Short-time Fourier transform x ∈ C N is The short-time Fourier transform of � N � � ψ ↓ s ( 1 − α ov ) ℓ x , � STFT [ ℓ ] ( � x )[ k , s ] := � , 0 ≤ k ≤ ℓ − 1 , 0 ≤ s ≤ ( 1 − α ov ) ℓ, k � � � i 2 π kj w [ ℓ ] ( j ) exp if 1 ≤ j ≤ ℓ � ℓ ψ k [ j ] := 0 otherwise Overlap between adjacent segments equals α ov ℓ
k = 3 s = 2 ( N := 500, ℓ := 128, α ov := 0 . 5) 1.0 0.5 0.0 Basis vector 0.5 1.0 0 100 200 300 400 500 Time 1.0 60 0.8 50 Frequency 40 0.6 STFT 30 0.4 coefficient 20 0.2 10 0 0.0 0 1 2 3 4 5 6 7 8 9 Time
k = 3 s = 3 ( N := 500, ℓ := 128, α ov := 0 . 5) 1.0 0.5 0.0 Basis vector 0.5 1.0 0 100 200 300 400 500 Time 1.0 60 0.8 50 Frequency 40 0.6 STFT 30 0.4 coefficient 20 0.2 10 0 0.0 0 1 2 3 4 5 6 7 8 9 Time
k = 8 s = 2 ( N := 500, ℓ := 128, α ov := 0 . 5) 1.0 0.5 0.0 Basis vector 0.5 1.0 0 100 200 300 400 500 Time 1.0 60 0.8 50 Frequency 40 0.6 STFT 30 0.4 coefficient 20 0.2 10 0 0.0 0 1 2 3 4 5 6 7 8 9 Time
k = 23 s = 6 ( N := 500, ℓ := 128, α ov := 0 . 5) 1.0 0.5 0.0 Basis vector 0.5 1.0 0 100 200 300 400 500 Time 1.0 60 0.8 50 Frequency 40 0.6 STFT 30 0.4 coefficient 20 0.2 10 0 0.0 0 1 2 3 4 5 6 7 8 9 Time
Matrix representation of STFT 0 0 0 0 0 0 F [ ℓ ] · · · � � · · · diag w [ ℓ ] 0 0 0 0 0 0 · · · · · · 0 0 0 0 0 0 F [ ℓ ] · · · � � · · · diag w [ ℓ ] 0 0 0 0 0 0 � x , · · · · · · 0 0 0 0 0 0 F [ ℓ ] · · · � � · · · diag w [ ℓ ] 0 0 0 0 0 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Computing the STFT Number of segments: n seg := N / ( 1 − α ov ) ℓ Complexity of multiplication by window: Complexity of applying DFT: Total complexity:
Computing the STFT Number of segments: n seg := N / ( 1 − α ov ) ℓ Complexity of multiplication by window: n seg ℓ Complexity of applying DFT: Total complexity:
Computing the STFT Number of segments: n seg := N / ( 1 − α ov ) ℓ Complexity of multiplication by window: n seg ℓ Complexity of applying DFT: n seg ℓ log ℓ (FFT) Total complexity:
Computing the STFT Number of segments: n seg := N / ( 1 − α ov ) ℓ Complexity of multiplication by window: n seg ℓ Complexity of applying DFT: n seg ℓ log ℓ (FFT) Total complexity: O ( N log ℓ ) (overlap is a fixed fraction)
Inverting the STFT Apply inverse DFT to each segment Combine segments Same complexity
Speech signal (window length = 62.5 ms) 10000 5000 0 5000 10000 0 1 2 3 4 5 6 Time (s)
Speech signal (window length = 62.5 ms) 4000 10 3 10 2 3000 Frequency (Hz) 10 1 2000 10 0 10 1 1000 10 2 10 3 0 0 1 2 3 4 5 6 Time (s)
Windowing Short-time Fourier transform Multiresolution analysis Denoising via thresholding
Image
Vertical line (column 135) 1.0 0.8 0.6 0.4 0.2 0 100 200 300 400 500
Multiresolution analysis Scale / resolution at which information is encoded is not uniform Goal: Decompose signals into components at different resolutions
Multiresolution decomposition Let N := 2 K for some K , a multiresolution decomposition of C N is a sequence of nested subspaces V K ⊂ V K − 1 ⊂ . . . ⊂ V 0 satisfying: ◮ V 0 = C N ◮ V k is invariant to translations of scale 2 k for 0 ≤ k ≤ K . If � x ∈ V k then x ↓ 2 k ∈ V k � for all l ∈ Z ◮ For any � x ∈ V j that is nonzero only between 1 and N / 2, the dilated vector � x ↔ 2 belongs to V j + 1
Dilation x ∈ C N be such that � Let � x [ j ] = 0 for all j ≥ N / M , where M is a positive integer The dilation of � x by a factor of M is �� j �� � x ↔ M [ j ] = � x M
How to build a multiresolution decomposition ◮ Set the coarsest subspace to be spanned by a low-frequency vector � ϕ , called a scaling vector or father wavelet V K := span ( � ϕ ) .
