design of robust steerable broadband beamformers incorporating ...
DESIGN OF ROBUST STEERABLE BROADBAND BEAMFORMERS INCORPORATING MICROPHONE GAIN AND PHASE ERROR CHARACTERISTICS Chiong-Ching Lai, Sven Nordholm, Yee-Hong Leung Department of Electrical and Computer Engineering Curtin University Kent Street, Bentley, WA 6102, Australia ABSTRACT Beamformers are known to be sensitive to errors and mismatches in their array elements. This paper proposes a robust steerable broadband beamformer design using the Farrow structure and for arbitrary array geometry. The design formulation includes stochastic models describing the microphone characteristics as random variables, thus allowing flexibility for microphone errors. This method establishes a direct relationship in controlling the robustness specification in the design from given microphone characteristics. The robust design procedure optimises the mean performance of the beamformer. Design examples show significant reduction in error sensitivity in the robust design formulation. Index Terms— steerable, broadband beamformer, robust, stochastic, microphone characteristics 1. INTRODUCTION Compared to fixed beam broadband beamformers, steerable broadband beamformers (SBBFs) offer an extra ability to steer its main beam on-the-fly. This steering capability offers dynamic beamforming which is extremely useful in applications that involve dynamic environments. Some examples of commercial applications where SBBFs are becoming more and more prevalent include audio-video conferencing, hands-free communication systems and audio surveillance systems, where the speaker is likely to move around [1]. Initial works on SBBFs include decoupling the beampattern spectro-spatial dependencies in order to steer the main beam using a Wigner rotation matrix [2], and using Farrow filters [3] for designing SBBFs [1, 4]. The SBBF design using Farrow filters is interesting as the main beam can be steered online with only a single parameter. However, these SBBFs, which resemble superdirective beamformers for low frequencies, are extremely sensitive to spatial white noise and errors in the array elements [5]. Although a robust design using norm constraint on the resulting filter weights is proposed in [4], there is no clear indication on the choice of the norm constraint parameter and the choice is made rather intuitively. In this paper, we propose a new robust SBBF design formulation. In the new formulation, the Farrow structure is used again as in [4], but it now admits any arbitrary array geometry. As for robustness, we follow one of the methods described in [6], which incorporates the probability density function of the microphone error characteristics into the design formulation and sets as the design objective, the mean performance. The proposed design formulation thus provides a direct relationship between the error characteristics and robustness control, resulting in better robustness specification decision. The robust SBBF design is formulated in both time domain (TD) and frequency domain (FD).
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This paper is organised as follows. Section 2 describes the spiral array geometry and SBBF structure with Farrow filters. Both the TD and FD design of robust SBBFs are formulated in Section 3, followed by design examples in Section 4. Finally, the conclusion is provided in Section 5. 2. FAR-FIELD SBBFS CONFIGURATION 2.1. Array geometry Consider the spiral array [4] shown in Fig. 1. It consists of P concentric rings with K microphones uniformly spaced in each ring. Successive rings are offset by 2π/(KP ) relative to each other. Under the far-field signal model, the array response vector d(ω, φ)1 for a signal with frequency ω and impinging the array at angle φ in the plane of the array is given by [d(ω, φ)]v = dpK+k (ω, φ) ωrp 2πk 2πp = exp j cos φ − − (1) c K PK th where p ∈ {0, ..., P −1} is the index of the p ring, k ∈ {0, ..., K− 1} is the index of the kth microphone in each ring, c is the speed of sound in the propagating medium (c = 343ms−1 in air) and v = pK + k ∈ {0, ..., P K − 1} is the index of the kth microphone in the pth ring. rp is the radius of the pth ring and is given by αc (2) rp = 2fp sin(π/K) where 0 < α ≤ 0.5 to prevent spatial aliasing, and fp is the highest frequency in Hz received by the pth ring. Note that the spacing between successive ring radii rp can be uniform or non-uniform, depending on the choice of fp . In (2), it is assumed that there are sufficient rings in the spiral array. For spiral array with only a few rings (P = 2, 3), r0 is obtained by setting f0 to be the maximum frequency of the received signal. Subsequent ring radii are then selected by trading off between having narrow beamwidth at low frequency and frequency invariant response. One example is rp = 2rp−1 for p ∈ {1, ..., P − 1}. 2.2. Beamformer structure For the TD Farrow filters shown in Fig. 2, its beampattern, steered to azimuth angle ψ, is given by −1 P −1 K−1 −1 N M G(ψ, ω, φ) = apK+k (ω, φ) p=0 k=0 m=0 n=0
× hp,k,m [n]e−jnωTs ψ m dpK+k (ω, φ) 1 All
(3)
vectors used in this paper are column vectors.
