Microphone-Array Signal Processing

Transcription

1 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 1/115 Microphone-Array Signal Processing José A. Apolinário Jr. and Marcello L. R. de Campos {apolin},{mcampos}@ieee.org IME Lab. Processamento Digital de Sinais

2 Outline

3 1. Introduction and Fundamentals Outline

4 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering

5 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming

6 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming

7 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 2/115 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming 5. DoA Estimation with Microphone Arrays

8 1. Introduction and Fundamentals Microphone-Array Signal Processing, c Apolinárioi & Campos p. 3/115

9 In this course, signals and noise... General concepts

10 General concepts In this course, signals and noise... have spatial dependence

11 General concepts In this course, signals and noise... have spatial dependence must be characterized as space-time processes

12 General concepts In this course, signals and noise... have spatial dependence must be characterized as space-time processes An array of sensors is represented in the figure below.

13 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 4/115 General concepts In this course, signals and noise... have spatial dependence must be characterized as space-time processes An array of sensors is represented in the figure below. signal #1 signal #2 interference SENSORS ARRAY PROCESSOR

14 1.2 Signals in Space and Time Microphone-Array Signal Processing, c Apolinárioi & Campos p. 5/115

15 Defining operators Let x ( ) and 2 x( ) be the gradient and Laplacian operators, i.e.,

16 Defining operators Let x ( ) and 2 x( ) be the gradient and Laplacian operators, i.e., x ( ) = ( ) x [ ( ) = x ı x + ( ) ı y + ( ) y z ( ) y ( ) z ] T ı z

17 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 6/115 Defining operators Let x ( ) and 2 x( ) be the gradient and Laplacian operators, i.e., and x ( ) = ( ) x [ ( ) = x ı x + ( ) ı y + ( ) y z ( ) y ( ) z ] T 2 x ( ) = 2 ( ) x + 2 ( ) 2 y + 2 ( ) 2 z 2 respectively. ı z

18 Wave equation From Maxwell s equations, 2 E = 1 c 2 2 E t 2

19 Wave equation From Maxwell s equations, 2 E = 1 c 2 2 E t 2 or, for s(x,t) a general scalar field, 2 s x + 2 s 2 y + 2 s 2 z = 1 2 s 2 c 2 t 2 where: c is the propagation speed, E is the electric field intensity, and x = [x y z] T is a position vector.

20 Wave equation From Maxwell s equations, 2 E = 1 c 2 2 E t 2 vector wave equation or, for s(x,t) a general scalar field, 2 s x + 2 s 2 y + 2 s 2 z = 1 2 s 2 c 2 t 2 where: c is the propagation speed, E is the electric field intensity, and x = [x y z] T is a position vector.

21 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 7/115 Wave equation From Maxwell s equations, 2 E = 1 c 2 2 E t 2 vector wave equation or, for s(x,t) a general scalar field, 2 s x + 2 s 2 y + 2 s 2 z = 1 2 s 2 c 2 t 2 scalar wave equation where: c is the propagation speed, E is the electric field intensity, and x = [x y z] T is a position vector. Note: From this point onwards the terms wave and field will be used interchangeably.

22 Monochromatic plane wave Now assume s(x, t) has a complex exponential form,

23 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 8/115 Monochromatic plane wave Now assume s(x, t) has a complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) where A is a complex constant and k x, k y, k z, and ω 0 are real constants.

24 Monochromatic plane wave Substituting the complex exponential form of s(x, t) into the wave equation, we have

25 Monochromatic plane wave Substituting the complex exponential form of s(x, t) into the wave equation, we have k 2 xs(x,t)+k 2 ys(x,t)+k 2 zs(x,t) = 1 c 2ω2 s(x,t)

26 Monochromatic plane wave Substituting the complex exponential form of s(x, t) into the wave equation, we have k 2 xs(x,t)+k 2 ys(x,t)+k 2 zs(x,t) = 1 c 2ω2 s(x,t) or, after canceling s(x, t),

27 Monochromatic plane wave Substituting the complex exponential form of s(x, t) into the wave equation, we have k 2 xs(x,t)+k 2 ys(x,t)+k 2 zs(x,t) = 1 c 2ω2 s(x,t) or, after canceling s(x, t), k 2 x +k 2 y +k 2 z = 1 c 2ω2

28 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 9/115 Monochromatic plane wave Substituting the complex exponential form of s(x, t) into the wave equation, we have k 2 xs(x,t)+k 2 ys(x,t)+k 2 zs(x,t) = 1 c 2ω2 s(x,t) or, after canceling s(x, t), k 2 x +k 2 y +k 2 z = 1 c 2ω2 constraints to be satisfied by the parameters of the scalar field

29 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is

30 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic

31 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane

32 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane

33 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane For example, take the position at the origin of the coordinate space:

34 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane For example, take the position at the origin of the coordinate space: x = [0 0 0] T

35 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 10/115 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane For example, take the position at the origin of the coordinate space: x = [0 0 0] T s(0,t) = Ae jωt

36 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane

37 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane The value of s(x,t) is the same for all points lying on the plane

38 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 11/115 Monochromatic plane wave From the constraints imposed by the complex exponential form, s(x,t) = Ae j(ωt k xx k y y k z z) is monochromatic plane The value of s(x,t) is the same for all points lying on the plane k x x+k y y +k z z = C where C is a constant.

39 Monochromatic plane wave Defining the wavenumber vector k as k = [k x k y k z ] T

40 Monochromatic plane wave Defining the wavenumber vector k as k = [k x k y k z ] T we can rewrite the equation for the monochromatic plane wave as s(x,t) = Ae j(ωt kt x)

41 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 12/115 Monochromatic plane wave Defining the wavenumber vector k as k = [k x k y k z ] T we can rewrite the equation for the monochromatic plane wave as s(x,t) = Ae j(ωt kt x) The planes where s(x,t) is constant are perpendicular to the wavenumber vector k

42 Monochromatic plane wave As the plane wave propagates, it advances a distance δx in δt seconds.

43 Monochromatic plane wave As the plane wave propagates, it advances a distance δx in δt seconds. Therefore, s(x,t) = s(x+δx,t+δt) Ae j(ωt kt x) = Ae j [ω(t+δt) k T (x+δx)]

44 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 13/115 Monochromatic plane wave As the plane wave propagates, it advances a distance δx in δt seconds. Therefore, s(x,t) = s(x+δx,t+δt) Ae j(ωt kt x) = Ae j [ω(t+δt) k T (x+δx)] = ωδt k T δx = 0

45 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e.,

46 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction.

47 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction. Therefore, k T δx = k δx

48 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction. Therefore, k T δx = k δx = ωδt = k δx

49 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction. Therefore, k T δx = k δx = ωδt = k δx or, equivalently,

50 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction. Therefore, k T δx = k δx = ωδt = k δx or, equivalently, δx δt = ω k

51 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 14/115 Monochromatic plane wave Naturally the plane wave propagates in the direction of the wavenumber vector, i.e., k and δx point in the same direction. Therefore, k T δx = k δx = ωδt = k δx or, equivalently, Remember the constraints? k 2 = ω 2 /c 2 δx δt = ω k

52 Monochromatic plane wave After T = 2π/ω seconds, the plane wave has completed one cycle and it appears as it did before, but its wavefront has advanced a distance of one wavelength, λ.

