Combined adaptive beamforming schemes for nonstationary ...
Signal Processing 93 (2013) 3306–3318
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Combined adaptive beamforming schemes for nonstationary interfering noise reduction$ Danilo Comminiello n, Michele Scarpiniti, Raffaele Parisi, Aurelio Uncini Department of Information Engineering, Electronics and Telecommunications (DIET), “Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy
a r t i c l e in f o
abstract
Article history: Received 27 July 2012 Received in revised form 27 March 2013 Accepted 9 May 2013 Available online 25 May 2013
This paper introduces new adaptive beamforming methods for nonstationary noise reduction, designed to be robust against broadband interfering signals. In particular, we propose combined beamforming schemes within a standard adaptive beamforming system, such as the generalized sidelobe canceller (GSC). The novelty of such combined adaptive beamformers relies on the use of different adaptive sidelobe cancelling structures which allow the system to achieve robustness in nonstationary noisy environments. The combined structures are based on the convex combination of two multiple-input singleoutput (MISO) adaptive systems with complementary capabilities. The whole beamformer benefits from such combination and results to be able to preserve the best properties of each system. We introduce two different adaptive schemes, whose difference lies in the way of combining the MISO systems. Moreover, we present a further adaptive beamforming scheme which generalizes the previous techniques, thus improving the robustness against nonstationary interfering signals in multisource environments. The effectiveness of the proposed systems is also assessed in a nonstationary dense multipath environment. The experiments show that the proposed combined beamforming schemes are capable of enhancing the desired signal even in the presence of nonstationary interfering signals. & 2013 Elsevier B.V. All rights reserved.
Keywords: Combined adaptive beamforming Combination of adaptive filters Nonstationary adaptive beamforming Interfering noise reduction
1. Introduction The presence of interfering signals may considerably deteriorate the quality of a desired signal received by a sensor array. Such problem occurs in several array processing applications and becomes even more evident when interfering signals are nonstationary [1–4]. Nonstationary interfering signals may be caused by moving interfering sources or abrupt changes in the propagation channel. Moreover, when interfering sources are located in dense multipath environments, such as acoustic environments, it
☆ This work has been partly supported by the Italian National Project: “Computational Analysis of Acoustic Scene for Machine Listening Systems and Augmented Reality Communications”, under grant number C26A11BC43. n Corresponding author. Tel.: +39 06 44585495; fax: +39 06 4873300. E-mail addresses: [email protected], [email protected] (D. Comminiello).
0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.05.014
is quite difficult to carry out a noise reduction without deteriorating the quality of the desired signal. In such conditions, the reduction of interfering signals often requires the use of FIR filters with a large number of taps, which involves high computational complexity and slow convergence rate [5,6]. In order to address this problem, broadband adaptive beamforming systems are widely used (see for example [5–9]), since they are highly effective in receiving a desired source signal while at the same time reducing the interfering components, even in dense multipath environments. The generalized sidelobe canceller (GSC) [10] is one of the most popular adaptive beamforming techniques for broadband signals. It is composed of a fixed beamformer (e.g. delay-and-sum or filter-and-sum [11]), and an adaptive path which carries out a reduction of noisy components produced by undesired interfering sources, thus leading to a lower noise power at the system output. In literature, many adaptive beamforming systems for nonstationary
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
noisy environments were based on the GSC [12–15]. The effectiveness of a GSC system strictly relies on the adaptive filtering algorithm used in the sidelobe cancelling path. Generally the adaptation of filters in time-domain may be performed by gradient-based adaptive algorithms, such as the least mean squares (LMS) algorithm (see for example [16]). Although this family of algorithms entails a low computational cost, when the filter length is quite large a rather slow convergence occurs [16], thus the adaptation of the filter weights becomes unpractical in real-time applications. Another time-domain standard approach is Hessian-based adaptive filtering, which is typical of algorithms such as recursive least squares (RLS) filter. The latter approach displays a faster convergence rate compared with gradient-based algorithms [16]. However, RLS adaptive filtering entails a high computational complexity, thus adaptation may become prohibitively expensive. Moreover, the RLS may perform worse than LMS algorithm in nonstationary environment, depending on the statistics of desired source signals [17]. A good compromise between performance and computational load may be obtained by using the family of affine projection algorithm (APA) [18], which is quite used in adaptive beamforming [19,20], since it shows better convergence rates and manageable computational complexity compared with other time-domain algorithms. Moreover, APA is the best suitable algorithm to process colored signals compared with other classic timedomain adaptive algorithms. Despite its good capabilities [18,16], APA suffers from adverse environment conditions, especially in the presence of multiple nonstationary sources which make the adaptation process unstable and reduce performance. In order to address this problem, we propose robust array beamforming schemes based on the adaptive combination of MISO filtering systems, that, in this case, are nothing but filter banks. Adaptive combination of adaptive filters is a very effective and flexible approach to balance the compromises inherent to the settings of adaptive filters [21,22]. Combined adaptive schemes are usually adopted with filters of the same family and complementary properties, e.g. using different step sizes or different filter lengths. They are also used even with filters of different families using different updating rules or different cost functions [23–27]. The combined scheme is capable of adaptively switching between filters according to the best performing filter, thus always providing the best possible filtering [21]. In this paper we propose beamforming architectures which exploit adaptive combination of filters to improve the tracking properties of the system in the presence of broadband nonstationary interfering signals. A first solution is to combine two MISO systems using the same updating algorithm but with different step size values. In fact, it is proved that a combination of a fast filter (i.e. using a large step size value) and a slow one (i.e. using a small step size value) results in faster convergence, lower residual misalignment and improved tracking capability compared with individual filters [28,29,21]. Another method to improve the tracking capability in nonstationary conditions is based on the combination of two filters with different updating approaches, i.e. one gradientbased and one Hessian-based [26,30]. This combination
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of filters exploits the fast convergence provided by the Hessian-based filter and the performance capabilities provided by the gradient-based filter which may outperform the Hessian-based one in nonstationary conditions [30,26]. Compared with the combination of filters with different step sizes, whose performance in terms of excess mean square error (EMSE) is never better than the performance of individual filters, the combination of filters with different updating approaches provides performance that may outperform the performance of correspondent individual filters in terms of EMSE [30]. As regards the adaptive combination, in this work we focus on the convex constrained combination using a sigmoid nonlinearity on the output stage, since it introduces less gradient noise compared to unconstrained and affine constrained combinations [27,25]. In particular, we propose two different beamforming architectures based on the combination of adaptive MISO systems using different updating approaches. The first scheme is based on a system-by-system (SS) combination, in which the overall output of the first MISO system is convexly combined with the overall output of the second MISO system. The second architecture is a filter-by-filter (FF) combination scheme, in which each adaptive filter of the first MISO system is convexly combined with the correspondent filter of the second MISO system. In both schemes, all the adaptive filters are updated using an APA. In order to differentiate the MISO systems, we use different projection orders for each system. Moreover, in order to use the best parameter setting for each filter and further improve the tracking performance, we propose a multistage combination scheme in which the filtering process is carried out in two steps. The paper is organized as follows: in Section 2 a problem formulation is derived by considering the overall array beamforming architecture. In Section 3 the adaptive combinations of MISO systems are introduced according to both the SS and the FF combination schemes, while the adaptive beamforming architecture based on the multi-stage combination scheme is derived in Section 4. Section 5 provides a mean-square performance analysis of the proposed multistage combination scheme. The effectiveness of the proposed beamforming systems is addressed by experimental results in Section 6 and, finally, in Section 7 our conclusions are drawn.
1.1. Notation In this paper, matrices are represented by boldface capital letters and vectors are denoted by boldface lowercase letters. Time-varying vectors and matrices show discrete-time index as a subscript index, while in time-varying scalar elements the time index is denoted in square brackets. A regression vector is represented as xn ∈RM ¼ ½x½n x½n−1 … x½n−M þ 1T , where M is the overall vector length and x½n−i is individual entry at the generic time instant n−i. A generic coefficient vector, in which all the elements depend on the same time instant, is denoted as wn ∈RM ¼ ½w0 ½n w1 ½n … wM−1 ½nT , where wi ½n is the generic i-th individual entry at the n-th time instant. The index related to a MISO systems is denoted
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as a superscript enclosed in brackets for vector and matrices and as subscripts for scalar values. The index related to each channel is denoted as subscript, after the time index for vectors and matrices, e.g. xn;p . The time index is omitted when the coefficient vector is a realization of a time-invariant process. All vectors are represented as column vectors. 2. Combined adaptive beamforming architecture The beamforming architecture adopted in this paper is a typical GSC configuration composed of a sensor array interface, a fixed beamformer (FBF), and an adaptive sidelobe cancelling path, as depicted in Fig. 1. Let us consider a sensor array interface composed of N elements. The signal ui ½n received by the i-th sensor, with i ¼ 0; …; N−1, is a delayed replica of the target signal s½n convolved with the impulse responses of the propagation channel. The FBF aligns the source signals with reference to the desired source direction, yielding the reference signal d½n. In this paper, a delay-and-sum beamformer is used as FBF, however, nothing prevents to consider other FBFs. In the adaptive path of the beamformer, the blocking matrix (BM) generates the noise references xp ½n, with p ¼ 0; …; P−1, being P ¼ N−1. In this paper, the blocking matrix, denoted with B∈RPN , is implemented by pairwise differences between sensor signals [11] (i.e. the sum of the elements of each row is null). This choice is due to the fact that the goal of the paper is to evaluate the effectiveness of the combined beamforming within a standard architecture. As an alternative, other BMs may be used to increase the overall performance of an adaptive beamformer (see for example [13,31,32,15]). The noise reference signals are then processed by means of the combined adaptive noise canceller (CANC), whose structure will be detailed in the next section. The goal of the CANC is to remove any residual noise components in the reference signal, minimizing the output power and yielding the beamformer output signal e½n.
adopted architecture, instead, results from combinations of adaptive filters. In particular, the structure is composed of two or more different MISO systems, each bringing different filtering capabilities to the whole beamformer. The MISO systems receive the same input signals, which are the noise reference signals resulting from the BM. However, due to the different projection orders, the input data matrices may be different from each other. Therefore, taking into account a number J of MISO systems, the p-th filter of the j-th MISO system, with j¼1,…,J, receives as input data a noise reference matrix XðjÞ n;p : 2
3 xTn;p 6 T 7 6 x 7 6 n−1;p 7 K j M XðjÞ ¼6 7 n;p ∈R ⋮ 6 7 4 T 5 xn−K j þ1;p 2 xp ½n 6 x ½n−1 6 p ¼6 6 ⋮ 4 xp ½n−K j þ 1
The trademark of the proposed beamforming approach is represented by the structure of the CANC. Generally, a conventional adaptive noise canceller (ANC) is composed of an adaptive filter bank forming an MISO system. The
u0 n
e n
+ -
z n x0 n Blocking Matrix
…
xp ½n−M
⋮
⋱
⋮
xp ½n K j
…
xp ½n−M−K j þ 2
ðjÞ ðjÞ ðjÞ T M wðjÞ n;p ∈R ¼ ½w0;p ½n w1;p ½n … wM−1;p ½n :
7 7 7 7 5
ð2Þ
All the filters of each MISO system, represented by (2), contain the same number M of coefficients and are adapted according to the affine projection algorithm (APA) [18]. The updating rule of the APA is ðjÞ ðjÞ;T ðjÞ ðjÞ;T −1 ðjÞ wðjÞ n;p ¼ wn−1;p þ μj Xn;p ðδj I þ Xn;p Xn;p Þ en
ð3Þ
Kj eðjÞ n ∈R
is the error vector of the j-th MISO system where containing the last Kj samples of the j-th error signal, which results from P−1
ð4Þ ðjÞ
uN 1 n
xp ½n−2
3
p¼0
3.1. Convex combination of adaptive filters using affine projection algorithm
d n
xp ½n−M þ 1
where Kj represents the projection order for all the filters of the j-th MISO system. We denote the coefficient vector of the p-th filter belonging to the j-th MISO system at the n-th time instant as
ðjÞ eðjÞ n ¼ dn − ∑ yn;p :
3. Adaptive combinations of MISO systems
Delay
…
ð1Þ
ðjÞ
Fixed Beamformert
xp ½n−1
Combined Adaptive Noise Canceller
Fig. 1. Combined adaptive beamforming architecture.
