1. Analysis of Audio Using PCA
SAMEER MESHRAM (M.S.)
VARUN RUPCHANDANI (M.S.)
VIDUR VIJ (M.S.)
June 4, 2018

2. CONTENTS Sound Waves - Basics Problem Statement / Proposed Algorithm
Overview VMD and PCA
Algorithm Implementation
Results
Conclusion i

3. SOUND WAVES - Basics A
Sound waves are disturbances in space caused by vibration from a source.
Usually represented in the form of a sinusoid:
𝑠 𝑡 =𝐴∗ 𝑠𝑖𝑛 2𝜋𝑓𝑡 + ∅ =𝐴∗𝑠𝑖𝑛 2𝜋 𝑇 𝑡+ ∅
Since analog signal A/D conversion is required for storage and processing
Audible range for humans: 20 Hz to 20 kHz
Fig : (A) Frequency and wavelength of a typical sound wave (F. Camastra et al 2015) 1

4. Algorithm for the proposed method
PROBLEM STATEMENT
Separation of two distinct audio sources from a single mixed audio signal
Applications: Bio-medical signal processing, speech recognition, etc. A
Fig : (A) Representation of individual and mixed sound signals
Algorithm for the proposed method
1. Input observed signal x(t)
2. Apply VMD on x(t) to get n modes : M 1 , M 2 , - up to M n
3. Apply PCA on M 1 to M n to select m principal components : P 1 , P 2 ,- up to P m
4. Extracted sources : P 1 , P 2 ,- up to P m 2

5. VARIATIONAL MODE DECOMPOSITION (VMD)
VMD decomposes a signal into a number of modes
Each mode is band-limited about its center frequency
It decomposes a real valued signal into a finite number of modes
Principal components of these modes are selected to reconstruct the original signal.
Fig : (A) VMD decomposition of noisy tri-harmonic.
(a) The noisy input signal.
(b)-(d) The three modes extracted by de-noising VMD, 3

6. PRINCIPAL COMPONENT ANALYSIS (PCA)
Principal Component Analysis transforms the data into a new domain
The new basis are perpendicular to each other
The new basis of the data is along the direction of max variance (PC1)
Data can be decomposed using eigenvalue decomposition of covariance matrix
Or SVD of data matrix
PC1
Fig : (A) Representation of the concept of PCA 4

7. ALGORITHM IMPLEMENTATION - I
Signal Input: 𝑥 𝑡 = 𝑥 1 𝑡 + 𝑥 2 𝑡
Case 1
Case 2
𝑥 1 𝑡 =sin(3t)
𝑥 1 𝑡 ="Cymbals audio“ [3 - 5 kHz]
𝑥 𝟐 𝑡 =sin(7t)
𝑥 𝟐 𝑡 ="Bass Drum audio" [ Hz]
Matlab function(s): audioread() A B
Fig : (A) Unmixed and mixed audio signal for case 1, (B) Unmixed and mixed audio signal for case 2 5

8. ALGORITHM IMPLEMENTATION - II
Variational Mode Decomposition on : 𝑥 𝑡 = 𝑥 1 𝑡 + 𝑥 2 𝑡
Input
𝒙 𝑡 =sin 3t +sin(7t)
𝒙 𝑡 ="Cymbals + Bass Drums"
Output
Modes: M 1 , M 2 , - M n
Matlab function(s): VMD()
Principal Component Analysis on VMD Modes : M 1 , M 2 , - M n
Input
Modes of 𝒙 𝑡
Output
Extracted Principal Components: P 1 , P 2 ,- P m
Matlab function(s): pca()
Signal Reconstruction:
𝐏𝐂𝐀 𝐫𝐞𝐜𝐨𝐧𝐬𝐭𝐫𝐮𝐜𝐭𝐢𝐨𝐧=𝐏𝐂 𝐬𝐜𝐨𝐫𝐞𝐬 × 𝐄𝐢𝐠𝐞𝐧𝐯𝐞𝐜𝐭𝐨𝐫𝐬 𝐓 +𝐌𝐞𝐚𝐧 6

9. RESULTS - Case 1 (Sinusoids)B
Fig : (A) Unmixed audio signal 1 for case 1,
(B) Unmixed audio signal 2 for case 1 7

10. RESULTS - Case 1 (Sinusoids)B
Fig : (A) Spectrogram of unmixed audio signal 1 for case 1, (B) Spectrogram of unmixed audio signal 2 for case 1 8

11. RESULTS - Case 1 (Sinusoids)
Comparison between original and separated signal
For 𝒙 𝟏 𝒕 =𝐬𝐢𝐧(𝟑𝐭)
Correlation coefficient
0.9872
Percentage error
6.3593
Root mean square error
0.0187
For 𝒙 𝟐 𝒕 =𝐬𝐢𝐧(𝟕𝐭)
Correlation coefficient
0.9987
Percentage error
4.5866
Root mean square error
0.0053 9

12. RESULTS - Case 2 (Cymbals & Bass Drums)
Fig : (A) Unmixed audio signal for cymbal,
(B) Unmixed audio signal for bass drum 10

13. RESULTS - Case 2 (Cymbals & Bass Drums)
Fig : (A) Spectrogram of unmixed audio signal for cymbal, (B) Spectrogram of unmixed audio signal for bass drum 11

14. RESULTS - Case 2 (Cymbals & Bass Drums)
Comparison between original and separated signal
For Cymbal
Correlation coefficient
0.6794
Percentage error
Root mean square error
0.1601
For Bass Drum
Correlation coefficient
0.9988
Percentage error
Root mean square error
0.0489 12

15. CONCLUSION
Algorithm based on VMD & PCA was implemented for audio separation
Mixture of pure sinusoids is separated very efficiently
Instruments cymbal and bass drum are separated satisfactorily
The algorithm has scope for further improvement 13

16. THE END
Thank You!
Questions?