Massive MIMO: Signal Structure, Efficient Processing, and Open Problems II

Transcription

1 Massive MIMO: Signal Structure, Efficient Processing, and Open Problems II Mahdi Barzegar Communications and Information Theory Group (CommIT) Technische Universität Berlin Heisenberg Communications and Information Theory Group Freie Universität Berlin CoSIP Retreat Berlin, December 2016 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

2 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

3 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

4 Overview The starting point h s (f N ) h s (f 1 ) f = 1 T s... Bandwidth W Focusing on the channel vector in each time-frequency resource block T s M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

5 Massive MIMO Regime Isotropic channel model for low-resolution antennas, or rich scattering channel 1, 2 h CN (0, σ 2 I M M ) 1 Emre Telatar. Capacity of Multi-antenna Gaussian Channels. In: European transactions on telecommunications 10.6 (1999), pp Lizhong Zheng and David NC Tse. Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel. In: Information Theory, IEEE Transactions on 48.2 (2002), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

6 Massive MIMO Regime Isotropic channel model for low-resolution antennas, or rich scattering channel 1, 2 h CN (0, σ 2 I M M ) In massive-mimo the base-station sees through the environment p scatterers, p M (M 1)d 4d 3d 2d d 0 θ i. User 1 Emre Telatar. Capacity of Multi-antenna Gaussian Channels. In: European transactions on telecommunications 10.6 (1999), pp Lizhong Zheng and David NC Tse. Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel. In: Information Theory, IEEE Transactions on 48.2 (2002), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

7 Massive MIMO Regime Isotropic channel model for low-resolution antennas, or rich scattering channel 1, 2 h CN (0, σ 2 I M M ) In massive-mimo the base-station sees through the environment This implies h CN (0, C), with rank (C) p M p scatterers, p M (M 1)d 4d 3d 2d d 0 θ i. User 1 Emre Telatar. Capacity of Multi-antenna Gaussian Channels. In: European transactions on telecommunications 10.6 (1999), pp Lizhong Zheng and David NC Tse. Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel. In: Information Theory, IEEE Transactions on 48.2 (2002), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

8 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

9 Sparse Channel Model for Massive MIMO Instantaneous sparse channel vector h = p i=1 w ia(θ i ), where a(θ) C M is the array response with [a(θ)] k = e jkπ sin(θ) sin(θmax) p scatterers, p M (M 1)d 4d 3d 2d d 0 θ i. User M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

10 Sparse Channel Model for Massive MIMO Instantaneous sparse channel vector h = p i=1 w ia(θ i ), where a(θ) C M is the array response with [a(θ)] k = e jkπ sin(θ) sin(θmax) Compressed sensing methods are used for channel estimation M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

11 Sparse Channel Model for Massive MIMO Instantaneous sparse channel vector h = p i=1 w ia(θ i ), where a(θ) C M is the array response with [a(θ)] k = e jkπ sin(θ) sin(θmax) Compressed sensing methods are used for channel estimation Selection of only a few array elements is sufficient to recover h 4d 3d 2d d 0 Antenna Selection 4d 3d 2d d 0 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

12 High-dimensional Estimation with Sparsity Constraints High-dim Data Low-dim Structure Low-dim Sketches (Antenna Selection) Sparse Channel Vector h = p i=1 w ia(θ i ) Efficient Implementation (less RF-chains, less power consumption,...) Compressed Sensing Techniques Beamformer Design M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

13 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

14 Subspace Estimation One can go beyond one-shot CS and think of subspace estimation. Observe the channel over time M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

15 Subspace Estimation One can go beyond one-shot CS and think of subspace estimation. Observe the channel over time Questions Q1: How to estimate the underlying subspace? What are the constraints? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

16 Subspace Estimation One can go beyond one-shot CS and think of subspace estimation. Observe the channel over time Questions Q1: How to estimate the underlying subspace? What are the constraints? Q2: Is subspace information particularly useful? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

17 Q1: How to estimate the subspace? Some Observations: The local geometry given by {(σi 2, θ i)} p i=1 is quasi-stationary with sharp transitions M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

