Practical Applications of the Wavelet Analysis

Transcription

1 Practical Applications of the Wavelet Analysis M. Bigi, M. Jacchia, D. Ponteggia ALMA International Europe (6- - Frankfurt)

2 Summary Impulse and Frequency Response Classical Time and Frequency Analysis Wavelet Analysis Theory Practical Implementation Examples of Application Q&A 2

3 Impulse and Frequency Response A linear system (LTI) can be fully described by means of its Impulse Response (IR) If the system is not linear we can easily measure its linear part Using small signals stimulus (MLS) LogChirp (Sinusoidal Sine Sweeps) 3

4 Impulse Response By means of the Fourier Transform pair it is possible to switch back and forth from time domain to frequency domain: x (t ) y (t ) LTI X ( j ω) h(t ) FFT Y ( j ω) H ( j ω) 1 j ωt h(t )= H ( j ω ) e dω 2 π H ( j ω )= h(t )e j ωt dt 4

5 Impulse Response LogChirp - Impulse Response CLIO Pa CH B dbspl Unsmoothed 48kHz 16K Rectangular Start 0.00ms Stop ms 17 FreqLO 2.93Hz ms Length ms 5

6 Frequency Response LogChirp - Frequency Response CLIO Deg CH B Unsmoothed 48kHz 16K Rectangular Start 8.02ms 1k Stop 27.15ms 2k FreqLO 52.29Hz Hz 5k 10k 20k Length 19.12ms 6

7 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 (ex. crossover alignment) unfortunately phase is buried into the propagation ter phase/time relationship is not simple (as may appear) 7

8 Temporal vs Spectral Analysis Temporal Analysis Spectral Analysis 8

9 Dual Domain From D.Davis, Sound System Engineering 9

10 Temporal Analysis Time Views: Impulse response Step Response ETC (envelope of analytic IR) Looking at the IR is not easy to infer the frequency components involved into the time distortions 10

11 Step Response LogChirp - Step Response CLIO Pa CH B dbspl Unsmoothed 48kHz 16K Rectangular Start 0.00ms Stop ms 21 ms FreqLO 2.93Hz Length ms 11

12 ETC LogChirp - ETC Plot CLIO db CH B dbspl Unsmoothed 48kHz 16K Rectangular Start 0.00ms 15 Stop ms 17 FreqLO 2.93Hz ms Length ms 12

13 Spectral Analysis Complex Frequency Response is usually shown as magnitude and phase components: j ω ϕ [H ( j ω )] H ( j ω )= H ( j ω ) e Phase component carry information on time behavior of the system, but it is difficult to interpret 13

14 Phase Response It is common engineering practice to check the time alignment of an electroacoustic system by looking at its phase response Only very experienced designers can handle the complexity of such approach Device phase response is buried into propagation delay Linear phase on log frequency scale might be difficult to read Direct time-phase relationship is valid only for all-pass systems 14

15 Phase Response LogChirp - Frequency Response CLIO Deg With propagation delay CH B k Unsmoothed 48kHz 16K Rectangular Start 0.00ms 2k Stop 23.92ms Hz 5k FreqLO 41.81Hz 10k 20k Length 23.92ms 15

16 Phase Response LogChirp - Frequency Response CLIO Deg Without propagation delay CH B k Unsmoothed 48kHz 16K Rectangular Start 0.00ms 2k Stop 23.92ms Hz 5k FreqLO 41.81Hz 10k 20k Length 23.92ms 16

17 Linear Phase on Log Freq. Axis 17

18 Linear Phase? An ideal perfect system exhibit a flat magnitude response and a linear phase response (in a linear frequency axis graph, while 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 18

19 Excess Phase Group Delay Hminimum_phase Hall_pass H Group delay = Time delay t (ω )= d ϕ (H all (ω) ) pass dω 19

20 Joint Time-Frequency Analysis Temporal Analysis Joint TF Analysis Spectral Analysis 20

21 Cumulative Spectral Decay The CSD is calculated by means of FT of progressively shorter sections of the IR 21

22 Cumulative Spectral Decay Waterfall db ms k 10k 20k Hz CLIO Cumulative Spectral Decay Rise 0.580ms Unsmoothed 22

23 Short Time Fourier Transform The STFT try to follow the temporal evolution of the IR and to apply FT to each section: j ωt F h ( τ, ω)= h (t ) g (t τ)e dt The STFT has fixed resolution over the time-frequency plane. The FFT size is linked to the section length. STFT is not suited to the analysis of wide-band longduration signals as the IR of an electro-acoustic system. 23

24 Short Time Fourier Transform Waterfall db ms k 10k 20k Hz CLIO Energy Time Frequency Unsmoothed 24

25 Wigner-Ville Distribution t t j ωt W h ( τ, ω)= x τ+ x τ+ e dt 2 2 ( ) ( ) The Wigner-Ville distribution has some very interesting properties (early implementation in MLSSA) Non limited time-frequency resolution Computationally not heavy But it has also some serious drawbacks Time-Frequency artifacts can easily appear Smoothed versions of the distribution can reduce the artifacts but they reduce also the resolution 25