How to build a multiresolution decomposition ◮ Decompose the finer subspaces into the direct sum V k := W k ⊕ V k + 1 , 0 ≤ k ≤ K − 1 , where W k captures the finest resolution available at level k ◮ Set W k to be spanned by shifts of a vector � µ dilated to have the appropriate resolution: V k := W k ⊕ V k + 1 , 0 ≤ k ≤ K − 1 , N − 1 2 k + 1 � µ ↓ m 2 k + 1 � � W K − 1 := span � . ↔ 2 k m = 0 The vector � µ is called a mother wavelet
Challenge How to choose mother and father wavelets? If chosen appropriately, basis vectors can be orthonormal
Haar wavelet basis The Haar father wavelet � ϕ is a constant vector, such that 1 √ ϕ [ j ] := � , 1 ≤ j ≤ N N The mother wavelet � µ satisfies − 1 2 , j = 1 , √ 1 µ [ j ] := � 2 , j = 2 , √ 0 , j > 2 Other options: Meyer, Daubechies, coiflets, symmlets, etc.
Haar wavelets Scale Basis functions 2 0
Haar wavelets Scale Basis functions 2 0
Haar wavelets Scale Basis functions 2 0
Haar wavelets Scale Basis functions 2 0 2 1
Haar wavelets Scale Basis functions 2 0 2 1
Haar wavelets Scale Basis functions 2 0 2 1
Haar wavelets Scale Basis functions 2 0 2 1 2 2
Haar wavelets Scale Basis functions 2 0 2 1 2 2
Haar wavelets Scale Basis functions 2 0 2 1 2 2 2 3
Haar wavelets Scale Basis functions 2 0 2 1 2 2 2 3
Multiresolution decomposition V K
Multiresolution decomposition V K W 2
Multiresolution decomposition V K W 2 W 1
Multiresolution decomposition V K W 2 W 1 W 0 x at scale 2 k P V k � x is an approximation of �
Vertical line (column 135) 1.0 0.8 0.6 0.4 0.2 0 100 200 300 400 500
Scale 2 9 Approximation Coefficients 1.0 Data Approximation 16 0.8 14 12 0.6 10 8 6 0.4 4 2 0.2 0 0.04 0.02 0.00 0.02 0.04 0 100 200 300 400 500
Scale 2 8 Approximation Coefficients 1.0 Data Approximation 4.0 0.8 3.5 3.0 2.5 0.6 2.0 1.5 0.4 1.0 0.5 0.2 0.0 0.04 0.02 0.00 0.02 0.04 0 100 200 300 400 500
Scale 2 7 Approximation Coefficients 1.0 Data Approximation 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.0 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500
Scale 2 6 Approximation Coefficients 1.0 Data Approximation 1.4 0.8 1.2 1.0 0.6 0.8 0.6 0.4 0.4 0.2 0.0 0.2 0 1 2 3 0 100 200 300 400 500
Scale 2 5 Approximation Coefficients 1.0 Data Approximation 0.8 0.6 0.5 0.6 0.4 0.3 0.2 0.4 0.1 0.0 0.2 0 2 4 6 0 100 200 300 400 500
Scale 2 4 Approximation Coefficients 1.0 Data Approximation 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.0 0.2 0.2 0 5 10 15 0 100 200 300 400 500
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