Energy Efficient ADC Bit Allocation and Hybrid Combining for Millimeter Wave MIMO Systems
Aryan Kaushik Christos Tsinos Evangelos Vlachos John Thompson






Energy Efficient ADC Bit Allocation and Hybrid Combining for Millimeter Wave MIMO Systems
    Aryan Kaushik^1, Christos Tsinos^2, Evangelos Vlachos^1, John Thompson^1
^1Institute for Digital Communications, The University of Edinburgh, United Kingdom.

^2Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Luxembourg.

Emails: {a.kaushik, e.vlachos, j.s.thompson}@ed.ac.uk, christos.tsinos@uni.lu
    2019
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Low resolution analog-to-digital converters (ADCs) can be employed to improve the energy efficiency (EE) of a wireless receiver since the power consumption of each ADC is exponentially related to its sampling resolution and the hardware complexity. In this paper, we aim to jointly optimize the sampling resolution, i.e., the number of ADC bits, and analog/digital hybrid combiner matrices which provides highly energy efficient solutions for millimeter wave multiple-input multiple-output systems. A novel decomposition of the hybrid combiner to three parts is introduced: the analog combiner matrix, the bit resolution matrix and the baseband combiner matrix. The unknown matrices are computed as the solution to a matrix factorization problem where the optimal, fully digital combiner is approximated by the product of these matrices. An efficient solution based on the alternating direction method of multipliers is proposed to solve this problem. The simulation results show that the proposed solution achieves high EE performance when compared with existing benchmark techniques that use fixed ADC resolutions.



energy efficient design, optimal bit resolution and hybrid combining, mmWave MIMO. 
















§ INTRODUCTION

The analog/digital (A/D) hybrid beamforming architectures for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems reduce the hardware complexity and the power consumption through fewer radio frequency (RF) chains and support multi-stream communication with good capacity performance <cit.>. Designing such systems for high energy efficiency (EE) gains would leverage their significance <cit.>. An alternative solution to reduce the power consumption and hardware complexity is by reducing the resolution sampling <cit.>. Some approaches have been applied in hybrid mmWave MIMO systems for EE maximization and low complexity with full resolution <cit.> and low resolution <cit.>.

The existing literature mostly discusses full or high resolution analog-to-digital converters (ADCs) with a small number of RF chains or low resolution ADCs with a large number of RF chains: either way only the fixed resolution ADCs are taken into account. References <cit.> consider EE optimization problems for A/D hybrid transceivers but with fixed and high resolution digital-to-analog converters (DACs)/ADCs. Reference <cit.> proposes a novel EE maximization transmission technique with subset selection optimization to find the best subset of the active RF chains and DAC resolution, which can be extended to low resolution ADCs at the receiver (RX). 
Reference <cit.> suggests implementing fixed and low resolution ADCs with few RF chains. Reference <cit.> studies the idea of a mixed-ADC architecture where a better energy-rate trade off is achieved by using mixed resolution ADCs but still with a fixed resolution for each ADC and it does not consider A/D hybrid beamforming. A hybrid beamforming system with fixed and low resolution ADCs has been analyzed for channel estimation in <cit.>. Varying resolution ADCs can be implemented at the RX <cit.> which may provide a better solution than fixed and low resolution ADCs. Extra care is needed when deciding the range of number of ADC bits as the total ADC power consumption can be dominated by only a few high resolution ADCs. Thus, a good trade-off between power consumption and performance is to consider the range of 1-8 bits for the varying number of ADC bits.






  §.§.§ Contributions

This paper designs an optimal EE solution for a mmWave A/D hybrid receiver MIMO system by introducing the novel decomposition of the A/D hybrid combiner to three parts representing the analog combiner matrix, the bit resolution matrix and baseband combiner matrix. Our aim is to minimize the distance between this decomposition, which is expressed as the product of three matrices, and the fully digital combiner matrix. The joint problem is decomposed into a series of sub-problems which are solved using an alternating optimization framework, i.e., alternating direction method of multipliers (ADMM) is developed to obtain the unknown matrices. The proposed design has high flexibility, given that the analog combiner is codebook-free, thus there is no restriction on the angular vectors and different bit resolutions can be assigned to each ADC. Our proposed solution optimizes the resolution on a packet-by-packet basis for each one of the ADCs unlike existing approaches that are based on fixed resolution sampling. We also implement an exhaustive search approach <cit.> for comparison which provides the upper bound for EE maximization.