In this equation dn ∈RK j ¼ ½d½n d½n−1 … d½n−K j þ 1T is the vector containing the last Kj samples of the desired ðjÞ ðjÞ Kj signal and yðjÞ n;p ∈R ¼ Xn;p wn−1;p is the vector containing the Kj projections of the output signal relative to the p-th filter of the j-th MISO system. Moreover, in (3) the parameters μj and δj are, respectively, the step size and the regularization factor, which are the same for all the filters of the j-th MISO system. Note that the APA described by (3) is an approximation of the multichannel APA (see for example [33]) since it neglects the correlation between channels. Using the updating rule described by (3) the considered MISO systems can be differentiated simply by changing the values of the filter parameters. To this end, and in order to improve the robustness of the beamformer against nonstationary signals, we can consider a number of J¼ 2 MISO systems (j¼1,2) that may have the same projection order but different step size values (or regularization factors as an alternative), and in particular a small step size value for the first MISO system and a large value for the second. It is also possible to fix the step size for both the MISO systems
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and to choose different projection orders (in particular K 1 ¼ 1 and K 2 4 1), thus, respectively, resulting in a gradient-based MISO system and an Hessian based one. In the following step, the two MISO systems can be combined under a convex constraint in order to exploit the different capabilities of both the systems. Aside from the chosen distinguishing parameters, there are two ways to combine the MISO systems. The first way is to convexly combine the outputs of the two MISO systems and the second one is to combine each filter of the first MISO system with the corresponding filter of the second MISO system under a convex constraint. We denote the former way as SS combination scheme and the latter as FF combination scheme, which are both described in the following two subsections.
The first proposed scheme is the system-by-system CANC, depicted in Fig. 2(a). The output of each MISO system, that we denote as yðjÞ ½n, is achieved by summing each individual filter output of the corresponding MISO system P−1
yðjÞ ½n ¼ ∑ yðjÞ p ½n:
ð5Þ
p¼0
As j¼1,2, Eq. (5) yields two system outputs that are then convexly combined to generate the overall CANC output z½n ¼ λ½nyð1Þ ½n þ ð1−λ½nÞyð2Þ ½n
parameter. This is the reason why the convex combination can be carried out by using a sigmoid function, as done in [21,34]. However, especially when J 42, any other convex constraint optimization algorithm can be applied (see for example [27]). Therefore, the mixing parameter in (6) can be adapted through the adaptation of an auxiliary parameter, a½n, related to λ½n by means of a sigmoidal function defined according to [34] λ½n ¼ βðsgmða½nÞ−αÞ ¼β
ð6Þ
where λ½n is the mixing parameter, which takes into account the convex constraints (0 ≤λ½n ≤1) [21]. Due to the fact that we have to combine only J ¼2 MISO systems, in (6) we need to adapt just one weight, that is the mixing
1 −α ð1 þ e−a½n Þ
ð7Þ
where β¼
3.2. System-by-system combination scheme
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1 1−2α
α ¼ sgmð−4Þ:
ð8Þ
The parameters (8) are introduced in (7) to prevent the stalling of the adaptation rule and to reduce gradient noise when the mixing parameter is close to 0 or to 1 [34]. To the same end, the value of a½n is kept within [4, −4]. The updating rule of a½n can be written as a½n þ 1 ¼ a½n−
μs ∂e2 ½n r½n ∂a½n
¼ a½n þ
μs e½nðyð1Þ ½n−yð2Þ ½nÞðλ½n þ αβÞðβ−αβ−λ½nÞ βr½n
ð9Þ where μs =r½n represents a normalized step size [35], r½n ¼ γr½n−1 þ ð1−γÞðyð1Þ ½n−yð2Þ ½nÞ2 is the estimated power of ðyð1Þ ½n−yð2Þ ½nÞ, and γ is a smoothing factor.
Fig. 2. Single-stage combined adaptive noise canceller architectures: (a) system-by-system and (b) filter-by-filter combination schemes.
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Finally, the beamformer output signal e½n, using the SS combination scheme, is achieved as e½n ¼ d½n−z½n:
ð10Þ
3.3. Filter-by-filter combination scheme The second proposed scheme is the filter-by-filter CANC in which the output signal z½n is built in a different way. As it can be seen in Fig. 2(b), the p-th filter output of the first MISO system is convexly combined with the correspondent p-th filter output of the second MISO system, thus generating P−1 outputs, each one relative to a noise reference ð2Þ yp ½n ¼ λp ½nyð1Þ p ½n þ ð1−λp ½nÞyp ½n
ð11Þ
where λp ½n is the p-th mixing parameter, adapted using the p-th auxiliary parameter, ap ½n, similarly to (9) ap ½n þ 1 ¼ ap ½n μ þ f e½nðypð1Þ ½n−yð2Þ p ½nÞλp ½nð1−λp ½nÞ r p ½n
ð12Þ
where μf is the step size value which may be chosen as the same for all combinations. Once computed the convex combinations, we achieve the CANC output signal z½n by summing the individual output contributions deriving from the combinations (see Fig. 2(b)) P−1
z½n ¼ ∑ yp ½n:
ð13Þ
p¼0
From Eq. (13) we derive the overall beamformer output signal e½n ¼ d½n−z½n, relative to the FF combination scheme. Both the combined architectures presented above improve the tracking capabilities of CANC giving robustness to the overall beamforming system in the presence of nonstationary interfering signals. Note that the computational increase is linear and, therefore, easily manageable for current processors.
and the projection orders. In particular, we choose a small step size μj ¼ μa for j¼1,3 and a large step size value μj ¼ μb for j¼2,4. Moreover, we update the first two MISO systems by using a gradient-based algorithm and the second two systems with an Hessian-based algorithm. This is obtained by setting a unitary projection order K j ¼ K A ¼ 1 for j ¼1, 2 and a higher projection order K j ¼ K B 41 for j¼3, 4. The choice of different step size values affects the convex combinations on the first stage, in which the first MISO system is combined with the second one and the third system with the fourth one. In this stage the convex combination may follow the SS scheme or the FF scheme. In Fig. 3 a multi-stage beamformer with a FF scheme on the first stage is depicted. On the other hand, the choice of different projection orders affects the convex combination on the second stage, in which the output signal resulting from the combination of the first and the second MISO systems is in turn combined with the output signal resulting from the combination of the third and the fourth MISO systems. Regarding the combination on the second stage, there is no distinction between SS and FF schemes. Output signals of the convex combinations on the first stage, denoted as zðAÞ ½n and zðBÞ ½n, may be achieved similarly to (6), according to an SS combination scheme, or similarly to (13), according to an FF combination scheme as depicted in Fig. 3. On the other hand, the convex combination on the second stage may yield the following output signal from the multi-stage CANC: z½n ¼ η½nzðAÞ ½n þ ð1−η½nÞzðBÞ ½n
ð14Þ
where η½n is the mixing parameter of the second stage, adapted using an auxiliary parameter, similar to (9). Once computed the second stage convex combination, we derive the overall multi-stage beamformer output signal e½n ¼ d½n−z½n, as done for the single-stage combination schemes in Section 3. The multi-stage beamforming architecture introduced above exploits the capabilities of each MISO system, thus improving the cancelling performance compared to conventional beamformers (e.g. using a single MISO ANC) and single-stage combined beamformers in the presence of nonstationary interfering signals.
4. Multi-stage combined beamforming The combined beamforming schemes described in the previous section are effective in the presence of multiple nonstationary sources both choosing different step size values (a small one and large one) and different projection orders (one of unitary order and one of higher order). Further improvements may be achieved if we consider the joint capabilities deriving from choosing both different step size values and projection orders. To this end we propose a multi-stage combination scheme in which the filtering process may involve more convex combinations of MISO systems. In particular, in order to yield an adaptive beamforming architecture that is robust against nonstationary conditions, we may consider a CANC composed of a number J ¼4 of MISO systems, as depicted in Fig. 3, each one bringing different capabilities to the whole architecture. We differentiate the four systems according to the step size values
5. Mean-square performance of the multi-stage combination scheme In this section we analyze the mean-square performance of the proposed multi-stage combination scheme. Before starting the analysis it is convenient to introduce some further notation. Let us denote each combination on the first stage with the index q¼{A,B}, where A and B are, respectively, referred to the first and the second combinations of the first stage. Each combination is composed of two MISO systems denoted by lower case letters a and b. The index variable associated to MISO systems is r ¼{a,b}. Hence, a generic MISO system of a generic first stage combination is denoted as qr ¼{Aa,Ab,Ba,Bb}. An important preliminary assumption must be made regarding the second stage combination. While the first stage of the multi-stage architecture combines MISO systems having different step size values, the second stage
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
2,
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,
Fig. 3. Multi-stage combined adaptive noise canceller.
