Apparatus and method for generating a sound field

1. An apparatus for generating a sound field on the basis of an input audio signal, wherein the apparatus comprises:

a plurality of transducers, wherein each transducer of the plurality of transducers is configured to be driven by a transducer driving signal q l of the respective transducer, wherein l∈{1, . . . , L} and wherein l denotes the l-th transducer;

a plurality of filters configured to generate for each transducer of the plurality of transducers the transducer driving signal q l of the respective transducer, wherein each of the filters of the plurality of filters is defined by a filter transfer function and wherein the transducer driving signal q l of the respective transducer is based on the filter transfer function of the respective transducer and the input audio signal; and

a control unit configured to provide or receive a first transducer driving signal vector q 0 of dimension L such that a gradient of J(q;ψ) with respect to q is zero in (q 0 ;ψ 0 ), wherein J(q;ψ) is a cost function having as variables a transducer driving signal vector q of dimension L and a weight matrix ψ of dimension M×M, and wherein ψ 0 is a first weight matrix of dimension M×M,

wherein the control unit is further configured to provide a second transducer driving signal vector q of dimension L such that a gradient of the cost function J(q;ψ) with respect to q is zero or approximately zero in ({tilde over (q)}; {tilde over (ψ)}), wherein {tilde over (ψ)} is a second weight matrix of dimension M×M, and wherein the control unit is configured to provide the second transducer driving signal vector {tilde over (q)} on the basis of:

the first transducer driving signal vector q 0 ,

the first weight matrix ψ 0 , and

the second weight matrix {tilde over (ψ)},

wherein the cost function is J(q;ψ)=∥{circumflex over (ψ)}({circumflex over (p)}−p)∥ 2 +β∥q∥ 2 , wherein {circumflex over (p)} is a target pressure vector of dimension M comprising M target pressure values {circumflex over (p)} m for a set of M control points, m∈{1, . . . , M}, p is a pressure vector of dimension M comprising M pressure values p m for the set of M control points, m∈{1, . . . , M}, and is a regularization parameter in the range of [0,∞).

2. The apparatus of claim 1 , wherein the control unit is configured to compute the second transducer driving signal vector {tilde over (q)} on the basis of a truncated Neumann series of order N as

{tilde over (q)}=Σ n=0 N (−( Z H ψ 0 Z+βI ) −1 Z H ΔψZ ) n ( q 0 +( Z H ψ 0 Z+βI ) −1 Z H Δψ{circumflex over (p)} ),

wherein Z is a transfer matrix of dimension M×L, I is the identity matrix of dimension L×L, Δψ denotes the difference between ψ 0 and {tilde over (ψ)} and the superscript H denotes Hermitian transposition.

3. The apparatus of claim 2 , wherein the sound field comprises an acoustically bright zone, an acoustically dark zone and an acoustically grey zone and wherein the cost function J(q;ψ) is given by the following equation:

∥ p B − p B ∥ 2 +ψ D ∥p D ∥ 2 +ψ G ∥p G ∥ 2 +β∥q∥ 2 ,

and wherein the gradient of J(q;ψ) with respect to q is zero in (q 0 ;ψ 0 ) under the constraint that |Σ l=1 L Z ml q l | 2 =|p m | 2 |p m,min | 2 for each m E B where B is the set of indices of control points in the bright zone and |p m,min | 2 is a positive real number associated with the respective desired minimum level of sound energy at a respective control point in the bright zone,

wherein P B denotes a sound pressure at a control point in the bright zone, p B denotes a desired sound pressure at the control point in the bright zone, p D denotes a respective sound pressure at a plurality of control points in the dark zone, p G denotes a respective sound pressure at a plurality of control points in the grey zone, Z ml denotes the element in the m-th row and the l-th column of the transfer matrix Z ψ D denotes a dark zone weighting parameter, ψ G denotes a grey zone weighting parameter and P B,min denotes a desired minimum level of sound energy at the control point in the bright zone.

4. The apparatus of claim 3 , wherein the control unit is configured to provide the second transducer driving signal vector {tilde over (q)} in response to an adjustment of the desired minimum level of sound energy at the control point in the bright zone.

5. The apparatus of claim 3 , wherein the truncated Neumann series of order N is defined by the following equation:

Σ n=0 N Δψ D n E n ,

wherein Δψ D denotes an adjustment of the dark zone weighting parameter ψ D and wherein the matrix E is defined by the following equation:

E=−A −1 Z D H Z D ,

wherein the matrix A is defined by the following equation:

A=Z B H Z B +ψ D Z D H Z D +ψ G Z G H Z G +βI,

wherein Z B denotes the transfer matrix for the bright zone, Z D denotes the transfer matrix for the dark zone, and Z G denotes the transfer matrix for the grey zone.

