Fast Online Source Steering Algorithm for Tracking Single Moving Source using Online Independent Vector Analysis
Taishi Nakashima (TMU)

Rintaro Ikeshita (NTT)

Nobutaka Ono (TMU)

Shoko Araki (NTT)Tomohiro Nakatani (NTT)ICASSP 2023, Paper ID: 2514

Conventional and Proposed Methods// “Predict” estimated sources𝒚𝑡̂ ← 𝑊𝑡−1 𝒙𝑡
for 𝑘 = 1, … , 𝐾

Fixed

𝑟𝑘,𝑡 ← √Σ𝑓 |𝑦𝑘,𝑓,𝑡̂Moved// Update covariance+• Online IP [Taniguchi 2014]∘ Must update all demixing vectors
� Not suitable for 1-ST+• Online ISS [Nakashima 2022]∘ Can directly update 𝒂ℓ,𝑡
🤔 Suitable for 1-ST yet not optimalContribution// Update “target” 𝒘ℓ,𝑡Complexity for 1-ST
𝒚𝑡 = 𝑊𝑡 𝒙𝑡 Optimize 𝑊𝑡
2Batch Optimal(*per freq.)
IP
ISS

𝑂(𝐾 𝑇 )𝑂(𝐾 2 𝑇 )𝑂(𝐾 2 𝑇 )Online Optimal(*per frame/freq.)
IP
ISS

2𝑂(𝐾 )𝑂(𝐾 2 )𝑂(𝐾 2 )OSS†
𝑂(𝐾 2 )
Proposed
FastOSS 𝑂(𝐾 2 )† online source steering⋆ 𝐾, 𝑇 : # of sources/framesℓ: target source index// Update estimated sources
3𝒚𝑡 ← 𝑊𝑡 𝒙𝑡
2� Calc. 𝑉𝑘,𝑡 , 𝑔𝑘,𝑡 , 𝐺𝑘,𝑡 : 𝑂(𝐾 ) for each 𝑘 � 𝑂(𝐾 ) in total

Derivation of OSS

Parameter:𝒂∈ℂℓ
𝑂(𝐾𝑇 ) �with𝒚,…
,𝒚3
1𝑇𝑂(𝐾 𝑇 )𝐾→0𝑂(𝐾𝑇 ) � Optimal
𝑂(𝐾) � Parameter: 𝒂ℓ,𝑡 ∈ ℂ
𝑂(𝐾 3 )𝑂(𝐾 3 ) � Not optimal...

𝐾𝑂(𝐾 2 ) � Faster than Online ISS
𝑂(𝐾) � Optimal

Online Blind source separation (BSS)
Estimated
BSS• BSS using past/current data
• Necessary for real-time app.
• Popular methods:∘ Online IVA [Kim2010][Taniguchi+ 2014]∘ Online AuxIVA

Observed

Estimated∗≔ 𝑦ℓ,𝑡̂

≔ 𝑦ℓ,𝑡̂𝐺𝑘,𝑡−1→𝐺ℓ,𝑡−1𝐺𝑘,𝑡−1= 𝛼𝐺+ (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )|𝑦ℓ,𝑡
̂ |2ℓ,𝑡−1Parameter� Covariance update
𝑉𝑘,𝑡 ← 𝛼𝑉𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝒙𝑡 𝒙H𝑡Forgetting factor Prev. cov.

Weight� Optimization of 𝑊𝑡 for batch AuxIVA directly applicable∘ Iterative projection (IP) [Ono2011][Scheibler+ 2020]∘ Iterative source steering (ISS)

Current obs.

was given in an oracle manner

Source 1
Update all

Source 2Update all

Source 3
Update all

Source 4
Update all⚠Outputs of online ISS, OSS, andFastOSS are theoretically identical

010Move00104030Time (s)205060 0OSS/FastOSS

1020Online IP

4030Time (s)50602• Proposed OSS/FastOSS updates only 𝐾 parameters (𝒂ℓ,𝑡 )� OSS/FastOSS performs as well as online IP*Experiments were run on AMD EPYC 7643 48-core processor@2.3 GHz, and signal length was 30 s.

OSS

Proposed

4.48

Method// “Predict” estimated sources𝒚𝑡̂ ← 𝑊𝑡−1 𝒙𝑡
for 𝑘 = 1, … , 𝐾// Update “weight”

𝑉𝑘,𝑡 ← 𝛼𝑉𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝒙𝑡 𝒙H𝑡// No covariance update!
forall 𝑘 ≠ ℓ// Update “non-target” 𝒘𝑘,𝑡∗𝑔𝑘,𝑡 ← (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝑦ℓ,𝑡
̂ 𝑦𝑘,𝑡̂𝐺𝑘,𝑡−1𝐺𝑘,𝑡 ← 𝛼 𝐺+ (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )|𝑦ℓ,𝑡
̂ |243FastOSS
OSS
Online ISS
Online IP

𝑔𝑘,𝑡𝒘𝑘,𝑡 ← 𝒘𝑘,𝑡−1 − 𝐺 𝒘ℓ,𝑡−1𝑘,𝑡𝐺ℓ,𝑡 ← 𝛼 + (1 − 𝛼)𝜑(𝑟ℓ,𝑡 )|𝑦ℓ,𝑡
̂ |2−1/2𝒘ℓ,𝑡 ← 𝐺ℓ,𝑡 𝒘ℓ,𝑡−1𝒚𝑡 ← 𝑊𝑡 𝒙𝑡// Update “target” 𝒘ℓ,𝑡// Update estimated sources