26 Wigner-Ville Distribution 26

27 Wavelet Analysis Wavelet Analysis of electro-acoustics systems is a relatively recent and hot topic: Research on wavelet analysis of signals has been extensively carried out in earth science. Use for loudspeaker Impulse Response analysis can be traced back to an early AES presentation by Keele [1]. A seminal paper on the use of the Wavelet Analysis has been published in JAES by Loutridis [2]. In recent AES conventions the topic has been discussed by Gunness [3], Brunet [4] and Ponteggia [5]. 27

28 Wavelet Analysis Basics The basic idea behind the wavelet analysis is to compare the impulse response with a set of functions with given temporal and spectral resolution (instead of sinusoids of infinite duration, as in FT) For each frequency to be analyzed a small portion of a wave ( wavelet ) is created In our case it will be an hann windowed sinusoid During analysis the wavelet is time-shifted and correlated with the impulse 28

29 Continuous Wavelet Transform The Continuous Wavelet Transform (mathematical definition) is the inner product between the IR and a scaled and translated version of a function ψ(t ) called mother wavelet : W h (a, b)= h (t ), ψa,b (t ) = h (t ) ψa, b (t )dt 1 t b ψa,b (t )= ψ a a ( ) 29

30 Continuous Wavelet Transform The Wavelet Transform coefficients are calculated using this equation: 1 t b W h (a, b)= h(t ) ψ dt a a Wavelet Coefficients Impulse Response Normalization ( ) Scaled and Translated Wavelets 30

31 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 is a constant-q analysis good tool to investigate long duration wide-band signals 31

32 Mother Wavelet The uncertainty principle states that the temporal and bandwidth resolutions product: 1 Δt Δω 2 ψ ψ The function with minimum time-frequency product is the Gaussian pulse. According to Loutridis [2] we choose as mother wavelet a modulated Gaussian pulse: t2 B 1 jω t ψ(t )= e e π B 0 32

33 Mother Wavelet The FT of the mother wavelet is: (ω ω0 )2 Ψ (ω)=e B 4 By adjusting B parameter in the mother wavelet we can exchange temporal and bandwidth resolution. 33

34 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: 1 2 da db E h = h (t )dt = W h (a, b) 2 C ψ 0 a 2 34

35 Scalogram Plot Perfect Dirac Pulse 0 db -20 Frequency Scale (a) Time (b) 35

36 Scalogram Plot frequency Joint TF Analysis time How energy flows into the system 36

37 CWT vs STFT Dirac pulse 3 sine (f, 2f,4f) 37

38 CWT, STFT and WVD Constant Q Wavelet Wigner-Ville smoothing STFT Fixed resolution artifacts 38

39 Wavelet Analysis Tool CLIO10 measurement system feature a Wavelet Analysis tool Open.mls impulse response files Calculate Wavelet Coefficient Matrix Parameters Q, start-stop frequency Create Scalogram colormap display 39

40 Wavelet Analysis Tool Impulse Response View 40

41 Wavelet Analysis Tool Wavelet Settings 41

42 Wavelet Analysis Tool Scalogram View 42

43 Scalogram Normalization Wavelet Analysis k Without Normalization db 10k Hz 1k Time-Frequency Energy Q BW octaves File: 20 ms CLIO 43

44 Scalogram Normalization Wavelet Analysis k With Normalization db 10k Hz 1k Time-Frequency Energy Normalized Q BW octaves File: ms CLIO 44

45 Wavelet Analysis Effect of Q 0 20k Q=3 10k Wavelet Analysis 0 20k Q=4.5 db 10k -5 db Hz -10 Hz 1k -15 1k Time-Frequency Energy Q BW octaves 273 ms Time-Frequency Energy Q BW octaves 273 ms 307 CLIO CLIO Wavelet Analysis Wavelet Analysis 0 20k Q=6 10k k Q=12 db 10k -5 db Hz -10 Hz 1k -15 1k Time-Frequency Energy Q BW octaves 273 ms CLIO Time-Frequency Energy Q BW octaves 273 ms CLIO 45

46 Wavelet as filter bank ETC Linear phase bandpass ETC... h (t ) Linear phase bandpass Scale Linear phase bandpass ETC 46

47 2-Way Professional 8'' 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. APN = All Pass Network LPC = Linear Phase Crossover 47

48 2-Way Professional 8'' LogChirp - Frequency Response CLIO dbv Deg APN LPC k 2k CH A dbv Unsmoothed 192kHz 65K Rectangular Start 1.28ms 5k Hz Stop 11.23ms 10k FreqLO Hz 20k Length 9.95ms 48