combines systems resulting from different projection orders. This leads to combine two different “virtual” weight vectors for the second stage. An exact analysis should be carried out by considering the higher projection
order as a reference and thus zero-padding the weight vector with the lower projection order. However, since it does not affect the result of our analysis we may choose a common projection order for a clearer exposition. Such
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value is chosen as the lower projection order value, that in our case is the unitary projection order. In order to derive the mean-square performance at steady-state, we consider the identification of an unknown model (see for example [16]). To this end, let us consider now the following linear regression model: ! P−1
∑ xTn;p wop
d½n ¼
þ v½n
ð15Þ
p¼0
where wop is the p-th unknown vector of length M to be identified and v½n is an independent identically distributed (i.i.d.) noise whose variance is s2v . We define now the “virtual” weight vectors resulting from the combination on the first and the second stages. In particular, for the first stage we can obtain different pairs of vectors for each channel, whether adopting an SS or an FF combination scheme SS :
ðqÞ ðqaÞ b n;p w ¼ λðqÞ ½nwn;p þ ð1−λðqÞ ½nÞwðqbÞ n;p
ð16Þ
FF :
ðqÞ ðqaÞ ðqÞ ðqbÞ b ðqÞ w n;p ¼ λp ½nw n;p þ ð1−λp ½nÞwn;p
ð17Þ
for q¼{A,B}. On the other hand, the overall virtual MISO system for the second stage can be described as b n;p ¼ w
b ðAÞ η½nw n;p
þ
b ðBÞ ð1−η½nÞw n;p :
ð18Þ
The weight error vectors are defined as the difference between the optimal solution and the virtual weights for the first and the second stage, respectively, as o b ðqÞ ~ ðqÞ w n;p ¼ wp −w n;p ;
q ¼ fA; Bg
b n;p : ~ n;p ¼ wop −w w
ð19Þ ð20Þ
We also define the a priori and a posteriori error signals, respectively, for each combination of the first stage P−1
ðqÞ
ð21Þ
ðqÞ
ð22Þ
ðqÞ ~ eðqÞ a;n ¼ ∑ Xn;p w n−1;p p¼0 P−1
ðqÞ ~ eðqÞ p;n ¼ ∑ Xn;p w n;p p¼0
with q¼{A,B}, and for the second stage P−1
~ n−1;p ea ½n ¼ ∑ xTn;p w
ð23Þ
p¼0 P−1
~ n;p : ep ½n ¼ ∑ xTn;p w
ð24Þ
p¼0
Note that the last two equations related to the second stage derive from a unitary projection order of the input data. In order to measure filtering performance of MISO systems the excess mean-square error (EMSE) can be adopted as measure, defined as the excess over the minimum mean-square error. Since the steady-state analysis is evaluated as n-∞, the EMSE of the second stage combination can be expressed as (see [21]) ξ ¼ lim Efjea ½nj2 g: n-∞
ð25Þ
With the aim of providing an extended expression for (25), it is useful to refer to an additional variable that measures the steady-state correlation for the second stage between
the a priori errors resulting from the combinations of the first stage. Such variable is denoted as cross-EMSE [21] and it can be expressed as ðBÞ ðBÞ;T ðAÞ ξcross ¼ lim EfeðAÞ;T a;n ea;n g ¼ lim Efea;n ea;n g: n-∞
n-∞
ð26Þ
Therefore, from [21] and taking into account (26), we can write the EMSE for the overall two-stage architecture ξ as ξ ¼ ξcross þ
ΔξðAÞ ΔξðBÞ ΔξðAÞ þ ΔξðBÞ
ð27Þ
where ΔξðqÞ ¼ ξðqÞ −ξcross ;
q ¼ fA; Bg:
ð28Þ
The expressions of the EMSE for both the combinations on the first stage, ξðqÞ , and the cross-EMSE for the second stage, ξcross can be easily derived according to [21,36]. The interested reader can find the analytical derivation of ξðqÞ and ξcross in Appendix A. 6. Experimental results In this section we carry out two different sets of experiments: the first set (in Subsection 6.1) aims at assessing the effectiveness of the described combined filtering schemes adopted in the proposed beamforming method; the second set of experiments, detailed in Section 6.2, is performed in order to evaluate the proposed combined beamforming architectures for noise reduction application in a multisource scenario. 6.1. Convergence performance of combined filtering schemes In the first set of experiments we prove the filtering effectiveness of proposed CANC schemes through a tracking analysis which describes the convergence performance. To this end we use conventional ANC MISO systems and the proposed combined schemes to identify an unknown nonstationary model and to compare their performance. In particular, the model to identify derives from (15) and is composed of a number M of transfer functions that, in the case of beamforming application, may represent the impulse responses between a source and M sensors. The initial optimal solution is formed with M¼7 independent random values between −1 and 1. In the following examples the initial system is wo1 ¼ ½0:4125 0:7632 − 0:5484 −0:6099 −0:4622 −0:4826 −0:5296T . The input signal is generated by means of a first-order pffiffiffiffiffiffiffiffiffiffiffi autoregressive model, whose transfer function is 1−α2 =ð1−αz−1 Þ, with α ¼ 0:8, fed with an i.i.d. Gaussian random process. The length of the input signal is of L¼ 8000 samples. In order to study the ability of combined schemes to react to nonstationary environments, at time instant n ¼ L=2 the system changes into wo2 ¼ ½−0:4223 0:0848 −0:1228 0:3876 0:9950 0:9806 −0:2700T . Furthermore, an additive i.i.d. noise signal ev ½n with variance s2v ¼ 0:01 is added to form the desired signal (it is the same for all the filtering architectures). In order to identify the unknown solutions wo1 and wo2 we use both conventional MISO systems and the adaptive combined filtering schemes described in Sections 3 and 4.
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
A similar result was achieved in the second experiment, where we compare the same filtering architectures of the first experiment, but we choose the same step size value μ ¼ 0:01 and different projection orders: K 1 ¼ 1 and K 2 ¼ 4. Similarly to the previous experiment, in Fig. 5(a) we have compared the performance of combined architectures with those of conventional ANC using both K1 and K2. Even in this case, both SS and FF schemes provide better convergence performance compared to conventional filtering. The mixing parameter behaviors in Fig. 5(b) confirm a slightly faster adaptation of the FF scheme compared to the SS one. Using the same scenario, we study now the convergence performance of the multi-stage combined architecture. As stated in Section 4, in a multi-stage combined scheme the combinations on the first stage may be performed in both SS or FF way. We consider two-stage combined schemes composed of four different MISO systems and we choose two different step size values, μa ¼ 0:01 and μb ¼ 0:1, and two different projection orders K A ¼ 1 and K B ¼ 4. A first comparison is performed between a multi-stage SS (MSS) scheme and a multi-stage FF (MFF) one. A first important result is shown in Fig. 6(a), where we can see that, using a multi-stage architecture, the superiority of the FF scheme over the SS one is noticeably more evident compared to the single-stage architectures (see Figs. 4(a) and 5(a)). Such performance improvement is strictly related to the coherence between the input signals to a convex combination scheme. In fact, in a multi-stage architecture, FF schemes on the first stage exploit the availability of a larger number of free parameters, thus increasing the coherence between the inputs on the second stage and leading to a performance improvement. In light of this result we take into account the performance of a multi-stage scheme whose combinations on the first stage are performed according to an FF scheme, as depicted in Fig. 3. In Fig. 6(b) the comparison between the MFF combined architecture and the individual conventional filterings shows that the multi-stage filtering results in the best performing architecture. Moreover, the performance
We compare their performance in terms of excess mean square error (EMSE) defined as EMSE½n ¼ Efðe½n−v½nÞ2 g, where e½n is the error signal of the filtering architecture, and the operator Efg is the mathematical expectation. The EMSE of each filtering structure is evaluated over 1000 independent runs. Moreover, in order to facilitate the visualization, the EMSE curves are filtered by a movingaverage filter. All the filtering architectures, included the conventional ones, use MISO systems with P ¼4 channels. In the first two experiments we compare a conventional MISO architecture and both single-stage combined architectures described in Section 3, i.e. the SS CANC and the FF CANC. Both the SS and the FF schemes are composed of two MISO systems, as depicted in Fig. 2. All the MISO systems use APA filters. In both the experiments for the adaptation of the mixing parameter of the SS filtering architecture (see Eq. (9)) we use a step size value of μs ¼ 102 , while a step size value of μf ¼ 103 is adopted for the adaptation of all the mixing parameters of the FF scheme (see Eq. (12)). Both the step size values provide the best performance in each case. In the first experiment, we evaluate the filtering architectures choosing the same projection order K¼2 for all the MISO systems and different step size values for the MISO systems of the combined schemes: a slower one μ1 ¼ 0:01 and a faster one μ2 ¼ 0:1. In Fig. 4(a) we have compared the performance of the combined architecture with those of the conventional ANC using both μ1 and μ2 . As it is possible to see, both SS and FF schemes take advantage from using the combined filtering with respect to conventional filtering. In fact, combined schemes always show the behavior of the best performing system and in transient state they behave even better than the best conventional filtering. Both the combined schemes provide good convergence performance, but the FF scheme is slightly better than the SS one due to the fact that the adaptation of the mixing parameters in the FF scheme is faster than in the SS one, as it can be seen in Fig. 4(b). This results in a quality improvement of the processed signal that can be decisive in many applications (e.g. speech applications).
−10
1
Conventional ANC,µ1 Conventional ANC,µ2 SS CANC FF CANC
−15
0.9 0.8
−20
0.7 Mixing parameter
EMSE [dB]
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−25 −30
0.6 0.5 0.4 0.3
−35
0.2 −40 0.1
SS CANC FF CANC
0
−45 0
1000
2000
3000
4000 Samples
5000
6000
7000
8000
0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
Fig. 4. (a) EMSE comparison between single-stage combined filtering architectures and conventional ones using the same projection order and different step size values. Combined architectures always provide the best performance. (b) Behavior comparison between the mixing parameter of the SS CANC and the mixing parameter relative to the first channel of the FF CANC.
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1
−10 Conventional ANC, K1 Conventional ANC, K2 SS CANC FF CANC
−15
0.9 0.8 0.7 Mixing parameter
EMSE [dB]
−20 −25 −30
0.6 0.5 0.4 0.3
−35
0.2 −40 0.1
SS CANC FF CANC
0
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1000
2000
3000
4000
5000
6000
7000
8000
0
1000
2000
3000
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4000
5000
6000
7000
8000
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Fig. 5. (a) EMSE comparison between single-stage combined filtering architectures and conventional ones using the same step size value and different projection orders. Combined architectures always provide the best performance. (b) Behavior comparison between the mixing parameter of the SS CANC and the mixing parameter relative to the first channel of the FF CANC.
−5
MSS CANC MFF CANC
−10
EMSE [dB]
−15 −20 −25 −30 −35 −40 −45 0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
−5 −10 −15 −20 −25 −30
FF,µa,µb, KA FF,µa,µb, KB FF,µa,KA, KB FF,µb,KA, KB MFF CANC
−15 −20 EMSE [dB]
EMSE [dB]
−10
Conventional ANC,µa,KA Conventional ANC,µb,KA Conventional ANC,µa,KB Conventional ANC,µb,KB MFF CANC
−25 −30 −35
−35
−40
−40 −45
−45 0
1000
2000
3000
4000
5000
6000
7000
8000
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0
1000
2000
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4000
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6000
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Fig. 6. EMSE comparison: (a) between MSS and MFF schemes; (b) between MFF architecture and conventional ANCs; and (c) between MFF architecture and single-stage FF ones. Multi-stage combined architectures always provide the best overall performance.
improvement of the MFF architecture results even from the comparison with the single-stage FF architectures, as depicted in Fig. 6(c).
Results described in this subsection show the filtering ability of proposed combined schemes compared to conventional filtering. For single-stage architectures both SS
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
and FF schemes can be used since they achieve same performance results. However, for multi-stage architectures a decisive preference is given to FF schemes which show better performance and reaction to abrupt changes in the environment. Moreover, FF schemes may exploit spatial diversity and thus different step size values for the adaptation of the mixing parameters may be chosen according to the scenario requirements. Finally, it has been shown that multi-stage combined filtering always achieves the best convergence performance.
6.2. Evaluation of combined beamformers in acoustic noise reduction In the second set of experiments we assess the effectiveness of the proposed combined beamforming architectures in a typical dense multipath environment, that is an acoustic environment. In particular, experiments take place in a 6 5 3:3 m room with a reverberation time of
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T 60 ≈120 ms for a hands-free speech communications application. The source of interest is a female speaker located 70 cm from the center of a microphone array, as depicted in Fig. 7. Two interfering sources are initially located, respectively, 1.9 m and 2.8 m from the center of the acoustic interface: the first source is a female speaker located on the left of the array, while on the right is located the second source which is a male speaker. White Gaussian noise is added at microphone signals as diffuse background noise, thus providing 20 dB of SNR (signal-tonoise ratio) with respect to the desired source. The overall input SNR level, measured for each sensor signal, is of about 3 dB. After 5 s from the start of the experiment the first source changes its position and after 10 s also the second source changes its position. The overall length of the experiment is 15 s. The impulse responses between sources and sensors are simulated by means of Roomsim, which is a Matlab tool [37]. Each impulse response is measured by using an 8 kHz sampling rate and it is truncated after M ¼340 samples,
Microphone Array
0,7 m
Desired Source
Interfering Source #2
Interfering Source #1
Background Noise
Microphone Array
Microphone Array
Interfering Source #1
Interfering Source #1 Desired Source
Interfering Source #2
Desired Source
Interfering Source #2
Background Noise
Background Noise
Fig. 7. Acoustic environment with nonstationary interfering sources. The source of interest is a female speaker located in front of a microphone array and two interfering speakers are located, respectively, on the left and on the right of the desired source (a). After 5 s from the start of the experiment the first interfering source moves to position 2 (b) and at second 10 also the second interfering source changes its position (c).