Proposed� Redefinition of OSS by multiplicative update of 𝑊𝑡
𝑊𝑡 = 𝑄𝑡 𝑊𝑡−1= 𝑄𝑡 (𝑄𝑡−1 𝑊𝑡−2 )⋮̃𝑊= 𝑄𝑡 (𝑄𝑡−1 ⋯ 𝑄1 )𝑊0 ≔ 𝑄𝑡 0
• Update one column of 𝑄𝑡 (𝐾 variables) � Calculated 𝑂(𝐾)� 𝑄𝑡 × 𝑄𝑡−1 is also 𝑂(𝐾) � Optimal cost! �

3.69

2.63

2.52

2

1.69
0.97

1

1.21

1.18

0.30 0.32

0.36 0.37

0.49 0.61

0.46 0.60

34560

The number of channelsℓ,𝑡−1� OSS requires 𝑂(𝐾 ) to update 𝒘𝑘,𝑡 for 𝑘 = 1, … , 𝐾

min. ∑𝑘 𝒘H𝑘,𝑡 𝑉𝑘,𝑡 𝒘𝑘,𝑡 − log|det 𝑊𝑡 |2� When/which source moved

Runtime performance� 𝑂(1)

2• Optimization problem

4• RT60 : approx. 300 ms0• Online IP updates 𝐾 parameters (𝒘1,𝑡 , … , 𝒘𝐾,𝑡 )𝐺𝑘,𝑡 = 𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡 𝒘ℓ,𝑡−1 = 𝒘Hℓ,𝑡−1 (𝛼𝑉𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝒙𝑡 𝒙H𝑡 ) 𝒘ℓ,𝑡−1= 𝛼𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡−1 𝒘ℓ,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )(𝒘Hℓ,𝑡−1 𝒙𝑡 )(𝒙H𝑡 𝒘ℓ,𝑡−1 )FastOSS

Online AuxIVA

4∗≔ 𝑦𝑘,𝑡̂

≔ 𝑦ℓ,𝑡̂� 𝑂(1)

𝑟𝑘,𝑡 ← √Σ𝑓 |𝑦𝑘,𝑓,𝑡̂|2Background
Observed

𝑔𝑘,𝑡 = 𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡 𝒘𝑘,𝑡−1 = 𝒘Hℓ,𝑡−1 (𝛼𝑉𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝒙𝑡 𝒙H𝑡 ) 𝒘𝑘,𝑡−1= 𝛼𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡−1 𝒘𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )(𝒘Hℓ,𝑡−1 𝒙𝑡 )(𝒙H𝑡 𝒘𝑘,𝑡−1 )∗= (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝑦ℓ,𝑡
̂ 𝑦𝑘,𝑡̂

1Moving

6.0 m10

� Construct recursive formulae for 𝑔𝑘,𝑡 , 𝐺𝑘,𝑡 to avoid 𝑉𝑘,𝑡
⋆

Fixed

Separation performance

𝑔𝑘,𝑡𝒘𝑘,𝑡 ← 𝒘𝑘,𝑡−1 − 𝐺 𝒘ℓ,𝑡−1𝑘,𝑡𝐺ℓ,𝑡 ← 𝒘Hℓ,𝑡−1 𝑉ℓ,𝑡 𝒘ℓ,𝑡−1−1/2𝒘ℓ,𝑡 ← 𝐺ℓ,𝑡 𝒘ℓ,𝑡−1

Conventional methods

3• 4 mic./sources∘ Source 4 moved instantaneously∘ The others were fixedMic.

2// Update “non-target” 𝒘𝑘,𝑡forall 𝑘 ≠ ℓ𝑔𝑘,𝑡 ← 𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡 𝒘𝑘,𝑡−1𝐺𝑘,𝑡 ← 𝒘Hℓ,𝑡−1 𝑉𝑘,𝑡 𝒘ℓ, 𝑡−1

� Fast online algorithm for 1-ST

Height = 2.75 m// Update “weight”|2𝑉𝑘,𝑡 ← 𝛼𝑉𝑘,𝑡−1 + (1 − 𝛼)𝜑(𝑟𝑘,𝑡 )𝒙𝑡 𝒙H𝑡• BSS under situations where only one source moves

MethodRoom layout

4.0 m

Online ISS

Conventional

SDR improvements (dB)Single source tracking (1-ST)

Experiments

Runtime (s)Overview

� FastOSS was 5x faster than Online ISS for 6 channels

Conclusion

References

Summary[Kim2010] T. Kim, IEEE Trans.
Circuits Syst. I, vol. 57, no. 7,
pp. 1431–1438, 2010.

• 1-ST: online BSS under a condition

where only one source moves• OSS/FastOSS∘ Fast online algorithm for 1-ST
� FastOSS achieved optimal cost

Future work
• Real-time implementation/evaluation• OSS for two source trakcking[Nakashima+ 2022] T. Nakashima
and N. Ono, in Proc. APSIPA,
pp. 750–754, Nov. 2022.
[Ono2011] N. Ono, in Proc.
WASPAA, pp. 189–192, Oct.
2011.
[Scheibler+ 2020] R. Scheibler andN. Ono, in Proc. ICASSP, pp.
236–240, 2020.
[Taniguchi+ 2014] T. Taniguchi, etal., in Proc. HSCMA, pp.
107–111, May 2014.