LINEAR MULTICHANNEL BLIND SOURCE SEPARATION BASED ON TIME-FREQUENCYMASK OBTAINED BY HARMONIC/PERCUSSIVE SOUND SEPARATION
Soichiro Oyabu† , Daichi Kitamura† , Kohei Yatabe‡†

National Institute of Technology, Kagawa College, Kagawa, Japan‡
Waseda University, Tokyo, Japan

ABSTRACT
Determined blind source separation (BSS) extracts the source signals by linear multichannel filtering. Its performance depends on theaccuracy of source modeling, and hence existing BSS methods haveproposed several source models. Recently, a new determined BSS algorithm that incorporates a time-frequency mask has been proposed.
It enables very flexible source modeling because the model is implicitly defined by a mask-generating function. Building up on thisframework, in this paper, we propose a unification of determined
BSS and harmonic/percussive sound separation (HPSS). HPSS is an
important preprocessing for musical applications. By incorporating
HPSS, both harmonic and percussive instruments can be accurately
modeled for determined BSS. The resultant algorithm estimates the
demixing filter using the information obtained by an HPSS method.
We also propose a stabilization method that is essential for the proposed algorithm. Our experiments showed that the proposed method
outperformed both HPSS and determined BSS methods including
independent low-rank matrix analysis.
Index Terms— Determined blind source separation (BSS), harmonic/percussive sound separation (HPSS), time-frequency masking, mask stabilization, plug-and-play scheme.
1. INTRODUCTION
Blind source separation (BSS) is a technique for separating individual sources from an observed mixture without any prior knowledge on the mixing system such as the positions of microphones andsource signals. In the (over-)determined situation (the number of microphones is greater than or equal to the number of sources), several
determined BSS algorithms have been developed based on the assumption of statistical independence between the sources, e.g., thefrequency-domain independent component analysis (FDICA) [1–3],
the independent vector analysis (IVA) [4–6], and the independent
low-rank matrix analysis (ILRMA) [7, 8]. This paper focuses onsuch a determined BSS algorithm.
Recently, a new class of determined BSS algorithms, which wecall the time-frequency-masking-based BSS (TFMBSS), has been
proposed [9]. Since source modeling is the key to success, TFMBSS
aims at integrating a source model that has not been utilized in determined BSS. To give some examples of conventional source modeling, IVA models co-occurrence among the frequency components ofeach source, and ILRMA models co-occurrence among the components via the low-rank time-frequency structure of the source signals.
These BSS methods have proposed such sophisticated source models
because the performance of BSS can be improved by a better model.
TFMBSS generalized a source model to a general time-frequency
mask so that it extends the class of source models to a wider one.
This work was partly supported by JSPS KAKENHI under Grants
19K20306 and 19H01116.

Some new models have successfully obtained well-performing BSS
algorithms using TFMBSS [9,10], and this strategy has capability offurther improvement by discovering a better model. In particular, anapplication-specific source model should be promising.
Harmonic/percussive sound separation (HPSS) [11–15] is one
application that requires specific modeling of source signals. Harmonic and percussive instruments have very different roles in music, and hence separating them is crucial for many applications including music analysis (estimation of chords, tempo, rhythm, notes,
genre, etc.) and remixing. HPSS separates them by modeling their
distinct spectral structures (i.e., smoothness of magnitude spectrograms along time or frequency axes). This is a concept that does
not exist in the conventional determined BSS methods handling all
source signals by the same source model. Combination of HPSS anddetermined BSS should be possible to realize a high-quality HPSS
method for multichannel observation because determined BSS performs linear filtering that do not cause nonlinear distortion. However, such combination has not been proposed yet owing to the difficulty of integrating the source models of HPSS into the conventional
BSS methods.
In this paper, by taking full advantage of the modeling capability of TFMBSS, we propose a multichannel HPSS method for spatially mixed harmonic and percussive instruments. The proposed
method incorporates a single-channel HPSS method into the iteration of TFMBSS so that the demixing filter is informed by the HPSS
method. A stabilization technique that is essential for the proposed
method is also developed in order to safely update the parameters.
By the experiments, it was shown that the proposed method outperformed both HPSS and determined BSS methods.
2. PRELIMINARIES
2.1. Harmonic/percussive sound separation (HPSS)HPSS [11–15] is a method for separating harmonic and percussive
sources based on the source models adapted for their distinct spectral patterns: the spectrograms of harmonic and percussive sources
have time- and frequency-continuous structures, respectively. Let
B ∈ CI×J be the complex-valued spectrogram of the observed signal. From this monaural mixture, HPSS in [12, 13] separates the
harmonic and percussive components, H ∈ CI×J and P ∈ CI×J ,respectively, by solving the following optimization problem:
min J (H, P ) s.t. |B|ξ = |H|ξ + |P |ξ ,