49 2-Way Professional 8'' LogChirp - Frequency Response CLIO dbv Deg APN LPC k 2k CH A dbv Unsmoothed 192kHz 65K Rectangular Start 1.28ms 5k Hz Stop 11.23ms 10k FreqLO Hz 20k Length 9.95ms 49

50 2-Way Professional 8'' Wavelet Analysis 0 APN case db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 50

51 2-Way Professional 8'' Wavelet Analysis 0 LPC case db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 51

52 2-Way Professional 8'' LogChirp - Frequency Response CLIO dbv Deg Correct Polarity Reversed Polarity k 2k CH A dbv Unsmoothed 192kHz 65K Rectangular Start 1.29ms 5k Hz Stop 11.24ms 10k FreqLO Hz 20k Length 9.95ms 52

53 2-Way Professional 8'' Wavelet Analysis 0 db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 53

54 3-Way Vertical Array Module 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. 54

55 3-Way Vertical Array Module LogChirp - Frequency Response CLIO dbspl Deg Original Filter Set Linear Phase Filter Set k 2k CH A dbspl Unsmoothed 48kHz 32K Rectangular Start 0.00ms 5k Hz Stop 15.67ms 10k FreqLO 63.83Hz k Length 15.67ms 55

56 3-Way Vertical Array Module LogChirp - Frequency Response CLIO dbspl Deg Original Filter Set Linear Phase Filter Set k 2k CH A dbspl Unsmoothed 48kHz 32K Rectangular Start 10.19ms 5k Hz Stop 25.65ms 10k FreqLO 64.69Hz k Length 15.46ms 56

57 3-Way Vertical Array Module Wavelet Analysis 0 Original Filter Set db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 57

58 3-Way Vertical Array Module Wavelet Analysis 0 Linear Phase Filter Set db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 58

59 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. 59

60 Compression Driver on CD Horn LogChirp - Frequency Response CLIO dbspl Deg k 2k CH A dbspl Unsmoothed 192kHz 131K Rectangular Start 0.82ms k Hz Stop 8.60ms 10k FreqLO Hz 20k Length 7.78ms 60

61 Compression Driver on CD Horn Wavelet Analysis 0 db 10k Hz -15 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 61

62 Electrostatic Loudspeaker MLS - Impulse Response 2.0 CLIO Pa CH A dbspl Unsmoothed 48kHz 32K Rectangular Start 0.00ms 6.0 Stop 10.48ms FreqLO 95.43Hz ms Length 10.48ms 62

63 Electrostatic Loudspeaker MLS - Frequency Response CLIO dbspl Deg Phase response k CH A dbspl Unsmoothed 48kHz 32K Rectangular Start 0.00ms 2k Hz Stop 10.48ms 5k 10k FreqLO 95.43Hz 20k Length 10.48ms 63

64 Electrostatic Loudspeaker Wavelet Analysis 0 20k db 10k Hz 1k Time-Frequency Energy Normalized Q BW octaves ms CLIO 64

65 Conclusions The Wavelet Analysis is a very powerful tool to inspect the response of an electro-acoustic system The Scalogram color plot helps to understand how the energy flows into the system from a joint time-frequency perspective The Wavelet Analysis can be very useful, if properly managed, side by side with the conventional analysis techniques 65

66 References [1] Keele, Jr., D. B. (Don), Time-Frequency Display of Electroacoustic Data Using Cycle-Octave Wavelet Transforms, 99th AES Convention, 1995 [2] Loutridis, Spyros J., Decomposition of Impulse Responses Using Complex Wavelets, JAES Volume 53 Issue 9 pp ; September 2005 [3] Gunness, David W.; Hoy, William R., A Spectrogram Display for Loudspeaker Transient Response, 119th AES Convention, 2005 [4] Brunet, Pascal; Rimkunas, Zachary; Temme, Steve, Evaluation of TimeFrequency Analysis Methods and Their Practical Applications, 123rd AES Convention, 2007 [5] Ponteggia, Daniele; Di Cola, Mario, Time-Frequency Characterization of Loudspeaker Responses Using Wavelet Analysis, 123rd AES Convention,

67 What Else? Due to its constant Q resolution and good representation of the energy flow into a system it is possible to use the Wavelet Analysis also in the inspection of Room Impulse Responses It is like having an enhanced ETC curve 67

68 Room Response Analysis Wavelet Analysis PM 0 20k Room Without Treatment db 10k Hz 1k ms Time-Frequency Energy Normalized Q BW octaves File: CLIO 68

69 Room Response Analysis Wavelet Analysis Increase in first reflections 20k PM 0 Room With Treatment db 10k -5 Reverberation and Modal Control Using Diffusers -10 Hz 1k ms Time-Frequency Energy Normalized Q BW octaves File: CLIO 69

70 Question Time Thank you for your attention More info available at our website Questions? 70