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Degrees of Freedom of Time Correlated MISO Broadcast Channel with Delayed CSIT

Sheng Yang, Mari Kobayashi, David Gesbert, Xinping Yi

arXiv:1203.2550v2cs.IT

TL;DR

The paper asks how to characterize degrees of freedom in a two-user time-correlated MISO broadcast channel with imperfect current and delayed CSIT. It derives the optimal region and proposes a scheme that quantizes and multicasts overheard interference while sending private messages, smoothly connecting MAT and zero-forcing strategies.

  • Problem

    The paper studies the optimal DoF region when the transmitter has delayed CSIT together with imperfect current CSIT in a time-correlated two-user MISO broadcast channel.

  • Method

    The scheme combines ZF spatial precoding based on imperfect current CSIT with MAT space-time alignment based on perfect past CSIT, digitally quantizing and multicasting overheard interference.

  • Results

    The proposed scheme achieves the converse’s upper bound for symmetric DoF and characterizes the optimal DoF region, including extensions with common messages and imperfect delayed CSIT.

  • Takeaways & Limitations

    The strategy interpolates between MAT when current-CSIT quality is poor and standard linear precoding when current-CSIT prediction becomes ideal.

Abstract

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We consider the time correlated multiple-input single-output (MISO) broadcast channel where the transmitter has imperfect knowledge on the current channel state, in addition to delayed channel state information. By representing the quality of the current channel state information as P^-α for the signal-to-noise ratio P and some constant α \geq 0, we characterize the optimal degree of freedom region for this more general two-user MISO broadcast correlated channel. The essential ingredients of the proposed scheme lie in the quantization and multicasting of the overheard interferences, while broadcasting new private messages. Our proposed scheme smoothly bridges between the scheme recently proposed by Maddah-Ali and Tse with no current state information and a simple zero-forcing beamforming with perfect current state information.

I. INTRODUCTION

The paper studies a two-user MISO broadcast channel with perfect delayed and imperfect current CSIT, seeking the optimal DoF when temporal correlation creates intermediate prediction quality. It establishes matching bounds and a scheme combining current-CSI precoding with delayed-CSI interference alignment and quantization.

  • I. INTRODUCTION: The model considers a transmitter with m ≥2 antennas sending private messages to two single-antenna receivers under channel uncertainty.The transmitter has delayed channel states and an estimate of the current state, while receivers know the channel states perfectly.
  • I. INTRODUCTION: For intermediate temporal correlation, the paper asks whether delayed CSIT and imperfect current CSIT can be exploited jointly for better DoF.The question concerns the range 0 < α < 1, between very fast and quasi-static channel behavior.
  • I. INTRODUCTION: The paper establishes an outer DoF-region bound using a genie-aided model and the extremal inequality.The genie provides one receiver’s received signal to the other, producing a degraded broadcast channel for the converse.
  • I. INTRODUCTION: The proposed scheme combines ZF spatial precoding from imperfect current CSIT with MAT space-time alignment from perfect past CSIT.It digitally transmits overheard interference instead of using the analog transmission initially considered in MAT.
  • I. INTRODUCTION: Current-CSI precoding reduces overheard-interference power, saving transmission resources through compression or quantization.It also permits two private messages to be transmitted in parallel with the multicast interference message.
  • I. INTRODUCTION: The scheme achieves the converse’s upper bound for symmetric DoF and extends the analysis to common messages and imperfect delayed CSIT.For other corner points, rate-splitting, spatial precoding, and superposition coding are used; exact achievable rate regions are also provided in the appendix.

III. CONVERSE

The converse derives the DoF outer bound using a genie-aided degraded broadcast channel, an extremal inequality, and the channel uncertainty's isotropic property. Symmetric reasoning completes the converse for the region shown in Fig. 1.

