A time-frequency technique for blind separation and localization of pure delayed sources Dimitri Nion, Bart Vandewoestyne, Siegfried Vanaverbeke, Koen Van Den Abeele, Herbert De Gersem, Lieven De Lathauwer K. U. Leuven Campus Kortrijk, Group Science, Engineering and Technology, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium
We present a time-frequency technique for the blind separation and localization of several sources, where a single scaled and delayed version of each source contributes to each sensor recording. The separation is performed in the time-frequency domain via an Alternating Least Squares (ALS) algorithm coupled with a Vandermonde structure enforcing strategy. The Time Differences Of Arrival (TDOAs) estimates are then exploited to localize the sources individually. Problem Formulation • Parameters
ALS algorithm with Vandermonde structure
and propagation model:
• Cost
function and optimization problem
– N : nb. of sources sn(t), n = 1, . . . , N . – M ≥ N : nb. of sensors; received signals rm(t), m = 1, . . . , M . – amn: attenuation factor between nth source and mth sensor. – τmn: Time Of Arrival (TOA), in seconds, between nth source and mth sensor. – Linear time-shift mixing model: line of sight propagation, no reflections [1].
F X def γ = kR(f ) − H(f ) · S(f )k2. f =1
min
{H(f ),S(f )}Ff=1
• Time-shift
model: rm(t) =
N X
amnsn(t − τmn)
(1) STEP 1: Time-frequency computation Build R(f ) ∈ CM ×P , f = 1, . . . , F from FFT of P overlapping windowed frames of recorded signals. (Typical parameters: F = 2048, Hanning window, 50% overlap).
Time-frequency reformulation • Parameters: – F : Length of each DFT frame, f = 1, . . . , F . – P : nb. of (possibly overlapping) frames, p = 1, . . . , P . – rm(p, f ): (p, f )th time-frequency sample of the mth recording. – sn(p, f ): (p, f )th time-frequency sample of the nth source. – Dmn: Time Of Arrival (TOA), in number of samples, between nth source and mth sensor. – ω = exp(−2jπ/F ).
discrete mixing model:
STEP 2: Blind separation —— Initialization ———ˆ )∈ stop=0, k = 1, Kmax (e.g., Kmax = 200) and ǫ (e.g., ǫ = 10−6). Randomly generate S(f CN ×P , f = 1, . . . , F . Possibly try several random starting points. —– Start alternating updates ——— while stop=0 k =k+1 ˆ (LS)(f ) = R(f ) · S(f ˆ )†, f = 1, . . . , F. (2.a). H (LS) ˆ˜ , aˆ˜ } ← periodogram(h ˆ (2.b). {D mn mn mn ), m = 1, . . . , M, n = 1, . . . , N., see [2]. ˆ˜ ˆ˜ (V DM ) D (F −1) D ˆ mn mn ], m = 1, . . . , M, n = 1, . . . , N. ˆ ˆ ˆ , . . . , a˜mnw hmn ← [a˜mn, a˜mn w ˆ (V DM )(f )† · R(f ), f = 1, . . . , F. ˆ )=H (2.c). S(f
– Analytic model: rm(p, f ) ≃
N X
γ
s.t. hmn defined in (4) is a Vandermonde vector, ∀m, ∀n,
n=1
• Time-Frequency
(8)
if (k = Kmax) or (|γ (k) − γ (k−1)| ≤ ǫ);stop=1;end amnω (f −1)Dmn sn(p, f ), f = 1, . . . , F.
(2)
end
(3)
STEP 3: Blind localization (rel) ˆ˜ ˆ˜ . ˜ and compute TDOAs D ˆ mn - Choose ref. sensor M =D − D mn ˜n M - Each source is localized individually on the basis of its TDOAs; its x and y coordinates can be estimated in the least squares sense, see [3,4].
n=1
– Matrix format: R(f ) ≃ H(f ) · S(f ), f = 1, . . . , F, where def [R(f )] m,p = rm(p, f ) is the M × P time-frequency observed matrix, * def
* [S(f )]n,p = sn(p, f ) is the N × P rank-N time-frequency source matrix, def (f −1)Dmn is the M × N rank-N mixing matrix. [H(f )] = a ω m,n mn *
Numerical experiments
– Additional structure: Vandermonde vectors
5
6
R=
N X
Sn •2 HTn .
(4)
(5)
n=1 M
F
5
2 sources
s4 (2.9, 4.1) 4
s3 (1.6, 3.7)
3
s2 (3.4, 3.7)
s1 (2.3, 1.9)
2
4
10
1
P
P
M
=
3
0 0
F F
4 sources 3 sources
TDOA MSE
– Tensor format:
= amn, amnω Dmn , . . . amnω (F −1)Dmn
iT
y coordinate (in meter)
hmn = [Hmn(1), Hmn(2), . . . , Hmn(F )]
T
h
10
1
2
3
4
5
10 −20
6
−15
−10
−5
x coordinate (in meter)
N X
0 SNR [dB]
5
10
15
20
F
(a) Spatial configuration
n=1
(b) MSE of TDOA 1
• Model
Hn (Vandermonde vectors)
ambiguities: N X
T · Z ). R= (Sn •2 Z−1 ) • (H n 2 n n n=1
(6) def
– To preserve the whole structure, Zn has to be diagonal and un = diag(Zn) has to be a Vandermonde vector: ˆ n = diag([αn, αnω φn , . . . , αnω (F −1)φn ])Hn, H
(7)
with unknown arbitrary scaling factor αn and phase factor φn. def
˜ mn def – In case of perfect separation, we get the estimates: a˜mn = amnαn and D = Dmn + φn. – The ambiguities {αn, φn} only depend on the source and can be removed by choosing a (rel) def a˜mn ˜ , and work with the relative attenuation factor amn = a˜ ˜ = aamn reference sensor, say M ˜ Mn
Mn
(rel) def ˜ ˜ ˜ = Dmn − D ˜ . and the relative Time Difference Of Arrival (TDOA) Dmn = D mn − D Mn Mn
4 sources 3 sources 2 sources
100 90
4 sources 3 sources 2 sources
0
10
80
MSE source coordinates
Sn (diagonal slices)
R
Percentage of non−perfectly estimated TDOAs
10
70 60 50 40 30 20
−1
10
−2
10
−3
10
−4
10
10 0 −20
−5
−15
−10
−5
0 SNR [dB]
5
10
15
(c) % of non-perfectly estimated TDOAs
20
10 −20
−15
−10
−5
0 SNR [dB]
5
10
15
20
(d) MSE of source coordinates
Figure 1: Spatial configuration and results of Monte-Carlo experiments. [1] A. Yeredor, “Blind Source Separation with Pure Delays Mixture”, ICA’01, 2001. [2] D. C. Rife, R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations”, IEEE Trans. Inform. Theory, IT-20(5), pp. 591–598, 1974 [3] K. W. Cheung, H. C. So, W. K. Ma, Y. T. Chan, “A constrained least squares approach to mobile positioning: algorithms and optimality”, EURASIP J. on Applied Sig. Proc., pp.1–23, 2006. [4] Y. Zhou, L. Lamont, “Constrained least squares approach for TDOA localization: a global optimum solution”, ICASSP’08, pp.2577–2580, 2008.