(1)

H,P

where | · | represents the element-wise absolute value,Xh2
κH |hi(j+1) |ρ − |hij |ρJ (H, P ) =i,j+κP |p(i+1)j |ρ − |pij |ρ

2 i

,

(2)

ξ and ρ are the domain parameters, hij and pij are the elements of
H and P , respectively, and κH > 0 and κP > 0 are the weight
coefficients for each term. The experiment in [13] confirmed thatξ = 2ρ = 1 provides a better separation performance. For ξ = 2ρ,(1) can be minimized by iterating the following update rules: [12,13]
dij |bij |2ρ,
cij + dij2cij = κ2H |h(i+1)j |ρ + |h(i−1)j |ρ ,
2dij = κ2P |pi(j+1) |ρ + |pi(j−1) |ρ ,|hij |2ρ =cij |bij |2ρ,
cij + dij|pij |2ρ =

(3)
(4)
(5)

where bij is the element of B.
2.2. Determined blind source separation (BSS)The determined BSS aims at separating each source from an observed multichannel mixture. Let the numbers of sources and microphones be denoted by N and M , respectively. The source, observed,
and estimated (separated) signals obtained via the short-time Fouriertransform (STFT) are respectively denoted as
sij = [sij1 , sij2 , · · · , sijn , · · · , sijN ]T ∈ CN ,
T

(6)

M

(7)

yij = [yij1 , yij2 , · · · , yijn , · · · , yijN ] ∈ C ,(8)

xij = [xij1 , xij2 , · · · , xijm , · · · , xijM ] ∈ C ,T

Nwhere i = 1, 2, · · · , I, j = 1, 2, · · · , J, n = 1, 2, · · · , N , and m =1, 2, · · · , M are the indices of frequency bins, time frames, sources,
and microphones (channels), respectively, and ·T denotes the transpose. We also denote the complex-valued spectrogram of the mthobserved signals in (7) as Xm ∈ CI×J , whose elements are xijm .
We assume the linear time-invariant mixing system as
xij = Ai sij ,(9)where Ai ∈ CM ×N is a frequency-wise mixing matrix, which depends on the mixing condition, such as locations of sources and microphones. This mixing model holds when the reverberation time issufficiently shorter than the window length used in STFT. If M = N
and Ai is invertible, the estimated signals can be obtained by multiplying a demixing matrix Wi ≈ A−1∈ CN ×M asiyij = Wi xij .

(10)

Since (10) is equivalent to linear filtering, artificial distortion in theestimated signals yij are minimized compared to those obtained by
nonlinear separation techniques, such as deep neural networks andsingle-channel source separation. This is a strong motivation of applying multichannel determined BSS. For this reason, in this paper,
we only focus on the determined case M = N = 2 (harmonic andpercussive sources). For the case M > N , a dimensionality reduction method can be utilized to set M = N .