Notation: A, a and a denote a matrix, a vector and a scalar, respectively. The complex conjugate transpose and transpose of A are denoted as A^H and A^T; |a| represents the determinant of a;

I_N represents N × N identity matrix; X∈ℂ^A × B and X∈ℝ^A × B denote A × B size 𝐗 matrix with complex and real entries, respectively; 𝒞𝒩 (a, A) denotes a complex Gaussian vector having mean a and covariance matrix A; 
[𝐀]_kl is the matrix entry at the k-th row and l-th column. The indicator function 1_𝒮{𝐀} of a set 𝒮 that acts over a matrix 𝐀  is defined as 0 ∀𝐀∈𝒮 and ∞∀𝐀∉𝒮. 




§ A/D HYBRID MMWAVE MIMO SYSTEM



 §.§ MmWave Channel Model

MmWave channels can be modeled by a narrowband clustered channel model due to different channel settings such as number of multipaths, amplitudes, etc., with N_cl clusters and N_ray propagation paths in each cluster <cit.>. Considering a single user mmWave system with N_T antennas at the transmitter (TX), transmitting N_s data streams to N_R antennas at the RX, the mmWave channel matrix can be written as follows:

    𝐇 = √(N_TN_R/N_cl N_ray)∑_i=1^N_cl∑_l=1^N_rayα_il𝐚_R(ϕ_il^r) 𝐚_T(ϕ_il^t)^H,

where α_il∈𝒞𝒩(0,σ_α,i^2) is the gain term with σ_α,i^2 being the average power of the i^th cluster. Furthermore, 𝐚_T(ϕ_il^t) and 𝐚_R(ϕ_il^r) represent the normalized transmit and receive array response vectors <cit.>, where ϕ_il^t and ϕ_il^r denote the azimuth angles of departure and arrival, respectively. We use uniform linear array (ULA) antennas for simplicity and model the antenna elements at the RX as ideal sectored elements <cit.>. However, the proposed technique is not limited to this setup and can be easily extended to the case of wideband channels and uniform planar/circular arrays. 



















    




 §.§ A/D Hybrid MIMO System Model

Based on the A/D hybrid beamforming scheme in the large-scale mmWave MIMO communication systems, the number of RX RF chains L_R follows the limitation N_s≤ L_R≤ N_R <cit.>. The matrices 𝐖_RF∈ℂ^N_R× L_R and 𝐖_BB∈ℂ^L_R× N_s denote the analog combiner and baseband (or digital) combiner matrices, respectively. The analog combiner matrix 𝐖_RF is based on phase shifters, i.e., the elements that have unit modulus and continuous phase. Thus, 𝐖_RF∈𝒲^N_R× L_R where the set 𝒲 represents the set of possible phase shifts in 𝐖_RF and for a variable a, is defined as, 𝒲 = {a ∈ℂ |  |a| = 1}. At the TX, with L_T RF chains, the analog precoder matrix is denoted as 𝐅_RF∈ℂ^N_T× L_T and the baseband precoder matrix is denoted as 𝐅_BB∈ℂ^L_T× N_s. The received signal y∈ℂ^N_R× 1 can be expressed as:
 
    𝐲 = 𝐇𝐅_RF𝐅_BB𝐱 + 𝐧,

 where 𝐱∈ℂ^N_s× 1 is the transmit symbol vector and 𝐧∈ℂ^N_R× 1 is a noise vector with independent and identically distributed entries and follow the complex Gaussian distribution with zero mean and σ_n^2 variance, i.e., 𝐧∼𝒞𝒩(0,σ_n^2 𝐈_N_R).
 
As widely used in the existing literature, we consider the linear additive quantization noise model (AQNM) to represent the distortion of quantization <cit.>.

Given that Q(·) denotes a uniform scalar quantizer then for the scalar complex input x ∈ℂ that is applied to both the real and imaginary parts, we have that,

    Q(x) ≈δ x + ϵ,

where δ = √(1-π√(3)/22^-2 b)∈ [m,M] is the multiplicative distortion parameter for a bit resolution equal to b <cit.> where m and M denote the minimum and maximum value of the range. Note that the introduced error in the linear approximation in (<ref>) decreases for larger resolutions. However, our proposed solution focuses on EE maximization and this linear approximation does not impact the performance significantly as observed from the simulation results in Section IV.



