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Channel Estimation for Hybrid Architecture Based Wideband Millimeter Wave Systems
Kiran Venugopal, Ahmed Alkhateeb, Nuria González Prelcic, Robert W. Heath
TL;DR
The paper addresses wideband frequency-selective mmWave channel estimation for hybrid architectures, extending beyond prior narrowband-focused work. It formulates estimation as sparse recovery in time and frequency domains and proposes explicit domain-specific algorithms. The proposed methods support SC-FDE and OFDM systems, achieve good estimation quality with small training overhead, and benefit from hybrid transceivers.
Problem
Hybrid mmWave channel-estimation methods had largely focused on narrowband channels, while wideband systems require frequency-selective estimation under hybrid hardware constraints.
Method
The paper uses angular- and delay-domain channel sparsity to formulate wideband estimation as sparse recovery and develops time-, frequency-, and combined-domain algorithms.
Results
The proposed algorithms achieve good channel-estimation quality with small training overhead, while multiple RF chains further improve estimation error and overhead.
Takeaways & Limitations
Compressed sensing with hybrid precoding and combining supports explicit wideband channel estimation and can support MIMO operation in mmWave systems.
Abstract
from arXiv · showhide
Hybrid analog and digital precoding allows millimeter wave (mmWave) systems to achieve both array and multiplexing gain. The design of the hybrid precoders and combiners, though, is usually based on knowledge of the channel. Prior work on mmWave channel estimation with hybrid architectures focused on narrowband channels. Since mmWave systems will be wideband with frequency selectivity, it is vital to develop channel estimation solutions for hybrid architectures based wideband mmWave systems. In this paper, we develop a sparse formulation and compressed sensing based solutions for the wideband mmWave channel estimation problem for hybrid architectures. First, we leverage the sparse structure of the frequency selective mmWave channels and formulate the channel estimation problem as a sparse recovery in both time and frequency domains. Then, we propose explicit channel estimation techniques for purely time or frequency domains and for combined time/frequency domains. Our solutions are suitable for both SC-FDE and OFDM systems. Simulation results show that the proposed solutions achieve good channel estimation quality, while requiring small training overhead. Leveraging the hybrid architecture at the transceivers gives further improvement in estimation error performance and achievable rates.
I. INTRODUCTION
The paper addresses the difficulty of estimating frequency-selective wideband mmWave channels under hybrid architectures, extending prior predominantly narrowband work. It formulates estimation as sparse recovery and proposes time-, frequency-, and combined-domain compressed-sensing approaches for SC-FDE and OFDM systems.
- Motivation: Prior mmWave channel-estimation work mostly assumed narrowband channels, despite mmWave systems’ wide bandwidths and frequency selectivity.The paper therefore targets efficient wideband frequency-selective channel estimation.
- Motivation: Hybrid architectures complicate channel estimation because the channel is observed through RF hardware and estimation occurs before beamforming under low SNR.Large mmWave channel matrices and low pre-beamforming SNR further increase the challenge.
- Sparse formulation: The proposed formulation represents wideband mmWave channels sparsely in both angular and delay domains using dictionaries based on array responses and pulse-shaping delays.The dictionary uses transmit and receive steering vectors on angle grids and a raised-cosine response on a delay grid.
- Sparse formulation: The paper formulates channel estimation as sparse recovery in the time domain and frequency domain while incorporating hybrid hardware and transmission-frame structure.This formulation jointly leverages frequency-selective channel sparsity and the assumed data-transmission frame.
- Algorithms: Explicit algorithms are proposed for purely time, purely frequency, and combined time-frequency estimation, with choices suited to different system-level constraints.The approaches support both SC-FDE and OFDM-based frequency-selective hybrid mmWave systems.
- Results: Simulations show good estimation quality with significantly less training than beam training, while multiple RF chains further reduce estimation error and training overhead.The combined time-frequency approach provides the best trade-off between low training overhead and minimum average estimation error.
II. SYSTEM AND CHANNEL MODELS
The paper considers hybrid mmWave MIMO systems with separate time- and frequency-domain estimation settings. Its assumed RF precoder and combiner use fully connected phase shifters.
- System model: The system model presents an SC-FDE hybrid architecture and a wideband mmWave channel model.The time-domain estimator operates on the SC-FDE system, whereas the frequency-domain approach applies to OFDM-based hybrid MIMO systems.
- Hybrid architecture: The RF precoder and combiner are implemented using a fully connected network of phase shifters.
A. System Model
The model is a single-user frequency-selective mmWave MIMO link with hybrid precoding and combining, geometric multipath, and delay- and angle-dependent channel structure.