ICASSP 2011
Fig. 1: Spiral array geometry Fig. 3: FD Farrow filter structure for SBBF [hf (ω)]q = Hp,k,m (ω) =
N −1
hp,k,m [n]e−jωnTs .
(12)
n=0
Index q = pK + kM + m describe how the transfer functions Hp,k,m (ω) are stacked to form hf (ω). Without microphone characteristics modelling, we have gf (ψ, ω, φ) = gof (ψ, ω, φ) = d(ω, φ) ⊗ f (ψ) . (13) 3. ROBUST SBBF DESIGN 3.1. Time domain formulation Following [6], the least-squares error function for specific microphone characteristics can be written as J(h, ζ,a0 , ..., aP K−1 ) = |G(ζ) − Hd (ζ)|2 = hT Q(ζ)h − 2hT gR (ζ) + |Hd (ζ)|2 Fig. 2: TD Farrow filter structure for SBBF where M − 1 is the order of Farrow filters indexed by m ∈ {0, ..., M − 1}, n ∈ {0, ..., N − 1} is the time index of the N -tap FIR filters, and Ts is the sampling period. av (ω, φ) specifies the microphone characteristics given by (4) av (ω, φ) = κv (ω, φ)ejγv (ω,φ) where both the gain κv (ω, φ) and phase γv (ω, φ) can be frequency and angle dependent. We keep the array response dv (ω, φ) arbitrary (in the azimuth plane) so that the robust design formulation in the next section is valid for any microphone array geometry. We can write (3) in matrix form given by G(ψ, ω, φ) = hT g(ψ, ω, φ) (5) where g(ψ, ω, φ) = (a(ω, φ) d(ω, φ)) ⊗ f (ψ) ⊗ e(ω), (6) T [a(ω, φ)]v = av (ω, φ) , f (ψ) = ψ 0 · · · ψ M −1 , (7) T [h]q = hp,k,m [n] , e(ω) = e−jωTs (0) · · · e−jωTs (N −1) . (8) The symbol denotes element-wise product and ⊗ denotes Kronecker product. Index q = pK + kM + mN + n describes how the real elements in h are stacked. Note that without modelling the microphone characteristics, we have g(ψ, ω, φ) = go (ψ, ω, φ) = d(ω, φ) ⊗ f (ψ) ⊗ e(ω) . (9) For the FD Farrow structure shown in Fig. 3 (ωi is the ith frequency bin), we have G(ψ, ω, φ) = hHf (ω)gf (ψ, ω, φ) (10) (11) where gf (ψ, ω, φ) = (a(ω, φ) d(ω, φ)) ⊗ f (ψ) ,
102
(14)
where Q(ζ) = g(ζ)gH (ζ) , (15) ∗ (16) gR (ζ) = Re{g(ζ)Hd (ζ)} , −jωNd ω ∈ Ωpb , φ ∈ Φpb (ψ) e . (17) Hd (ζ) = 0 otherwise Nd gives the desired delay, Ωpb specifies the spectral passband and Φpb (ψ), which depends on steering angle ψ, defines the spatial passband. Note that ω and φ are dropped from a, κ and γ (their dependencies are understood from the context), and (ψ, ω, φ) is replaced with ζ for notational convenience. The total cost function, which is defined as the sum of the cost functions for all feasible microphone characteristics, weighted by the probability density functions (PDFs) of the microphone is given by characteristics Jtot (h, ζ) =
···
J(h, ζ, a0 , ..., aP K−1 )fA0 (a0 )
× · · · fAP K−1 (aP K−1 )da0 · · · daP K−1 (18) where fA (a) = f,ρ (κ, γ) is the PDF for the random variable A (microphone characteristics), which is a joint PDF for the random variables (gain) and ρ (phase). To simplify the design model, we assume all microphones have similar characteristics, such that av for v ∈ {0, ..., P K − 1} can be described by the same PDF fA (a). Furthermore, we assume and ρ are independent such that fA (a) = f (κ)fρ (γ), where f (κ) and fρ (γ) are the PDFs of the gain κ and phase γ respectively. Hence, (18) reduces to Jtot (h, ζ) =
···
J(h, ζ, a0 , ..., aP K−1 )fA (a0 ) × · · · fA (aP K−1 )da0 · · · daP K−1
¯ R (ζ)h − 2hT g =h Q ¯R (ζ) + |Hd (ζ)|2 T
(19)
(20) where g ¯R (ζ) = μκ μcγ goR (ζ) − μsγ goI (ζ) with goR (ζ) and goI (ζ) the real and imaginary parts of goi (ζ) = go (ζ)Hd∗ (ζ), and κv f (κv )dκv , μcγ = cos(γv )fρ (γv )dγv , μsγ = sin(γv )fρ (γv )dγv .