53 Monochromatic plane wave After T = 2π/ω seconds, the plane wave has completed one cycle and it appears as it did before, but its wavefront has advanced a distance of one wavelength, λ. For δx = λ and δt = T = 2π ω, T = λ k ω = λ = 2π k

54 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 15/115 Monochromatic plane wave After T = 2π/ω seconds, the plane wave has completed one cycle and it appears as it did before, but its wavefront has advanced a distance of one wavelength, λ. For δx = λ and δt = T = 2π ω, T = λ k ω = λ = 2π k The wavenumber vector, k, may be considered a spatial frequency variable, just as ω is a temporal frequency variable.

55 Monochromatic plane wave We may rewrite the wave equation as s(x,t) = Ae j(ωt kt x) = Ae jω(t αt x) where α = k/ω is the slowness vector.

56 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 16/115 Monochromatic plane wave We may rewrite the wave equation as s(x,t) = Ae j(ωt kt x) = Ae jω(t αt x) where α = k/ω is the slowness vector. As c = ω/ k, vector α has a magnitude which is the reciprocal of c.

57 Periodic propagating periodic waves Any arbitrary periodic waveform s(x,t) = s(t α T x) with fundamental period ω 0 can be represented as a sum: s(x,t) = s(t α T x) = n= S n e jnω 0(t α T x)

58 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 17/115 Periodic propagating periodic waves Any arbitrary periodic waveform s(x,t) = s(t α T x) with fundamental period ω 0 can be represented as a sum: s(x,t) = s(t α T x) = n= S n e jnω 0(t α T x) The coefficients are given by S n = 1 T T 0 s(u)e jnω 0u du

59 Periodic propagating periodic waves Based on the previous derivations, we observe that:

60 Periodic propagating periodic waves Based on the previous derivations, we observe that: The various components of s(x, t) have different frequencies ω = nω 0 and different wavenumber vectors, k.

61 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 18/115 Periodic propagating periodic waves Based on the previous derivations, we observe that: The various components of s(x, t) have different frequencies ω = nω 0 and different wavenumber vectors, k. The waveform propagates in the direction of the slowness vector α = k/ω.

62 Nonperiodic propagating waves More generally, any function constructed as the integral of complex exponentials who also have a defined and converged Fourier transform can represent a waveform

63 Nonperiodic propagating waves More generally, any function constructed as the integral of complex exponentials who also have a defined and converged Fourier transform can represent a waveform s(x,t) = s(t α T x) = 1 2π S(ω)e jω(t αt x) dω

64 Nonperiodic propagating waves More generally, any function constructed as the integral of complex exponentials who also have a defined and converged Fourier transform can represent a waveform where s(x,t) = s(t α T x) = 1 2π S(ω)e jω(t αt x) dω S(ω) = s(u)e jωu du

65 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 19/115 Nonperiodic propagating waves More generally, any function constructed as the integral of complex exponentials who also have a defined and converged Fourier transform can represent a waveform where s(x,t) = s(t α T x) = 1 2π S(ω)e jω(t αt x) dω S(ω) = s(u)e jωu du We will come back to this later...

66 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 20/115 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming 5. DoA Estimation with Microphone Arrays

67 2. Sensor Arrays and Spatial Filtering Microphone-Array Signal Processing, c Apolinárioi & Campos p. 21/115

68 2.1 Wavenumber-Frequency Space Microphone-Array Signal Processing, c Apolinárioi & Campos p. 22/115

69 Space-time Fourier Transform The four-dimensional Fourier transform of the space-time signal s(x,t) is given by

70 Space-time Fourier Transform The four-dimensional Fourier transform of the space-time signal s(x,t) is given by S(k,ω) = s(x,t)e j(ωt kt x) dxdt

71 Space-time Fourier Transform The four-dimensional Fourier transform of the space-time signal s(x,t) is given by S(k,ω) = s(x,t)e j(ωt kt x) dxdt temporal frequency

72 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 23/115 Space-time Fourier Transform The four-dimensional Fourier transform of the space-time signal s(x,t) is given by S(k,ω) = s(x,t)e j(ωt kt x) dxdt temporal frequency spatial frequency: wavenumber

73 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 24/115 Space-time Fourier Transform The four-dimensional Fourier transform of the space-time signal s(x,t) is given by S(k,ω) = s(x,t)e j(ωt kt x) dx dt s(x,t) = 1 (2π) 4 S(k,ω)e j(ωt kt x) dk dω

74 Space-time Fourier Transform We have already concluded that if the space-time signal is a propagating waveform such that s(x,t) = s(t α T 0x), then its Fourier transform is equal to S(k,ω) = S(ω)δ(k ωα 0 )

75 Space-time Fourier Transform We have already concluded that if the space-time signal is a propagating waveform such that s(x,t) = s(t α T 0x), then its Fourier transform is equal to S(k,ω) = S(ω)δ(k ωα 0 ) Remember the nonperiodic propagating wave Fourier transform?

76 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 25/115 Space-time Fourier Transform We have already concluded that if the space-time signal is a propagating waveform such that s(x,t) = s(t α T 0x), then its Fourier transform is equal to S(k,ω) = S(ω)δ(k ωα 0 ) Remember the nonperiodic propagating wave Fourier transform? This means that s(x,t) only has energy along the direction of k = k 0 = ωα 0 in the wavenumber-frequency space.

77 2.2 Frequency-Wavenumber (WN) Response and Beam patterns (BP) Microphone-Array Signal Processing, c Apolinárioi & Campos p. 26/115

78 Signals at the sensors An array is a set of N (isotropic) sensors located at positions p n,n = 0,1,,N 1

79 Signals at the sensors An array is a set of N (isotropic) sensors located at positions p n,n = 0,1,,N 1 The sensors spatially sample the signal field at locations p n

80 Signals at the sensors An array is a set of N (isotropic) sensors located at positions p n,n = 0,1,,N 1 The sensors spatially sample the signal field at locations p n At the sensors, the set of N signals are denoted by

81 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 27/115 Signals at the sensors An array is a set of N (isotropic) sensors located at positions p n,n = 0,1,,N 1 The sensors spatially sample the signal field at locations p n At the sensors, the set of N signals are denoted by f(t,p) = f(t,p 0 ) f(t,p 1 ). f(t,p N 1 )

82 Array output f (t, p ) 0 h (t) 0 f (t, p 1 ) h (t) 1 Σ.... y(t) f (t, p N 1 ) h (t) N 1

83 Array output f (t, p ) 0 h (t) 0 f (t, p 1 ) h (t) 1 Σ.... y(t) f (t, p N 1 ) h (t) N 1 y(t) = = N 1 n=0 h n (t τ)f n (τ,p n )dτ h T (t τ)f(τ,p)dτ

84 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 28/115 Array output f (t, p ) 0 h (t) 0 f (t, p 1 ) h (t) 1 Σ.... y(t) f (t, p N 1 ) h (t) N 1 y(t) = = N 1 n=0 h n (t τ)f n (τ,p n )dτ h T (t τ)f(τ,p)dτ where h(t) = [h o (t) h 1 (t) h N 1 (t)] T

