The FT of the mother wavelet is:

• By adjusting B parameter in the mother wavelet we can exchange temporal and bandwidth resolution.


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• The computation of the coefficients directly from the equation:

is very expensive.

• An alternative approach based on conventional FT can be used. For every scale a it is possible to calculate CWT coefficients:


32

Computational IssuesComputational Issues

• We made a set of speed tests to check the computational time of the previous calculation algorithm:


33

Scalogram PlotScalogram Plot

• Once the coefficient matrix is calculated we need to graphically represent the results.

• The Spectrogram is a well known tool to show the energy of a signal in the time-frequency plane, it is defined as the squared modulus of the STFT.

• The Scalogram is defined in a similar way as the squared modulus of the CWT. The energy of the signal is mapped in a time-scale plane:

• It is possible to apply a transformation to get the usual time-frequency plane.


34

0

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dB

Time (b)

Fre

qu

en

cy

Scalogram PlotScalogram Plot

• Scalogram of a Dirac pulse:

Sca

le (

a)


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Wavelet vs STFTWavelet vs STFT

• Comparison of CWT and STFT resolutions: region of influence of a Dirac pulse and three sinusoids.


36

Wavelet, STFT and WignerWavelet, STFT and Wigner

• There is a strong link between Wigner-Ville distribution, spectrograms and scalograms. The latter two can be seen as “smoothed” versions of the first.


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• Scalogram of the CWT of a Dirac pulse. We notice the energy spread at low frequencies.

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0 34 68 102 137 171 205 239 273 307 341 ms

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CLIO

Wavelet Analysis

Time-Frequency Energy Q 3.000 BW 0.333 octaves


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• It is possible to apply a “scale normalization” that lead to an easy to read modified scalogram:

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0 34 68 102 137 171 205 239 273 307 341 ms

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CLIO

Wavelet Analysis



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• Wavelet Analysis of two way loudspeaker system

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5.1 7.0 8.8 11 13 14 16 18 20 22 24 ms

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Wavelet Analysis 27-07-2007 14.43.00


File:


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• Plot of the “peak energy” arrival curve:

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File:


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• “level” normalization (better energy decay view):

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Time-Frequency Energy Normalized Q 3.000 BW 0.333 octaves

File:


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Trading BW and Time resolutionTrading BW and Time resolution0

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0 34 68 102 137 171 205 239 273 307 341 ms

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Wavelet Analysis


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0 34 68 102 137 171 205 239 273 307 341 ms

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Wavelet Analysis


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Wavelet Analysis


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Wavelet Analysis


Q=3 Q=4.5

Q=6 Q=12


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Real World ExamplesReal World Examples

• We will show some examples of wavelet analysis on real world loudspeaker systems– 2 way professional 8” loudspeaker box– 3 way vertical array element– compression driver on CD horn– Hi-Fi electrostatic loudspeaker– Hi-Fi loudspeaker box with passive radiator


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2 way professional 8”2 way professional 8”

• This is a simple two way system equipped with a 8’’ cone woofer and 1’’ compression driver.

• We analyze how two different crossover strategies affect the time alignment between drivers and which of the two perform better in term of time coherence.


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• Frequency response:

100 200 500 1k 2k 5k 10k 20k100 Hz

-20.0

dBV

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Deg

-30.0 108.0

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CLIO

LogChirp - Frequency Response 01-08-2007 16.39.29

CH A dBV Unsmoothed 192kHz 65K Rectangular Start 1.28ms Stop 11.23ms FreqLO 100.47Hz Length 9.95ms

APN

LPC


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• Phase response:

100 200 500 1k 2k 5k 10k 20k100 Hz

-20.0

dBV

180.0

Deg

-30.0 108.0

-40.0 36.0

-50.0 -36.0

-60.0 -108.0

-70.0 -180.0

CLIO



APN

LPC


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• APN wavelet analysis:

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0 1.5 3.0 4.5 6.0 7.5 9.0 10 12 13 15 ms

1k

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CLIO

Wavelet Analysis



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• LPC wavelet analysis:

0

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0 1.5 3.0 4.5 6.1 7.6 9.1 11 12 14 15 ms

1k

10k

200

Hz

CLIO

Wavelet Analysis



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• Reverse polarity, frequency response:

100 200 500 1k 2k 5k 10k 20k100 Hz

-20.0

dBV

180.0

Deg

-30.0 108.0

-40.0 36.0

-50.0 -36.0

-60.0 -108.0

-70.0 -180.0

CLIO



Correct Polarity

Reversed Polarity


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• Reverse polarity, wavelet analysis:

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0 1.5 3.0 4.5 6.0 7.5 9.0 10 12 13 15 ms

1k

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Wavelet Analysis



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3 way VA element3 way VA element

• Big format vertical array element.