18 Q1: How to estimate the subspace? Some Observations: The local geometry given by {(σi 2, θ i)} p i=1 is quasi-stationary with sharp transitions In the traditional periodic training scheme, we have t training samples: y i = s i h i + n i x i = By i, i [t] s 0 s 1 s t 1 s t τ τ T M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

19 Q1: How to estimate the subspace? Some Observations: There is an intermediate regime τ T T containing 1 ν training samples, in which the subspace information can be exploited s 0 s 1 s t 1 s t τ τ T s 0 s 1 s ν 1 s ν τ τ T T M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

20 Q1: How to estimate the subspace? Some Observations: There is an intermediate regime τ T T containing 1 ν training samples, in which the subspace information can be exploited s 0 s 1 s ν 1 s ν τ τ T T M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

21 Q1: How to estimate the subspace? Some Observations: There is an intermediate regime τ T T containing 1 ν training samples, in which the subspace information can be exploited s 0 s 1 s ν 1 s ν τ τ T T Typical Window Size Subspace Estimation should be done with around ν samples! M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

22 Q2: How to exploit the subspace information? Example 1 If signal subspace U M q is known, then h span(u) approximately: this can be used to improve estimation of h h M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

23 Q2: How to exploit the subspace information? Example 1 If signal subspace U M q is known, then h span(u) approximately: this can be used to improve estimation of h h This can be seen as a support estimate for the sparse recovery problem M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

24 Q2: How to exploit the subspace information? Example 2 This solves the aging problem in mm-wave channels s i s i+1 τ r h (δ) 1 τ δ M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

25 Q2: How to exploit the subspace information? Example 2 This solves the aging problem in mm-wave channels Reject the interference by zero-forcing to signal subspace rather than zero-forcing to instantaneous channel vector s i s i+1 τ h i 2 h i r h (δ) 1 h i 1 τ δ M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

26 How to exploit the subspace information? Example 3 This can be used to cluster the user based on their signal subspace, and is suitable for HDA implementations 3 (M 1)d 4d Digital Base-band Processing ADC (uplink) DAC (downlink) M M Analog RF-Chain 3d 2d d 0 3 Ansuman Adhikary et al. Joint spatial division and multiplexing for mm-wave channels. In: IEEE J. on Sel. Areas on Commun. (JSAC) 32.6 (2014), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

27 How to exploit the subspace information? Example 3 This can be used to cluster the user based on their signal subspace, and is suitable for HDA implementations 3 (M 1)d 4d Digital Base-band Processing ADC (uplink) DAC (downlink) M M Analog RF-Chain 3d 2d d 0 Analog Beamforming Subspace Information Digital Beamforming Instantaneous Channel Information 3 Ansuman Adhikary et al. Joint spatial division and multiplexing for mm-wave channels. In: IEEE J. on Sel. Areas on Commun. (JSAC) 32.6 (2014), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

28 Algorithm Design Signal Model Process {h i } ν i=1 with a local geometry given by C h = p i=1 σ2 i a(θ i)a(θ i ) H, and unknown time-variation M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

29 Algorithm Design Signal Model Process {h i } ν i=1 with a local geometry given by C h = p i=1 σ2 i a(θ i)a(θ i ) H, and unknown time-variation Noisy observations y i = h i + n i M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

30 Algorithm Design Signal Model Process {h i } ν i=1 with a local geometry given by C h = p i=1 σ2 i a(θ i)a(θ i ) H, and unknown time-variation Noisy observations y i = h i + n i Available data size ν M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

31 Algorithm Design Signal Model Process {h i } ν i=1 with a local geometry given by C h = p i=1 σ2 i a(θ i)a(θ i ) H, and unknown time-variation Noisy observations y i = h i + n i Available data size ν Input Data Low-dim sketches x i = By i with B m M typically antenna selection M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

32 Algorithm Design Signal Model Process {h i } ν i=1 with a local geometry given by C h = p i=1 σ2 i a(θ i)a(θ i ) H, and unknown time-variation Noisy observations y i = h i + n i Available data size ν Input Data Low-dim sketches x i = By i with B m M typically antenna selection Objective Design a robust algorithm for estimating the signal subspace with ν noisy sketches, and by exploiting spatial sparsity ɛ-efficiency criterion ν i=1 P U(h i ) 2 (1 ɛ) ν i=1 h i 2 for the q-dim signal subspace given by U M q such that U H U = I q q M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