  • Converse tools: A genie-aided model gives receiver 1 user 2’s received signal, creating a degraded broadcast channel for bounding the rates.The same reasoning applies with the receiver roles swapped.
  • Converse tools: The extremal inequality bounds a weighted difference of differential entropies through a Gaussian distribution under covariance constraints.The proof uses Gaussian entropy maximization and covariance ordering to obtain a closed-form upper bound.
  • Converse tools: The channel’s isotropic uncertainty is used at the end to bound the expectation of the logarithmic term and tighten the pre-log factor.This yields the desired DoF scaling after division by log P and taking P →∞.
  • Converse result: The resulting outer bounds include the single-user constraints and the weighted sum-rate constraints defining the DoF region.The converse is completed by applying the genie argument symmetrically to both receivers.

IV. ACHIEVABILITY

Achievability reduces to showing the corner points of the DoF region, with the proof simplified to two transmit antennas because the region is unchanged for m ≥2.

  • IV. ACHIEVABILITY: The extreme points (1, 0) and (0, 1) are achieved by serving only one receiver.The remaining proof targets (1, α), (α, 1), and the symmetric corner point.
  • IV. ACHIEVABILITY: The DoF region does not depend on the number of transmit antennas m for m ≥2, so achievability is proved assuming m = 2.The exact achievable rate region is given in the appendix.

A. Achieving (1, α) and (α, 1)

Rate-splitting achieves the asymmetric corner points by sending a common message alongside spatially precoded private messages using only current CSIT. This gives a sum DoF of 1+α without delayed CSIT.

  • A. Achieving (1, α) and (α, 1): A superposition scheme sends one common signal and two private signals, with private power Pp ∼P^α and common power Pc ∼P.Each receiver decodes the common message first, then its own private message.
  • A. Achieving (1, α) and (α, 1): The common message provides DoF 1−α, while each private message provides DoF α after common-message removal.The corresponding SINRs scale as P^(1−α) for the common signal and P^α for private signals.
  • A. Achieving (1, α) and (α, 1): Splitting one user’s message into common and private parts achieves d1 ≤1 and d2 ≤α simultaneously, with the roles reversed for (α, 1).The same construction applied to user 2 achieves the opposite corner point.
  • A. Achieving (1, α) and (α, 1): Rate-splitting achieves (1, α) and (α, 1) using only current CSIT, without delayed CSIT.The common message is intended for one receiver but decodable by both receivers.

B. Achieving the symmetric corner point

The symmetric-corner scheme combines a MAT-style three-slot alignment with imperfect-current-CSIT precoding. It reduces overheard interference, digitizes it, and multicasts it while transmitting new private messages.

  • B. Achieving the symmetric corner point: MAT alignment places each user’s interference in a one-dimensional subspace while preserving a two-dimensional useful-signal subspace.Each user therefore obtains two useful observations over the three-slot scheme.
  • B. Achieving the symmetric corner point: The proposed scheme optimally exploits both perfect delayed CSIT and imperfect current CSIT by building on a variant of MAT alignment.The construction targets the symmetric corner point, which is strictly larger than 1+α for α < 1.
  • B. Achieving the symmetric corner point: Spatial precoding and power allocation broadcast 1+(1−α)=2−α streams in the first slot instead of two full-power streams.The estimated-channel direction uses power P^(1−α), while the orthogonal direction uses power proportional to P.
  • B. Achieving the symmetric corner point: The overheard interferences η1 and η2 are digitized using approximately 2(1−α) log P bits because precoding reduces their power to O(P^(1−α)).Quantization replaces the analog delivery used in the original MAT scheme.
  • B. Achieving the symmetric corner point: The digitized interferences are broadcast as a common message alongside two new private messages of α log P bits each in the second and third slots.This superposition simultaneously resolves overheard interference and delivers new private information.

Digitizing the overheard interferences

The scheme quantizes overheard interference to the noise level, multicasts the digitized information, and sends new private messages simultaneously. It combines MAT alignment with imperfect-current-CSIT precoding to improve DoF in intermediate α regimes.