The parameter ϵ is the additive quantization noise with ϵ∼𝒞𝒩(0, σ_ϵ^2) , where  σ_ϵ = √(1-π√(3)/22^-2 b)√(π√(3)/22^-2 b). Based on AQNM, the vector containing the complex output of all the ADCs can be expressed as follows:

    Q(𝐖_RF^H𝐲 ) ≈Δ^H 𝐖_RF^H 𝐲 + ϵ,

where Q(𝐖_RF^H 𝐲) ∈ℂ^L_R× 1 and Δ = Δ^H ∈ℂ^L_R× L_R is a diagonal matrix with values depending on the ADC resolution b_i of each ADC. Specifically, each diagonal entry of Δ is given by: 

    [Δ]_ii = √(1-π√(3)/22^-2 b_i)∈ [m,M] ∀ i=1,…, L_R,

where, for simplicity, we assume that the range [m,M] is the same for each one of the ADCs. The second term of (<ref>) expresses the additive quantization noise for all RF chains, with ϵ∈𝒞𝒩(0, 𝐂_ϵ) <cit.> 
where 𝐂_ϵ is a diagonal covariance matrix with entries as follows: 


    [𝐂_ϵ]_ii = (1-π√(3)/22^-2 b_i) (π√(3)/22^-2 b_i) ∀ i=1,…, L_R.




    
























After the effect of the quantization and application of the baseband combining matrix, the output 𝐫∈ℂ^N_s× 1 at the RX can be expressed as:

    𝐫 =  
    𝐖_BB^H Δ^H 𝐖_RF^H 𝐲 + 𝐖_BB^H ϵ.

Based on the received signal expression in (<ref>), we can express (<ref>) as follows:

    𝐫 =  
    𝐖_BB^H Δ^H 𝐖_RF^H 𝐇𝐅_RF𝐅_BB𝐱 + 𝐖_BB^H Δ^H 𝐖_RF^H 𝐧 + 𝐖_BB^H ϵ_η,

where η is the combined effect of the Gaussian and the quantization noise with η∼𝒞𝒩(0, 𝐑_η). Here 𝐑_η∈ℂ^L_R× L_R is the combined noise covariance matrix with, 

    𝐑_η = σ_n^2 𝐖_BB^HΔ^H𝐖_RF^H𝐖_RFΔ𝐖_BB + 𝐖_BB^H𝐂_ϵ𝐖_BB.




§ BIT ALLOCATION AND HYBRID COMBINER DESIGN




 §.§ Problem Formulation

Let us consider a point-to-point MIMO system with the linear quantization model. We define the EE as the ratio of the information rate and the total consumed power as,

    EE(𝐖_RF, Δ, 𝐖_BB) ≜R(𝐖_RF, Δ, 𝐖_BB)/P(Δ)  (bits/Joule),

where the information rate is defined as,

    R(𝐖_RF, Δ, 𝐖_BB) ≜log_2 |𝐈_L_R + 𝐑^-1_η/N_s𝐖_BB^H Δ^H𝐖_RF^H𝐇𝐅×
    𝐅^H𝐇^H 𝐖_RFΔ𝐖_BB|(bits/s),

where the A/D hybrid precoder 𝐅=𝐅_RF𝐅_BB∈ℂ^N_T× N_s. 



Similar to the power model at the TX in <cit.>, the total consumed power at the RX is expressed as:

    P(Δ) = P_D + N_RP_R +  N_RL_RP_PS + P_CP  (W),

where P_PS is the power per phase shifter, P_R is the power per antenna, P_D is the power associated with the total quantization operation, and following (<ref>) and <cit.>, we have 

    P_D= P_ADC∑_i=1^L_R 2^b_i= P_ADC∑_i=1^L_R( π√(3)/2 (1 - [Δ]_ii^2))^1/2  (W),

where P_ADC is the power consumed per bit in the ADC and P_CP is the power required by all circuit components. 


Considering the rate and power model in (<ref>) and (<ref>), respectively, we can express the following fractional problem:

    (𝒫_1): max_𝐖_RF, Δ, 𝐖_BB  R(𝐖_RF, Δ, 𝐖_BB)/P(Δ)
     subject to 𝐖_RF∈𝒲^N_R× L_R, 
    Δ∈𝒟^L_R× L_R,

where the set 𝒟 represents the finite states of the quantizer and is defined as,


    𝒟 = {Δ∈ℝ^L_R× L_R| m ≤ [Δ]_ii≤ M ∀ i = 1,..., L_R}.