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which is also the length of each filter. The microphone interface is a classic uniform linear array (ULA) composed of 8 omnidirectional sensors equally spaced with a distance of 5 cm, thus having a good spatial resolution even at mid-low frequencies. The noise reduction performed by the beamformer is usually associated with an SNR improvement, defined as [11]
7. Conclusions
"
# Efs2in ½ng SNR ¼ 10 log Efs2in ½ng−Efs2out ½ng
ð29Þ
where sin ½n is the generic input clean signal and sout ½n is the processed signal. We compute the SNR level over the total length of the experiment (0–15 s) and also in three different sub-intervals of time: the initial state, from 0 to 5 s, when the two interfering sources are located in their initial position; the first change, from 5 to 10 s, which includes the position change of the first interfering source and the consequent readaptation of the filtering system; the second change, from 10 to 15 s, when also the second interfering source changes its position. We compare GSC beamformers having different ANC: conventional ANCs with different parameter settings, the single-stage FF combined ANC with different parameter settings, and the two-stage combined ANC. Filter parameters μa , μb , K A , K B , μs and μf are the same used in the first set of experiments. Results are collected in Table 1, in which it is possible to notice the behavior of the different beamformers taken into account and their contribution to the noise reduction in terms of SNR improvement. We could have shown performance of both SS and FF combination schemes and both varying the step size values and the projection order. For a better ease of reading results, we only show the performance relative to FF combination schemes, which achieve the more relevant results, and we only vary the step size value for the single-stage combined ANCs. From Table 1 one can infer that all the conventional ANCs show difficulties when a source change its position, thus decreasing speech enhancement performance. The more stable conventional ANC is the one having a large step size value and a large projection order, but its performance is the poorest in terms of SNR. A significant enhancement is achieved by means of the FF CANC and a further improvement is provided by the MFF CANC which achieves the best performance in each time interval in terms of SNR. Table 1 SNR comparison in dB. We evaluate the beamformers over three subintervals of time, 0–5, 5–10 and 10–15 s, and over the total length of the experiment, 0–15 s. Multi-stage combined beamformer always performs the best reduction of interfering signals. GSC Conventional ANC, μa , Conventional ANC, μa , Conventional ANC, μb , Conventional ANC, μb , FF CANC, μa , K A , K B FF CANC, μb , K A , K B MFF CANC
KA KB KA KB
SNR values obtained from this experiment are not definitely the best achievable values, since better results may be obtained using more sophisticated GSC beamformers, e.g. involving any voice activity detectors (VADs), adaptive BMs or post-filters. The obtained results are sufficient to show the effectiveness of the proposed combined beamformers compared to conventional methods.
0–5 s
5–10 s
10–15 s
0–15 s
17.2 17.8 18.1 13.4 18.4 18.2 18.8
14.2 16.7 16.3 13.2 17.0 17.5 18.1
14.9 16.8 16.5 13.4 18.1 18.0 19.1
15.6 16.9 16.8 13.4 18.0 17.9 18.7
In this paper we have introduced combined beamforming methods whose goal is to improve the performance, in terms of noise reduction, in the presence of nonstationary interfering sources. The trademark of proposed methods relies on the use of combined filtering schemes in the ANC block. These combined schemes are based on the adaptive combination of MISO systems with different parameter settings thus involving complementary capabilities. The whole beamforming system benefits from the different capabilities of each MISO systems, yielding improved performance. We introduced two different ways of combining the MISO systems which are the SS scheme and the FF one. Both the combined architectures provide better performance compared to conventional beamformers. A multi-stage combined beamformer has been also introduced in which the adaptive combination of MISO systems can be performed in subsequent stages. In particular, we have taken into account a two-stage combined beamformer which outperforms the single-stage schemes, thus always providing the best performance when nonstationary sources interfere with the enhancement of a desired signal. Among the multi-stage architectures, the ones using FF schemes are decisively preferable due to the fact that the adaptive combination is performed for each channel. This allows FF beamformers to better react to abrupt changes in the environment. In conclusion, the goal of this work was to introduce novel schemes for adaptive beamforming that can improve the overall performance, thus paving the way for future developments. In particular, simply changing some aspects of the multi-stage combined architecture (e.g. the adaptive algorithms used for the MISO systems, the kind of adaptive combination, the number of combinations, the number of combination stages, the general adaptive beamforming technique) may lead to several combined architectures well-suited for specific beamforming applications. Appendix A. Analytical derivation of the EMSE for the two-stage combination scheme In order to achieve ξ in (27) it is necessary to derive the expressions of the EMSE for both the combinations on the first stage, ξðqÞ , and the cross-EMSE for the second stage, ξcross . As represented in Fig. 3, the first stage is composed of two (or even more) convex combinations of MISO systems, each of which may be characterized by an SS or an FF combination scheme. Therefore, in theory each combination may require a different EMSE measurement. However, since from [21] the EMSE measure depends only on the step size values and on the input signal, we can derive the
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
same expression both for SS and FF schemes. Therefore, EMSEs for the first stage can be distinguished just according to the projection order of the input data. In particular, the two combinations on the first stage are, respectively, characterized by the projection orders K j ¼ K A ¼ 1 for j¼1,2 and K j ¼ K B 4 1 for j¼ 3,4 as stated in Section 4, thus the EMSE measures for the first stage of combinations can be described by the following expression (similarly to (27) [21]): ξðqÞ ¼ ξðqÞ cross þ
ΔξðqaÞ ΔξðqbÞ ðqaÞ
Δξ
þ Δξ
ðqbÞ
ðA:1Þ
for q¼ {A,B} and ðqrÞ
Δξ
ðqrÞ
¼ξ
−ξðqrÞ cross
ðA:2Þ
the EMSE of a generic MISO system, for q ¼{A,B} being ξ and r ¼{a,b}. In order to compute (A.2) we must derive the expressions of ξðqrÞ and ξðqrÞ cross . EMSE values ξðqrÞ are related to individual MISO systems of each combination on the first stage and can be derived according to [36] as ξðqrÞ ¼
μr s2vq TrðEfR nðqÞ gÞ ð2−μr ÞTrðS EfR ðqÞ n gÞ
we get the following relation: ðqÞ ðqbÞ ðqbÞ ðqaÞ;T ðqÞ ðqbÞ μa μb EfeðqaÞ;T RðqÞ Rn ea;n g þ μb EfeðqaÞ;T n en g ¼ μa Efen n a;n R n en g:
ðA:7Þ Then, by substituting we obtain
ðA:3Þ
ðqÞ ðqÞ;T −1 where RðqÞ n ¼ ðδq I þ Xn;p Xn;p Þ . The parameter S derives
from the assumption in [36], according to which EfeðqrÞ;T a;n ðqrÞ 2 T eðqrÞ a;n g ¼ Efjea j g S, where S≈I for small μr , and S≈ð1 1 Þ
for large μr , being 1 ¼ ½1 0 ⋯ 0T . The regularization factor δq is assumed to be a small value, the same for each combined pair of MISO systems. Note also that in (A.3) the step size value is distinguished according to the only index r ¼{a,b}. This is due to the fact that each pair of combined MISO system on the first stage is characterized by the same two values for the step size parameters. On the other hand, to derive the expression of ξðqrÞ cross , we consider the energy conservation relation (see [16]) that relates the weight error vectors and the a priori and a posteriori error signals of each combination on the first stage ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ~ ðqrÞ ~ ðqrÞ w n;p þ Xn;p ðXn;p Xn;p Þ ea;n ¼ w n−1;p þ Xn;p ðXn;p Xn;p Þ ep;n
eðqrÞ n
¼
eðqrÞ a;n
þ
vðqÞ n ,
for r ¼ a; b, in (A.7),
ðqÞ ðqbÞ μa μb EfeðqaÞ;T a;n R n ea;n g ðqaÞ;T ðqÞ ðqbÞ R nðqÞ vðqÞ þ μa μb EfvðqÞ;T n n g ¼ ðμa þ μb ÞEfea;n R n ea;n g:
ðA:8Þ
Solving each term of (A.8) according to [36] leads to ðqÞ ðqÞ 2 ðμa þ μb −μa μb Þξcross TrðS EfR ðqÞ n gÞ ¼ μa μb svq TrðEfR n gÞ
ξðqÞ cross
ðqrÞ
3317
ðA:9Þ
ðqbÞ;T ðqaÞ Efea;n ea;n g
¼ for n-∞. Therefore, from (A.9) where we can derive the expression of the cross-EMSE ξðqÞ cross for the combinations on the first stage ðqÞ ξcross ¼
μ′s2vq TrðEfR nðqÞ gÞ ð1−μ′ÞTrðS EfR nðqÞ gÞ
where μa μb μ′ ¼ μa þ μb
ðA:10Þ
ðA:11Þ
for each combination q of the first stage. By substituting (A.(11) and A.3) in (A.2), we can solve the EMSE values ξðqÞ for the first stage of combinations, described by (A.1). Once obtained the expression of ξðqÞ , it is now necessary to compute the value of the cross-EMSE for the second stage ξcross in order to derive the complete expression (27) of the overall EMSE of the two-stage combined architecture. Repeating the procedure applied for ξðqÞ cross , we get ξcross ¼
μ″s2v TrðEfR n gÞ ð1−μ″ÞTrðS EfRn gÞ
ðA:12Þ
in which each term is referred to a unitary projection order, and where, after few passages, μa μb : ðA:13Þ μ″ ¼ 2ðμa þ μb Þ Therefore, substituting (A.(12) and A.10) in (28), the expression of the EMSE (27) for the two-stage combined architecture described in Section 4 can be derived. References
ðA:4Þ ðqÞ;T −1 is similarly to [36, Eq. (8)], where the term ðXðqÞ n;p Xn;p Þ assumed to be invertible. Multiplying the transpose of (A.4) for r ¼a by (A.4) for r ¼ b, and cancelling terms, the following generalized energy conservation relation for each q can be achieved: ðqÞ ðqÞ;T −1 ðqbÞ ðqaÞ;T ~ ðqaÞ;T ~ ðqbÞ ~ ðqaÞ;T ~ ðqbÞ w n;p w n;p þ ea;n ðXn;p Xn;p Þ ea;n ¼ w n−1;p w n−1;p ðqÞ;T −1 ðqbÞ ðqaÞ;T þ ep;n ðXðqÞ n;p Xn;p Þ ep;n :
ðA:5Þ
Taking the expectations of both sides of (A.5) and considering that [16,21] ðqÞ ðqÞ;T −1 ðqbÞ ðqÞ;T −1 ðqbÞ ðqaÞ;T ðqaÞ;T Efea;n ðXn;p Xn;p Þ ea;n g ¼ Efep;n ðXðqÞ n;p Xn;p Þ ep;n g;
n-∞
and ðqÞ ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ðqrÞ eðqrÞ p;n ¼ ea;n −μr Xn;p Xn;p ðδq I þ Xn;p Xn;p Þ en ;
ðA:6Þ
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Combined adaptive beamforming schemes for nonstationary interfering noise reduction$ Danilo Comminiello n, Michele Scarpiniti, Raffaele Parisi, Aurelio Uncini Department of Information Engineering, Electronics and Telecommunications (DIET), “Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy
a r t i c l e in f o
abstract
Article history: Received 27 July 2012 Received in revised form 27 March 2013 Accepted 9 May 2013 Available online 25 May 2013
This paper introduces new adaptive beamforming methods for nonstationary noise reduction, designed to be robust against broadband interfering signals. In particular, we propose combined beamforming schemes within a standard adaptive beamforming system, such as the generalized sidelobe canceller (GSC). The novelty of such combined adaptive beamformers relies on the use of different adaptive sidelobe cancelling structures which allow the system to achieve robustness in nonstationary noisy environments. The combined structures are based on the convex combination of two multiple-input singleoutput (MISO) adaptive systems with complementary capabilities. The whole beamformer benefits from such combination and results to be able to preserve the best properties of each system. We introduce two different adaptive schemes, whose difference lies in the way of combining the MISO systems. Moreover, we present a further adaptive beamforming scheme which generalizes the previous techniques, thus improving the robustness against nonstationary interfering signals in multisource environments. The effectiveness of the proposed systems is also assessed in a nonstationary dense multipath environment. The experiments show that the proposed combined beamforming schemes are capable of enhancing the desired signal even in the presence of nonstationary interfering signals. & 2013 Elsevier B.V. All rights reserved.
