Mathematics in Acoustics

Transcription

1 Mathematics in Acoustics Peter Balazs Acoustics Research Institute (ARI) Austrian Academy of Sciences Peter Balazs (ARI) Mathematics in Acoustics 1 / 21

2 Overview: 1 Applied Mathematics Peter Balazs (ARI) Mathematics in Acoustics 2 / 21

3 Overview: 1 Applied Mathematics 2 Numerical Mathematics Peter Balazs (ARI) Mathematics in Acoustics 2 / 21

4 Overview: 1 Applied Mathematics 2 Numerical Mathematics 3 Application-oriented Mathematics Peter Balazs (ARI) Mathematics in Acoustics 2 / 21

5 Applied Mathematics, part 1 Peter Balazs (ARI) Mathematics in Acoustics 3 / 21

6 Applied Mathematics: Vibrations (Numerical Acoustics) δ t x L = z L dxdt = 0, [( ) 1 2(ν + 1) G(x, z, θ) (ux 2 + wz 2 ) + 4ν 2 1 2ν 1 2ν G(x, z, θ)u xw z + + G(x, z, θ)(u z + w x ) 2 ] 1 2 ρ(u2 t + w 2 t )dz f Ext w z=0. (1) Peter Balazs (ARI) Mathematics in Acoustics 4 / 21

7 Applied Mathematics: Vibrations (Numerical Acoustics) 3 mathematicians Peter Balazs (ARI) Mathematics in Acoustics 5 / 21

8 Signal Processing : Time Frequency Analysis Peter Balazs (ARI) Mathematics in Acoustics 6 / 21

9 Short Time Fourier Transformation (STFT) Definition Let f,g 0 in L 2 ( R d), then we call V g f (τ, ω) = R d f (x)g(x τ)e 2πiωx dx. the Short Time Fourier Transformation (STFT) of the signal f with the window g. Peter Balazs (ARI) Mathematics in Acoustics 7 / 21

10 Short Time Fourier Transformation (STFT) Peter Balazs (ARI) Mathematics in Acoustics 8 / 21

11 Applied Mathematics, part 2 Peter Balazs (ARI) Mathematics in Acoustics 9 / 21

12 Applied Mathematics: System Identification Multiple Exponentiell Sweeps Method Peter Balazs (ARI) Mathematics in Acoustics 10 / 21

13 Numerical Mathematics Peter Balazs (ARI) Mathematics in Acoustics 11 / 21

14 Perfect Reconstruction Resynthesis I Commonly used windows and their spectra: Peter Balazs (ARI) Mathematics in Acoustics 12 / 21

15 Perfect Reconstruction Resynthesis II Overlap Add: Comparison of Errors for N win = 1024 and overlap = 50%.: window \ error max. rel. error rel. err. rand. sig. rel. err. audio sig. Hanning Hamming Rectangular e e 008 Bartlett e e 008 Blackman Harris Trunc. Gaussian Kaiser (β = 0.5) Tukeywin Peter Balazs (ARI) Mathematics in Acoustics 13 / 21

16 Perfect Reconstruction Resynthesis III Frame theory = perfect reconstruction window \ error rel. err. audio sig. (50%) rel. err. audio sig. (25%) rel. err. audio sig. (12.5%) Hanning e e e 008 Hamming e e e 008 Rectangular 2.092e e e 016 Bartlett e e e 008 Blackman Harris e e e 008 Trunc. Gaussian e e e 008 Kaiser (β = 0.5) e e e 009 Tukeywin e e e 009 Table: dual method: Comparison of relative Errors for N win = 1024 and different overlaps. Peter Balazs (ARI) Mathematics in Acoustics 14 / 21

17 Double Preconditioning I To find dual window efficiently: P = C 1 D (S) 1 S D(S) 1 Figure: The double preconditioning matrix - Parameter: g, a,b - Initialization: B = block(g, a, b) - Preconditioning : P 1 = inv block (diag block (B)) S 1 = P 1 block B P 2 = inv block (circ block (S 1)) S 2 = P 2 block S 1 Figure: The double preconditioning algorithm Peter Balazs (ARI) Mathematics in Acoustics 15 / 21

18 Double Preconditioning II 1 Original window Canonical dual Diagonal dual Circulant dual Double dual Peter Balazs (ARI) Mathematics in Acoustics 16 / 21

19 Application-oriented Mathematics Abstract Nonsense with Motivation in Applications Peter Balazs (ARI) Mathematics in Acoustics 17 / 21

20 Frames I : definition Definition The sequence (g k k K) is called a frame for the Hilbert space H, if constants A, B > 0 exist, such that A f 2 H k f, g k 2 B f 2 H f H Peter Balazs (ARI) Mathematics in Acoustics 18 / 21

21 Frames I : definition Definition The sequence (g k k K) is called a frame for the Hilbert space H, if constants A, B > 0 exist, such that A f 2 H k f, g k 2 B f 2 H f H Gabor frame : (g m,n ) = (M nb T ma g) for some a, b. frames = spanning systems in H frames = generalization of bases frame condition = generalization of Parseval s theorem Perfect reconstruction is guaranteed with the canonical dual frame g k = S 1 g k with S the frame operator (i.e. combined analysis/resynthesis operator). Peter Balazs (ARI) Mathematics in Acoustics 18 / 21

22 Frame Multiplier I : definition Definition Let H 1, H 2 be Hilbert-spaces, let (g k ) k K be a frame in H 1, (f k ) k K in H 2. Define the operator M m,(fk ),(g k ) : H 1 H 2, the frame multiplier for these frames as the operator M m,(fk ),(g k )f = k m k f, g k f k where m l (K) is called the symbol. Peter Balazs (ARI) Mathematics in Acoustics 19 / 21

23 Frame Multiplier II : exemplary new theoretical result Theorem Let M m,fk,g k be a frame multiplier for {g k } and {f k } with the upper frame bounds B and B respectively. Then 1 If m l M is a well defined bounded operator. M Op B B m. 2 M m,f k,g k = M m,gk,f k. Therefore if m is real-valued and f k = g k, M is self-adjoint. 3 If m c 0, M is compact. 4 If m l 1, M is a trace class operator with M trace B B m 1. And tr(m) = m k f k, g k. k 5 If m l 2, M is a Hilbert Schmidt operator with M HS B B m 2. Peter Balazs (ARI) Mathematics in Acoustics 20 / 21

24 Personal References: P. Balazs, Regular and Irregular Gabor Multipliers with Application to Psychoacoustic Masking, PhD Thesis, Universität Wien (2005) P. Balazs, H. G. Feichtinger, M. Hampejs, G. Kracher, Double preconditioning for Gabor frames, accepted for IEEE Trans. Signal Processing (2005) P. Balazs, Basic Definition and Properties of Bessel Multipliers, Journal of Mathematical Analysis and Applications (in press, available online) P. Balazs, W. Kreuzer, H. Waubke, A stochastic 2D-model for calculating vibrations in liquids and soils,accepted for Journal of Computational Acoustics (2006) P. Majdak, P. Balazs, Multiple Exponential Sweep Method with Application to HRTF measurements, preprint P. Balazs, J.-P. Antoine, Weighted and controlled frames, submitted Peter Balazs (ARI) Mathematics in Acoustics 21 / 21