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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis D. Ponteggia 1 M. Di Cola 2 1 Audiomatica, Firenze, ITALY 2 Audio Labs Systems, Milano, ITALY 123rd AES Convention, 2007 October 5-8 New York, NY

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 2 Outline Introduction Loudspeaker Characterization The Continuous Wavelet Transform Practical Examples Conclusions

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 3 Motivation 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.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 4 Motivation 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.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 5 Loudspeaker 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.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 6 Fourier 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:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 7 Dual Domain Impulse response Complex Frequency Response From D.Davis, “Sound System Engineering”

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 8 The Impulse Response (IR) Impulse Response of a two way loudspeaker system: 5.77.69.51113151719212224ms 0.100 0.060 0.020 -0.020 -0.060 -0.100 Pa CLIO LogChirp - Impulse Response21-09-2006 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 9 Complex Frequency Response Complex Frequency Response of a two way loudspeaker system: 20501002005001k2k5k10k20k20Hz 110.0360.0 Deg 100.0216.0 90.072.0 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 10 IR 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-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 11 Time 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 12 Step Response 5.77.69.51113151719212224ms 0.20 0.12 0.040 -0.040 -0.12 -0.20 Pa CLIO LogChirp - Step Response21-09-2006 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 13 ETC 5.77.69.51113151719212224ms 0 -10.0 -20 -30 -40 -50 dB CLIO LogChirp - ETC Plot21-09-2006 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 14 Spectral 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:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 15 A Closer Look To The Measurement 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 16 Phase Frequency Response (as measured) 20501002005001k2k5k10k20k20Hz 100.0360.0 Deg 80.0216.0 60.072.0 40.0-72.0 20.0-216.0 0.0-360.0 CLIO LogChirp - Frequency Response CH B Unsmoothed 48kHz 16K Rectangular Start 0.00ms Stop 23.92ms FreqLO 41.81Hz Length 23.92ms

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 17 Delay 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 18 Phase Frequency Response (delay removed) 20501002005001k2k5k10k20k20Hz 100.0360.0 Deg 80.0216.0 60.072.0 40.0-72.0 20.0-216.0 0.0-360.0 CLIO LogChirp - Frequency Response CH B Unsmoothed 48kHz 16K Rectangular Start 0.00ms Stop 23.92ms FreqLO 41.81Hz Length 23.92ms

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 19 Linear 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

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 20 Linear Phase Response

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 21 Joint 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.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 22 Cumulative Spectral Decay The CSD is calculated by means of FT of progressively shorter sections of the IR.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 23 Cumulative Spectral Decay 0 -10 -20 -30 -40 -50 dB 0.0 12.5 25.0 37.5 ms 1001k10k20k20 Hz CLIO Waterfall26-07-2007 14.41.48 Cumulative Spectral Decay Rise 0.580ms Unsmoothed

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 24 Short 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 25 Short Time Fourier Transform 0 -10 -20 -30 -40 -50 dB 0.0 12.5 25.0 37.5 ms 1001k10k20k20 Hz CLIO Waterfall26-07-2007 14.42.16 Energy Time Frequency Unsmoothed

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 26 Wigner-Ville Distribution The Wigner was already used for loudspeaker IR analysis, but it exhibits cross-components artifact.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 27 Continuous 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 28 Continuous Wavelet Transform 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 29 Continuous Wavelet Transform 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 30 Continuous Wavelet Transform 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 31 Continuous Wavelet Transform 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:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 32 Computational Issues We made a set of speed tests to check the computational time of the previous calculation algorithm:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 33 Scalogram 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.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 34 0 -10 -20 -30 -40 -50 dB Time (b) Frequency Scalogram Plot Scalogram of a Dirac pulse: Scale (a)

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 35 Wavelet vs STFT Comparison of CWT and STFT resolutions: region of influence of a Dirac pulse and three sinusoids.