Keywords: Combined adaptive beamforming Combination of adaptive filters Nonstationary adaptive beamforming Interfering noise reduction
1. Introduction The presence of interfering signals may considerably deteriorate the quality of a desired signal received by a sensor array. Such problem occurs in several array processing applications and becomes even more evident when interfering signals are nonstationary [1–4]. Nonstationary interfering signals may be caused by moving interfering sources or abrupt changes in the propagation channel. Moreover, when interfering sources are located in dense multipath environments, such as acoustic environments, it
☆ This work has been partly supported by the Italian National Project: “Computational Analysis of Acoustic Scene for Machine Listening Systems and Augmented Reality Communications”, under grant number C26A11BC43. n Corresponding author. Tel.: +39 06 44585495; fax: +39 06 4873300. E-mail addresses: [email protected], [email protected] (D. Comminiello).
0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.05.014
is quite difficult to carry out a noise reduction without deteriorating the quality of the desired signal. In such conditions, the reduction of interfering signals often requires the use of FIR filters with a large number of taps, which involves high computational complexity and slow convergence rate [5,6]. In order to address this problem, broadband adaptive beamforming systems are widely used (see for example [5–9]), since they are highly effective in receiving a desired source signal while at the same time reducing the interfering components, even in dense multipath environments. The generalized sidelobe canceller (GSC) [10] is one of the most popular adaptive beamforming techniques for broadband signals. It is composed of a fixed beamformer (e.g. delay-and-sum or filter-and-sum [11]), and an adaptive path which carries out a reduction of noisy components produced by undesired interfering sources, thus leading to a lower noise power at the system output. In literature, many adaptive beamforming systems for nonstationary
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noisy environments were based on the GSC [12–15]. The effectiveness of a GSC system strictly relies on the adaptive filtering algorithm used in the sidelobe cancelling path. Generally the adaptation of filters in time-domain may be performed by gradient-based adaptive algorithms, such as the least mean squares (LMS) algorithm (see for example [16]). Although this family of algorithms entails a low computational cost, when the filter length is quite large a rather slow convergence occurs [16], thus the adaptation of the filter weights becomes unpractical in real-time applications. Another time-domain standard approach is Hessian-based adaptive filtering, which is typical of algorithms such as recursive least squares (RLS) filter. The latter approach displays a faster convergence rate compared with gradient-based algorithms [16]. However, RLS adaptive filtering entails a high computational complexity, thus adaptation may become prohibitively expensive. Moreover, the RLS may perform worse than LMS algorithm in nonstationary environment, depending on the statistics of desired source signals [17]. A good compromise between performance and computational load may be obtained by using the family of affine projection algorithm (APA) [18], which is quite used in adaptive beamforming [19,20], since it shows better convergence rates and manageable computational complexity compared with other time-domain algorithms. Moreover, APA is the best suitable algorithm to process colored signals compared with other classic timedomain adaptive algorithms. Despite its good capabilities [18,16], APA suffers from adverse environment conditions, especially in the presence of multiple nonstationary sources which make the adaptation process unstable and reduce performance. In order to address this problem, we propose robust array beamforming schemes based on the adaptive combination of MISO filtering systems, that, in this case, are nothing but filter banks. Adaptive combination of adaptive filters is a very effective and flexible approach to balance the compromises inherent to the settings of adaptive filters [21,22]. Combined adaptive schemes are usually adopted with filters of the same family and complementary properties, e.g. using different step sizes or different filter lengths. They are also used even with filters of different families using different updating rules or different cost functions [23–27]. The combined scheme is capable of adaptively switching between filters according to the best performing filter, thus always providing the best possible filtering [21]. In this paper we propose beamforming architectures which exploit adaptive combination of filters to improve the tracking properties of the system in the presence of broadband nonstationary interfering signals. A first solution is to combine two MISO systems using the same updating algorithm but with different step size values. In fact, it is proved that a combination of a fast filter (i.e. using a large step size value) and a slow one (i.e. using a small step size value) results in faster convergence, lower residual misalignment and improved tracking capability compared with individual filters [28,29,21]. Another method to improve the tracking capability in nonstationary conditions is based on the combination of two filters with different updating approaches, i.e. one gradientbased and one Hessian-based [26,30]. This combination
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of filters exploits the fast convergence provided by the Hessian-based filter and the performance capabilities provided by the gradient-based filter which may outperform the Hessian-based one in nonstationary conditions [30,26]. Compared with the combination of filters with different step sizes, whose performance in terms of excess mean square error (EMSE) is never better than the performance of individual filters, the combination of filters with different updating approaches provides performance that may outperform the performance of correspondent individual filters in terms of EMSE [30]. As regards the adaptive combination, in this work we focus on the convex constrained combination using a sigmoid nonlinearity on the output stage, since it introduces less gradient noise compared to unconstrained and affine constrained combinations [27,25]. In particular, we propose two different beamforming architectures based on the combination of adaptive MISO systems using different updating approaches. The first scheme is based on a system-by-system (SS) combination, in which the overall output of the first MISO system is convexly combined with the overall output of the second MISO system. The second architecture is a filter-by-filter (FF) combination scheme, in which each adaptive filter of the first MISO system is convexly combined with the correspondent filter of the second MISO system. In both schemes, all the adaptive filters are updated using an APA. In order to differentiate the MISO systems, we use different projection orders for each system. Moreover, in order to use the best parameter setting for each filter and further improve the tracking performance, we propose a multistage combination scheme in which the filtering process is carried out in two steps. The paper is organized as follows: in Section 2 a problem formulation is derived by considering the overall array beamforming architecture. In Section 3 the adaptive combinations of MISO systems are introduced according to both the SS and the FF combination schemes, while the adaptive beamforming architecture based on the multi-stage combination scheme is derived in Section 4. Section 5 provides a mean-square performance analysis of the proposed multistage combination scheme. The effectiveness of the proposed beamforming systems is addressed by experimental results in Section 6 and, finally, in Section 7 our conclusions are drawn.
1.1. Notation In this paper, matrices are represented by boldface capital letters and vectors are denoted by boldface lowercase letters. Time-varying vectors and matrices show discrete-time index as a subscript index, while in time-varying scalar elements the time index is denoted in square brackets. A regression vector is represented as xn ∈RM ¼ ½x½n x½n−1 … x½n−M þ 1T , where M is the overall vector length and x½n−i is individual entry at the generic time instant n−i. A generic coefficient vector, in which all the elements depend on the same time instant, is denoted as wn ∈RM ¼ ½w0 ½n w1 ½n … wM−1 ½nT , where wi ½n is the generic i-th individual entry at the n-th time instant. The index related to a MISO systems is denoted
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as a superscript enclosed in brackets for vector and matrices and as subscripts for scalar values. The index related to each channel is denoted as subscript, after the time index for vectors and matrices, e.g. xn;p . The time index is omitted when the coefficient vector is a realization of a time-invariant process. All vectors are represented as column vectors. 2. Combined adaptive beamforming architecture The beamforming architecture adopted in this paper is a typical GSC configuration composed of a sensor array interface, a fixed beamformer (FBF), and an adaptive sidelobe cancelling path, as depicted in Fig. 1. Let us consider a sensor array interface composed of N elements. The signal ui ½n received by the i-th sensor, with i ¼ 0; …; N−1, is a delayed replica of the target signal s½n convolved with the impulse responses of the propagation channel. The FBF aligns the source signals with reference to the desired source direction, yielding the reference signal d½n. In this paper, a delay-and-sum beamformer is used as FBF, however, nothing prevents to consider other FBFs. In the adaptive path of the beamformer, the blocking matrix (BM) generates the noise references xp ½n, with p ¼ 0; …; P−1, being P ¼ N−1. In this paper, the blocking matrix, denoted with B∈RPN , is implemented by pairwise differences between sensor signals [11] (i.e. the sum of the elements of each row is null). This choice is due to the fact that the goal of the paper is to evaluate the effectiveness of the combined beamforming within a standard architecture. As an alternative, other BMs may be used to increase the overall performance of an adaptive beamformer (see for example [13,31,32,15]). The noise reference signals are then processed by means of the combined adaptive noise canceller (CANC), whose structure will be detailed in the next section. The goal of the CANC is to remove any residual noise components in the reference signal, minimizing the output power and yielding the beamformer output signal e½n.
adopted architecture, instead, results from combinations of adaptive filters. In particular, the structure is composed of two or more different MISO systems, each bringing different filtering capabilities to the whole beamformer. The MISO systems receive the same input signals, which are the noise reference signals resulting from the BM. However, due to the different projection orders, the input data matrices may be different from each other. Therefore, taking into account a number J of MISO systems, the p-th filter of the j-th MISO system, with j¼1,…,J, receives as input data a noise reference matrix XðjÞ n;p : 2
3 xTn;p 6 T 7 6 x 7 6 n−1;p 7 K j M XðjÞ ¼6 7 n;p ∈R ⋮ 6 7 4 T 5 xn−K j þ1;p 2 xp ½n 6 x ½n−1 6 p ¼6 6 ⋮ 4 xp ½n−K j þ 1
The trademark of the proposed beamforming approach is represented by the structure of the CANC. Generally, a conventional adaptive noise canceller (ANC) is composed of an adaptive filter bank forming an MISO system. The
u0 n
e n
+ -
z n x0 n Blocking Matrix
…
xp ½n−M
⋮
⋱
⋮
xp ½n K j
…
xp ½n−M−K j þ 2
ðjÞ ðjÞ ðjÞ T M wðjÞ n;p ∈R ¼ ½w0;p ½n w1;p ½n … wM−1;p ½n :
7 7 7 7 5
ð2Þ
All the filters of each MISO system, represented by (2), contain the same number M of coefficients and are adapted according to the affine projection algorithm (APA) [18]. The updating rule of the APA is ðjÞ ðjÞ;T ðjÞ ðjÞ;T −1 ðjÞ wðjÞ n;p ¼ wn−1;p þ μj Xn;p ðδj I þ Xn;p Xn;p Þ en
ð3Þ
Kj eðjÞ n ∈R
is the error vector of the j-th MISO system where containing the last Kj samples of the j-th error signal, which results from P−1
ð4Þ ðjÞ
uN 1 n
xp ½n−2
3
p¼0
3.1. Convex combination of adaptive filters using affine projection algorithm
d n
xp ½n−M þ 1
where Kj represents the projection order for all the filters of the j-th MISO system. We denote the coefficient vector of the p-th filter belonging to the j-th MISO system at the n-th time instant as
ðjÞ eðjÞ n ¼ dn − ∑ yn;p :
3. Adaptive combinations of MISO systems
Delay
…
ð1Þ
ðjÞ
Fixed Beamformert
xp ½n−1
Combined Adaptive Noise Canceller
Fig. 1. Combined adaptive beamforming architecture.