85 In the frequency domain, Y(ω) = y(t)e jωt dt = H T (ω)f(ω)

86 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 29/115 In the frequency domain, Y(ω) = where H(ω) = y(t)e jωt dt = H T (ω)f(ω) F(ω) = F(ω,p) = h(t)e jωt dt f(t,p)e jωt dt

87 Plane wave propagating Consider a plane wave propagating in the direction of vector a: sinθcosφ a = sinθsinφ cosθ

88 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 30/115 Plane wave propagating Consider a plane wave propagating in the direction of vector a: sinθcosφ a = sinθsinφ cosθ If f(t) is the signal that would be received at the origin, then: f(t τ 0 ) f(t τ 1 ) f(t,p) =. f(t τ N 1 )

89 Plane wave (assuming φ = 90 o ) sinθ z θ 1 y z a cos θ plane wave p n α u e y a

90 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? plane wave p n α u e y a

91 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave p n α u e y a

92 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave τ n = e c p n α u e y a

93 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave τ n = e c BUT p n α u e y a

94 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave τ n = e c BUT p n α e = p n cos(α) = u p n cos(α) }{{} =1 u e y a

95 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave τ n = e c BUT p n α e = p n cos(α) = u p n cos(α) }{{} =1 u e y a τ n = ut p n c = at p n c

96 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 31/115 Plane wave (assuming φ = 90 o ) z sinθ a z θ 1 cos θ y e =? e = cτ n plane wave τ n = e c BUT p n α e = p n cos(α) = u p n cos(α) }{{} =1 u e y a τ n = ut p n c = at p n c is the time since the plane wave hits the sensor at location p n until it reaches point (0,0).

97 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 32/115 Back to the frequency domain Then, we have: F(ω) = = e jωt f(t τ 0 )dt e jωt f(t τ 1 )dt. e jωt f(t τ N 1 )dt e jωτ 0 e jωτ 1. F(ω) e jωτ N 1

98 Definition of Wavenumber For plane waves propagating in a locally homogeneous medium: k = ω c a = 2π c/f a = 2π λ a = 2π λ u

99 Definition of Wavenumber For plane waves propagating in a locally homogeneous medium: k = ω c a = 2π c/f a = 2π λ a = 2π λ u Wavenumber Vector ("spatial frequency")

100 Definition of Wavenumber For plane waves propagating in a locally homogeneous medium: k = ω c a = 2π c/f a = 2π λ a = 2π λ u Note that k = 2π λ Wavenumber Vector ("spatial frequency")

101 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 33/115 Definition of Wavenumber For plane waves propagating in a locally homogeneous medium: k = ω c a = 2π c/f a = 2π λ a = 2π λ u Note that k = 2π λ Therefore Wavenumber Vector ("spatial frequency") ωτ n = ω c at p n = k T p n

102 And we have F(ω) = e jkt p 0 e jkt p 1. e jkt p N 1 Array Manifold Vector F(ω) = F(ω)v k (k)

103 Array Manifold Vector And we have F(ω) = e jkt p 0 e jkt p 1. F(ω) = F(ω)v k (k) e jkt p N 1 Array Manifold Vector

104 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 34/115 Array Manifold Vector And we have F(ω) = e jkt p 0 e jkt p 1. F(ω) = F(ω)v k (k) e jkt p N 1 Array Manifold Vector In this particular example, we can use h n (t) = 1 N δ(t+τ n) such that y(t) = f(t) Following, we have the delay-and-sum beamformer.

105 Delay-and-sum Beamformer f (t τ ) 0 +τ 0 f (t τ ) 1 +τ Σ 1 N y(t) f (t τ ) N 1 +τ N 1

106 Delay-and-sum Beamformer f (t τ ) 0 +τ 0 f (t τ ) 1 +τ Σ 1 N y(t) f (t τ ) N 1 +τ N 1 A common delay is added in each channel to make the operations physically realizable

107 Delay-and-sum Beamformer f (t τ ) 0 +τ 0 f (t τ ) 1 +τ Σ 1 N y(t) f (t τ ) N 1 +τ N 1 A common delay is added in each channel to make the operations physically realizable Since F {h n (t)} = F { 1 N δ(t+τ n) } = e jωτ n

108 Delay-and-sum Beamformer f (t τ ) 0 +τ 0 f (t τ ) 1 +τ Σ 1 N y(t) f (t τ ) N 1 +τ N 1 A common delay is added in each channel to make the operations physically realizable Since F {h n (t)} = F { 1 N δ(t+τ n) } = e jωτ n We can write H T (ω) = 1 N vh k (k)

109 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 35/115 Delay-and-sum Beamformer f (t τ ) 0 +τ 0 f (t τ ) 1 +τ Σ 1 N y(t) f (t τ ) N 1 +τ N 1 A common delay is added in each channel to make the operations physically realizable Since F {h n (t)} = F { 1 N δ(t+τ n) } = e jωτ n We can write H T (ω) = 1 N vh k (k) Array Manifold Vector

110 e jωt h(t) H(ω)e jωt LTI System

111 LTI System e jωt h(t) H(ω)e jωt Space-time signals (base functions): f n (t,p) = e jω(t τ n) = e j(ωt kt p n ) Note that ωτ n = k T p n

112 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 36/115 LTI System e jωt h(t) H(ω)e jωt Space-time signals (base functions): f n (t,p) = e jω(t τ n) = e j(ωt kt p n ) f(t,p) = e jωt v k (k) Note that ωτ n = k T p n

113 Frequency-Wavenumber Response Function The response of the array to this plane wave is: y(t,k) = H T (ω)v k (k)e jωt

114 Frequency-Wavenumber Response Function The response of the array to this plane wave is: y(t,k) = H T (ω)v k (k)e jωt After taking the Fourier transform, we have: Y(ω,k) = H T (ω)v k (k)

115 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 37/115 Frequency-Wavenumber Response Function The response of the array to this plane wave is: y(t,k) = H T (ω)v k (k)e jωt After taking the Fourier transform, we have: Y(ω,k) = H T (ω)v k (k) And we define the Frequency-Wavenumber Response Function: Upsilon Υ(ω,k) H T (ω)v k (k) Υ(ω,k) describes the complex gain of an array to an input plane wave with wavenumber k and temporal frequency ω.

116 Beam Pattern and Bandpass Signal BEAM PATTERN is the Frequency Wavenumber Response Function evaluated versus the direction: B(ω : θ,φ) = Υ(ω,k) Note that k = 2π a(θ,φ), and a is the unit vector with λ spherical coordinates angles θ and φ

117 Beam Pattern and Bandpass Signal BEAM PATTERN is the Frequency Wavenumber Response Function evaluated versus the direction: B(ω : θ,φ) = Υ(ω,k) Note that k = 2π a(θ,φ), and a is the unit vector with λ spherical coordinates angles θ and φ Let s write a bandpass signal: f(t,p n ) = 2Re{ f(t,p n )e jω ct },n = 0,1,,N 1

118 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 38/115 Beam Pattern and Bandpass Signal BEAM PATTERN is the Frequency Wavenumber Response Function evaluated versus the direction: B(ω : θ,φ) = Υ(ω,k) Note that k = 2π a(θ,φ), and a is the unit vector with λ spherical coordinates angles θ and φ Let s write a bandpass signal: f(t,p n ) = 2Re{ f(t,p n )e jω ct },n = 0,1,,N 1 2πB s ω c ω c corresponds to the carrier frequency and the complex envelope f(t,p n ) is bandlimited to the region ω ω c }{{} ω L 2πB s /2