33 Multiple Measurement Vector (MMV) Observations H = [h 1, h 2,..., h ν] = [a(θ 1), a(θ 2)... a(θ G )]... := A M G Γ G ν M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

34 Multiple Measurement Vector (MMV) Observations H = [h 1, h 2,..., h ν] = [a(θ 1), a(θ 2)... a(θ G )]... := A M G Γ G ν (M 1)d 4d 3d 2d d 0 θ i. User Only p out of G grid elements are active M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

35 Atomic-norm Denoising for MMV 4, 5, 6 Atomic-norm can be used to estimate the channel vectors Atomic-norm Regularizer for MMV Underlying dictionary D = {a(θ)γ H : θ [ θ max, θ max ], γ C ν } 4 Venkat Chandrasekaran et al. The convex geometry of linear inverse problems. In: Foundations of Computational mathematics 12.6 (2012), pp Badri Narayan Bhaskar, Gongguo Tang, and Benjamin Recht. Atomic norm denoising with applications to line spectral estimation. In: Signal Processing, IEEE Transactions on (2013), pp Yuanxin Li and Yuejie Chi. Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors. In: arxiv preprint arxiv: (2014). M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

36 Atomic-norm Denoising for MMV 4, 5, 6 Atomic-norm can be used to estimate the channel vectors Atomic-norm Regularizer for MMV Underlying dictionary D = {a(θ)γ H : θ [ θ max, θ max ], γ C ν } Spatial-sparsity promoting, channel-variation ignoring regularizer for the channel vectors H M ν : { } H D = inf λi : λ i > 0, θ i, γ i s.t. λi a(θ i )γ H i = H 4 Venkat Chandrasekaran et al. The convex geometry of linear inverse problems. In: Foundations of Computational mathematics 12.6 (2012), pp Badri Narayan Bhaskar, Gongguo Tang, and Benjamin Recht. Atomic norm denoising with applications to line spectral estimation. In: Signal Processing, IEEE Transactions on (2013), pp Yuanxin Li and Yuejie Chi. Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors. In: arxiv preprint arxiv: (2014). M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

37 Atomic-norm Denoising for MMV Atomic-norm can be used to estimate the channel vectors Atomic-norm Regularizer for MMV Underlying dictionary D = {a(θ)γ H : θ [ θ max, θ max ], γ C ν } Spatial-sparsity promoting, channel-variation ignoring regularizer for the channel vectors H M ν : { } H D = inf λi : λ i > 0, θ i, γ i s.t. λi a(θ i )γ H i = H Atomic-norm Denoising for MMV We have the matrix of sketches X = [x 1, x 2,..., x ν ], where x i = B(h i + n i ) M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

38 Atomic-norm Denoising for MMV Atomic-norm can be used to estimate the channel vectors Atomic-norm Regularizer for MMV Underlying dictionary D = {a(θ)γ H : θ [ θ max, θ max ], γ C ν } Spatial-sparsity promoting, channel-variation ignoring regularizer for the channel vectors H M ν : { } H D = inf λi : λ i > 0, θ i, γ i s.t. λi a(θ i )γ H i = H Atomic-norm Denoising for MMV We have the matrix of sketches X = [x 1, x 2,..., x ν ], where x i = B(h i + n i ) The channel vectors are estimated via Ĥ = arg min H D s.t. X BH δ H M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

39 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

40 Spatio-Temporal Correlations and Sparse Scattering First Extreme: Fast-varying Channel Vectors Although channel vector is randomly varying with time its underlying subspace remains invariant h 1 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

41 Spatio-Temporal Correlations and Sparse Scattering First Extreme: Fast-varying Channel Vectors Although channel vector is randomly varying with time its underlying subspace remains invariant h 2 h 1 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

42 Spatio-Temporal Correlations and Sparse Scattering First Extreme: Fast-varying Channel Vectors Although channel vector is randomly varying with time its underlying subspace remains invariant h 2 h 1 h 3 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