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 36 Wavelet, 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 37 Wavelet Analysis Scalogram of the CWT of a Dirac pulse. We notice the energy spread at low frequencies. 0 -10 -20 -30 -40 -50 dB 03468102137171205239273307341 ms 100 1k 10k 20k 100 Hz CLIO Wavelet Analysis Time-Frequency Energy Q 3.000 BW 0.333 octaves

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 38 Wavelet Analysis It is possible to apply a “scale normalization” that lead to an easy to read modified scalogram: 0 -10 -20 -30 -40 -50 dB 03468102137171205239273307341 ms 100 1k 10k 20k 100 Hz CLIO Wavelet Analysis Time-Frequency Energy Q 3.000 BW 0.333 octaves

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 39 Wavelet Analysis Wavelet Analysis of two way loudspeaker system

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 40 Wavelet Analysis Plot of the “peak energy” arrival curve:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 41 Wavelet Analysis “level” normalization (better energy decay view):

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 42 Trading BW and Time resolution Q=3Q=4.5 Q=6Q=12

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 43 Real 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 44 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 45 2 way professional 8” Frequency response: 1002005001k2k5k10k20k100Hz -20.0 dBV 180.0 Deg -30.0108.0 -40.036.0 -50.0-36.0 -60.0-108.0 -70.0-180.0 CLIO LogChirp - Frequency Response01-08-2007 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 46 2 way professional 8” Phase response: 1002005001k2k5k10k20k100Hz -20.0 dBV 180.0 Deg -30.0108.0 -40.036.0 -50.0-36.0 -60.0-108.0 -70.0-180.0 CLIO LogChirp - Frequency Response01-08-2007 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 47 2 way professional 8” APN wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 48 2 way professional 8” LPC wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 49 2 way professional 8” Reverse polarity, frequency response: 1002005001k2k5k10k20k100Hz -20.0 dBV 180.0 Deg -30.0108.0 -40.036.0 -50.0-36.0 -60.0-108.0 -70.0-180.0 CLIO LogChirp - Frequency Response01-08-2007 16.41.11 CH A dBV Unsmoothed 192kHz 65K Rectangular Start 1.29ms Stop 11.24ms FreqLO 100.47Hz Length 9.95ms Correct Polarity Reversed Polarity

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 50 2 way professional 8” Reverse polarity, wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 51 3 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 52 3 way VA element Frequency response: 2005001k2k5k10k20k200Hz 110.0 dBSPL 180.0 100.0108.0 90.036.0 80.0-36.0 70.0-108.0 60.0-180.0 CLIO LogChirp - Frequency Response25-04-2006 12.41.02 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 53 3 way VA element Phase response: 2005001k2k5k10k20k200Hz 110.0 dBSPL 180.0 Deg 100.0108.0 90.036.0 80.0-36.0 70.0-108.0 60.0-180.0 CLIO LogChirp - Frequency Response23-04-2006 16.47.22 CH A dBSPL Unsmoothed 48kHz 32K Rectangular Start 10.19ms Stop 25.65ms FreqLO 64.69Hz Length 15.46ms Original Filter Set Linear Phase Filter Set

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 54 3 way VA element Original filter wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 55 3 way VA element Linear phase wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 56 Compression 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 57 Compression driver on CD horn Frequency response: 1002005001k2k5k10k20k100Hz 110.0 dBSPL 180.0 100.0108.0 90.036.0 80.0-36.0 70.0-108.0 60.0-180.0 CLIO LogChirp - Frequency Response24-07-2007 13.01.16 CH A dBSPL Unsmoothed 192kHz 131K Rectangular Start 0.82ms Stop 8.60ms FreqLO 128.51Hz Length 7.78ms Deg

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 58 Compression driver on CD horn Wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 59 Hi-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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 60 Hi-Fi electrostatic loudspeaker Impulse response: 01.02.03.04.05.06.07.08.09.010ms 2.0 1.2 0.40 -0.40 -1.2 -2.0 Pa CLIO MLS - Impulse Response CH A dBSPL Unsmoothed 48kHz 32K Rectangular Start 0.00ms Stop 10.48ms FreqLO 95.43Hz Length 10.48ms

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 61 Hi-Fi Electrostatic Loudspeaker Phase response: 20501002005001k2k5k10k20k20Hz 120.0 dBSPL 180.0 Deg 110.0108.0 100.036.0 90.0-36.0 80.0-108.0 70.0-180.0 CLIO MLS - Frequency Response CH A dBSPL Unsmoothed 48kHz 32K Rectangular Start 0.00ms Stop 10.48ms FreqLO 95.43Hz Length 10.48ms

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 62 Hi-Fi Electrostatic Loudspeaker Wavelet analysis:

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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 63 Conclusions 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 64 Further 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 65 Available 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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Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis - D. Ponteggia, M. Di Cola 66 Thank you for your attention!

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