μκ =
(21) (22)
¯ R (ζ), we have As for Q ¯ R (ζ) = Q κu κw f (κu )f (κw )dκu dκw [QoR (ζ) × cos(γu − γw )fρ (γu )fρ (γw )dγu dγw + QoI (ζ) sin(γu − γw )fρ (γu ) fρ (γw )dγu dγw ] (23) with QoR (ζ) and QoI (ζ) the real and imaginary parts of Qo (ζ) = go (ζ)goH (ζ). Note that we have dropped ω and φ in all μ’s and σ’s for notational convenience. For u = w, we have, from (23), ¯ R (ζ) = QoR (ζ) Q where
σκ2
κ2 f (κ)dκ = σκ2 QoR (ζ)
(24)
is the second moment of gain PDF, i.e. σκ2 =
κ2 f (κ)dκ .
On the other hand, for u = w, we have
¯ R (ζ) = μ2κ σγc QoR (ζ) + σγs QoI (ζ) Q σγc = (μcγ )2 + (μsγ )2 , σγs = μsγ μcγ − μcγ μsγ = 0 . Combining the results from (23)-(28), we have ¯ R (ζ) = A QoR (ζ) Q
where
(μ2κ σγc (1L1
(25) (26) (27) (28) (29)
σκ2 IL1 )
A= − I L1 ) + ⊗ 1 L2 (30) where I is the identity matrix and 1 is a square matrix with all elements equal to 1. Their dimensions are given by the subscripts L1 = P · K and L2 = M · N (L2 = M for FD design). We are now ready to define the final cost function for designing robust SBBFs in the weighted least-squares (WLS) sense. Weighting the cost function in (19) with F (ζ) and integrating it across steering angle Ψ, frequency Ω andazimuth angle Φ results in JW LS (h) =
where QW LS gW LS dW LS
Ψ T
Ω
Φ
F (ζ)Jtot (h, ζ)dφdωdψ T
= h QW LS h − 2h gW LS + dW LS ¯ R (ζ)dφdωdψ , = F (ζ)Q Ψ Ω Φ = F (ζ)¯ gR (ζ)dφdωdψ , Ψ Ω Φ = F (ζ)|Hd (ζ)|2 dφdωdψ . Ψ
Ω
(31) (32) (33) (34)
Φ
Minimising (31), we have the robust SBBF weights given by hW LS = Q−1 W LS gW LS .
(35)
3.2. Frequency domain formulation
= hHf (ω)Qf (ζ)hf (ω) − 2Re{hHf (ω)gfi (ζ)} + |Hd (ζ)|2 gfi (ζ)
− 2Re{hTef (ω)gef (ζ)} + |Hd (ζ)|2 Re{Qf (ζ)} −Im{Qf (ζ)} where Qef (ζ) = , Im{Qf (ζ)} Re{Qf (ζ)} i Re{gf (ζ)} Re{hf (ω)} . gef (ζ) = , hef (ω) = i Im{hf (ω)} Im{gf (ζ)}
(37) (38) (39)
Following a similar procedure and the assumptions as in Section 3.1, the total cost functionin FDformulation is given by Jtot (hef (ω), ζ) =
···
J(hef (ω), ζ, a0 , ..., aP K−1 )
× fA (a0 ) · · · fA (aP K−1 )da0 · · · daP K−1 ¯ f (ζ)hef (ω) − 2hTef (ω)¯ gf (ζ) + |Hd (ζ)|2 (40) = hTef (ω)Q where ¯ ¯ g ¯f R (ζ) ¯ f (ζ) = Qf R (ζ) −Qf I (ζ) , g Q . (41) (ζ) = ¯ f ¯ f R (ζ) ¯ f I (ζ) g ¯f I (ζ) Q Q Following the same procedure as we evaluate (20) and (29), we have g ¯f R (ζ) = μκ μcγ gof R (ζ) − μsγ gof I (ζ) (42)
c s (43) g ¯f I (ζ) = μκ μγ gof I (ζ) + μγ gof R (ζ) ¯ f R (ζ) = A Qof R (ζ) (44) Q ¯ f I (ζ) = A Qof I (ζ) Q (45) where gof R (ζ) and gof I (ζ) are the real and imaginary parts of i gof (ζ) = gof (ζ)Hd∗ (ζ), and Qof R (ζ) and Qof I (ζ) are the parts H of Qof (ζ) = gof (ζ)gof (ζ). Hence, the final cost function in the WLS sense for SBBFs design in FD is given by f JW LS (hef (ω)) =
=
where
F (ζ)Jtot (hef (ω), ζ)dφdψ Ψ Φ hTef (ω)QfW LS (ω)hef (ω)
f f − 2hTef (ω)gW LS (ω) + dW LS (ω) ¯ f (ζ)dφdψ , QfW LS (ω) = F (ζ)Q Ψ Φ f F (ζ)¯ gf (ζ)dφdψ , gW LS (ω) = Ψ Φ F (ζ)|Hd (ζ)|2 dφdψ . dfW LS (ω) = Ψ
(46) (47) (48) (49)
Φ
Minimising (46) on a finite set of frequency points ω ∈ Ωpb gives