Algorithm 1 TFMBSS [9, 10]
Input: X, w[1] , y[1] , µ1 , µ2 , α
Output: w[K+1]
1: for k = 1, · · · , K do
e = proxµ1 I w[k] − µ1 µ2 X H y[k]2:w
e − w[k] )3:z = y[k] + X(2w4:
M = generateMask(z)e =z−M⊙z5:y6:y[k+1] = αey + (1 − α)y[k][k+1]
e + (1 − α)w[k]
7:
w
= αw
8: end for
Let w be a vectorized form of the demixing matrices (Wi )Ii=1 ,and X be the corresponding matrix composed of the observed spectrograms (Xm )M
m=1 , so that, for all i, j, (10) can be compactly written as y = Xw. By this vectorization and matricization, TFMBSS
is given as in Algorithm 1, where ⊙ is the element-wise product,proxµ1 I is the singular-value operation defined in [16], and µ1 , µ2
and α are easily determined step-size parameters (see [10] for detailsof the algorithm). The function generateMask(·) in the fourth line
generates a time-frequency mask M. When this mask enhances thesource signals, the TFMBSS algorithm works as if the enhanced signals are obtained by the source models utilized for determined BSS.
In other words, the source model of determined BSS is implicitly defined through the mask. Any mask aiming at the target signals can beinserted into TFMBSS, and hence the problem of source modelingis reduced to design of a mask-generating function.
3. PROPOSED METHOD
3.1. Motivation
The single-channel HPSS as in Section 2.1 is frequently used as apreprocessing for music analysis. However, due to the nonlinear separation mechanism, the separated harmonic and percussive sounds
are contaminated by artificial distortions, e.g., musical noise. Such
distortions can be harmful for the following process and degrade theartistic value of music signals. In contrast, the linear spatial separation (10) can minimize the distortion in the separated signals. Hence,
combination with determined BSS should be valuable for HPSS.
To achieve distortion-less separation of harmonic and percussive instruments, we propose a unification of single-channel HPSSand determined BSS. The proposed method is based on TFMBSS
whose source model is implicitly defined via HPSS. At each iteration of TFMBSS, HPSS is performed for updating the mask M.
Then, TFMBSS updates the demixing filter w based on the mask.
To reduce instability of mask generation, we also propose a masksmoothing process that is essential for the proposed method. Thissmoothing process can stabilize the update of demixing filter, resulting in better estimation of the separated signals.
3.2. Determined BSS algorithm based on HPSS

2.3. Time-frequency-masking-based BSS (TFMBSS)
To blindly estimate a demixing matrix Wi in (10), TFMBSS has
been proposed in [9] as a new determined BSS framework. It generalized a BSS algorithm in [16] that is based on a proximal splitting
algorithm [17–20]. The important feature of TFMBSS is that thesource model for determined BSS is implicitly defined via a timefrequency mask. Therefore, this algorithm enables collaboration ofdetermined BSS with a time-frequency masking method.

A block diagram of the proposed algorithm is shown in Fig. 1. As
usual in determined BSS, the observed signal is firstly converted by
STFT. Then, a BSS algorithm is applied to estimate the demixing
filter. After separation by (10), the back projection technique [23,24]
is applied, and the inverse STFT gives the separated signals.
In the proposed method, HPSS is utilized as a mask-generating
function in the fourth line of Algorithm 1. Below, we explain thisprocess, from input z to output M, in a step-by-step manner.

Initialize

andHPSS-based mask-generating functionin the fourth line of Algorithm 1

with

HPSS
Smoothedmask

SmoothingParameters in HPSS

Smoothing
HPSS

Initialize

Mask

and

Table 1. Experimental conditions
Window function in STFT
Window length in STFT
Shift length in STFT

Parameters in TFMBSS
Number of iterations in TFMBSS

with

Hann window
128 ms64 ms
κH = 1.02, κP = 1.01
ξ = 2, ρ = 1
α = 0.25µ1 = µ2 = 1.0
500

Mask (old)Back projection
Back projection
Harmonic estimate
STFT

TFMBSS

Back
projection

Inverse
STFT
Percussive estimate

Observed signal

Fig. 1. Block diagram of the proposed method.

algorithm unstable. To stabilize the algorithm regardless of the
choice of mask-generating function, we propose a mask smoothing
technique for TFMBSS.
The algorithm can be stabilized by avoiding huge difference between the current and previous masks. Therefore, we propose thefollowing smoothing rule that is independently applied to MH andMP right after the calculation of these masks in (13):M = Mβ ⊙ Mold βold

(14)

2IJ

Firstly, the input auxiliary variable z ∈ C
is split into two
matrices, ZH ∈ CI×J and ZP ∈ CI×J . Note that this splitting ismerely reshaping of the variable, and no processing is applied. The
half of z is always assigned to ZH , and the other half is assigned to
ZP for every iteration. Then, the back projection technique [23, 24]
is applied to each of them in order to fix the frequency-wise scales.
This scale alignment is crucial for HPSS to work properly.
Secondly, two independent HPSS processes are performed.
Each HPSS is initialized by ZH and ZP as follows:|B (H) |ξ = |ZH |ξ ,|B(P ) ξ