In this equation dn ∈RK j ¼ ½d½n d½n−1 … d½n−K j þ 1T is the vector containing the last Kj samples of the desired ðjÞ ðjÞ Kj signal and yðjÞ n;p ∈R ¼ Xn;p wn−1;p is the vector containing the Kj projections of the output signal relative to the p-th filter of the j-th MISO system. Moreover, in (3) the parameters μj and δj are, respectively, the step size and the regularization factor, which are the same for all the filters of the j-th MISO system. Note that the APA described by (3) is an approximation of the multichannel APA (see for example [33]) since it neglects the correlation between channels. Using the updating rule described by (3) the considered MISO systems can be differentiated simply by changing the values of the filter parameters. To this end, and in order to improve the robustness of the beamformer against nonstationary signals, we can consider a number of J¼ 2 MISO systems (j¼1,2) that may have the same projection order but different step size values (or regularization factors as an alternative), and in particular a small step size value for the first MISO system and a large value for the second. It is also possible to fix the step size for both the MISO systems
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and to choose different projection orders (in particular K 1 ¼ 1 and K 2 4 1), thus, respectively, resulting in a gradient-based MISO system and an Hessian based one. In the following step, the two MISO systems can be combined under a convex constraint in order to exploit the different capabilities of both the systems. Aside from the chosen distinguishing parameters, there are two ways to combine the MISO systems. The first way is to convexly combine the outputs of the two MISO systems and the second one is to combine each filter of the first MISO system with the corresponding filter of the second MISO system under a convex constraint. We denote the former way as SS combination scheme and the latter as FF combination scheme, which are both described in the following two subsections.
The first proposed scheme is the system-by-system CANC, depicted in Fig. 2(a). The output of each MISO system, that we denote as yðjÞ ½n, is achieved by summing each individual filter output of the corresponding MISO system P−1
yðjÞ ½n ¼ ∑ yðjÞ p ½n:
ð5Þ
p¼0
As j¼1,2, Eq. (5) yields two system outputs that are then convexly combined to generate the overall CANC output z½n ¼ λ½nyð1Þ ½n þ ð1−λ½nÞyð2Þ ½n
parameter. This is the reason why the convex combination can be carried out by using a sigmoid function, as done in [21,34]. However, especially when J 42, any other convex constraint optimization algorithm can be applied (see for example [27]). Therefore, the mixing parameter in (6) can be adapted through the adaptation of an auxiliary parameter, a½n, related to λ½n by means of a sigmoidal function defined according to [34] λ½n ¼ βðsgmða½nÞ−αÞ ¼β
ð6Þ
where λ½n is the mixing parameter, which takes into account the convex constraints (0 ≤λ½n ≤1) [21]. Due to the fact that we have to combine only J ¼2 MISO systems, in (6) we need to adapt just one weight, that is the mixing
1 −α ð1 þ e−a½n Þ
ð7Þ
where β¼
3.2. System-by-system combination scheme
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1 1−2α
α ¼ sgmð−4Þ:
ð8Þ
The parameters (8) are introduced in (7) to prevent the stalling of the adaptation rule and to reduce gradient noise when the mixing parameter is close to 0 or to 1 [34]. To the same end, the value of a½n is kept within [4, −4]. The updating rule of a½n can be written as a½n þ 1 ¼ a½n−
μs ∂e2 ½n r½n ∂a½n
¼ a½n þ
μs e½nðyð1Þ ½n−yð2Þ ½nÞðλ½n þ αβÞðβ−αβ−λ½nÞ βr½n
ð9Þ where μs =r½n represents a normalized step size [35], r½n ¼ γr½n−1 þ ð1−γÞðyð1Þ ½n−yð2Þ ½nÞ2 is the estimated power of ðyð1Þ ½n−yð2Þ ½nÞ, and γ is a smoothing factor.
Fig. 2. Single-stage combined adaptive noise canceller architectures: (a) system-by-system and (b) filter-by-filter combination schemes.
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Finally, the beamformer output signal e½n, using the SS combination scheme, is achieved as e½n ¼ d½n−z½n:
ð10Þ
3.3. Filter-by-filter combination scheme The second proposed scheme is the filter-by-filter CANC in which the output signal z½n is built in a different way. As it can be seen in Fig. 2(b), the p-th filter output of the first MISO system is convexly combined with the correspondent p-th filter output of the second MISO system, thus generating P−1 outputs, each one relative to a noise reference ð2Þ yp ½n ¼ λp ½nyð1Þ p ½n þ ð1−λp ½nÞyp ½n
ð11Þ
where λp ½n is the p-th mixing parameter, adapted using the p-th auxiliary parameter, ap ½n, similarly to (9) ap ½n þ 1 ¼ ap ½n μ þ f e½nðypð1Þ ½n−yð2Þ p ½nÞλp ½nð1−λp ½nÞ r p ½n
ð12Þ
where μf is the step size value which may be chosen as the same for all combinations. Once computed the convex combinations, we achieve the CANC output signal z½n by summing the individual output contributions deriving from the combinations (see Fig. 2(b)) P−1
z½n ¼ ∑ yp ½n:
ð13Þ
p¼0
From Eq. (13) we derive the overall beamformer output signal e½n ¼ d½n−z½n, relative to the FF combination scheme. Both the combined architectures presented above improve the tracking capabilities of CANC giving robustness to the overall beamforming system in the presence of nonstationary interfering signals. Note that the computational increase is linear and, therefore, easily manageable for current processors.
and the projection orders. In particular, we choose a small step size μj ¼ μa for j¼1,3 and a large step size value μj ¼ μb for j¼2,4. Moreover, we update the first two MISO systems by using a gradient-based algorithm and the second two systems with an Hessian-based algorithm. This is obtained by setting a unitary projection order K j ¼ K A ¼ 1 for j ¼1, 2 and a higher projection order K j ¼ K B 41 for j¼3, 4. The choice of different step size values affects the convex combinations on the first stage, in which the first MISO system is combined with the second one and the third system with the fourth one. In this stage the convex combination may follow the SS scheme or the FF scheme. In Fig. 3 a multi-stage beamformer with a FF scheme on the first stage is depicted. On the other hand, the choice of different projection orders affects the convex combination on the second stage, in which the output signal resulting from the combination of the first and the second MISO systems is in turn combined with the output signal resulting from the combination of the third and the fourth MISO systems. Regarding the combination on the second stage, there is no distinction between SS and FF schemes. Output signals of the convex combinations on the first stage, denoted as zðAÞ ½n and zðBÞ ½n, may be achieved similarly to (6), according to an SS combination scheme, or similarly to (13), according to an FF combination scheme as depicted in Fig. 3. On the other hand, the convex combination on the second stage may yield the following output signal from the multi-stage CANC: z½n ¼ η½nzðAÞ ½n þ ð1−η½nÞzðBÞ ½n
ð14Þ
where η½n is the mixing parameter of the second stage, adapted using an auxiliary parameter, similar to (9). Once computed the second stage convex combination, we derive the overall multi-stage beamformer output signal e½n ¼ d½n−z½n, as done for the single-stage combination schemes in Section 3. The multi-stage beamforming architecture introduced above exploits the capabilities of each MISO system, thus improving the cancelling performance compared to conventional beamformers (e.g. using a single MISO ANC) and single-stage combined beamformers in the presence of nonstationary interfering signals.
4. Multi-stage combined beamforming The combined beamforming schemes described in the previous section are effective in the presence of multiple nonstationary sources both choosing different step size values (a small one and large one) and different projection orders (one of unitary order and one of higher order). Further improvements may be achieved if we consider the joint capabilities deriving from choosing both different step size values and projection orders. To this end we propose a multi-stage combination scheme in which the filtering process may involve more convex combinations of MISO systems. In particular, in order to yield an adaptive beamforming architecture that is robust against nonstationary conditions, we may consider a CANC composed of a number J ¼4 of MISO systems, as depicted in Fig. 3, each one bringing different capabilities to the whole architecture. We differentiate the four systems according to the step size values
5. Mean-square performance of the multi-stage combination scheme In this section we analyze the mean-square performance of the proposed multi-stage combination scheme. Before starting the analysis it is convenient to introduce some further notation. Let us denote each combination on the first stage with the index q¼{A,B}, where A and B are, respectively, referred to the first and the second combinations of the first stage. Each combination is composed of two MISO systems denoted by lower case letters a and b. The index variable associated to MISO systems is r ¼{a,b}. Hence, a generic MISO system of a generic first stage combination is denoted as qr ¼{Aa,Ab,Ba,Bb}. An important preliminary assumption must be made regarding the second stage combination. While the first stage of the multi-stage architecture combines MISO systems having different step size values, the second stage
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
2,
3311
,
Fig. 3. Multi-stage combined adaptive noise canceller.
combines systems resulting from different projection orders. This leads to combine two different “virtual” weight vectors for the second stage. An exact analysis should be carried out by considering the higher projection
order as a reference and thus zero-padding the weight vector with the lower projection order. However, since it does not affect the result of our analysis we may choose a common projection order for a clearer exposition. Such
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value is chosen as the lower projection order value, that in our case is the unitary projection order. In order to derive the mean-square performance at steady-state, we consider the identification of an unknown model (see for example [16]). To this end, let us consider now the following linear regression model: ! P−1
∑ xTn;p wop
d½n ¼
þ v½n
ð15Þ
p¼0
where wop is the p-th unknown vector of length M to be identified and v½n is an independent identically distributed (i.i.d.) noise whose variance is s2v . We define now the “virtual” weight vectors resulting from the combination on the first and the second stages. In particular, for the first stage we can obtain different pairs of vectors for each channel, whether adopting an SS or an FF combination scheme SS :
ðqÞ ðqaÞ b n;p w ¼ λðqÞ ½nwn;p þ ð1−λðqÞ ½nÞwðqbÞ n;p
ð16Þ
FF :
ðqÞ ðqaÞ ðqÞ ðqbÞ b ðqÞ w n;p ¼ λp ½nw n;p þ ð1−λp ½nÞwn;p
ð17Þ
for q¼{A,B}. On the other hand, the overall virtual MISO system for the second stage can be described as b n;p ¼ w
b ðAÞ η½nw n;p
þ
b ðBÞ ð1−η½nÞw n;p :
ð18Þ
The weight error vectors are defined as the difference between the optimal solution and the virtual weights for the first and the second stage, respectively, as o b ðqÞ ~ ðqÞ w n;p ¼ wp −w n;p ;
q ¼ fA; Bg
b n;p : ~ n;p ¼ wop −w w
ð19Þ ð20Þ
We also define the a priori and a posteriori error signals, respectively, for each combination of the first stage P−1
ðqÞ
ð21Þ
ðqÞ
ð22Þ
ðqÞ ~ eðqÞ a;n ¼ ∑ Xn;p w n−1;p p¼0 P−1
ðqÞ ~ eðqÞ p;n ¼ ∑ Xn;p w n;p p¼0
with q¼{A,B}, and for the second stage P−1
~ n−1;p ea ½n ¼ ∑ xTn;p w
ð23Þ
p¼0 P−1
~ n;p : ep ½n ¼ ∑ xTn;p w
ð24Þ
p¼0
Note that the last two equations related to the second stage derive from a unitary projection order of the input data. In order to measure filtering performance of MISO systems the excess mean-square error (EMSE) can be adopted as measure, defined as the excess over the minimum mean-square error. Since the steady-state analysis is evaluated as n-∞, the EMSE of the second stage combination can be expressed as (see [21]) ξ ¼ lim Efjea ½nj2 g: n-∞
ð25Þ
With the aim of providing an extended expression for (25), it is useful to refer to an additional variable that measures the steady-state correlation for the second stage between
the a priori errors resulting from the combinations of the first stage. Such variable is denoted as cross-EMSE [21] and it can be expressed as ðBÞ ðBÞ;T ðAÞ ξcross ¼ lim EfeðAÞ;T a;n ea;n g ¼ lim Efea;n ea;n g: n-∞
n-∞
ð26Þ
Therefore, from [21] and taking into account (26), we can write the EMSE for the overall two-stage architecture ξ as ξ ¼ ξcross þ
ΔξðAÞ ΔξðBÞ ΔξðAÞ þ ΔξðBÞ
ð27Þ
where ΔξðqÞ ¼ ξðqÞ −ξcross ;
q ¼ fA; Bg:
ð28Þ
The expressions of the EMSE for both the combinations on the first stage, ξðqÞ , and the cross-EMSE for the second stage, ξcross can be easily derived according to [21,36]. The interested reader can find the analytical derivation of ξðqÞ and ξcross in Appendix A. 6. Experimental results In this section we carry out two different sets of experiments: the first set (in Subsection 6.1) aims at assessing the effectiveness of the described combined filtering schemes adopted in the proposed beamforming method; the second set of experiments, detailed in Section 6.2, is performed in order to evaluate the proposed combined beamforming architectures for noise reduction application in a multisource scenario. 6.1. Convergence performance of combined filtering schemes In the first set of experiments we prove the filtering effectiveness of proposed CANC schemes through a tracking analysis which describes the convergence performance. To this end we use conventional ANC MISO systems and the proposed combined schemes to identify an unknown nonstationary model and to compare their performance. In particular, the model to identify derives from (15) and is composed of a number M of transfer functions that, in the case of beamforming application, may represent the impulse responses between a source and M sensors. The initial optimal solution is formed with M¼7 independent random values between −1 and 1. In the following examples the initial system is wo1 ¼ ½0:4125 0:7632 − 0:5484 −0:6099 −0:4622 −0:4826 −0:5296T . The input signal is generated by means of a first-order pffiffiffiffiffiffiffiffiffiffiffi autoregressive model, whose transfer function is 1−α2 =ð1−αz−1 Þ, with α ¼ 0:8, fed with an i.i.d. Gaussian random process. The length of the input signal is of L¼ 8000 samples. In order to study the ability of combined schemes to react to nonstationary environments, at time instant n ¼ L=2 the system changes into wo2 ¼ ½−0:4223 0:0848 −0:1228 0:3876 0:9950 0:9806 −0:2700T . Furthermore, an additive i.i.d. noise signal ev ½n with variance s2v ¼ 0:01 is added to form the desired signal (it is the same for all the filtering architectures). In order to identify the unknown solutions wo1 and wo2 we use both conventional MISO systems and the adaptive combined filtering schemes described in Sections 3 and 4.