119 Bandlimited and Narrowband Signals Bandlimited plane wave: f(t,p n ) = 2Re{ f(t τ n )e jω c(t τ n ) },n = 0,1,,N 1

120 Bandlimited and Narrowband Signals Bandlimited plane wave: f(t,p n ) = 2Re{ f(t τ n )e jω c(t τ n ) },n = 0,1,,N 1 Maximum travel time ( T max ) across the (linear) array: travel time between the two sensors at the extremities (signal arriving along the end-fire)

121 Bandlimited and Narrowband Signals Bandlimited plane wave: f(t,p n ) = 2Re{ f(t τ n )e jω c(t τ n ) },n = 0,1,,N 1 Maximum travel time ( T max ) across the (linear) array: travel time between the two sensors at the extremities (signal arriving along the end-fire) Assuming the origin is at the array s center of gravity: N 1 n=0 p n = 0 τ n T max

122 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 39/115 Bandlimited and Narrowband Signals Bandlimited plane wave: f(t,p n ) = 2Re{ f(t τ n )e jω c(t τ n ) },n = 0,1,,N 1 Maximum travel time ( T max ) across the (linear) array: travel time between the two sensors at the extremities (signal arriving along the end-fire) Assuming the origin is at the array s center of gravity: N 1 n=0 p n = 0 τ n T max In Narrowband (NB) signals, B s T max 1 f(t τ n ) f(t) and f(t,p n ) = 2Re{ f(t)e jω cτ n e jω ct }

123 For NB signals, the delay is approximated by a phase-shift: delay&sum beamformer PHASED ARRAY Phased-Array

124 For NB signals, the delay is approximated by a phase-shift: delay&sum beamformer PHASED ARRAY Phased-Array f (t τ ) 0 j 0 0 e + ω τ f(t) f (t τ ) 1 j 0 1 e + ω τ. f(t)... Σ 1 N y(t) f (t τ ) N 1 j 0 N 1 e + ω τ f(t)

125 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 40/115 For NB signals, the delay is approximated by a phase-shift: delay&sum beamformer PHASED ARRAY Phased-Array f (t τ ) 0 j 0 0 e + ω τ f(t) f (t τ ) 1 j 0 1 e + ω τ. f(t)... Σ 1 N y(t) f (t τ ) N 1 j 0 N 1 e + ω τ f(t) The phased array can be implemented adjusting the gain and phase to achieve a desired beam pattern

126 NB Beamformers In narrowband beamformers: y(t,k) = w H v k (k)e jωt

127 NB Beamformers In narrowband beamformers: y(t,k) = w H v k (k)e jωt f (t τ ) 0 w * 0 y (t) 0 f (t τ ) 1 w * 1 y (t) 1... Σ y(t). f (t τ ) N 1 w * N 1 y (t) N 1

128 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 41/115 NB Beamformers In narrowband beamformers: y(t,k) = w H v k (k)e jωt f (t τ ) 0 w * 0 y (t) 0 f (t τ ) 1 w * 1 y (t) 1... Σ y(t). f (t τ ) N 1 w * N 1 y (t) N 1 Υ(ω,k) = }{{} w H v k (k) H T (ω)

129 2.3 Uniform Linear Arrays (ULA) Microphone-Array Signal Processing, c Apolinárioi & Campos p. 42/115

130 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 43/115 Uniformly Spaced Linear Arrays z (array axis = "endfire") (grazing angle) d θ r _ θ = π _ θ 2 (broadside angle) y φ x

131 z An ULA along axis z: N 1 (polar angle) θ ULA d y x 1 0 φ (azimuth angle)

132 z An ULA along axis z: N 1 (polar angle) θ ULA d y x 1 0 φ (azimuth angle) Location of the elements: { pzn = (n N 1 )d, for n = 0,1,,N 1 2 p xn = p yn = 0

133 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 44/115 z An ULA along axis z: N 1 (polar angle) θ ULA d y x 1 0 φ (azimuth angle) Location of the elements: { pzn = (n N 1 )d, for n = 0,1,,N 1 2 p xn = p yn = 0 Therefore, p n = 0 0 (n N 1)d 2

134 ULA Array manifold vector: v k (k) = e jkt p 0. e jkt p n. e jkt p N 1 [k x k y k z ] 0 0 [ n N 1 2 ] d

135 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 45/115 ULA Array manifold vector: v k (k) = [e jkt p 0 e jkt p 1 e jkt p N 1 ] T v k (k) = v k (k z ) = (N 1) +j k e 2 z d e +j(n 1 2 1)k zd. e j(n 1 2 )k zd

136 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 46/115 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming 5. DoA Estimation with Microphone Arrays

137 3. Optimal Beamforming Microphone-Array Signal Processing, c Apolinárioi & Campos p. 47/115

138 3.1 Introduction Microphone-Array Signal Processing, c Apolinárioi & Campos p. 48/115

139 Introduction Scope: use the statistical representation of signal and noise to design array processors that are optimal in a statistical sense.

140 Introduction Scope: use the statistical representation of signal and noise to design array processors that are optimal in a statistical sense. We assume that the appropriate statistics are known.

141 Introduction Scope: use the statistical representation of signal and noise to design array processors that are optimal in a statistical sense. We assume that the appropriate statistics are known. Our objective of interest is to estimate the waveform of a plane-wave impinging on the array in the presence of noise and interfering signals.

142 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 49/115 Introduction Scope: use the statistical representation of signal and noise to design array processors that are optimal in a statistical sense. We assume that the appropriate statistics are known. Our objective of interest is to estimate the waveform of a plane-wave impinging on the array in the presence of noise and interfering signals. Even if a particular beamformer developed in this chapter has good performance, it does not guarantee that its adaptive version (next chapter) will. However, if the performance is poor, it is unlikely that the adaptive version will be useful.

143 3.2 Optimal Beamformers Microphone-Array Signal Processing, c Apolinárioi & Campos p. 50/115

144 Snapshot model in the frequency domain: MVDR Beamformer

145 Snapshot model in the frequency domain: MVDR Beamformer In many applications, we implement a beamforming in the frequency domain (ω m = ω c +m 2π and M varies T from M 1 to M 1 if odd and from M to M 1 if even). X ( ω T (M 1)/2,k) NB Beamformer Y T ( ω,k) (M 1)/2 X(t) Fourier Transform at M Frequencies X T ( ω m,k ) NB Beamformer Y ( T ω m,k ) Inverse Discrete y(t) Fourier Transform X T ( ω,k) (M 1)/2 NB Beamformer Y ( ω,k) T (M 1)/2

146 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 51/115 Snapshot model in the frequency domain: MVDR Beamformer In many applications, we implement a beamforming in the frequency domain (ω m = ω c +m 2π and M varies T from M 1 to M 1 if odd and from M to M 1 if even). X ( ω T (M 1)/2,k) NB Beamformer Y T ( ω,k) (M 1)/2 X(t) Fourier Transform at M Frequencies X T ( ω m,k ) NB Beamformer Y ( T ω m,k ) Inverse Discrete y(t) Fourier Transform X T ( ω,k) (M 1)/2 NB Beamformer Y ( ω,k) T (M 1)/2 In order to generate these vectors, divide the observation interval T in K disjoint intervals of duration T : (k 1) T t < k T,k = 1,,K.