43 Spatio-Temporal Correlations and Sparse Scattering First Extreme: Fast-varying Channel Vectors Although channel vector is randomly varying with time its underlying subspace remains invariant This subspace information depends on C h = p i=1 σ2 i a(θ i)a(θ i ) H, which encodes the local geometry of the user given by {(σi 2, θ i)} p i=1 h 2 h 1 h 3 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

44 Spatio-Temporal Correlations and Sparse Scattering Second Extreme: Slowly-varying Channel Vectors Now consider C h = p i=1 σ2 i a(θ i)a(θ i ) H, and suppose h i is slowly varying M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

45 Spatio-Temporal Correlations and Sparse Scattering Second Extreme: Slowly-varying Channel Vectors Now consider C h = p i=1 σ2 i a(θ i)a(θ i ) H, and suppose h i is slowly varying In the extreme case, we have h i = h M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

46 Spatio-Temporal Correlations and Sparse Scattering Second Extreme: Slowly-varying Channel Vectors Now consider C h = p i=1 σ2 i a(θ i)a(θ i ) H, and suppose h i is slowly varying In the extreme case, we have h i = h h 1 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

47 Spatio-Temporal Correlations and Sparse Scattering Second Extreme: Slowly-varying Channel Vectors Now consider C h = p i=1 σ2 i a(θ i)a(θ i ) H, and suppose h i is slowly varying In the extreme case, we have h i = h h 2 h 1 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

48 Spatio-Temporal Correlations and Sparse Scattering Second Extreme: Slowly-varying Channel Vectors Now consider C h = p i=1 σ2 i a(θ i)a(θ i ) H, and suppose h i is slowly varying In the extreme case, we have h i = h h 2 h 1 h 3 M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

49 Spatio-Temporal Correlations and Sparse Scattering Observations For the process {h i } t i=1 with covariance matrix C h = p i=1 σ2 i a(θ i)a(θ i ) H, as far as the local geometry remains invariant: M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

50 Spatio-Temporal Correlations and Sparse Scattering Observations For the process {h i } t i=1 with covariance matrix C h = p i=1 σ2 i a(θ i)a(θ i ) H, as far as the local geometry remains invariant: In the fast-varying regime: the process lies on a p-dim subspace M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

51 Spatio-Temporal Correlations and Sparse Scattering Observations For the process {h i } t i=1 with covariance matrix C h = p i=1 σ2 i a(θ i)a(θ i ) H, as far as the local geometry remains invariant: In the fast-varying regime: the process lies on a p-dim subspace In the slowly-varying regime: the process lies on a 1-dim subspace M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

52 Spatio-Temporal Correlations and Sparse Scattering Observations For the process {h i } t i=1 with covariance matrix C h = p i=1 σ2 i a(θ i)a(θ i ) H, as far as the local geometry remains invariant: In the fast-varying regime: the process lies on a p-dim subspace In the slowly-varying regime: the process lies on a 1-dim subspace Depending on the channel dynamics, the effective dimension q is between 1 and p M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

53 Spatio-Temporal Correlations and Sparse Scattering Observations For the process {h i } t i=1 with covariance matrix C h = p i=1 σ2 i a(θ i)a(θ i ) H, as far as the local geometry remains invariant: In the fast-varying regime: the process lies on a p-dim subspace In the slowly-varying regime: the process lies on a 1-dim subspace Depending on the channel dynamics, the effective dimension q is between 1 and p There is a nice sparse subspace structure that can be exploited! M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

54 Spatio-Temporal Correlations and Sparse Scattering Observations Fast-varying Channel Γ G ν = Slowly-varying Channel Γ G ν = M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

55 Spatio-Temporal Correlations and Sparse Scattering Observations Fast-varying Channel Γ G ν = Slowly-varying Channel Γ G ν = Open Problem I What is a good MMV algorithm, which is indifferent towards the temporal behavior? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

56 Notes We identified an underlying signal subspace sparsity that can be exploited on top of one-shot sparsity to boost the system performance M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

57 Notes We identified an underlying signal subspace sparsity that can be exploited on top of one-shot sparsity to boost the system performance Subspace estimation can be done with ν 50 number of samples for snr 0 10 db M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