• Comparison between APN and LPC crossover strategies.

• Frequency response almost identical (small differences), while phase response shows remarkably different responses.


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200 500 1k 2k 5k 10k 20k200 Hz

110.0

dBSPL

180.0

100.0 108.0

90.0 36.0

80.0 -36.0

70.0 -108.0

60.0 -180.0

CLIO


CH A dBSPL Unsmoothed 48kHz 32K Rectangular Start 0.00ms Stop 15.67ms FreqLO 63.83Hz Length 15.67ms

Deg

Original Filter Set

Linear Phase Filter Set


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• Phase response:

200 500 1k 2k 5k 10k 20k200 Hz

110.0

dBSPL

180.0

Deg

100.0 108.0

90.0 36.0

80.0 -36.0

70.0 -108.0

60.0 -180.0

CLIO



Original Filter Set

Linear Phase Filter Set


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• Original filter wavelet analysis:

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0 3.0 6.0 9.0 12 15 18 21 24 27 30 ms

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Hz

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Wavelet Analysis



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• Linear phase wavelet analysis:

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10.0 13 16 19 22 25 28 31 34 37 40 ms

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Wavelet Analysis



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Compression driver on CD hornCompression driver on CD horn

• A common feature of a constant directivity horn is the diffraction slot used at the horn throat.

• In large format horns it is common practice to couple the drivers to an exponential portion of the horn that ends up in a very narrow slot that is forced to diffract in a subsequent section of the horn. This generates reflected waves.

• The wavelet analysis can show how much energy is reflected back and forward inside the horn, and which frequency bands are affected.


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100 200 500 1k 2k 5k 10k 20k100 Hz

110.0

dBSPL

180.0

100.0 108.0

90.0 36.0

80.0 -36.0

70.0 -108.0

60.0 -180.0

CLIO



Deg


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• Wavelet analysis:

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1k

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Wavelet Analysis



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Hi-Fi electrostatic loudspeakerHi-Fi electrostatic loudspeaker

• We measured an HI-FI electrostatic loudspeaker that is “time aligned” by its principle of operation.

• This is confirmed by the almost flat phase response.

• The wavelet analysis confirm the result.


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Hi-Fi electrostatic loudspeakerHi-Fi electrostatic loudspeaker

• Impulse response:

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10ms

2.0

1.2

0.40

-0.40

-1.2

-2.0

Pa

CLIO

MLS - Impulse Response



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Hi-Fi Electrostatic LoudspeakerHi-Fi Electrostatic Loudspeaker

• Phase response:

20 50 100 200 500 1k 2k 5k 10k 20k20 Hz

120.0

dBSPL

180.0

Deg

110.0 108.0

100.0 36.0

90.0 -36.0

80.0 -108.0

70.0 -180.0

CLIO

MLS - Frequency Response



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Hi-Fi Electrostatic LoudspeakerHi-Fi Electrostatic Loudspeaker

• Wavelet analysis:

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0 2.0 4.0 6.0 8.0 10 12 14 16 18 20 ms

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10k

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CLIO

Wavelet Analysis



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ConclusionsConclusions

The Wavelet Analysis:

• is a useful tool to inspect loudspeaker impulse responses.

• gives a system time-frequency energy footprint that is easily readable.

• It could be used into the daily work of the loudspeaker or transducer designer side by side with other well-known tools.


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Further DevelopmentsFurther Developments

• Enhance computational speed by using a different calculation algorithm. In the future we can move towards a “real time” wavelet analysis.

• Explore alternative mappings, such as Wavelet Coefficient Phase color-maps or 3D time-frequency-angle plots.


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Available LiteratureAvailable Literature

• O.Rioul, M.Vetterli, “Wavelets and Signal Processing” IEEE SP magazine, vol. 4, no. 4, pp. 12-38, Oct. 1991

• D.B.Keele, “Time Frequency Display of Electroacustic Data Using Cycle-Octave Wavelet Transforms” AES 99th, New York, NY, USA, 1995

• S.J.Loutridis, “Decomposition of Impulse Responses Using Complex Wavelets” JAES, vol. 53, No. 9, pp. 796–811 (2005 September)

• D.W.Gunness, W.R.Hoy, “A Spectrogram Display for Loudspeaker Transient Response” AES 119th, New York, NY, USA, 2006


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Thank you for your attention!Thank you for your attention!

Documents

Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis D. Ponteggia 1 M. Di Cola 2 1 Audiomatica, Firenze, ITALY 2 Audio Labs