147 MVDR Beamformer T must be significantly greater than the propagation time across the array.

148 MVDR Beamformer T must be significantly greater than the propagation time across the array. T also depends on the bandwidth of the input signal.

149 MVDR Beamformer T must be significantly greater than the propagation time across the array. T also depends on the bandwidth of the input signal. Assume an input signal with BW B s centered in f c

150 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 52/115 MVDR Beamformer T must be significantly greater than the propagation time across the array. T also depends on the bandwidth of the input signal. Assume an input signal with BW B s centered in f c In order to develop the frequency-domain snapshot model for the case in which the desired signals and the interfering signals can de modeled as plane waves, we have two cases: desired signals are deterministic or samples of a random process.

151 MVDR Beamformer Let s assume the case where the signal is nonrandom but unknown; we initially consider the case of single plane-wave signal.

152 MVDR Beamformer Let s assume the case where the signal is nonrandom but unknown; we initially consider the case of single plane-wave signal. Frequency-domain snapshot consists of signal plus noise: X(ω) = X s (ω)+n(ω)

153 MVDR Beamformer Let s assume the case where the signal is nonrandom but unknown; we initially consider the case of single plane-wave signal. Frequency-domain snapshot consists of signal plus noise: X(ω) = X s (ω)+n(ω) The signal vector can be written as X s (ω) = F(ω)v(ω : k s ) where F(ω) is the frequency-domain snapshot of the source signal and v(ω : k s ) is the array manifold vector for a plane-wave with wavenumber k s.

154 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 53/115 MVDR Beamformer Let s assume the case where the signal is nonrandom but unknown; we initially consider the case of single plane-wave signal. Frequency-domain snapshot consists of signal plus noise: X(ω) = X s (ω)+n(ω) The signal vector can be written as X s (ω) = F(ω)v(ω : k s ) where F(ω) is the frequency-domain snapshot of the source signal and v(ω : k s ) is the array manifold vector for a plane-wave with wavenumber k s. The noise snapshot is a zero-mean random vector N(ω) with spectral matrix given by Sn(ω) = Sc(ω)+σ 2 ω I

155 MVDR Beamformer We process X(ω) with the 1 N operator W H (ω): X ( ω ) ( ω) ( ω) W H Y

156 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 54/115 MVDR Beamformer We process X(ω) with the 1 N operator W H (ω): X ( ω ) ( ω) ( ω) W H Y Distortionless criterion (in the absence of noise): Y(ω) = F(ω) = W H (ω)x s (ω) = F(ω)W H (ω)v(ω : k s ) = W H (ω)v(ω : k s ) = 1

157 MVDR Beamformer In the presence of noise, we have: Y(ω) = F(ω)+Y n (ω)

158 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 55/115 MVDR Beamformer In the presence of noise, we have: Y(ω) = F(ω)+Y n (ω) The mean square of the output noise is: E[ Y n (ω) 2 ] = W H (ω)s n (ω)w(ω)

159 MVDR Beamformer In the MVDR beamformer, we want to minimize E[ Y n (ω) 2 ] subject to W H (ω)v(ω : k s ) = 1

160 MVDR Beamformer In the MVDR beamformer, we want to minimize E[ Y n (ω) 2 ] subject to W H (ω)v(ω : k s ) = 1 Using the method of Lagrange multipliers, we define the following cost function to be minimized F = W H (ω)s n (ω)wω +λ [ W H (ω)v(ω : k s ) 1 ] +λ [ v H (ω : k s )W(ω) 1 ]

161 MVDR Beamformer In the MVDR beamformer, we want to minimize E[ Y n (ω) 2 ] subject to W H (ω)v(ω : k s ) = 1 Using the method of Lagrange multipliers, we define the following cost function to be minimized F = W H (ω)s n (ω)wω +λ [ W H (ω)v(ω : k s ) 1 ] +λ [ v H (ω : k s )W(ω) 1 ]...and the result (suppressing ω and k s ) is W H mvdr = Λ s v H S 1 n where Λ s = [ v H S 1 n v ] 1

162 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 56/115 MVDR Beamformer In the MVDR beamformer, we want to minimize E[ Y n (ω) 2 ] subject to W H (ω)v(ω : k s ) = 1 Using the method of Lagrange multipliers, we define the following cost function to be minimized F = W H (ω)s n (ω)wω +λ [ W H (ω)v(ω : k s ) 1 ] +λ [ v H (ω : k s )W(ω) 1 ]...and the result (suppressing ω and k s ) is W H mvdr = Λ s v H S 1 n where Λ s = [ v H S 1 n v ] 1 This result is referred to as MVDR or Capon Beamformer.

163 Constrained Optimal Filtering The gradient of ξ with respect to w (real case): wξ = ξ w 0 ξ w 1. ξ w N 1

164 Constrained Optimal Filtering The gradient of ξ with respect to w (real case): wξ = ξ w 0 ξ w 1. ξ w N 1 From the definition above, it is easy to show that: w(b T w) = w(w T b) = b

165 Constrained Optimal Filtering The gradient of ξ with respect to w (real case): wξ = ξ w 0 ξ w 1. ξ w N 1 From the definition above, it is easy to show that: w(b T w) = w(w T b) = b Also w(w T Rw) = R T w+rw

166 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 57/115 Constrained Optimal Filtering The gradient of ξ with respect to w (real case): wξ = ξ w 0 ξ w 1. ξ w N 1 From the definition above, it is easy to show that: w(b T w) = w(w T b) = b Also w(w T Rw) = R T w+rw which, when R is symmetric, leads to w(w T Rw) = 2Rw

167 Constrained Optimal Filtering We now assume the complex case w = a+jb.

168 Constrained Optimal Filtering We now assume the complex case w = a+jb. The gradient becomes w ξ = ξ a 0 +j ξ b 0 ξ a 1 +j ξ b 1. ξ ξ a N 1 +j b N 1

169 Constrained Optimal Filtering We now assume the complex case w = a+jb. The gradient becomes w ξ = ξ a 0 +j ξ b 0 ξ a 1 +j ξ b 1. ξ ξ a N 1 +j b N 1 which corresponds to wξ = aξ +j b ξ

170 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 58/115 Constrained Optimal Filtering We now assume the complex case w = a+jb. The gradient becomes w ξ = ξ a 0 +j ξ b 0 ξ a 1 +j ξ b 1. ξ ξ a N 1 +j b N 1 which corresponds to wξ = aξ +j b ξ Let us define the derivative a 0 j b 0 = 1 a w 2 1 j b 1. a N 1 j b N 1 w (with respect to w):

171 Constrained Optimal Filtering The conjugate derivative with respect to w is a 0 +j b 0 = 1 a 1 +j b 1 w 2. a N 1 +j b N 1

172 Constrained Optimal Filtering The conjugate derivative with respect to w is a 0 +j b 0 = 1 a 1 +j b 1 w 2. a N 1 +j b N 1 Therefore, w ξ = a ξ +j b ξ is equivalent to 2 ξ w.

173 Constrained Optimal Filtering The conjugate derivative with respect to w is a 0 +j b 0 = 1 a 1 +j b 1 w 2. a N 1 +j b N 1 Therefore, w ξ = a ξ +j b ξ is equivalent to 2 ξ w. The complex gradient may be slightly tricky if compared to the simple real gradient. For this reason, we exemplify the use of the complex gradient by calculating w E[ e(k) 2 ].