−1 f gW (50) hef (ω) = QfW LS (ω) LS (ω) . Restructuring hef (ω) into its real and imaginary parts, we obtain the complex frequency weights for the robust SBBF given by hf (ω) = [hef (ω)]1:L − j [hef (ω)](L+1):2L (51) where L = P · K · M . Note that the design problem size in (50) is smaller than (35) since the dimension of QfW LS (ω) is much smaller than QW LS . However, in (50), we need to solve for each frequency point ω ∈ Ωpb in order to get a complete set of complex weights for the FD robust SBBF. 4. DESIGN EXAMPLES
Consider a least-squares error function similar to (14), except we now formulate the design in FD. We have J(hf (ω), ζ, a0 , ..., aP K−1 ) = |Gf (ζ) − Hd (ζ)|2 gf (ζ)gfH (ζ)
their corresponding real and imaginary parts such that (36) can be expressed in real matrices and vectors as follows. J(hef (ω), ζ, a0 , ..., aP K−1 ) = hTef (ω)Qef (ζ)hef (ω)
(36)
gf (ζ)Hd∗ (ζ).
and = Following where Qf (ζ) = [7], we express the matrices and vectors in (36) as the direct sum of
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Two design examples are given in order to illustrate the design procedure in Section 3. The designs are for the spiral array of Section 2.1 with the following design parameters: P = 4, K = 5, M = 5, N = 16 (for TD), Nd = 8, Ts = 1/8000, Ωpb = [0.05π, 0.95π], Ψ = [−0.2π, 0.2π], Φ = [−π, π], α = 0.45, 0.15π beam width, 64 frequency, 64 azimuth angle and 16 steering angle points. The
ring radii are rp = [0.033, 0.089, 0.242, 0.657]m which correspond to fp = [4000, 1474, 543, 200]Hz. For illustration purposes, we assume (without loss of generality) that the microphone characteristics are independent of azimuth angle. The microphone gain and phase characteristics are assumed to be Gaussian, N (μκ (ω), σκ2 (ω)) and N (μγ (ω), σγ2 (ω)), with μκ (ω) = (π − ω + 9)/10, μγ (ω) = (π − ω − 1)/10 and σκ (ω) = σγ (ω) = (π − ω + 10)/20. Figs. 4, 5 and 6 show the beampatterns, steered to 20◦ , for the designed non-robust TD, robust TD and robust FD SBBF respectively. In order to demonstrate the designs’ robustness, we have injected both frequency dependent gain and phase errors into the designed SBBFs. Both gain and phase errors follow zero mean Gaussian distributions with standard deviation σ(ω) = (2 − ω/π)/10. From Figs. 4 and 5, it is clear that the beampatterns of the TD robust design, especially at low frequency, are maintained even in the presence of errors. Similar improvement is obtained from the FD design. Significant reduction towards error sensitivity is clear from Table 1, which shows the values of the cost function in (31) and (46), evaluated for the non-robust and robust designs, with and without microphones errors. Table 1: Cost function values for different SBBFs designs TD design FD design Non-robust Robust Non-robust Robust Without error 6.22 6.97 0.60 0.91 With error 1.93 × 103 7.05 1.07 × 109 0.97
in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 77–80, Apr. 2009. [6] S. Doclo and M. Moonen, “Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics,” IEEE Trans. Signal Processing, vol. 51, no. 10, pp. 2511–2526, Oct. 2003. [7] R.G. Lorenz and S.P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Processing, vol. 53, no. 5, pp. 1684–1696, May 2005.
