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Design of Massive-MIMO-NOMA with Limited Feedback

Zhiguo Ding, H. Vincent Poor

arXiv:1511.05583v1cs.IT

TL;DR

Massive-MIMO-NOMA requires a practical alternative to perfect transmitter CSI, whose acquisition can be difficult and bandwidth-intensive. The paper proposes a limited-feedback scheme using spatial channel structure to decompose the system into separated SISO-NOMA channels, then develops analytical performance results for perfect ordering and one-bit feedback.

  • Problem

    Existing MIMO-NOMA work often assumes perfect transmitter CSI, which is difficult to realize and can consume excessive bandwidth.

  • Method

    The proposed limited-feedback downlink scheme exploits users’ spatial channel structure to decompose massive-MIMO-NOMA into separated SISO-NOMA channels.

  • Results

    Analytical results evaluate the proposed scheme for perfect user ordering and one-bit feedback, including exact outage probabilities and achievable diversity order.

  • Takeaways & Limitations

    The scheme significantly simplifies massive-MIMO-NOMA design while avoiding feedback of users’ channel matrices to the base station.

Abstract

from arXiv · show

In this letter, a low-feedback non-orthogonal multiple access (NOMA) scheme using massive multiple-input multiple-output (MIMO) transmission is proposed. In particular, the proposed scheme can decompose a massive-MIMO-NOMA system into multiple separated single-input single-output NOMA channels, and analytical results are developed to evaluate the performance of the proposed scheme for two scenarios, with perfect user ordering and with one-bit feedback, respectively.

I. INTRODUCTION

The paper addresses practical CSI-feedback challenges in massive-MIMO-NOMA by proposing a limited-feedback downlink protocol based on users’ spatial channel structure. The protocol decomposes the system into separated SISO-NOMA channels and analyzes perfect ordering and one-bit-feedback scenarios.

  • Existing MIMO-NOMA designs commonly assume perfect transmitter CSI, which is difficult to realize and can consume excessive bandwidth.
  • The proposed downlink protocol does not require users to feed their channel matrices back to the base station.
  • The method exploits users’ spatial channel clustering and applies MIMO-NOMA concepts to decompose massive-MIMO-NOMA into separated SISO-NOMA channels.
  • Analytical performance results are developed for perfect user ordering and one-bit feedback, including exact outage probabilities and achievable diversity order from high-SNR approximations.
  • The system model uses K spatial clusters whose users share a spatial correlation matrix, while the base station has M antennas and each user has N antennas.

A. Inter-Cluster Interference Cancellation

The inter-cluster interference cancellation design uses cluster-specific processing so that the received system model removes interference between clusters.

  • The base station uses a precoding matrix P_k to avoid inter-cluster interference for cluster k.
  • The effective transmit dimension is defined as M̃ = M − r(K − 1) in the proposed construction.
  • Using P_k, the resulting system model has the inter-cluster interference removed.

B. The Application of the NOMA Approach

The NOMA stage applies within each interference-separated cluster, grouping users and assigning shared precoding vectors with different power coefficients to accommodate multiple users.

  • The massive-MIMO system is decomposed into K separated clusters, each with M̃ effective transmit antennas.
  • The resulting decomposition creates two difficulties: accommodating L users when L > M̃ and lacking instantaneous transmitter CSI.
  • Each cluster’s L users are divided into Q groups of size P, satisfying L = PQ.
  • Within each group, users share the same precoding vector but receive different power allocation coefficients.
  • Without transmitter CSI, the scheme selects the precoding vectors using the available design specified in the paper.

C. Receiver Detection

Receiver processing further transforms the massive-MIMO-NOMA model into separated SISO-NOMA channels without requiring the fast-fading channel realization at the base station.

  • The received signal for each user is formulated after applying the cluster and group transmission design.
  • Under N ≥ M̃, zero-forcing processing is applied at the receiver.
  • The receiver model accounts for the covariance of the noise vector in the detected signal formulation.
  • The resulting massive-MIMO-NOMA system decomposes into separated SISO-NOMA channels without knowing G_k,q,p at the base station.

A. With Perfect Knowledge of User Ordering

With perfect knowledge of user ordering, the proposed scheme derives the effective-channel distribution and outage performance for the ordered users. The resulting diversity order for user U_k,q,p is p(N−M̃+1).