174 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 59/115 Constrained Optimal Filtering The conjugate derivative with respect to w is a 0 +j b 0 = 1 a 1 +j b 1 w 2. a N 1 +j b N 1 Therefore, w ξ = a ξ +j b ξ is equivalent to 2 ξ w. The complex gradient may be slightly tricky if compared to the simple real gradient. For this reason, we exemplify the use of the complex gradient by calculating w E[ e(k) 2 ]. w E[e(k)e (k)] = E{e (k)[ w e(k)]+e(k)[ w e (k)]}

175 Constrained Optimal Filtering We compute each gradient... w e(k) = a [d(k) w H x(k)]+j b [d(k) w H x(k)] = x(k) x(k) = 2x(k)

176 Constrained Optimal Filtering We compute each gradient... w e(k) = a [d(k) w H x(k)]+j b [d(k) w H x(k)] = x(k) x(k) = 2x(k) and w e (k) = a [d (k) w T x (k)]+j b [d (k) w T x (k)] = x (k)+x (k) = 0

177 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 60/115 We compute each gradient... Constrained Optimal Filtering w e(k) = a [d(k) w H x(k)]+j b [d(k) w H x(k)] and = x(k) x(k) = 2x(k) w e (k) = a [d (k) w T x (k)]+j b [d (k) w T x (k)] = x (k)+x (k) = 0 such that the final result is w E[e(k)e (k)] = 2E[e (k)x(k)] = 2E[x(k)[d(k) w H x(k)] } = 2E[x(k)d (k)] +2E[x(k)x H (k)] w }{{}}{{} p R

178 Constrained Optimal Filtering Which results in the Wiener solution w = R 1 p.

179 Constrained Optimal Filtering Which results in the Wiener solution w = R 1 p. When a set of linear constraints involving the coefficient vector of an adaptive filter is imposed, the resulting problem (LCAF) admitting the MSE as the objective function can be stated as minimizing E[ e(k) 2 ] subject to C H w = f.

180 Constrained Optimal Filtering Which results in the Wiener solution w = R 1 p. When a set of linear constraints involving the coefficient vector of an adaptive filter is imposed, the resulting problem (LCAF) admitting the MSE as the objective function can be stated as minimizing E[ e(k) 2 ] subject to C H w = f. The output of the processor is y(k) = w H x(k).

181 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 61/115 Constrained Optimal Filtering Which results in the Wiener solution w = R 1 p. When a set of linear constraints involving the coefficient vector of an adaptive filter is imposed, the resulting problem (LCAF) admitting the MSE as the objective function can be stated as minimizing E[ e(k) 2 ] subject to C H w = f. The output of the processor is y(k) = w H x(k). It is worth mentioning that the most general case corresponds to having a reference signal, d(k). It is, however, usual to have no reference signal as in Linearly-Constrained Minimum-Variance (LCMV) applications. In LCMV, if f = 1, the system is often referred to as Minimum-Variance Distortionless Response (MVDR).

182 Using Lagrange multipliers, we form Constrained Optimal Filtering ξ(k) = E[e(k)e (k)]+l T R Re[CH w f]+l T I Im[CH w f]

183 Constrained Optimal Filtering Using Lagrange multipliers, we form ξ(k) = E[e(k)e (k)]+l T R Re[CH w f]+l T I Im[CH w f] We can also represent the above expression with a complex L given by L R +jl I such that ξ(k) = E[e(k)e (k)]+re[l H (C H w f)] = E[e(k)e (k)]+ 1 2 LH (C H w f)+ 1 2 LT (C T w f )

184 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 62/115 Constrained Optimal Filtering Using Lagrange multipliers, we form ξ(k) = E[e(k)e (k)]+l T R Re[CH w f]+l T I Im[CH w f] We can also represent the above expression with a complex L given by L R +jl I such that ξ(k) = E[e(k)e (k)]+re[l H (C H w f)] = E[e(k)e (k)]+ 1 2 LH (C H w f)+ 1 2 LT (C T w f ) Noting that e(k) = d(k) w H x(k), we compute: w ξ(k) = w {E[e(k)e (k)]+ 1 2 LH (C H w f)+ 1 } 2 LT (C T w f ) = E[ 2x(k)e (k)]+0+cl = 2E[x(k)d (k)]+2e[x(k)x H (k)]w+cl

185 Constrained Optimal Filtering By using R = E[x(k)x H (k)] and p = E[d (k)x(k)], the gradient is equated to zero and the results can be written as (note that stationarity was assumed for the input and reference signals): 2p+2Rw+CL = 0

186 Constrained Optimal Filtering By using R = E[x(k)x H (k)] and p = E[d (k)x(k)], the gradient is equated to zero and the results can be written as (note that stationarity was assumed for the input and reference signals): 2p+2Rw+CL = 0 Which leads to w = 1 2 R 1 (2p CL)

187 Constrained Optimal Filtering By using R = E[x(k)x H (k)] and p = E[d (k)x(k)], the gradient is equated to zero and the results can be written as (note that stationarity was assumed for the input and reference signals): 2p+2Rw+CL = 0 Which leads to w = 1 2 R 1 (2p CL) If we pre-multiply the previous expression by C H and use C H w = f, we find L: L = 2(C H R 1 C) 1 (C H R 1 p f)

188 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 63/115 Constrained Optimal Filtering By using R = E[x(k)x H (k)] and p = E[d (k)x(k)], the gradient is equated to zero and the results can be written as (note that stationarity was assumed for the input and reference signals): 2p+2Rw+CL = 0 Which leads to w = 1 2 R 1 (2p CL) If we pre-multiply the previous expression by C H and use C H w = f, we find L: L = 2(C H R 1 C) 1 (C H R 1 p f) By replacing L, we obtain the Wiener solution for the linearly constrained adaptive filter: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p)

189 Constrained Optimal Filtering The optimal solution for LCAF: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p)

190 Constrained Optimal Filtering The optimal solution for LCAF: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p) Note that if d(k) = 0, then p = 0, and we have (LCMV): w opt = R 1 C(C H R 1 C) 1 f

191 Constrained Optimal Filtering The optimal solution for LCAF: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p) Note that if d(k) = 0, then p = 0, and we have (LCMV): w opt = R 1 C(C H R 1 C) 1 f Yet with d(k) = 0 but f = 1 (MVDR) w opt = R 1 C(C H R 1 C) 1

192 Constrained Optimal Filtering The optimal solution for LCAF: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p) Note that if d(k) = 0, then p = 0, and we have (LCMV): w opt = R 1 C(C H R 1 C) 1 f Yet with d(k) = 0 but f = 1 (MVDR) w opt = R 1 C(C H R 1 C) 1 For this case, d(k) = 0, the cost function is termed minimum output energy (MOE) and is given by E[ e(k) 2 ] = w H Rw

193 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 64/115 Constrained Optimal Filtering The optimal solution for LCAF: w opt = R 1 p+r 1 C(C H R 1 C) 1 (f C H R 1 p) Note that if d(k) = 0, then p = 0, and we have (LCMV): w opt = R 1 C(C H R 1 C) 1 f Yet with d(k) = 0 but f = 1 (MVDR) w opt = R 1 C(C H R 1 C) 1 For this case, d(k) = 0, the cost function is termed minimum output energy (MOE) and is given by E[ e(k) 2 ] = w H Rw Also note that in case we do not have constraints (C and f are nulls), the optimal solution above becomes the unconstrained Wiener solution R 1 p.