ξ| = |ZP | ,|H (H) | = |P (H) | = |ZH |ξ /2,
|H

(P )

| = |P(P )ξ

| = |ZP | /2,
(H)

(11)
(12)

where the HPSS for ZH is performed with B , H
and P (H) ,
(P )
(P )
(P )
and that for ZP is performed with B , H
and P . By iterating the update rules in (3)–(5), two pairs of separated signals
are obtained: |H (H) |2ρ and |P (H) |2ρ are separated from ZH , and|H (P ) |2ρ and |P (P ) |2ρ are separated from ZP .
Thirdly, two Wiener-like masks [10], MH and MP , are constructed using the results of HPSS as follows:MH =|H (H) |2|H (H) |2 + |P (H) |2

,

MP =

where Mold is the mask obtained in the previous iterate, and β ≥ 0
and βold ≥ 0 are the smoothing parameters such that β + βold =1. The degree of smoothness is decided by this parameter. When
βold = 0 (i.e., β = 1), (14) does nothing to the current mask. By
increasing βold toward 1, the smoothing effect becomes stronger.
The effect of this smoothing will be discussed via an experiment.
Note that the applicability of this smoothing technique is not
limited to the proposed method. It can be applied to any BSS method
based on TFMBSS. Therefore, it should be beneficial for TFMBSSbased algorithms already proposed in the literature [9, 10].
4. EXPERIMENTS(H)|P (P ) |2
|H (P ) |2 + |P (P ) |2

, (13)

where all the operations are performed element-wise. These masksenhance the harmonic or percussive components by eliminating theother components. Therefore, the demixing filter is informed about
which component to reduce.
Finally, these masks MH ∈ [0, 1]I×J and MP ∈ [0, 1]I×J are
concatenated and vectorized to form a mask M ∈ [0, 1]2IJ that canbe applied to z ∈ C2IJ as in the fifth line of Algorithm 1. These four
steps define the function generateMask(·), which maps z to M, forthe proposed method. Note that the quality of masks depend on the
degree of success of HPSS. Since HPSS may fail to correctly separate the signals, we additionally propose a stabilization techniquethat is separately explained in the next subsection.
3.3. Mask smoothing technique for TFMBSS
Since TFMBSS is built upon a proximal algorithm [10], its updateshould stay in the proximity of the previous state. However, a maskgenerating function may violate this requirement, which makes the

4.1. Conditions
We conducted experiments of separating drums and other musicalinstruments. For the testing data, “drums” and “other” sources of 20
songs in the SiSEC 2016 MUS dataset [25] were used. To produce
multichannel observed signals, these dry sources were convolved
with the impulse response “E2A” in the RWCP database [26] (reverberation time was 300 ms) with 5.66 cm microphone spacing andsource orientations of 50◦ &130◦ (90◦ corresponds to the normal direction of the two microphones), as in [27]. The improvement ofsource-to-distortion ratio (SDR) [28] was used as an evaluation scorebecause it is in good agreement with both degree of separation andabsence of artificial noise. The other experimental conditions are
summarized in Table 1.
4.2. Effect of numbers of iterations in HPSSThe number of HPSS iterations (3)–(5) affects the separation performance of the proposed method. Here, we compare the performanceof the proposed method with various numbers of iterations in HPSS.
Table 2 shows the SDR improvements of the proposed method averaged over harmonic and percussive sources and 20 songs, where
the smoothing parameters were set to β = 0.25 and βold = 0.75.
Since the update rules (3)–(5) of HPSS rapidly converges to a local
minimum of (1), the performance of the proposed method saturates
when the number of iterations in HPSS is set to around 15. On thebasis of this result, we set the number of iterations of HPSS for the
proposed method to 15 in the following experiments.