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
A similar result was achieved in the second experiment, where we compare the same filtering architectures of the first experiment, but we choose the same step size value μ ¼ 0:01 and different projection orders: K 1 ¼ 1 and K 2 ¼ 4. Similarly to the previous experiment, in Fig. 5(a) we have compared the performance of combined architectures with those of conventional ANC using both K1 and K2. Even in this case, both SS and FF schemes provide better convergence performance compared to conventional filtering. The mixing parameter behaviors in Fig. 5(b) confirm a slightly faster adaptation of the FF scheme compared to the SS one. Using the same scenario, we study now the convergence performance of the multi-stage combined architecture. As stated in Section 4, in a multi-stage combined scheme the combinations on the first stage may be performed in both SS or FF way. We consider two-stage combined schemes composed of four different MISO systems and we choose two different step size values, μa ¼ 0:01 and μb ¼ 0:1, and two different projection orders K A ¼ 1 and K B ¼ 4. A first comparison is performed between a multi-stage SS (MSS) scheme and a multi-stage FF (MFF) one. A first important result is shown in Fig. 6(a), where we can see that, using a multi-stage architecture, the superiority of the FF scheme over the SS one is noticeably more evident compared to the single-stage architectures (see Figs. 4(a) and 5(a)). Such performance improvement is strictly related to the coherence between the input signals to a convex combination scheme. In fact, in a multi-stage architecture, FF schemes on the first stage exploit the availability of a larger number of free parameters, thus increasing the coherence between the inputs on the second stage and leading to a performance improvement. In light of this result we take into account the performance of a multi-stage scheme whose combinations on the first stage are performed according to an FF scheme, as depicted in Fig. 3. In Fig. 6(b) the comparison between the MFF combined architecture and the individual conventional filterings shows that the multi-stage filtering results in the best performing architecture. Moreover, the performance
We compare their performance in terms of excess mean square error (EMSE) defined as EMSE½n ¼ Efðe½n−v½nÞ2 g, where e½n is the error signal of the filtering architecture, and the operator Efg is the mathematical expectation. The EMSE of each filtering structure is evaluated over 1000 independent runs. Moreover, in order to facilitate the visualization, the EMSE curves are filtered by a movingaverage filter. All the filtering architectures, included the conventional ones, use MISO systems with P ¼4 channels. In the first two experiments we compare a conventional MISO architecture and both single-stage combined architectures described in Section 3, i.e. the SS CANC and the FF CANC. Both the SS and the FF schemes are composed of two MISO systems, as depicted in Fig. 2. All the MISO systems use APA filters. In both the experiments for the adaptation of the mixing parameter of the SS filtering architecture (see Eq. (9)) we use a step size value of μs ¼ 102 , while a step size value of μf ¼ 103 is adopted for the adaptation of all the mixing parameters of the FF scheme (see Eq. (12)). Both the step size values provide the best performance in each case. In the first experiment, we evaluate the filtering architectures choosing the same projection order K¼2 for all the MISO systems and different step size values for the MISO systems of the combined schemes: a slower one μ1 ¼ 0:01 and a faster one μ2 ¼ 0:1. In Fig. 4(a) we have compared the performance of the combined architecture with those of the conventional ANC using both μ1 and μ2 . As it is possible to see, both SS and FF schemes take advantage from using the combined filtering with respect to conventional filtering. In fact, combined schemes always show the behavior of the best performing system and in transient state they behave even better than the best conventional filtering. Both the combined schemes provide good convergence performance, but the FF scheme is slightly better than the SS one due to the fact that the adaptation of the mixing parameters in the FF scheme is faster than in the SS one, as it can be seen in Fig. 4(b). This results in a quality improvement of the processed signal that can be decisive in many applications (e.g. speech applications).
−10
1
Conventional ANC,µ1 Conventional ANC,µ2 SS CANC FF CANC
−15
0.9 0.8
−20
0.7 Mixing parameter
EMSE [dB]
3313
−25 −30
0.6 0.5 0.4 0.3
−35
0.2 −40 0.1
SS CANC FF CANC
0
−45 0
1000
2000
3000
4000 Samples
5000
6000
7000
8000
0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
Fig. 4. (a) EMSE comparison between single-stage combined filtering architectures and conventional ones using the same projection order and different step size values. Combined architectures always provide the best performance. (b) Behavior comparison between the mixing parameter of the SS CANC and the mixing parameter relative to the first channel of the FF CANC.
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1
−10 Conventional ANC, K1 Conventional ANC, K2 SS CANC FF CANC
−15
0.9 0.8 0.7 Mixing parameter
EMSE [dB]
−20 −25 −30
0.6 0.5 0.4 0.3
−35
0.2 −40 0.1
SS CANC FF CANC
0
−45 0
1000
2000
3000
4000
5000
6000
7000
8000
0
1000
2000
3000
Samples
4000
5000
6000
7000
8000
Samples
Fig. 5. (a) EMSE comparison between single-stage combined filtering architectures and conventional ones using the same step size value and different projection orders. Combined architectures always provide the best performance. (b) Behavior comparison between the mixing parameter of the SS CANC and the mixing parameter relative to the first channel of the FF CANC.
−5
MSS CANC MFF CANC
−10
EMSE [dB]
−15 −20 −25 −30 −35 −40 −45 0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
−5 −10 −15 −20 −25 −30
FF,µa,µb, KA FF,µa,µb, KB FF,µa,KA, KB FF,µb,KA, KB MFF CANC
−15 −20 EMSE [dB]
EMSE [dB]
−10
Conventional ANC,µa,KA Conventional ANC,µb,KA Conventional ANC,µa,KB Conventional ANC,µb,KB MFF CANC
−25 −30 −35
−35
−40
−40 −45
−45 0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
0
1000
2000
3000
4000
5000
6000
7000
8000
Samples
Fig. 6. EMSE comparison: (a) between MSS and MFF schemes; (b) between MFF architecture and conventional ANCs; and (c) between MFF architecture and single-stage FF ones. Multi-stage combined architectures always provide the best overall performance.
improvement of the MFF architecture results even from the comparison with the single-stage FF architectures, as depicted in Fig. 6(c).
Results described in this subsection show the filtering ability of proposed combined schemes compared to conventional filtering. For single-stage architectures both SS
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
and FF schemes can be used since they achieve same performance results. However, for multi-stage architectures a decisive preference is given to FF schemes which show better performance and reaction to abrupt changes in the environment. Moreover, FF schemes may exploit spatial diversity and thus different step size values for the adaptation of the mixing parameters may be chosen according to the scenario requirements. Finally, it has been shown that multi-stage combined filtering always achieves the best convergence performance.
6.2. Evaluation of combined beamformers in acoustic noise reduction In the second set of experiments we assess the effectiveness of the proposed combined beamforming architectures in a typical dense multipath environment, that is an acoustic environment. In particular, experiments take place in a 6 5 3:3 m room with a reverberation time of
3315
T 60 ≈120 ms for a hands-free speech communications application. The source of interest is a female speaker located 70 cm from the center of a microphone array, as depicted in Fig. 7. Two interfering sources are initially located, respectively, 1.9 m and 2.8 m from the center of the acoustic interface: the first source is a female speaker located on the left of the array, while on the right is located the second source which is a male speaker. White Gaussian noise is added at microphone signals as diffuse background noise, thus providing 20 dB of SNR (signal-tonoise ratio) with respect to the desired source. The overall input SNR level, measured for each sensor signal, is of about 3 dB. After 5 s from the start of the experiment the first source changes its position and after 10 s also the second source changes its position. The overall length of the experiment is 15 s. The impulse responses between sources and sensors are simulated by means of Roomsim, which is a Matlab tool [37]. Each impulse response is measured by using an 8 kHz sampling rate and it is truncated after M ¼340 samples,
Microphone Array
0,7 m
Desired Source
Interfering Source #2
Interfering Source #1
Background Noise
Microphone Array
Microphone Array
Interfering Source #1
Interfering Source #1 Desired Source
Interfering Source #2
Desired Source
Interfering Source #2
Background Noise
Background Noise
Fig. 7. Acoustic environment with nonstationary interfering sources. The source of interest is a female speaker located in front of a microphone array and two interfering speakers are located, respectively, on the left and on the right of the desired source (a). After 5 s from the start of the experiment the first interfering source moves to position 2 (b) and at second 10 also the second interfering source changes its position (c).