  • A. With Perfect Knowledge of User Ordering: The base station assumes users are ordered and derives the SINR needed to evaluate each ordered user's outage probability.The analysis characterizes the effective channel gain through Wishart and inverse-Wishart distributions.
  • A. With Perfect Knowledge of User Ordering: The proposed scheme provides an outage-performance expression for user U_k,q,p under perfect user ordering.The theorem applies to the p-th user in the q-th group of the k-th cluster.
  • A. With Perfect Knowledge of User Ordering: The theorem states that user U_k,q,p achieves diversity order p(N−M̃+1).The result is stated for the proposed massive-MIMO-NOMA scheme with the targeted rates and incomplete-gamma expression specified in the theorem.
  • A. With Perfect Knowledge of User Ordering: The density of the p-th ordered effective channel is obtained from the marginal distribution of the inverse-Wishart matrix diagonal element.The order-statistics density uses the CDF powers (F(x))^(P−p) and (1−F(x))^(p−1).
  • A. With Perfect Knowledge of User Ordering: The ordered user's outage probability depends on the minimum decoding threshold among messages that the user must decode.For user p, the relevant threshold is ξ*_k,q,p = min{ξ_k,q,n, 1 ≤ n ≤ p}.

2) Calculating the outage probability:

The outage probability is computed from the event that user U_k,q,p cannot decode at least one required message, using the effective-channel density and its CDF.

  • 2) Calculating the outage probability:: Outage occurs when user U_k,q,p cannot decode message s_k,q,n for at least one n satisfying 1 ≤ n ≤ p.The relevant threshold is ξ*_k,q,p = min{ξ_k,q,n, 1 ≤ n ≤ p}.

3) Obtaining the diversity order:

The diversity-order analysis uses a high-SNR approximation of the outage probability derived from the fixed-rate CDF approximation.

  • 3) Obtaining the diversity order:: For fixed R_k,q,n, the CDF approximation is used to evaluate the outage probability at high SNR.The proof states that the resulting approximations follow from equation (20).

B. With One-Bit Feedback

With one-bit feedback, users are split by whether their effective channel gains exceed a threshold, and users within each subgroup are randomly ordered. For two users per group, one-bit feedback achieves the same diversity order as perfect ordering.

  • B. With One-Bit Feedback: The scheme does not require global CSI about H_k,q,p at the base station, but the base station still needs to order scalar effective channel gains.This motivates replacing full ordering information with one-bit feedback.
  • B. With One-Bit Feedback: Each user compares its effective channel gain with threshold τ and feeds back one bit indicating whether the gain is below or above τ.Users are divided into subgroups S1 and S2 according to this feedback.
  • B. With One-Bit Feedback: When a subgroup contains multiple users, the base station randomly orders them.The analysis focuses on the special case P = 2 users in each group, with predefined targeted rates and power-allocation coefficients.
  • B. With One-Bit Feedback: The weak-user outage probability is decomposed into the cases where the user belongs to S1 or S2.The derivation evaluates these cases using joint and marginal distributions of the effective channel gains.
  • B. With One-Bit Feedback: The subgroup outage calculations use threshold conditions involving C1, C2, τ̃, and φ_i, together with the joint density 2f(x)f(y).The marginal CDF of the weak channel gain is F_[C1]_q,q(x) = (F(x))^2.
  • B. With One-Bit Feedback: For two users in each group, one-bit feedback achieves the same diversity order as perfect user ordering.The high-SNR analysis gives diversity order 2(N−M̃+1), and the same conclusion holds for other choices of τ and P1.

IV. NUMERICAL STUDIES

Numerical studies show that the proposed massive-MIMO-NOMA scheme improves throughput over OMA, matches analytical predictions, and preserves diversity order with one-bit feedback.

  • The proposed massive-MIMO-NOMA scheme significantly improves system throughput over OMA.
  • Three times greater sum rate than OMA is achieved at ρ = 20dB with N = 2.
  • Analytical results and simulations match perfectly for perfect user ordering.
  • One-bit feedback achieves the same diversity order as perfect user ordering.
  • With one-bit feedback, the strong-CSI user experiences less outage because it may receive a smaller targeted data rate.

V. CONCLUSIONS

The proposed limited-feedback massive-MIMO-NOMA scheme decomposes the system into separated SISO-NOMA channels and develops analytical results for perfect ordering and one-bit feedback.

  • The proposed scheme uses limited feedback for massive-MIMO-NOMA transmission.
  • The massive-MIMO-NOMA system is decomposed into multiple separated SISO-NOMA channels.
  • Analytical results evaluate performance with perfect user ordering and one-bit feedback.
  • The numerical studies include outage sum-rates for three users, analytical accuracy with perfect ordering, and outage performance under different feedback amounts.
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