58 Notes We identified an underlying signal subspace sparsity that can be exploited on top of one-shot sparsity to boost the system performance Subspace estimation can be done with ν 50 number of samples for snr 0 10 db Low-complexity algorithms exist that can exploit the spatial sparsity, are robust to channel dynamics, and extract the signal subspace quite efficiently M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

59 Outline 1 Overview 2 One-Shot Channel Vector Estimation 3 Subspace Estimation: Exploiting Spatial Sparsity 4 Exploiting Spatio-Temporal Sparsity 5 Instances of Other Interesting Problems M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

60 Antenna Configuration in Massive MIMO Figure: Some possible configurations of a Massive MIMO BS 7 7 Erik G Larsson et al. Massive MIMO for next generation wireless systems. In: IIEEE Communications Magazine 52.2 (2014), pp M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

61 Antenna Configuration in Massive MIMO Question Q1: What is a suitable antenna arrangement? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

62 Antenna Configuration in Massive MIMO Two Examples A Linear Array: A Toeplitz covariance matrix = [a( κ)] n = e jκy dy (n 1) = C aa H : M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

63 Antenna Configuration in Massive MIMO Two Examples A Rectangular Array: A Block-Toeplitz covariance matrix = [a( κ)] m,n j(κy dy (n 1)+κx dx (m 1)) = e = C aa H : M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

64 Bias-Variance Trade-off More Structure Efficient channel estimation algorithms Less variance! More bias! Less Structure More DoF Less bias! More variance! M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

65 Bias-Variance Trade-off More Structure Efficient channel estimation algorithms Less variance! More bias! Less Structure More DoF Less bias! More variance! Open Problem II What is the the best configuration? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

66 FDD Channel Feedback Problem The user feeds back the downlink channel to BS (M 1)d 4d 3d 2d d User M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

67 FDD Channel Feedback Problem M 1 feedback is too time-consuming Question Is there a way to estimate the downlink channel from the uplink channel? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

68 FDD Channel Feedback Problem M 1 feedback is too time-consuming Question Is there a way to estimate the downlink channel from the uplink channel? General channel model (uplink) a f1 (θ) = h up = 1 p i=1 e j 2π c f1d sin(θ). e j 2π c (M 1)f1d sin(θ) w i a f1 (θ i ) + ρ(dθ)a f1 (θ) + n, w i CN (0, σi 2 ), n CN (0, σni 2 M ) M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

69 FDD Channel Feedback Problem M 1 feedback is too time-consuming Question Is there a way to estimate the downlink channel from the uplink channel? General channel model (uplink) a f1 (θ) = h up = 1 p i=1 e j 2π c f1d sin(θ). e j 2π c (M 1)f1d sin(θ) w i a f1 (θ i ) + h : a parametric stochastic process over frequency Parameters: {w i, θ i } p i=1 Good news: M p ρ(dθ)a f1 (θ) + n, w i CN (0, σi 2 ), n CN (0, σni 2 M ) M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

70 A Possible Approach: Wold s Decomposition Theorem Wold s Decomposition Theorem Any zero-mean WSS process {x f ; f Z} can be uniquely decomposed as x f = b i ε f i + d f i=0 where b 0 = 1 and i=1 b i 2 < ε f is a white noise process. {d f ; f Z} is a deterministic process. E{d f ε f } = 0 f, f Definition A stationary process {d f : f Z} is called deterministic if its current value (d f ) can be predicted using the entire past (d f 1, d f 2,...). M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

71 FDD Channel Feedback Problem h up = p w i a f1 (θ i ) i=1 }{{} deterministic process + ρ(dθ) a f1 (θ) + n }{{} innovation process M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

72 FDD Channel Feedback Problem h up = p w i a f1 (θ i ) i=1 }{{} deterministic process + ρ(dθ) a f1 (θ) + n }{{} innovation process The deterministic process may contain a substantial amount of the energy. M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

73 FDD Channel Feedback Problem h up = p w i a f1 (θ i ) i=1 }{{} deterministic process + ρ(dθ) a f1 (θ) + n }{{} innovation process The deterministic process may contain a substantial amount of the energy. Open Problem III Is there a way to extract the deterministic component? M. Barzegar (TU-Berlin) Massive MIMO II CoSIP / 36

74 Questions?