TimeFrequency Characterization of Loudspeaker Responses Using Wavelet Analysis

1
Time-Frequency Characterization of Loudspeaker
Responses Using Wavelet Analysis

• D. Ponteggia1 M. Di Cola2
• 1Audiomatica, Firenze, ITALY
• 2Audio Labs Systems, Milano, ITALY
• 123rd AES Convention, 2007 October 5-8 New York,
  NY

2
Outline

• Introduction
• Loudspeaker Characterization
• The Continuous Wavelet Transform
• Practical Examples
• Conclusions

3
Motivation

• This work is a direct spin-off of a previous work
  presented at AES 121th in San Francisco last
  yearM. 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
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.

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.

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

7
Dual Domain
From D.Davis, Sound System Engineering
8
The Impulse Response (IR)

• Impulse Response of a two way loudspeaker system

LogChirp - Impulse Response
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Complex Frequency Response

• Complex Frequency Response of a two way
  loudspeaker system

LogChirp - Frequency Response
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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

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

12
Step Response
LogChirp - Step Response
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ETC
LogChirp - ETC Plot
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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

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

16
Phase Frequency Response(as measured)
LogChirp - Frequency Response
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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!

18
Phase Frequency Response (delay removed)
LogChirp - Frequency Response
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Length 23.92ms
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

20
Linear Phase Response
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.

22
Cumulative Spectral Decay

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

23
Cumulative Spectral Decay
Waterfall
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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.

25
Short Time Fourier Transform
Waterfall
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26
Wigner-Ville Distribution

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

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

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

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

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.

31
Continuous Wavelet Transform

• The computation of the coefficients directly from
  the equationis 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 Issues

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

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.

34
Scalogram Plot

• Scalogram of a Dirac pulse

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Wavelet vs STFT

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

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.

37
Wavelet Analysis

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

38
Wavelet Analysis

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

39
Wavelet Analysis

• Wavelet Analysis of two way loudspeaker system

40
Wavelet Analysis

• Plot of the peak energy arrival curve

41
Wavelet Analysis

• level normalization (better energy decay view)

42
Trading BW and Time resolution
Q3
Q4.5
Q6
Q12
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

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.

45
2 way professional 8

• Frequency response

LogChirp - Frequency Response
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2 way professional 8

• Phase response

LogChirp - Frequency Response
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2 way professional 8

• APN wavelet analysis

48
2 way professional 8

• LPC wavelet analysis

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2 way professional 8

• Reverse polarity, frequency response

LogChirp - Frequency Response
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2 way professional 8

• Reverse polarity, wavelet analysis

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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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3 way VA element

• Frequency response

LogChirp - Frequency Response
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3 way VA element

• Phase response

LogChirp - Frequency Response
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3 way VA element

• Original filter wavelet analysis

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3 way VA element

• Linear phase wavelet analysis

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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.

57
Compression driver on CD horn

• Frequency response

LogChirp - Frequency Response
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Compression driver on CD horn

• Wavelet analysis

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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Hi-Fi electrostatic loudspeaker

• Impulse response

MLS - Impulse Response
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Hi-Fi Electrostatic Loudspeaker

• Phase response

MLS - Frequency Response
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Hi-Fi Electrostatic Loudspeaker

• Wavelet analysis

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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.

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.

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. 796811 (2005 September)
• D.W.Gunness, W.R.Hoy, A Spectrogram Display for
  Loudspeaker Transient Response AES 119th, New
  York, NY, USA, 2006

66
Thank you for your attention!