Table 2. Average SDR improvements of the proposed method withvarious numbers of iterations in HPSS
Number of iterationsin HPSS
1
3
5
7
9
11
13
15
20

Average SDR
improvement [dB]
8.29
10.40
10.87
10.98
11.08
10.79
11.09
11.29
11.06

Table 4. Conditions for other HPSS and BSS algorithms
Parameters for multichannel
HPSS described in [21]Fine-tuned parameters
for multichannel HPSS
Number of bases in ILRMA
Number of iterations

128 ms Hann window with 1/2 shift
αh = αp = 10, mh = mp = 5
γ1 = 0.5, γ2 = 1
16 ms Hann window with 1/2 shift
αh = αp = 5, mh = mp = 10
γ1 = γ2 = 1
10 for each source
20 for single-channel HPSS
15 for multichannel HPSS
30 for AuxIVA
100 for ILRMA

Table 5. Average SDR improvement for HPSS and BSS algorithms
SDR improvement [dB]1614HPSS/BSS algorithm

12Single-channel HPSS
Multichannel HPSS withthe parameters in [21]Multichannel HPSS withthe fine-tuned parameters
AuxIVA
ILRMA
Proposed method1086βold=0040.5
βold=0.52βold=0.75
0.75
0.875
βold=0.875

0Average SDR
improvement [dB]
4.80
-0.47
1.25
7.91
7.76
11.29

βold=0.9375
0.9375

-2

0

100

200

300

400

500

Number of iterations in TFMBSS
Fig. 2. Typical example of convergence behaviors in the proposed
method with various β and βold .

Table 3. Average SDR improvements of the proposed method withvarious smoothing parameters β and βold
β
1
0.5
0.25
0.125
0.0625

βold
0
0.5
0.75
0.875
0.9375

Average SDR improvement [dB]
7.40
10.14
11.29
11.27
11.01

4.3. Effect of smoothing parameters
Here, we investigate the effect of the proposed mask-smoothing
technique in Section 3.3. A typical example of the convergence
behaviors of SDR improvements averaged over harmonic and percussive sources is shown in Fig. 2. Note that β = 1 − βold , andβold = 0 corresponds to the algorithm without the smoothing technique. From this figure, we can confirm that the smoothing process
clearly stabilizes the proposed algorithm. This tendency was thesame for the other songs.
As in the figure, there is a trade-off between stability and convergence speed. Therefore, for a fixed number of iterations, the smoothing parameter should affect the separation performance. Table 3
summarizes the SDR improvements averaged over 20 songs. This
table shows that excess amount of smoothing results in degradationof performance. A preferable condition of the smoothing process
seems around β = 0.25 or 0.125 for the proposed method. The
comparison in the next subsection is made by β = 0.25.
4.4. Comparison with existing HPSS and BSS algorithms
We compared five methods: (a) single-channel HPSS [12, 13], (b)multichannel HPSS [21], (c) IVA based on the auxiliary function

technique (AuxIVA) [6], (d) ILRMA [7, 8], and (e) the proposed
method. For the multichannel HPSS, we used the MATLAB implementation1 provided by the authors of [21], where two conditionsof parameters (given and fine-tuned) were used as shown in Table 4(see [21] for details of these parameters). For single-channel HPSS,
AuxIVA, ILRMA, and the proposed method, we used 128-ms-long
Hann window with half shifting. The other conditions for the conventional methods are summarized in Table 4.
Table 5 shows the average SDR improvements for all the methods. In this experiment, the single-channel HPSS outperformed themultichannel one. While multichannel HPSS can reduce the artificial distortions in the estimated signals using spatial information, thedegree of separation tends to be sacrificed. Also, multichannel HPSS
aims to separate the typical stereo music with panning mixtures.
Although the spatial covariance model [29] in multichannel HPSS
can handle the convolutive mixture (9), its parameter optimization
was unstable and the performance was not high in this experiment.
AuxIVA and ILRMA provide better performance compared to theconventional HPSS methods. This is because the source models assumed in IVA and ILRMA work well for modeling the drums andinstrumental signals. The proposed method outperformed the other
HPSS and BSS techniques because the suitable source models forseparating the harmonic and percussive sources are utilized.
5. CONCLUSION
In this paper, a new algorithm that unifies TFMBSS and HPSS-based
time-frequency mask estimation was proposed. Also, we revealed
the importance of the mask-smoothing process to greatly improve
the performance of TFMBSS. The proposed method outperformed
the conventional HPSS-based and FDICA-based techniques.
The utilization of another time-frequency mask is our future
work. We expect that the proposed mask-smoothing technique is
useful even in such applications with various time-frequency masks.
1 Retrieved from https://www.irisa.fr/metiss/ngoc/sw/hpss.rar

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