- System model: The transmitter and receiver use hybrid precoders and combiners with N_RF RF chains, while training uses time-domain precoders and combiners.The transmitter precoder maps N_s data streams through analog and digital stages; the receiver applies an analogous hybrid combiner.
- System model: The channel has N_c delay taps and is modeled as frequency selective between N_t transmit and N_r receive antennas.
- Channel model: The geometric channel consists of N_p paths whose gains are shaped by a raised-cosine pulse response at each path delay.Each path has complex gain α_l and delay τ_l.
- Channel model: Each path is characterized by receive and transmit angles with corresponding array response vectors known to the transmitter and receiver.The estimation algorithm applies to arbitrary antenna-array configurations.
- Channel model: Vectorization expresses the channel through columns of the Khatri-Rao structure, including terms of the form ¯a_T(θ_l) ⊗ a_R(φ_l).The resulting vectorized channel is the unknown estimated by the proposed algorithms.
III. TIME-DOMAIN CHANNEL ESTIMATION VIA COMPRESSED SENSING
The time-domain estimator uses wideband mmWave channel sparsity while accounting for training hardware constraints, frame structure, and beam-pattern constraints.
- Time-domain estimation: The proposed time-domain algorithm performs explicit channel estimation by exploiting sparsity in the wideband mmWave channel.
- Time-domain estimation: The method incorporates hardware constraints on the training frame and on the precoding-combining beam patterns.
A. Sparse Formulation in the Time Domain
The time-domain formulation exploits angular and delay sparsity to represent wideband mmWave channel measurements as a sparse recovery problem under hybrid precoding and combining.
- System model: The model uses block transmission with zero padding of at least Nc −1 samples for a frequency-selective channel with Nc taps.The padding supports the assumed frame structure and prevents inter-frame interference during RF reconfiguration.
- System model: Hybrid transceivers use RF precoders and combiners that can be reconfigured across training frames, with one pair per frame.The phase angles are chosen uniformly at random from a finite set, while different settings per symbol are considered impractical at high symbol rates.
- Sparse representation: The unknown gains and delays, together with the angular dictionaries, determine the explicit multi-tap MIMO channel.The complex gains and delays remain unknown and must be estimated from the measurements.
- Sparse representation: The channel is represented with dictionaries over quantized transmit angles, receive angles, and delays, producing a sparse vector of time-domain channel gains.The resulting vector has dimension GcGrGt and contains Np nonzero entries.
- Recovery formulation: Stacking measurements from M training frames yields a time-domain sparse recovery formulation with a known sensing matrix and dictionary.The receiver can therefore apply sparse recovery using the measured signals and the frame-specific RF processing.
B. AoA/AoD and Channel Gain Estimation in the Time Domain
The time-domain estimator first recovers sparse angular and delay support, then estimates the associated channel gains, with OMP and least squares as the described tools.
- Support recovery: Compressed sensing recovers support corresponding to AoA, AoD, and path delay, so each selected dictionary entry represents a channel path.The corresponding nonzero coefficient represents that path’s gain.
- Grid design: Increasing or decreasing the angular and delay grid sizes provides a way to reduce quantization error in the time-domain dictionary.The sensing matrix is known at the receiver, enabling sparse recovery for AoA and AoD estimation.
- Support recovery: OMP is used to solve the sparse recovery problem, with stopping based either on known sparsity or a residual threshold.In noise, the threshold is set to the noise power, ϵ = E[e∗e].
- Gain estimation: After support estimation, the channel gains are estimated using least squares on the selected support.The paper presents least squares as one approach among several possible gain-estimation methods.
IV. FREQUENCY-DOMAIN CHANNEL ESTIMATION VIA COMPRESSED SENSING
The frequency-domain approach formulates channel estimation separately across subcarriers, adapting the sparse representation to frequency-selective mmWave channels.
- Formulation: The section compares frequency-domain compressed-sensing formulation choices and their associated advantages and disadvantages.
- Formulation: The frequency-domain channel matrix is derived from the geometric channel model and then vectorized for estimation.The vectorized representation parallels the time-domain formulation.
- Formulation: Unlike the time-domain approach, the unknown channel vectors for the K subcarriers can be estimated separately.This per-subcarrier structure is the key distinction identified between the two domains.
A. Sparse Formulation in the Frequency Domain
The frequency-domain estimator converts received training signals into per-subcarrier sparse recovery problems while retaining frequency-flat RF processing and supporting SC-FDE and OFDM.
- Signal processing: For SC-FDE with zero padding, overlapping and sum processing followed by a K-point FFT produces the frequency-domain sparse problem for k = 1, 2, · · · , K.
- System processing: Different baseband precoders and combiners can be used across subcarriers, while the RF precoders and combiners remain frequency flat.
