SLIDE 1
Matching the Analysis Scheme to the Signal Fritz Menzer - - PowerPoint PPT Presentation
Matching the Analysis Scheme to the Signal Fritz Menzer - - PowerPoint PPT Presentation
Time-Frequency Analysis for Audio Workshop Matching the Analysis Scheme to the Signal Fritz Menzer (fritz.menzer@epfl.ch) Communication Systems, 5 th year Ecole Polytechnique F ed erale de Lausanne 15th April, 2004 Overview 1
SLIDE 2
SLIDE 3
1 Introduction
- If you know what you’re looking at, you can examine it more
precisely.
3
SLIDE 4
2 Perfect Reconstruction - who cares?
2.1 Definition of perfect reconstruction
- Definition: Perfect Reconstruction (PR) method: method
providing direct and inverse transforms T and T −1 such that for any signal s, T −1(T (s)) = s
- FFT based methods, Cosine Modulated Filterbanks and Wavelet
transforms are usually PR methods.
- Simple operations like filtering or distortion do not necessarily
allow PR (i.e. it may be impossible to find T −1). Example: Quantisation obviously does not allow to reconstuct the
- riginal signal perfectly.
4
SLIDE 5
2.2 Do we need perfect reconstruction?
5 10 15 20 −2 −1.5 −1 −0.5 0.5 1 1.5 5 10 15 20 −2 −1.5 −1 −0.5 0.5 1 1.5 samples 5 10 15 20 50 100 150 5 10 15 20 20 40 60 80 frequency [kHz]
Noise Noise, down- and upsampled by 4
5
SLIDE 6
Do we need perfect reconstruction?
- Not needed for:
– Modifying a signal – Handling noise – If the nature of the signal is known
- Why use PR methods for compression?
– Generality (ideally any signal can be treated) – Localising the source of errors!
6
SLIDE 7
3 Harmonic Band Wavelet Transform (Polotti and Evangelista, 2000)
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
❄
2
✲ ✲ ❄ ✲ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲
x(n) ... ... ... P φ(k) ψ(k) φ(k) ψ(k) ... P φ(k) ψ(k) φ(k) ψ(k) ... P φ(k) ψ(k) φ(k) ψ(k) ... DC Comp. g1(k) gP−1(k) Sinusoidal Part Sinusoidal Part g0(k)
7
SLIDE 8
frequency [Hz] time [sec] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
SLIDE 9
time [sec] frequency [Hz] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
SLIDE 10
3.1 Coefficient modeling
- Wavelet Transform
- Model the scale residual sinusoidally
- Model the wavelet coefficients using LPC
0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7
- n=1
|
1,0( )
|
2,0( )
|
3,0( )|
|
4,0( )|
|
4,0( )
n=2 n=3 N=4 scale residual ω ω ω ω ω Φ | Ψ Ψ Ψ Ψ | |
10
SLIDE 11
3.2 Advantages / Drawbacks
+ Meaningful adaptation of frequency and time resolution = ⇒ Visually better resolution + Reasonable model for the coefficients − Works only for monophonic, harmonic sounds − No model for the transients
11
SLIDE 12
4 From HBWT to inharmonic sound modeling
time [sec] frequency [Hz] 1 2 3 4 5 6 7 200 400 600 800 1000 1200 1400 1600 1800
SLIDE 13
4.1 Taking filters from different PR filterbanks
ω 1st partial 2nd partial 3rd partial . . . ω 1st partial 2nd partial 3rd partial . . .
SLIDE 14
4.2 Why aliasing is not a problem
If a sinusoid of the form sin
ˆ
kπ P t + ϕ
is the input to a P-channel cosine modulated filterbank, only two bands will output nonzero coefficients: |Hk(ej
ˆ kπ P )| = 0 ⇔ k ∈ {ˆ
k − 1, ˆ k}
ω partial’s frequency
= ⇒ there is no aliasing of the sinusoidal part, but only of the part that we model as noise!
14
SLIDE 15
SLIDE 16
SLIDE 17
4.3 Method Overview
Analysis
analyse signal → find N partials → determine filterbank ↓ calculate 2N sets of filterbank coefficients + residual ↓ calculate wavelet transform (WT) of filterbank coefficients ↓ model the WT coefficients sinusoidally and with LPC
Synthesis
reconstruct WT coefficients ↓ perform inverse wavelet transform → get filterbank coefficients ↓ inverse filterbank ↓ add residual (or not)
17
SLIDE 18
4.4 Sounds
- Original Gong
- Reconstructed from the Filterbank Coefficients
- Synthesized from model parameters
- 1 octave pitch-shifted Gong
- Time-stretched Gong
- Sinusoidal-only Gong
- First wavelet scale only
- Harmonic Gong
18
SLIDE 19
5 Time-Frequency Analysis and Granular Synthesis
- Any Time-Frequency Transform implements a sort of Granular
Synthesis.
- Each coefficient corresponds to a grain
- Grains are played at precise instants (instead of randomly)
- To produce a grain, set all coefficients to zero, except one that
will be set to one. Then perform the inverse transform.
19
SLIDE 20
Windowed FFT (STFT) grain
2 4 6 8 10 12 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x 10−3 time [msec]
play
20
SLIDE 21
Cosine Modulated Filterbank grain
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 time [msec]
play
21
SLIDE 22
Full-tree wavelet “grain”
10 20 30 40 50 60 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 time [msec]
play
22
SLIDE 23
HBWT grain (noise part)
2 4 6 8 10 12 14 16 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 time [msec]
play
23
SLIDE 24
HBWT grain (sinusoidal part)
20 40 60 80 100 120 140 160 −0.05 −0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04 0.05 time [msec]
play
24
SLIDE 25
5.1 Time-domain effects
1 2 3 4 5 6 7 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 time [msec] 1 2 3 4 5 6 7 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 time [msec]
Channel 8:
- ne grain
played continuously Channel 9:
- ne grain
played continuously
25
SLIDE 26
5.2 Scale of all grains in a 1024-band full-tree wavelet decomposition
time [sec] frequency [Hz] 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 104
play
26
SLIDE 27
A References
- Article on Harmonic Band Wavelet Transform by Polotti and
Evangelista http://lcavwww.epfl.ch/publications/publications/2000/PolottiE00b.pdf
- DAFx 2002 paper on adaptation to inharmonic sounds
http://lcavwww.epfl.ch/publications/publications/2002/PolottiE02.pdf
- Some material (presentation slides, Matlab functions and pure data
- bjects)