6. The apparatus of claim 5 , wherein the control unit is configured to determine the adjustment Δψ D of the dark zone weighting parameter ψ D by determining the root of the following equation within the interval −0.5≤Δψ D ≤0.5:

Σ n=0 N |Δψ D | n |z B T E n q|−|p B,min |=0,

wherein z B T denotes portion of the transfer matrix defining a vector and p B,min denotes a desired minimum level of sound energy at the control point in the bright zone.

7. The apparatus of claim 2 , wherein the order N of the truncated Neumann series depends on frequency.

8. The apparatus of claim 7 , wherein the order N of the truncated Neumann series decreases with increasing frequency.

9. The apparatus of claim 7 , wherein the control unit is configured to determine the order N of the truncated Neumann series on the basis of the following equation:

N

=

min

N

⁢

{

ɛ

≤

ɛ

MAX

}

,

wherein ε MAX denotes an error threshold and ε denotes an error measure defined by the following equation:

ɛ

=

10

⁢

⁢

log

10

(



q

~

N

-

q

~



2



q

~



2

)

,

wherein {tilde over (q)} N denotes the transducer driving signal vector determined on the basis of the truncated Neumann series.

10. The apparatus of claim 1 , wherein the first transducer driving signal vector q 0 is

q 0 =( Z H ψ 0 Z+βI ) −1 Z H ψ 0 p ,

wherein Z is a transfer matrix of dimension M×L.

11. The apparatus of claim 1 , wherein the control unit is configured to determine the regularization factor β on the basis of a normalized Tikhonov regularization.

12. The apparatus of claim 1 , wherein the apparatus further comprises a memory configured to store the first transducer driving signal vector q 0 .

13. A method for generating a sound field on the basis of an input audio signal, wherein the method comprises the steps of:

providing or receiving a first transducer driving signal vector q 0 of dimension L such that a gradient of J(q;ψ) with respect to q is zero in (q 0 ;ψ 0 ), wherein J(q;ψ) is a cost function having as variables a transducer driving signal vector q of dimension L and a weight matrix ψ of dimension M×M, and wherein ψ 0 is a first weight matrix of dimension M×M;

providing a second transducer driving signal vector {tilde over (q)} of dimension L such that a gradient of the cost function J(q;ψ) with respect to q is zero in ({tilde over (q)}; {tilde over (ψ)}), wherein {tilde over (ψ)} is a second weight matrix of dimension M×M, and wherein the second transducer driving signal vector {tilde over (q)} is provided on the basis of:

the first transducer driving signal vector q 0 ,

the first weight matrix ψ 0 , and

the second weight matrix {tilde over (ψ)}; and

driving each transducer of a plurality of L transducers by a respective component {tilde over (q)} l , l∈{1, . . . , L}, of the second transducer driving signal vector {tilde over (q)};

wherein the cost function is J(q;ψ)=∥{tilde over (ψ)}( p −p)∥ 2 +β∥q∥ 2 , wherein p is a target pressure vector of dimension M comprising M target pressure values p m for a set of M control points, m∈{1, . . . , M}, p is a pressure vector of dimension M comprising M pressure values p m for the set of M control points, m∈{1, . . . , M}, and β is a regularization parameter in the range of [0,∞).

14. A non-transitory storage medium carrying a program code which when executed by one or more processors of a computer causes the computer to perform a method of generating a sound field on the basis of an input audio signal, wherein the method comprises the steps of:

providing or receiving a first transducer driving signal vector q 0 of dimension L such that a gradient of J(q;ψ) with respect to q is zero in (q 0 ;ψ 0 ), wherein J(q;ψ) is a cost function having as variables a transducer driving signal vector q of dimension L and a weight matrix ψ of dimension M×M, and wherein ψ 0 is a first weight matrix of dimension M×M;

providing a second transducer driving signal vector {tilde over (q)} of dimension L such that a gradient of the cost function J(q;ψ) with respect to q is zero in ({tilde over (q)}; {tilde over (ψ)}), wherein {tilde over (ψ)} is a second weight matrix of dimension M×M, and wherein the second transducer driving signal vector {tilde over (q)} is provided on the basis of:

the first transducer driving signal vector q 0 ,

the first weight matrix ψ 0 , and

the second weight matrix {tilde over (ψ)}; and

driving each transducer of a plurality of L transducers by a respective component {tilde over (q)} l , l∈{1, . . . , L}, of the second transducer driving signal vector {tilde over (q)};

wherein the cost function is J(q;ψ)=∥{tilde over (ψ)}( p −p)∥ 2 +β∥q∥ 2 , wherein p is a target pressure vector of dimension M comprising M target pressure values p m for a set of M control points, m∈{1, . . . , M}, p is a pressure vector of dimension M comprising M pressure values P m for the set of M control points, m∈{1, . . . , M}, and β is a regularization parameter in the range of [0,∞).