  • Digitizing the overheard interferences: Quantizing overheard interference replaces analog transmission with digital multicast when interference power no longer matches available transmit power.The quantization rate depends on interference power, which is determined by current-CSIT quality.
  • Digitizing the overheard interferences: Setting distortion to the noise level avoids DoF loss from quantization.The scheme uses D1 = D2 = 1 and multicasts the combined quantization index to both users.
  • Multicasting digitized interferences and broadcasting new private messages: The common-message DoF is dc = 1 −α while digitized interferences and fresh private messages are transmitted together.The digitized interferences require approximately 2(1−α) log P bits and are broadcast in two slots.
  • Achievability: The proposed scheme achieves the converse upper bound for symmetric DoF and completes achievability of the whole region.The construction decodes MAT-aligned messages together with fresh private messages over three slots.
  • Comparison with existing schemes: Strictly larger DoF than max{2/3, α} is obtained by jointly exploiting imperfect current CSIT and perfect delayed CSIT for α ∈ (0, 1).At α near 0, MAT is optimal; at α ≥ 1, delayed CSIT is unnecessary and ZF is asymptotically optimal.
  • Numerical examples: For α = 0.5, the reported ergodic sum-rate comparison includes the proposed scheme, rate-splitting, TDMA, ZF, and MAT alignment.The cited result passage reports values 3, 3/2, and 5/3 as expected from the DoF results.

V. DISCUSSIONS

The paper extends the DoF characterization to common messages and examines limited-feedback delayed CSIT. Imperfect delayed CSIT reduces the effective prediction quality and can make MAT alignment unhelpful below a feedback-accuracy threshold.

  • A. DoF with common message: The optimal common-message DoF region is characterized as a polyhedron whose vertices determine the entire region by time sharing.The region is expressed in terms of common and private-message DoF variables.
  • A. DoF with common message: The mixed vertex (1 −α, α, α) is achievable with the proposed scheme.Time sharing among the listed vertices achieves the complete region.
  • B. Imperfect delayed CSI: Limited feedback: Limited feedback is modeled by quantizing delayed channel states with precision exponent β.The transmitter uses quantized state predictions and quantized past states for precoding and MAT alignment.
  • B. Imperfect delayed CSI: Limited feedback: When β falls below α, residual interference becomes the dominating source of interference and the corner points become (1, β) and (β, 1).As β decreases to α, the symmetric DoF drops while the corner points remain unchanged until this transition.
  • B. Imperfect delayed CSI: Limited feedback: The aggregated prediction-error exponent is α′ = min{α, β}.It combines channel variation, characterized by α, with quantization error, characterized by β.
  • B. Imperfect delayed CSI: Limited feedback: Optimal schemes for the imperfect delayed-CSIT case remain an open problem.The paper states that it is unclear whether the naive extension is optimal.

C. Bandwidth-limited Doppler process

A bandwidth-limited Doppler model provides a practical interpretation of current-CSIT quality through channel prediction, estimation, and feedback. Under the model, the paper relates α to Doppler bandwidth and reports agreement between non-ergodic and ergodic DoF.

  • C. Bandwidth-limited Doppler process: The fading process follows a Doppler model with channel coefficients band-limited to [−F, F].The normalized Doppler parameter is expressed using mobile speed, carrier frequency, slot duration, and light speed.
  • C. Bandwidth-limited Doppler process: Receivers estimate the current channel using pilot-based downlink training and noisy observations.The current channel is decomposed into an estimate and an estimation error.
  • C. Bandwidth-limited Doppler process: The transmitter and the other receiver obtain next-slot predictions from noisy feedback observations.The prediction is based on the sequence of observations available up to the preceding slot.
  • C. Bandwidth-limited Doppler process: For the modeled channel, the proposed analysis applies with α = 1 − 2F and β = 1.This relation is stated for the prediction-error scaling in the imperfect-delayed-CSIT setting.
  • C. Bandwidth-limited Doppler process: The non-ergodic DoF coincides with the ergodic DoF.The paper obtains this by redefining non-ergodic DoF in the same manner as multiplexing gain.