194 The GSC We start by doing a transformation in the coefficient vector. Let T = [C B] such that [ ] wu w = T w = [C B] = C w U B w L w L

195 The GSC We start by doing a transformation in the coefficient vector. Let T = [C B] such that [ ] wu w = T w = [C B] w L = C w U B w L Matrix B is usually called the Blocking Matrix and we recall that C H w = g such that C H w = C H C w U C H B w L = g.

196 The GSC We start by doing a transformation in the coefficient vector. Let T = [C B] such that [ ] wu w = T w = [C B] w L = C w U B w L Matrix B is usually called the Blocking Matrix and we recall that C H w = g such that C H w = C H C w U C H B w L = g. If we impose the condition B H C = 0 or, equivalently, C H B = 0, we will have w U = (C H C) 1 g.

197 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 65/115 The GSC We start by doing a transformation in the coefficient vector. Let T = [C B] such that [ ] wu w = T w = [C B] w L = C w U B w L Matrix B is usually called the Blocking Matrix and we recall that C H w = g such that C H w = C H C w U C H B w L = g. If we impose the condition B H C = 0 or, equivalently, C H B = 0, we will have w U = (C H C) 1 g. w U is fixed and termed the quiescent weight vector; the minimization process will be carried out only in the lower part, also designated w GSC = w L.

198 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 66/115 The GSC It is shown below how to split the transformation matrix into two parts: a fixed path and an adaptive path. input signal MN 1 w(k) reference signal Σ + MN MN T MN 1 w(k) + Σ MN p C MN p B 1 w U (k) w (k) L p MN p Σ + (a) (b) MN MN p C MN p B p MN p 1 w U (k) 1 w (k) L + Σ + Σ (c) input signal F B w(k) GSC + Σ reference signal + Σ (d)

199 This structure (detailed below) was named the Generalized Sidelobe Canceller (GSC). x (k) F B x GSC H (k) = B x(k) w(k 1) GSC H F x(k) + Σ y (k) GSC y(k) + e(k) d(k) Σ The GSC

200 This structure (detailed below) was named the Generalized Sidelobe Canceller (GSC). x (k) F B x GSC H (k) = B x(k) w(k 1) GSC H F x(k) + Σ y (k) GSC y(k) + e(k) d(k) Σ The GSC It is always possible to have the overall equivalent coefficient vector which is given by w = F Bw GSC.

201 This structure (detailed below) was named the Generalized Sidelobe Canceller (GSC). x (k) F B x GSC H (k) = B x(k) w(k 1) GSC H F x(k) + Σ y (k) GSC y(k) + e(k) d(k) Σ The GSC It is always possible to have the overall equivalent coefficient vector which is given by w = F Bw GSC. If we pre-multiply last equation by B H and isolate w GSC, we find w GSC = (B H B) 1 B H w.

202 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 67/115 This structure (detailed below) was named the Generalized Sidelobe Canceller (GSC). x (k) F B x GSC H (k) = B x(k) w(k 1) GSC H F x(k) + Σ y (k) GSC y(k) + e(k) d(k) Σ The GSC It is always possible to have the overall equivalent coefficient vector which is given by w = F Bw GSC. If we pre-multiply last equation by B H and isolate w GSC, we find w GSC = (B H B) 1 B H w. Knowing that T = [C B] and that T H T = I, it follows that P = I C(C H C) 1 C H = B(B H B)B H.

203 The GSC A simple procedure to find the optimal GSC solution comes from the unconstrained Wiener solution applied to the unconstrained filter: w GSC OPT = R 1 GSC p GSC

204 The GSC A simple procedure to find the optimal GSC solution comes from the unconstrained Wiener solution applied to the unconstrained filter: w GSC OPT = R 1 GSC p GSC From the figure, it is clear that: R GSC = E[x GSC x H GSC ] = E[BH xx H B] = B H RB

205 The GSC A simple procedure to find the optimal GSC solution comes from the unconstrained Wiener solution applied to the unconstrained filter: w GSC OPT = R 1 GSC p GSC From the figure, it is clear that: R GSC = E[x GSC x H GSC ] = E[BH xx H B] = B H RB The cross-correlation vector is given as: p GSC = E[d GSCx GSC ] = E{[F H x d] [B H x]} = E[ B H d x+b H xx H F] = B H p+b H RF

206 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 68/115 The GSC A simple procedure to find the optimal GSC solution comes from the unconstrained Wiener solution applied to the unconstrained filter: w GSC OPT = R 1 GSC p GSC From the figure, it is clear that: R GSC = E[x GSC x H GSC ] = E[BH xx H B] = B H RB The cross-correlation vector is given as: p GSC = E[d GSCx GSC ] = E{[F H x d] [B H x]} = E[ B H d x+b H xx H F] = B H p+b H RF and w GSC OPT = (B H RB) 1 ( B H p+b H RF)

207 A common case is when d(k) = 0: x (k) F B x GSC H (k) = B x(k) H d (k) = F x(k) GSC w(k 1) GSC + Σ e (k) GSC y (k) GSC The GSC

208 A common case is when d(k) = 0: x (k) F B x GSC H (k) = B x(k) H d (k) = F x(k) GSC w(k 1) GSC + Σ e (k) GSC y (k) GSC The GSC We have dropped the negative sign that should exist according to the notation used. Although we define e GSC (k) = e(k), the inversion of the sign in the error signal actually results in the same results because the error function is always based on the absolute value.

209 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 69/115 A common case is when d(k) = 0: x (k) F B x GSC H (k) = B x(k) H d (k) = F x(k) GSC w(k 1) GSC + Σ e (k) GSC y (k) GSC The GSC We have dropped the negative sign that should exist according to the notation used. Although we define e GSC (k) = e(k), the inversion of the sign in the error signal actually results in the same results because the error function is always based on the absolute value. In this case, the optimum filter w OPT is: F Bw GSC OPT = F B(B H RB) 1 B H RF = R 1 C(C H R 1 C) 1 f (LCMV solution)

210 The GSC Choosing the blocking matrix: B plays an important role since its choice determines computational complexity and even robustness against numerical instability.

211 The GSC Choosing the blocking matrix: B plays an important role since its choice determines computational complexity and even robustness against numerical instability. Since the only need for B is having its columns forming a basis orthogonal to the constraints, B H C = 0, a myriad of options are possible.

212 The GSC Choosing the blocking matrix: B plays an important role since its choice determines computational complexity and even robustness against numerical instability. Since the only need for B is having its columns forming a basis orthogonal to the constraints, B H C = 0, a myriad of options are possible. Let us recall the paper by Griffiths and Jim where the term GSC was coined; let C T =

213 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 70/115 The GSC Choosing the blocking matrix: B plays an important role since its choice determines computational complexity and even robustness against numerical instability. Since the only need for B is having its columns forming a basis orthogonal to the constraints, B H C = 0, a myriad of options are possible. Let us recall the paper by Griffiths and Jim where the term GSC was coined; let C T = With simple constraint matrices, simple blocking matrices satisfying B T C = 0 are possible.