Fig. 4: Beampattern of non-robust TD design
5. CONCLUSION A robust SBBF design using the Farrow structure and for arbitrary microphone array is proposed. Robustness is achieved by modelling the microphone characteristics as random variables, which provides flexibility for errors and mismatches in the microphones. The design procedure optimises the mean performance of the SBBF in order to achieve a desired level of robustness. This error modelling method allows the SBBF robustness level to be controlled during design to match the given microphone characteristics. The design has been formulated in both TD and FD. Lastly, design examples show that the proposed robust design has significantly reduced the sensitivity of SBBFs towards microphone errors.
Fig. 5: Beampattern of robust TD design
6. REFERENCES [1] M. Kajala and M. Hamalainen, “Filter-and-sum beamformer with adjustable filter characteristics,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, vol. 5, pp. 2917–2920, May 2001. [2] L. C. Parra, “Steerable frequency-invariant beamforming for arbitrary arrays,” J. Acoust. Soc. Amer., vol. 119, pp. 3839– 3847, 2006. [3] C.W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits and Syst., vol. 3, pp. 2641–2645, Jun. 1988. [4] C. C. Lai, S. Nordholm, and Y. H. Leung, “Design of robust steerable broadband beamformers with spiral arrays and the farrow filter structure,” in Proc. Int. Workshop on Acoust. Echo and Noise Control, Aug. 2010. [5] E. Mabande, A. Schad, and W. Kellermann, “Design of robust superdirective beamformers as a convex optimization problem,”
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Fig. 6: Beampattern of robust FD design
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This paper is organised as follows. Section 2 describes the spiral array geometry and SBBF structure with Farrow filters. Both the TD and FD design of robust SBBFs are formulated in Section 3, followed by design examples in Section 4. Finally, the conclusion is provided in Section 5. 2. FAR-FIELD SBBFS CONFIGURATION 2.1. Array geometry Consider the spiral array [4] shown in Fig. 1. It consists of P concentric rings with K microphones uniformly spaced in each ring. Successive rings are offset by 2π/(KP ) relative to each other. Under the far-field signal model, the array response vector d(ω, φ)1 for a signal with frequency ω and impinging the array at angle φ in the plane of the array is given by [d(ω, φ)]v = dpK+k (ω, φ) ωrp 2πk 2πp = exp j cos φ − − (1) c K PK th where p ∈ {0, ..., P −1} is the index of the p ring, k ∈ {0, ..., K− 1} is the index of the kth microphone in each ring, c is the speed of sound in the propagating medium (c = 343ms−1 in air) and v = pK + k ∈ {0, ..., P K − 1} is the index of the kth microphone in the pth ring. rp is the radius of the pth ring and is given by αc (2) rp = 2fp sin(π/K) where 0 < α ≤ 0.5 to prevent spatial aliasing, and fp is the highest frequency in Hz received by the pth ring. Note that the spacing between successive ring radii rp can be uniform or non-uniform, depending on the choice of fp . In (2), it is assumed that there are sufficient rings in the spiral array. For spiral array with only a few rings (P = 2, 3), r0 is obtained by setting f0 to be the maximum frequency of the received signal. Subsequent ring radii are then selected by trading off between having narrow beamwidth at low frequency and frequency invariant response. One example is rp = 2rp−1 for p ∈ {1, ..., P − 1}. 2.2. Beamformer structure For the TD Farrow filters shown in Fig. 2, its beampattern, steered to azimuth angle ψ, is given by −1 P −1 K−1 −1 N M G(ψ, ω, φ) = apK+k (ω, φ) p=0 k=0 m=0 n=0
× hp,k,m [n]e−jnωTs ψ m dpK+k (ω, φ) 1 All
(3)
vectors used in this paper are column vectors.
ICASSP 2011
Fig. 1: Spiral array geometry Fig. 3: FD Farrow filter structure for SBBF [hf (ω)]q = Hp,k,m (ω) =
N −1
hp,k,m [n]e−jωnTs .