- System processing: The approach processes received signals per subcarrier and therefore works for both SC-FDE and OFDM systems.
- Sparse model: The frequency-domain formulation assumes a sparse angular representation for each subcarrier’s channel vector.The sparse vector has dimension GrGt and Np nonzero entries.
- Measurement model: Measurements from M training frames use different RF precoder–combiner pairs and are stacked to form the sparse formulation for each subcarrier.
B. AoA/AoD and Channel Gain Estimation per Subcarrier
The frequency-domain approach estimates dominant AoAs/AoDs and channel coefficients for each subcarrier using sparse recovery followed by coefficient estimation. Repeating this process across all K subcarriers characterizes the frequency-selective channel, but requires per-subcarrier processing.
- Frequency-domain sparse recovery: The estimated support sets identify specific AoAs and AoDs associated with the non-zero entries of the sparse channel representation.
- Frequency-domain sparse recovery: OMP estimates the support of the sparse vector and therefore the dominant angles of arrival and departure for the kth subcarrier.The stopping threshold is ϵ = E[˘e[k]∗˘e[k]].
- Channel-gain estimation: Least square estimation then recovers the channel coefficients corresponding to the estimated sparse support.
- Channel-gain estimation: Repeating the estimation for all K subcarriers fully characterizes the frequency-selective mmWave channel.
- Computational cost: Although the frequency-domain matrices are smaller than their time-domain counterparts, channel estimation must be invoked K times.Additional pre-processing and FFT operations are also required.
V. COMBINED TIME-FREQUENCY COMPRESSIVE CHANNEL ESTIMATION
The combined method estimates common angular support through frequency-domain compressed sensing, then recovers channel gains and path delays in the time domain. This reduces matrix dimensions and avoids separate channel estimation for every subcarrier.
- V. COMBINED TIME-FREQUENCY COMPRESSIVE CHANNEL ESTIMATION: The proposed technique uses compressed sensing to estimate AoAs and AoDs in the frequency domain before recovering the full channel in time.
- Frequency-domain support recovery: Because the AoA/AoD information is common across subcarriers, sparse recovery can concatenate estimates from P subcarriers to obtain shared support sets.OMP may be applied to P subcarriers in parallel.
- Time-domain channel recovery: The time-domain formulation is conditioned on the estimated support, and least squares or MMSE recovers channel coefficients across all delay taps.
- Advantages: The combined approach uses smaller frequency-domain measurement and dictionary matrices than the corresponding time-domain matrices.Their dimensions are MNRF × NrNt and NrNt × GrGt, respectively.
- Advantages: Unlike the frequency-domain approach, the combined method evaluates channel estimates once in the time domain rather than separately per subcarrier.This further reduces computation.
VI. SIMULATION RESULTS
Simulations evaluate the proposed time-, frequency-, and combined time/frequency-domain estimators using NMSE and spectral efficiency. Performance depends on training length, RF-chain count, antenna-array size, angle quantization, path count, and SNR.
- Hybrid configurations: Multiple RF chains improve NMSE and reduce the number of training-frame transmissions by providing more effective measurements and beam patterns.Additional receiver RF combiners provide more effective combining beam patterns, while transmitter RF chains provide more random precoders for compressed sensing.
- Array and quantization effects: Increasing the antenna-array size can worsen NMSE unless the angle grid is sufficiently large relative to the array, in which case finer grids improve estimation accuracy.The reported degradation is attributed to greater ambiguity in the array response, while larger Gr and Gt narrow the quantization-related error gap.
- Domain selection: At low SNR, the combined time-frequency approach has the lowest average NMSE, whereas the three proposed approaches have similar performance at higher SNR.The combined approach is also reported to provide the best error performance in the relevant comparison.
- Channel sparsity: More channel paths increase estimation error for a fixed training budget, while the time-domain approach performs best for fewer paths and the combined approach for more paths.More compressive measurements are required as Np increases because the number of unknown channel parameters also increases.
VII. CONCLUSION
The paper concludes that sparse-recovery algorithms enable wideband channel estimation for hybrid mmWave systems across time, frequency, and combined domains. Simulations indicate low training overhead and improved estimation performance when multiple RF chains are used.
- The proposed algorithms estimate frequency-selective mmWave channels using sparse recovery and support MIMO operation after beam training.
- Three approaches operate in purely time, purely frequency, and combined time-frequency domains for SC-FDE and OFDM systems.
- The hybrid transceiver architecture and frame structure allow compressed sensing tools to achieve low training overhead for explicit channel estimation.
- More compressive measurements improve estimation error performance but increase signaling overhead.
- The combined time-frequency approach provides the best trade-off between low training overhead and minimum average estimation error.
- Using multiple RF chains at the transceivers further reduces estimation error and training overhead.