Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation L. Cottatellucci laura.cottatellucci@eurecom.fr joint work with R. M¨ uller, and M. Vehkaper¨ a
I. Outline 2 Outline 1. Motivations 2. System Model 3. Subspace Approach 4. Subspace Method in Practical Systems: Eigenvalue Separation 5. Performance Simulations 6. Conclusions L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 3 MIMO Cellular Systems Cooperative approach: • Space division multiple access inside a cell • Channel sharing among cells is spectral efficient but... Data sharing; • ...interference management highly costly Channel state information acquisition; Signalling. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 4 A General System Model y ( m ) = Hx ( m ) + n ( m ) K + = N • Multiuser CDMA; • Multiuser SIMO; • Single/Multiuser MIMO. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 5 Capacity per Received Signal Dimension 1.4 Verdu et Shamai, ’99 E b /N 0 =10 dB 1.2 MMSE Capacity per received signal 1 Optimal Decorrelator 0.8 0.6 0.4 Matched Filter 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 System Load (K/N) • At very low loads all detectors have equal performance. • Matched filter: only knowledge of channel for user of interest needed. • MMSE detector: statistical knowledge of all channel required. At very low load matched filter optimally combats interference without coordination/cooperation L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 6 Massive MIMO Concept • Huge antenna arrays ( R ≫ 1 antennas) at the base stations serving a few users ( T ≪ R users) • Under assumption of perfect channel knowledge and T/R → 0 , beams can be made sharper and sharper and interference vanishes. Interference management without coordination or cooperation! L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 7 Pilot Contamination for TDD Systems Simple scenario • Users send orthogonal pilots within a cell, but the same training sequences are used in adjacent cells. • By channel reciprocity, the channel estimates are useful for both uplink detection and downlink precoding. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 8 Pilot Contamination Simple channel estimation (Marzetta ’10) • Linear channel estimation by decorrelator/matched filter is limited by copilot inter- ference. • Subsequent detection or precoding based on the low quality channel estimates degrade significantly the system spectral efficiency. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 9 Proposed Countermeasures: State of Art • Coordinated scheduling among cells. • Coordinated training sequence assignment. ...but coordination very costly and complex in terms of signaling! L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
II. Motivations 10 A Deeper Look at the Impairment • In the simple Marzetta’s scheme, array gain is utilized for data detection but not for channel estimation. • Linear channel estimation does not exploit the array gain. Guidelines for Countermeasures • General channel estimation that utilizes the array gain. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
III. System Model 11 System Model I L interfering cells R receive antennas R>>T R>>T(L+1) T transmitters L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
III. System Model 12 System Model II Received power I I R>>T R>>T(L+1) Received power P Assumptions • Power control such that in-cell users’ signals are received with equal power P. • Handover to guarantee that P > I. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
III. System Model 13 System Model for Channel Estimation C C T LT C P + = R I Received Signals Noise Pilots + Data Y = HX + W • C : coherence time. • Y : R × C matrix of received signals. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
IV. Subspace Approach 14 Projection Subspace � In absence of noise and interference, Y Y H 1−T/R is a matrix with T positive eigenvalues and R − T zero eigenvalues. T/R P 0 � Let S be the R × T matrix of eigenvectors corresponding to the nonzero eigenvalues: – S spans the signal subspace; – Y ′ = S H Y is the projection of the received signal into the signal space; – We can estimate the equivalent channel in the T dimensional signal subspace S using Y ′ without performance loss. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
IV. Subspace Approach 15 Projection Subspace In the presence of additive Gaussian noise and C sufficiently large � The matrix S consisting of the Y Y H eigenvectors corresponding to the T largest eigenvalues is still a basis of the signal subspace; � By using the projection Y ′ = S H Y , the white noise impairing the observed signal is reduced from Rσ 2 to Tσ 2 • In massive MIMO, since R ≫ T and T/R → 0 the noise is negligible compared to the signal power. – S spans the signal subspace; – Y ′ = S H Y is the projection of the received signal into the signal space; – We can estimate the equivalent channel in the T dimensional signal subspace S using Y ′ without performance loss. Fully blind method to obtain array gain! L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
IV. Subspace Approach 16 Projection Subspace Method � In the presence of additive Gaussian noise and intercell interference – If T/R → 0 and P > I k the signals of interest and the interferences are almost orthogonal. – There will be two disjoint clusters of eigenvalues with the T highest eigenvalues associated to the signal of interest. � The same projection method can be applied also in this case. � Interference power subspace and withe noise become negligible! Pilot contamination is not a fundamental issue in massive MIMO! L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
17 How this method can be extended to practical systems with a finite number of receive antennas and finite coherence time? L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
V. Subspace Method in Practical Systems 18 Eigenvalue Spectrum of Y Y H for Practical Systems If the eigenvalue spectrum of Y Y H consists of disjoint bulks associated to the interference and desired signals, the subspace method can still be applied and suppresses the most of interference and noise also when T/R = α > 0 and C/R = κ < + ∞ . Fundamental to study the eigenvalue spectrum! We approximate a system with finite T, R, C by a system with T, R, C → + ∞ and T/R → α and R/C → κ. L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
V. Subspace Method in Practical Systems 19 Eigenvalue Distribution of Observation Signal Covariance α =1/100, κ = 10/3, r=1/100, t=4/100, T=3, R=300, C=1000, P=0.1, I=0.025, W=1 −3 3 x 10 Interference Signal of Interest 2.5 2 1.5 1 0.5 0 0 20 40 60 80 100 120 140 s (eigenvalue) Solid red line: Asymptotic eigenvalue distribution by random matrix theory L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
V. Subspace Method in Practical Systems 20 Analysis of the Eigenvalue Bulk Gap � Assume worst case with interferers received at the maximum power I < P. � Let β = I/P. � Approximate the eigenvalue distribution finite systems by asymptotic eigenvalue dis- tribution. Conservative condition for a nonzero gap btw interference and signal bulks (1 − β ) 2 ( Lβ 2 + 3( L + 1) β + 1 − 2(1 + β ) √ 3 Lβ ) T ( Lβ 2 − 1)( Lβ 2 + 6( L − 1) β − 1) + (9 L 2 − 2 L + 9) β 2 . C ≤ � Dependent only on the ratio T/C ! � Independent of R ! L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
V. Subspace Method in Practical Systems 21 Separability Region Region of separability for signal and interference subspaces 0.5 L=2 0.45 L=4 L=7 0.4 0.35 0.3 T/C 0.25 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I/P L. Cottatellucci et al., Low Complexity Pilot Decontamination via Blind Signal Subspace Estimation ⃝ Eurecom January 2014 c
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