(12)
n=0
Index q = pK + kM + m describe how the transfer functions Hp,k,m (ω) are stacked to form hf (ω). Without microphone characteristics modelling, we have gf (ψ, ω, φ) = gof (ψ, ω, φ) = d(ω, φ) ⊗ f (ψ) . (13) 3. ROBUST SBBF DESIGN 3.1. Time domain formulation Following [6], the least-squares error function for specific microphone characteristics can be written as J(h, ζ,a0 , ..., aP K−1 ) = |G(ζ) − Hd (ζ)|2 = hT Q(ζ)h − 2hT gR (ζ) + |Hd (ζ)|2 Fig. 2: TD Farrow filter structure for SBBF where M − 1 is the order of Farrow filters indexed by m ∈ {0, ..., M − 1}, n ∈ {0, ..., N − 1} is the time index of the N -tap FIR filters, and Ts is the sampling period. av (ω, φ) specifies the microphone characteristics given by (4) av (ω, φ) = κv (ω, φ)ejγv (ω,φ) where both the gain κv (ω, φ) and phase γv (ω, φ) can be frequency and angle dependent. We keep the array response dv (ω, φ) arbitrary (in the azimuth plane) so that the robust design formulation in the next section is valid for any microphone array geometry. We can write (3) in matrix form given by G(ψ, ω, φ) = hT g(ψ, ω, φ) (5) where g(ψ, ω, φ) = (a(ω, φ) d(ω, φ)) ⊗ f (ψ) ⊗ e(ω), (6) T [a(ω, φ)]v = av (ω, φ) , f (ψ) = ψ 0 · · · ψ M −1 , (7) T [h]q = hp,k,m [n] , e(ω) = e−jωTs (0) · · · e−jωTs (N −1) . (8) The symbol denotes element-wise product and ⊗ denotes Kronecker product. Index q = pK + kM + mN + n describes how the real elements in h are stacked. Note that without modelling the microphone characteristics, we have g(ψ, ω, φ) = go (ψ, ω, φ) = d(ω, φ) ⊗ f (ψ) ⊗ e(ω) . (9) For the FD Farrow structure shown in Fig. 3 (ωi is the ith frequency bin), we have G(ψ, ω, φ) = hHf (ω)gf (ψ, ω, φ) (10) (11) where gf (ψ, ω, φ) = (a(ω, φ) d(ω, φ)) ⊗ f (ψ) ,
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(14)
where Q(ζ) = g(ζ)gH (ζ) , (15) ∗ (16) gR (ζ) = Re{g(ζ)Hd (ζ)} , −jωNd ω ∈ Ωpb , φ ∈ Φpb (ψ) e . (17) Hd (ζ) = 0 otherwise Nd gives the desired delay, Ωpb specifies the spectral passband and Φpb (ψ), which depends on steering angle ψ, defines the spatial passband. Note that ω and φ are dropped from a, κ and γ (their dependencies are understood from the context), and (ψ, ω, φ) is replaced with ζ for notational convenience. The total cost function, which is defined as the sum of the cost functions for all feasible microphone characteristics, weighted by the probability density functions (PDFs) of the microphone is given by characteristics Jtot (h, ζ) =
···
J(h, ζ, a0 , ..., aP K−1 )fA0 (a0 )
× · · · fAP K−1 (aP K−1 )da0 · · · daP K−1 (18) where fA (a) = f,ρ (κ, γ) is the PDF for the random variable A (microphone characteristics), which is a joint PDF for the random variables (gain) and ρ (phase). To simplify the design model, we assume all microphones have similar characteristics, such that av for v ∈ {0, ..., P K − 1} can be described by the same PDF fA (a). Furthermore, we assume and ρ are independent such that fA (a) = f (κ)fρ (γ), where f (κ) and fρ (γ) are the PDFs of the gain κ and phase γ respectively. Hence, (18) reduces to Jtot (h, ζ) =
···
J(h, ζ, a0 , ..., aP K−1 )fA (a0 ) × · · · fA (aP K−1 )da0 · · · daP K−1
¯ R (ζ)h − 2hT g =h Q ¯R (ζ) + |Hd (ζ)|2 T
(19)
(20) where g ¯R (ζ) = μκ μcγ goR (ζ) − μsγ goI (ζ) with goR (ζ) and goI (ζ) the real and imaginary parts of goi (ζ) = go (ζ)Hd∗ (ζ), and κv f (κv )dκv , μcγ = cos(γv )fρ (γv )dγv , μsγ = sin(γv )fρ (γv )dγv .
μκ =
(21) (22)
¯ R (ζ), we have As for Q ¯ R (ζ) = Q κu κw f (κu )f (κw )dκu dκw [QoR (ζ) × cos(γu − γw )fρ (γu )fρ (γw )dγu dγw + QoI (ζ) sin(γu − γw )fρ (γu ) fρ (γw )dγu dγw ] (23) with QoR (ζ) and QoI (ζ) the real and imaginary parts of Qo (ζ) = go (ζ)goH (ζ). Note that we have dropped ω and φ in all μ’s and σ’s for notational convenience. For u = w, we have, from (23), ¯ R (ζ) = QoR (ζ) Q where
σκ2
κ2 f (κ)dκ = σκ2 QoR (ζ)
(24)
is the second moment of gain PDF, i.e. σκ2 =
κ2 f (κ)dκ .