A. Proof of Lemma 1

The proof of Lemma 1 transforms the relevant random variables using isotropic and unitary-invariance properties, then applies entropy inequalities and phase averaging.

  • A. Proof of Lemma 1: A unitary eigenvector transformation preserves the isotropic distribution needed for the proof.The transformed vector has the same distribution as the original error vector, and its scalar components are rotation-invariant.
  • A. Proof of Lemma 1: Lemma 3 averages over a uniformly distributed phase to establish the required rotational-invariance identity.The proof reduces the variables to non-negative real magnitudes before direct phase integration.
  • A. Proof of Lemma 1: The final derivation combines the phase-averaging lemma with the preceding transformed-variable bounds.The proof then completes the target inequality for Lemma 1.
  • A. Proof of Lemma 1: The proof uses concavity of the logarithm and Jensen’s inequality to bound an intermediate expectation.These steps appear in the derivation of the stated inequalities.

B. Proof of Lemma 2

Lemma 2 constructs an achievable rate region using a common message alongside two privately precoded messages, with decoding based on treating the other signal layers as noise.

  • Coding scheme: The scheme transmits one common message and two private messages using time-varying linear precoders based only on estimated current channels.The transmitted signal is xt = Ωt ˜xc,t + Ξt ˜up,t + Γt˜vp,t.
  • Achievable region: The achievable rate region is the union of rate triples (Rc, Rp1, Rp2) satisfying Proposition 1.The region is expressed through rate constraints involving common-message and private-message components.
  • Covariance design: The covariance matrices are constrained to be positive semidefinite with total trace at most P and depend only on the channel estimate.The policy is Q( ˆS) ≜ {Qc, Qp1, Qp2 ⪰ 0 : tr(Qc + Qp1 + Qp2) ≤ P}.
  • Decoding: The common message is decoded first by treating private signals as noise, after which each private message is decoded after removing the common signal.Private interference is treated as noise during the second decoding step.
  • Covariance design: Choosing Qc ∼ PI, Qp1 ∼ P^αΨˆg⊥, and Qp2 ∼ P^αΨˆh⊥ yields the stated Lemma 2 result.The private covariance choices align the private signals with estimated channel null spaces.

C. Achievable rate region of the sum-DoF optimal scheme

The sum-DoF-optimal scheme uses two phases: it sends private MIMO messages, quantizes the resulting interferences, and then multicasts those quantized interferences alongside new private messages.

  • Two-phase scheme: The scheme divides transmission into Phase 1 and Phase 2, with durations n1 and n2 channel uses.Phase 2 reuses the same codebooks and precoders with codeword length n2.
  • Phase 1: In Phase 1, two private codewords are transmitted using time-varying precoders that depend on the estimated current state.The signal is xt = Θt ˜ut + Φt˜vt.
  • Phase 1: At the end of Phase 1, the transmitter quantizes the normalized overheard interferences using source codebooks.The quantized interferences are represented by n1(Rη1 + Rη2) bits and form the common message Wc.
  • Phase 2: In Phase 2, the quantized interferences are multicast as a common message while two new private messages are transmitted simultaneously.The transmitted signal is xt = Ωt ˜xc,t + Ξt ˜up,t + Γt˜vp,t.
  • Decoding: Each user first recovers the pair consisting of the quantized-interference message and its private message, then decodes its original MIMO message.The first step requires the triple (Rc, Rp1, Rp2) to lie in Proposition 1's region.
  • Achievable region: The resulting achievable rate region is the union of rate pairs over channel-estimate-dependent policies and positive semidefinite covariance matrices.Proposition 2 defines compression and MIMO rates before combining them into the user rates.
  • Power allocation: Private-signal power scales as P^α while unintended received power remains at P^0, preventing DoF loss for the unintended receiver.The chosen distortions keep reconstruction errors at the noise level.
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