Microphone-Array Signal Processing

Transcription

1 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 1/27 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 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 2/27 Outline 1. Introduction and Fundamentals 2. Sensor Arrays and Spatial Filtering 3. Optimal Beamforming 4. Adaptive Beamforming 5. DoA Estimation with Microphone Arrays

3 2. Sensor Arrays and Spatial Filtering Microphone-Array Signal Processing, c Apolinárioi & Campos p. 3/27

4 2.1 Wavenumber-Frequency Space Microphone-Array Signal Processing, c Apolinárioi & Campos p. 4/27

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

6 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

7 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

8 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 5/27 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

9 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 6/27 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

10 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 )

11 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?

12 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 7/27 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.

13 2.2 Frequency-Wavenumber (WN) Response and Beam patterns (BP) Microphone-Array Signal Processing, c Apolinárioi & Campos p. 8/27

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

15 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

16 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

17 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 9/27 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 )

18 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

19 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

20 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 10/27 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

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

22 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 11/27 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

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

24 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 12/27 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 )

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

26 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

27 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

28 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

29 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

30 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

31 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

32 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 13/27 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).

33 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 14/27 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

34 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

35 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")

36 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")

37 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 15/27 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

38 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)

39 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

40 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 16/27 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.

41 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

42 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

43 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

44 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)

45 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 17/27 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

46 e jωt h(t) H(ω)e jωt LTI System

47 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

48 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 18/27 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

49 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

50 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)

51 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 19/27 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.

52 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

53 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

54 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 20/27 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

55 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

56 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)

57 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

58 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 21/27 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 }

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

60 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)

61 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 22/27 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

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

63 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

64 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 23/27 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)

65 2.3 Uniform Linear Arrays (ULA) Microphone-Array Signal Processing, c Apolinárioi & Campos p. 24/27

66 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 25/27 Uniformly Spaced Linear Arrays z (array axis = "endfire") (grazing angle) d θ r _ θ = π _ θ 2 (broadside angle) y φ x

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

68 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

69 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 26/27 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

70 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

71 Microphone-Array Signal Processing, c Apolinárioi & Campos p. 27/27 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