On the other hand, for u = w, we have
¯ R (ζ) = μ2κ σγc QoR (ζ) + σγs QoI (ζ) Q σγc = (μcγ )2 + (μsγ )2 , σγs = μsγ μcγ − μcγ μsγ = 0 . Combining the results from (23)-(28), we have ¯ R (ζ) = A QoR (ζ) Q
where
(μ2κ σγc (1L1
(25) (26) (27) (28) (29)
σκ2 IL1 )
A= − I L1 ) + ⊗ 1 L2 (30) where I is the identity matrix and 1 is a square matrix with all elements equal to 1. Their dimensions are given by the subscripts L1 = P · K and L2 = M · N (L2 = M for FD design). We are now ready to define the final cost function for designing robust SBBFs in the weighted least-squares (WLS) sense. Weighting the cost function in (19) with F (ζ) and integrating it across steering angle Ψ, frequency Ω andazimuth angle Φ results in JW LS (h) =
where QW LS gW LS dW LS
Ψ T
Ω
Φ
F (ζ)Jtot (h, ζ)dφdωdψ T
= h QW LS h − 2h gW LS + dW LS ¯ R (ζ)dφdωdψ , = F (ζ)Q Ψ Ω Φ = F (ζ)¯ gR (ζ)dφdωdψ , Ψ Ω Φ = F (ζ)|Hd (ζ)|2 dφdωdψ . Ψ
Ω
(31) (32) (33) (34)
Φ
Minimising (31), we have the robust SBBF weights given by hW LS = Q−1 W LS gW LS .
(35)
3.2. Frequency domain formulation
= hHf (ω)Qf (ζ)hf (ω) − 2Re{hHf (ω)gfi (ζ)} + |Hd (ζ)|2 gfi (ζ)
− 2Re{hTef (ω)gef (ζ)} + |Hd (ζ)|2 Re{Qf (ζ)} −Im{Qf (ζ)} where Qef (ζ) = , Im{Qf (ζ)} Re{Qf (ζ)} i Re{gf (ζ)} Re{hf (ω)} . gef (ζ) = , hef (ω) = i Im{hf (ω)} Im{gf (ζ)}
(37) (38) (39)
Following a similar procedure and the assumptions as in Section 3.1, the total cost functionin FDformulation is given by Jtot (hef (ω), ζ) =
···
J(hef (ω), ζ, a0 , ..., aP K−1 )
× fA (a0 ) · · · fA (aP K−1 )da0 · · · daP K−1 ¯ f (ζ)hef (ω) − 2hTef (ω)¯ gf (ζ) + |Hd (ζ)|2 (40) = hTef (ω)Q where ¯ ¯ g ¯f R (ζ) ¯ f (ζ) = Qf R (ζ) −Qf I (ζ) , g Q . (41) (ζ) = ¯ f ¯ f R (ζ) ¯ f I (ζ) g ¯f I (ζ) Q Q Following the same procedure as we evaluate (20) and (29), we have g ¯f R (ζ) = μκ μcγ gof R (ζ) − μsγ gof I (ζ) (42)
c s (43) g ¯f I (ζ) = μκ μγ gof I (ζ) + μγ gof R (ζ) ¯ f R (ζ) = A Qof R (ζ) (44) Q ¯ f I (ζ) = A Qof I (ζ) Q (45) where gof R (ζ) and gof I (ζ) are the real and imaginary parts of i gof (ζ) = gof (ζ)Hd∗ (ζ), and Qof R (ζ) and Qof I (ζ) are the parts H of Qof (ζ) = gof (ζ)gof (ζ). Hence, the final cost function in the WLS sense for SBBFs design in FD is given by f JW LS (hef (ω)) =
=
where
F (ζ)Jtot (hef (ω), ζ)dφdψ Ψ Φ hTef (ω)QfW LS (ω)hef (ω)
f f − 2hTef (ω)gW LS (ω) + dW LS (ω) ¯ f (ζ)dφdψ , QfW LS (ω) = F (ζ)Q Ψ Φ f F (ζ)¯ gf (ζ)dφdψ , gW LS (ω) = Ψ Φ F (ζ)|Hd (ζ)|2 dφdψ . dfW LS (ω) = Ψ
(46) (47) (48) (49)
Φ
Minimising (46) on a finite set of frequency points ω ∈ Ωpb gives
−1 f gW (50) hef (ω) = QfW LS (ω) LS (ω) . Restructuring hef (ω) into its real and imaginary parts, we obtain the complex frequency weights for the robust SBBF given by hf (ω) = [hef (ω)]1:L − j [hef (ω)](L+1):2L (51) where L = P · K · M . Note that the design problem size in (50) is smaller than (35) since the dimension of QfW LS (ω) is much smaller than QW LS . However, in (50), we need to solve for each frequency point ω ∈ Ωpb in order to get a complete set of complex weights for the FD robust SBBF. 4. DESIGN EXAMPLES
Consider a least-squares error function similar to (14), except we now formulate the design in FD. We have J(hf (ω), ζ, a0 , ..., aP K−1 ) = |Gf (ζ) − Hd (ζ)|2 gf (ζ)gfH (ζ)
their corresponding real and imaginary parts such that (36) can be expressed in real matrices and vectors as follows. J(hef (ω), ζ, a0 , ..., aP K−1 ) = hTef (ω)Qef (ζ)hef (ω)
(36)
gf (ζ)Hd∗ (ζ).