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which is also the length of each filter. The microphone interface is a classic uniform linear array (ULA) composed of 8 omnidirectional sensors equally spaced with a distance of 5 cm, thus having a good spatial resolution even at mid-low frequencies. The noise reduction performed by the beamformer is usually associated with an SNR improvement, defined as [11]
7. Conclusions
"
# Efs2in ½ng SNR ¼ 10 log Efs2in ½ng−Efs2out ½ng
ð29Þ
where sin ½n is the generic input clean signal and sout ½n is the processed signal. We compute the SNR level over the total length of the experiment (0–15 s) and also in three different sub-intervals of time: the initial state, from 0 to 5 s, when the two interfering sources are located in their initial position; the first change, from 5 to 10 s, which includes the position change of the first interfering source and the consequent readaptation of the filtering system; the second change, from 10 to 15 s, when also the second interfering source changes its position. We compare GSC beamformers having different ANC: conventional ANCs with different parameter settings, the single-stage FF combined ANC with different parameter settings, and the two-stage combined ANC. Filter parameters μa , μb , K A , K B , μs and μf are the same used in the first set of experiments. Results are collected in Table 1, in which it is possible to notice the behavior of the different beamformers taken into account and their contribution to the noise reduction in terms of SNR improvement. We could have shown performance of both SS and FF combination schemes and both varying the step size values and the projection order. For a better ease of reading results, we only show the performance relative to FF combination schemes, which achieve the more relevant results, and we only vary the step size value for the single-stage combined ANCs. From Table 1 one can infer that all the conventional ANCs show difficulties when a source change its position, thus decreasing speech enhancement performance. The more stable conventional ANC is the one having a large step size value and a large projection order, but its performance is the poorest in terms of SNR. A significant enhancement is achieved by means of the FF CANC and a further improvement is provided by the MFF CANC which achieves the best performance in each time interval in terms of SNR. Table 1 SNR comparison in dB. We evaluate the beamformers over three subintervals of time, 0–5, 5–10 and 10–15 s, and over the total length of the experiment, 0–15 s. Multi-stage combined beamformer always performs the best reduction of interfering signals. GSC Conventional ANC, μa , Conventional ANC, μa , Conventional ANC, μb , Conventional ANC, μb , FF CANC, μa , K A , K B FF CANC, μb , K A , K B MFF CANC
KA KB KA KB
SNR values obtained from this experiment are not definitely the best achievable values, since better results may be obtained using more sophisticated GSC beamformers, e.g. involving any voice activity detectors (VADs), adaptive BMs or post-filters. The obtained results are sufficient to show the effectiveness of the proposed combined beamformers compared to conventional methods.
0–5 s
5–10 s
10–15 s
0–15 s
17.2 17.8 18.1 13.4 18.4 18.2 18.8
14.2 16.7 16.3 13.2 17.0 17.5 18.1
14.9 16.8 16.5 13.4 18.1 18.0 19.1
15.6 16.9 16.8 13.4 18.0 17.9 18.7
In this paper we have introduced combined beamforming methods whose goal is to improve the performance, in terms of noise reduction, in the presence of nonstationary interfering sources. The trademark of proposed methods relies on the use of combined filtering schemes in the ANC block. These combined schemes are based on the adaptive combination of MISO systems with different parameter settings thus involving complementary capabilities. The whole beamforming system benefits from the different capabilities of each MISO systems, yielding improved performance. We introduced two different ways of combining the MISO systems which are the SS scheme and the FF one. Both the combined architectures provide better performance compared to conventional beamformers. A multi-stage combined beamformer has been also introduced in which the adaptive combination of MISO systems can be performed in subsequent stages. In particular, we have taken into account a two-stage combined beamformer which outperforms the single-stage schemes, thus always providing the best performance when nonstationary sources interfere with the enhancement of a desired signal. Among the multi-stage architectures, the ones using FF schemes are decisively preferable due to the fact that the adaptive combination is performed for each channel. This allows FF beamformers to better react to abrupt changes in the environment. In conclusion, the goal of this work was to introduce novel schemes for adaptive beamforming that can improve the overall performance, thus paving the way for future developments. In particular, simply changing some aspects of the multi-stage combined architecture (e.g. the adaptive algorithms used for the MISO systems, the kind of adaptive combination, the number of combinations, the number of combination stages, the general adaptive beamforming technique) may lead to several combined architectures well-suited for specific beamforming applications. Appendix A. Analytical derivation of the EMSE for the two-stage combination scheme In order to achieve ξ in (27) it is necessary to derive the expressions of the EMSE for both the combinations on the first stage, ξðqÞ , and the cross-EMSE for the second stage, ξcross . As represented in Fig. 3, the first stage is composed of two (or even more) convex combinations of MISO systems, each of which may be characterized by an SS or an FF combination scheme. Therefore, in theory each combination may require a different EMSE measurement. However, since from [21] the EMSE measure depends only on the step size values and on the input signal, we can derive the
D. Comminiello et al. / Signal Processing 93 (2013) 3306–3318
same expression both for SS and FF schemes. Therefore, EMSEs for the first stage can be distinguished just according to the projection order of the input data. In particular, the two combinations on the first stage are, respectively, characterized by the projection orders K j ¼ K A ¼ 1 for j¼1,2 and K j ¼ K B 4 1 for j¼ 3,4 as stated in Section 4, thus the EMSE measures for the first stage of combinations can be described by the following expression (similarly to (27) [21]): ξðqÞ ¼ ξðqÞ cross þ
ΔξðqaÞ ΔξðqbÞ ðqaÞ
Δξ
þ Δξ
ðqbÞ
ðA:1Þ
for q¼ {A,B} and ðqrÞ
Δξ
ðqrÞ
¼ξ
−ξðqrÞ cross
ðA:2Þ
the EMSE of a generic MISO system, for q ¼{A,B} being ξ and r ¼{a,b}. In order to compute (A.2) we must derive the expressions of ξðqrÞ and ξðqrÞ cross . EMSE values ξðqrÞ are related to individual MISO systems of each combination on the first stage and can be derived according to [36] as ξðqrÞ ¼
μr s2vq TrðEfR nðqÞ gÞ ð2−μr ÞTrðS EfR ðqÞ n gÞ
we get the following relation: ðqÞ ðqbÞ ðqbÞ ðqaÞ;T ðqÞ ðqbÞ μa μb EfeðqaÞ;T RðqÞ Rn ea;n g þ μb EfeðqaÞ;T n en g ¼ μa Efen n a;n R n en g:
ðA:7Þ Then, by substituting we obtain
ðA:3Þ
ðqÞ ðqÞ;T −1 where RðqÞ n ¼ ðδq I þ Xn;p Xn;p Þ . The parameter S derives
from the assumption in [36], according to which EfeðqrÞ;T a;n ðqrÞ 2 T eðqrÞ a;n g ¼ Efjea j g S, where S≈I for small μr , and S≈ð1 1 Þ
for large μr , being 1 ¼ ½1 0 ⋯ 0T . The regularization factor δq is assumed to be a small value, the same for each combined pair of MISO systems. Note also that in (A.3) the step size value is distinguished according to the only index r ¼{a,b}. This is due to the fact that each pair of combined MISO system on the first stage is characterized by the same two values for the step size parameters. On the other hand, to derive the expression of ξðqrÞ cross , we consider the energy conservation relation (see [16]) that relates the weight error vectors and the a priori and a posteriori error signals of each combination on the first stage ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ~ ðqrÞ ~ ðqrÞ w n;p þ Xn;p ðXn;p Xn;p Þ ea;n ¼ w n−1;p þ Xn;p ðXn;p Xn;p Þ ep;n
eðqrÞ n
¼
eðqrÞ a;n
þ
vðqÞ n ,
for r ¼ a; b, in (A.7),
ðqÞ ðqbÞ μa μb EfeðqaÞ;T a;n R n ea;n g ðqaÞ;T ðqÞ ðqbÞ R nðqÞ vðqÞ þ μa μb EfvðqÞ;T n n g ¼ ðμa þ μb ÞEfea;n R n ea;n g:
ðA:8Þ
Solving each term of (A.8) according to [36] leads to ðqÞ ðqÞ 2 ðμa þ μb −μa μb Þξcross TrðS EfR ðqÞ n gÞ ¼ μa μb svq TrðEfR n gÞ
ξðqÞ cross
ðqrÞ
3317
ðA:9Þ
ðqbÞ;T ðqaÞ Efea;n ea;n g
¼ for n-∞. Therefore, from (A.9) where we can derive the expression of the cross-EMSE ξðqÞ cross for the combinations on the first stage ðqÞ ξcross ¼
μ′s2vq TrðEfR nðqÞ gÞ ð1−μ′ÞTrðS EfR nðqÞ gÞ
where μa μb μ′ ¼ μa þ μb
ðA:10Þ
ðA:11Þ
for each combination q of the first stage. By substituting (A.(11) and A.3) in (A.2), we can solve the EMSE values ξðqÞ for the first stage of combinations, described by (A.1). Once obtained the expression of ξðqÞ , it is now necessary to compute the value of the cross-EMSE for the second stage ξcross in order to derive the complete expression (27) of the overall EMSE of the two-stage combined architecture. Repeating the procedure applied for ξðqÞ cross , we get ξcross ¼
μ″s2v TrðEfR n gÞ ð1−μ″ÞTrðS EfRn gÞ
ðA:12Þ
in which each term is referred to a unitary projection order, and where, after few passages, μa μb : ðA:13Þ μ″ ¼ 2ðμa þ μb Þ Therefore, substituting (A.(12) and A.10) in (28), the expression of the EMSE (27) for the two-stage combined architecture described in Section 4 can be derived. References
ðA:4Þ ðqÞ;T −1 is similarly to [36, Eq. (8)], where the term ðXðqÞ n;p Xn;p Þ assumed to be invertible. Multiplying the transpose of (A.4) for r ¼a by (A.4) for r ¼ b, and cancelling terms, the following generalized energy conservation relation for each q can be achieved: ðqÞ ðqÞ;T −1 ðqbÞ ðqaÞ;T ~ ðqaÞ;T ~ ðqbÞ ~ ðqaÞ;T ~ ðqbÞ w n;p w n;p þ ea;n ðXn;p Xn;p Þ ea;n ¼ w n−1;p w n−1;p ðqÞ;T −1 ðqbÞ ðqaÞ;T þ ep;n ðXðqÞ n;p Xn;p Þ ep;n :
ðA:5Þ
Taking the expectations of both sides of (A.5) and considering that [16,21] ðqÞ ðqÞ;T −1 ðqbÞ ðqÞ;T −1 ðqbÞ ðqaÞ;T ðqaÞ;T Efea;n ðXn;p Xn;p Þ ea;n g ¼ Efep;n ðXðqÞ n;p Xn;p Þ ep;n g;
n-∞
and ðqÞ ðqÞ;T ðqÞ ðqÞ;T −1 ðqrÞ ðqrÞ eðqrÞ p;n ¼ ea;n −μr Xn;p Xn;p ðδq I þ Xn;p Xn;p Þ en ;
ðA:6Þ
[1] A.B. Gershman, U. Nickel, F. Böhme, Adaptive beamforming algorithms with robustness against jammer motion, IEEE Transactions on Signal Processing 45 (1997) 1878–1885. [2] A.B. Gershman, L. Stanković, V. Katkovnik, Sensor array signal tracking using a data-driven window approach, Signal Processing 80 (12) (2000) 2507–2515. [3] S.A. Vorobyov, A.B. Gershman, Z.-Q. Luo, N. Ma, Adaptive beamforming with joint robustness against mismatched signal steering vector and interference nonstationarity, IEEE Signal Processing Letters 11 (2) (2004) 108–111. [4] S. Markovich, S. Gannot, I. Cohen, Multichannel eigenspace beamforming in a reverberant noisy environment with multiple interfering speech signals, IEEE Transactions on Audio Speech and Language Processing 17 (6) (2009) 1071–1086. [5] J. Li, P. Stoica (Eds.), Robust Adaptive Beamforming, Telecommunications and Signal Processing, Wiley & Sons, Inc., Hoboken, NJ, 2005. [6] S. Haykin, K.J.R. Liu (Eds.), Handbook on Array Processing and Sensor Networks, Wiley & Sons, Inc., Hoboken, NJ, 2010. [7] M. Agrawal, S. Prasad, Robust adaptive beamforming for wide-band, moving, and coherent jammers via uniform linear arrays, IEEE Transactions on Antennas and Propagation 47 (8) (1999) 1267–1275.
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