214 The GSC For this particular example, the paper presents two possibilities. The first one (orthogonal) is:

215 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 71/115 The GSC For this particular example, the paper presents two possibilities. The first one (orthogonal) is: B T 1 =

216 And the second possibility (non-orthogonal) is: The GSC

217 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 72/115 The GSC And the second possibility (non-orthogonal) is: B T 2 =

218 The GSC SVD: the blocking matrix can be produced with the following Matlab command lines, [U,S,V]=svd(C); B3=U(:,p+1:M*N); % p=n in this case

219 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 73/115 The GSC SVD: the blocking matrix can be produced with the following Matlab command lines, [U,S,V]=svd(C); B3=U(:,p+1:M*N); % p=n in this case B T 3 is given by:

220 QRD: the blocking matrix can be produced with the following Matlab command lines, [Q,R]=qr(C); B4=Q(:,p+1:M*N); The GSC

221 QRD: the blocking matrix can be produced with the following Matlab command lines, [Q,R]=qr(C); B4=Q(:,p+1:M*N); B 4 was identical to B 3 (SVD). The GSC

222 QRD: the blocking matrix can be produced with the following Matlab command lines, [Q,R]=qr(C); B4=Q(:,p+1:M*N); B 4 was identical to B 3 (SVD). The GSC Two other possibilities are: the one presented in [Tseng Griffiths 88] where a decomposition procedure is introduced in order to offer an effective implementation structure and the other one concerned to a narrowband BF implemented with GSC where B is combined with a wavelet transform [Chu Fang 99].

223 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 74/115 QRD: the blocking matrix can be produced with the following Matlab command lines, [Q,R]=qr(C); B4=Q(:,p+1:M*N); B 4 was identical to B 3 (SVD). The GSC Two other possibilities are: the one presented in [Tseng Griffiths 88] where a decomposition procedure is introduced in order to offer an effective implementation structure and the other one concerned to a narrowband BF implemented with GSC where B is combined with a wavelet transform [Chu Fang 99]. Finally, a new efficient linearly constrained adaptive scheme which can also be visualized as a GSC structure can be found in [Campos&Werner&Apolinário IEEE-TSP Sept. 2002].

224 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 75/115 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming 5. DoA Estimation with Microphone Arrays

225 4. Adaptive Beamforming Microphone-Array Signal Processing, c Apolinárioi & Campos p. 76/115

226 4.1 Introduction Microphone-Array Signal Processing, c Apolinárioi & Campos p. 77/115

227 Introduction Scope: instead of assuming knowledge about the statistical properties of the signals, beamformers are designed based on statistics gathered online.

228 Introduction Scope: instead of assuming knowledge about the statistical properties of the signals, beamformers are designed based on statistics gathered online. Different algorithms may be employed for iteratively approximating the desired solution.

229 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 78/115 Introduction Scope: instead of assuming knowledge about the statistical properties of the signals, beamformers are designed based on statistics gathered online. Different algorithms may be employed for iteratively approximating the desired solution. We will briefly cover a small subset of algorithms for constrained adaptive filters.

230 Introduction Linearly constrained adaptive filters (LCAF) have found application in numerous areas, such as spectrum analysis, spatial-temporal processing, antenna arrays, interference suppression, among others.

231 Introduction Linearly constrained adaptive filters (LCAF) have found application in numerous areas, such as spectrum analysis, spatial-temporal processing, antenna arrays, interference suppression, among others. LCAF algorithms incorporate into the solution application-specific requirements translated into a set of linear equations to be satisfied by the coefficients.

232 Introduction Linearly constrained adaptive filters (LCAF) have found application in numerous areas, such as spectrum analysis, spatial-temporal processing, antenna arrays, interference suppression, among others. LCAF algorithms incorporate into the solution application-specific requirements translated into a set of linear equations to be satisfied by the coefficients. For example, if direction of arrival of the signal of interest is known, jammer suppression can take place through spatial filtering without the need of training signal, or

233 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 79/115 Introduction Linearly constrained adaptive filters (LCAF) have found application in numerous areas, such as spectrum analysis, spatial-temporal processing, antenna arrays, interference suppression, among others. LCAF algorithms incorporate into the solution application-specific requirements translated into a set of linear equations to be satisfied by the coefficients. For example, if direction of arrival of the signal of interest is known, jammer suppression can take place through spatial filtering without the need of training signal, or in systems with constant-envelope modulation (e.g., M-PSK), a constant-modulus constraint can mitigate multipath propagation effects.

234 4.2 Constrained FIR Filters Microphone-Array Signal Processing, c Apolinárioi & Campos p. 80/115

235 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 81/115 Broadband Array Beamformer x (k) 1 w (k) 1 x (k) 2 w (k) 2 Σ y(k) x (k) M w (k) M Algorithm: min ξ(k) w H s.t. C w=f e(k) Σ - d(k) +

236 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 82/115 Optimal Constrained MSE Filter We look for where min w ξ(k) s.t. CH w = f, ξ(k) = E[ e(k) 2 ] C is the MN p constraint matrix f is the p 1 gain vector

237 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 83/115 Optimal Constrained MSE Filter The optimal beamformer is w(k) = R 1 p+r 1 C ( C H R 1 C ) 1( f C H R 1 p ) where: R = E [ x(k)x H (k) ] and p = E[d (k)x(k)] w(k) = [ w1(k) T w2(k) T wm T (k)] T x(k) = [ x T 1(k) x T 2(k) x T M (k)] T x T i (k) = [x i (k) x i (k 1) x i (k N +1)]

238 The Constrained LS Beamformer In the absence of statistical information, we may choose [ ] k min ξ(k) = λ k i d(i) w H x(i) 2 s.t. C H w = f w with λ (0,1], i=0

239 The Constrained LS Beamformer In the absence of statistical information, we may choose [ ] k min ξ(k) = λ k i d(i) w H x(i) 2 s.t. C H w = f w i=0 with λ (0,1], which gives, as solution,

240 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 84/115 The Constrained LS Beamformer In the absence of statistical information, we may choose [ ] k min ξ(k) = λ k i d(i) w H x(i) 2 s.t. C H w = f w i=0 with λ (0,1], which gives, as solution, where w(k) = R 1 (k)p(k) +R 1 (k)c ( C H R 1 (k)c ) 1[ f C H R 1 (k)p(k) ], R(k) = k i=0 λk i x(i)x H (i), and p(k) = k i=0 λk i d (i)x(i).

241 The Constrained LMS Algorithm A (cheaper) alternative cost function is [ min ξ(k) = w(k) w(k 1) 2 +µ e(k) 2] s.t.c H w(k) = f, w

242 The Constrained LMS Algorithm A (cheaper) alternative cost function is min w [ ξ(k) = w(k) w(k 1) 2 +µ e(k) 2] s.t.c H w(k) = f, which gives, as solution,

243 The Constrained LMS Algorithm A (cheaper) alternative cost function is min w [ ξ(k) = w(k) w(k 1) 2 +µ e(k) 2] s.t.c H w(k) = f, which gives, as solution, w(k) = w(k 1)+µe (k) [ I C ( C H C ) ] 1 C H x(k), where e(k) = d(k) w H (k 1)x(k), µ is a positive small constant called step size.