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CRB-Based Resource Allocation in Multi-User Uplink Transmissions

Xue Zhang, Abla Kammoun, Mohamed-Slim Alouini

Abstract

In this work, we study the design of receivers for uplink multi-user systems, aiming to estimate both the channel and the transmitted symbols. We consider two estimation strategies: (i) a joint estimation approach, where the channel and symbols are estimated simultaneously, and (ii) a sequential estimation approach, where the channel is first estimated and then used for symbol detection. For both strategies, we derive the Cramér-Rao Bound (CRB) for symbol estimation to characterize fundamental performance limits. When efficient receivers achieving the CRB exist, these bounds provide accurate lower bounds on the mutual information. In general, however, such receivers may not be available, and we instead use these same CRB-based metrics as practical proxies for achievable throughput. Leveraging tools from random matrix theory (RMT), we analyze the asymptotic behavior of these lower bounds under various asymptotic regimes for both estimation strategies. This analysis enables the derivation of generic power allocation guidelines that asymptotically maximize the proxy metrics. Simulation results confirm the accuracy of the asymptotic expressions and their effectiveness in guiding resource allocation decisions.

CRB-Based Resource Allocation in Multi-User Uplink Transmissions

Abstract

In this work, we study the design of receivers for uplink multi-user systems, aiming to estimate both the channel and the transmitted symbols. We consider two estimation strategies: (i) a joint estimation approach, where the channel and symbols are estimated simultaneously, and (ii) a sequential estimation approach, where the channel is first estimated and then used for symbol detection. For both strategies, we derive the Cramér-Rao Bound (CRB) for symbol estimation to characterize fundamental performance limits. When efficient receivers achieving the CRB exist, these bounds provide accurate lower bounds on the mutual information. In general, however, such receivers may not be available, and we instead use these same CRB-based metrics as practical proxies for achievable throughput. Leveraging tools from random matrix theory (RMT), we analyze the asymptotic behavior of these lower bounds under various asymptotic regimes for both estimation strategies. This analysis enables the derivation of generic power allocation guidelines that asymptotically maximize the proxy metrics. Simulation results confirm the accuracy of the asymptotic expressions and their effectiveness in guiding resource allocation decisions.
Paper Structure (18 sections, 8 theorems, 110 equations, 7 figures)

This paper contains 18 sections, 8 theorems, 110 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Joint Channel and Symbol Estimation
  4. Symbol CRB Matrix
  5. Mutual Information for Gaussian Symbols
  6. Sequential channel and symbol estimation
  7. Channel estimation using the LMMSE
  8. Symbol CRB matrix
  9. Mutual Information for Gaussian Symbols
  10. Simulation results
  11. Accuracy assessment of the asymptotic expressions $\overline{\mathcal{I}}_{\rm joint}$ and $\overline{\mathcal{I}}_{\rm seq}$
  12. Empirical validation of the derived power allocation schemes
  13. Conclusion
  14. Proof of Theorem \ref{['theorem_complex_CRB']}
  15. Proof of Theorem \ref{['theorem_MI_joint']}
  16. ...and 3 more sections

Key Result

Theorem 1

The CRB for the symbol and the channel parameters is obtained as: where $\pmb{\mathcal{J}}_{ss}={\bf I}_{N-L}\otimes \frac{1}{\sigma_v^2}{\bf G}^{H}{\bf G}$, $\pmb{\mathcal{J}}_{gg}=\frac{1}{\sigma_v^2}{\bf X}^{\ast}\otimes {\bf I}_M$, and Accordingly, the CRB for the symbol matrix $\mathbf{S}_d$ is given by

Figures (7)

  • Figure 1: Receiver strategies for data-symbol recovery: (i) joint channel–symbol estimation and (ii) sequential channel and symbol estimation.
  • Figure 2: $\mathcal{I}_{\mathrm{joint}}/(K(N-L))$ versus $N$ under different SNR levels for strategy (i) with $\alpha=\beta=c=1/2$.
  • Figure 3: $\mathcal{I}_{\mathrm{seq}}/(K(N-L))$ versus $N$ under different SNR levels for strategy (ii) with $\alpha=\beta=c=1/2$.
  • Figure 4: $\mathrm{NLAE}$ versus $N$ under different values of $\alpha$ and $\beta$ for strategy (i) with $c=1/2$ and $\mathrm{SNR}=10$ dB.
  • Figure 5: $\mathrm{NLAE}$ versus $N$ under different values of $\alpha$ and $\beta$ for strategy (ii) with $c=1/2$ and $\mathrm{SNR}=10$ dB.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Theorem 5: Derivation of the BCRB
  • proof
  • Theorem 6: Asymptotic derivation of $\overline{\mathcal{I}}_{\rm seq}$
  • ...and 5 more