and = Following where Qf (ζ) = [7], we express the matrices and vectors in (36) as the direct sum of
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Two design examples are given in order to illustrate the design procedure in Section 3. The designs are for the spiral array of Section 2.1 with the following design parameters: P = 4, K = 5, M = 5, N = 16 (for TD), Nd = 8, Ts = 1/8000, Ωpb = [0.05π, 0.95π], Ψ = [−0.2π, 0.2π], Φ = [−π, π], α = 0.45, 0.15π beam width, 64 frequency, 64 azimuth angle and 16 steering angle points. The
ring radii are rp = [0.033, 0.089, 0.242, 0.657]m which correspond to fp = [4000, 1474, 543, 200]Hz. For illustration purposes, we assume (without loss of generality) that the microphone characteristics are independent of azimuth angle. The microphone gain and phase characteristics are assumed to be Gaussian, N (μκ (ω), σκ2 (ω)) and N (μγ (ω), σγ2 (ω)), with μκ (ω) = (π − ω + 9)/10, μγ (ω) = (π − ω − 1)/10 and σκ (ω) = σγ (ω) = (π − ω + 10)/20. Figs. 4, 5 and 6 show the beampatterns, steered to 20◦ , for the designed non-robust TD, robust TD and robust FD SBBF respectively. In order to demonstrate the designs’ robustness, we have injected both frequency dependent gain and phase errors into the designed SBBFs. Both gain and phase errors follow zero mean Gaussian distributions with standard deviation σ(ω) = (2 − ω/π)/10. From Figs. 4 and 5, it is clear that the beampatterns of the TD robust design, especially at low frequency, are maintained even in the presence of errors. Similar improvement is obtained from the FD design. Significant reduction towards error sensitivity is clear from Table 1, which shows the values of the cost function in (31) and (46), evaluated for the non-robust and robust designs, with and without microphones errors. Table 1: Cost function values for different SBBFs designs TD design FD design Non-robust Robust Non-robust Robust Without error 6.22 6.97 0.60 0.91 With error 1.93 × 103 7.05 1.07 × 109 0.97
in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 77–80, Apr. 2009. [6] S. Doclo and M. Moonen, “Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics,” IEEE Trans. Signal Processing, vol. 51, no. 10, pp. 2511–2526, Oct. 2003. [7] R.G. Lorenz and S.P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Processing, vol. 53, no. 5, pp. 1684–1696, May 2005.
Fig. 4: Beampattern of non-robust TD design
5. CONCLUSION A robust SBBF design using the Farrow structure and for arbitrary microphone array is proposed. Robustness is achieved by modelling the microphone characteristics as random variables, which provides flexibility for errors and mismatches in the microphones. The design procedure optimises the mean performance of the SBBF in order to achieve a desired level of robustness. This error modelling method allows the SBBF robustness level to be controlled during design to match the given microphone characteristics. The design has been formulated in both TD and FD. Lastly, design examples show that the proposed robust design has significantly reduced the sensitivity of SBBFs towards microphone errors.
Fig. 5: Beampattern of robust TD design
6. REFERENCES [1] M. Kajala and M. Hamalainen, “Filter-and-sum beamformer with adjustable filter characteristics,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, vol. 5, pp. 2917–2920, May 2001. [2] L. C. Parra, “Steerable frequency-invariant beamforming for arbitrary arrays,” J. Acoust. Soc. Amer., vol. 119, pp. 3839– 3847, 2006. [3] C.W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits and Syst., vol. 3, pp. 2641–2645, Jun. 1988. [4] C. C. Lai, S. Nordholm, and Y. H. Leung, “Design of robust steerable broadband beamformers with spiral arrays and the farrow filter structure,” in Proc. Int. Workshop on Acoust. Echo and Noise Control, Aug. 2010. [5] E. Mabande, A. Schad, and W. Kellermann, “Design of robust superdirective beamformers as a convex optimization problem,”
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Fig. 6: Beampattern of robust FD design