The channel's singular value decomposition (SVD) is written as 𝐇 = 𝐔_HΣ_H𝐕_H^H, where 𝐔_H∈ℂ^N_R× N_R and 𝐕_H∈ℂ^N_T× N_T are unitary matrices, and Σ_H∈ℝ^N_R× N_T is a rectangular matrix of singular values in decreasing order whose diagonal elements are non-negative real numbers and whose non-diagonal elements are zero. The optimal, fully digital combiner matrix 𝐖_opt consists of the N_s columns of the left singular matrix 𝐔_H. Our goal, by solving (𝒫_1), is to obtain the combiner matrices and the bit resolution matrix in an optimal manner. We introduce the novel decomposition of the A/D hybrid combiner to three parts representing the analog combiner matrix, the bit resolution matrix and digital combiner matrix, 

i.e., 𝐖_RFΔ𝐖_BB. So the Euclidean distance 𝐖_opt - 𝐖_RFΔ𝐖_BB_F^2 should be as small as possible for a maximum throughput combiner design. Note that we optimize over the bit resolution matrix with varying resolutions and the choice of combiner matrices at the RX.


The maximization of the fractional problem (𝒫_1) is equivalent with the solution of the following problem:

    (𝒫_2):    min_𝐖_RF,Δ,𝐖_BB1/2𝐖_opt-𝐖_RFΔ𝐖_BB_F^2 + γ P(Δ),
       subject to𝐖_RF∈𝒲^N_R× L_R, Δ∈𝒟^L_R× L_R,
 

where the parameter γ∈ℝ^+ denotes the trade-off between the rate and the power consumption.

 The main idea to prove the equivalence is first to apply the Dinkelbach approach to transform the fractional problem into an affine one <cit.>. Afterwards, based on <cit.>, the maximization of the rate R can be expressed as minimization of the Euclidean distance between the computed A/D hybrid combiner and the optimal, fully digital combiner 𝐖_opt. The details of this proof are omitted due to space limitations. 

Parameter γ also determines how close is the solution of (𝒫_2) to (𝒫_1). In this work, γ is selected after an exhaustive search over all the possible values in the range of [0.001, 0.1] and the value which gives the best result for (𝒫_2) is selected. Problem (𝒫_2) is non-convex due to the constraints on the structure of matrix 𝐖_RF. Similar non-convex problems have been recently addressed in the literature via alternating direction method of multipliers (ADMM) based solutions <cit.>. 




 §.§ Proposed ADMM Solution

In the following we develop an iterative procedure for solving (𝒫_2) based on the ADMM approach <cit.>. This method, is a variant of the standard augmented Lagrangian method that uses partial updates (similar to the Gauss-Seidel method for the solution of linear equations) to solve constrained optimization problems. This method replaces a constrained minimization problem by a series of unconstrained problems and add a penalty term to the objective function. This penalty improves robustness compared to other optimization methods for constrained problems (for example, the dual ascent method) and in particular achieves convergence without the need of specific assumptions for the objective function, i.e., strict convexity and finiteness. 

The interested reader may refer to <cit.> for further information. 

We first transform (𝒫_2) into a form that can be addressed via ADMM. By using the auxiliary variable 𝐙, (𝒫_2) can be written in the following form:


    (𝒫_3):    min_𝐙,𝐖_RF, Δ,𝐖_BB1/2‖𝐖_opt-𝐙‖_F^2 + 1_𝒲^N_R× L_R{𝐖_RF}
        + 1_𝒟^L_R× L_R{Δ} + γ P(Δ), 
       subject to𝐙 = 𝐖_RFΔ𝐖_BB.
 












Problem (𝒫_3) formulates the A/D hybrid combiner matrix design as a matrix factorization problem. That is, the overall combiner 𝐙 is sought so that it minimizes the Euclidean distance to the optimal, fully digital combiner 𝐖_opt while supporting decomposition into three factors: the analog combiner matrix 𝐖_RF, the matrix Δ which is related to the resolution of each ADC and the digital combiner matrix 𝐖_BB.  



The augmented Lagrangian function of (𝒫_3) is given by, 


    ℒ(𝐙,𝐖_RF,Δ,𝐖_BB,Λ)  = 1/2‖𝐖_opt-𝐙‖^2_F  +1_𝒲^N_R× L_R{𝐖_RF}
    +1_𝒟^L_R× L_R{Δ}+α/2‖𝐙+Λ/α - 𝐖_RFΔ𝐖_BB‖_F^2 +γ P(Δ),


where α is a scalar penalty parameter and Λ∈ℂ^N_R× L_R is the Lagrange Multiplier matrix. 


According to ADMM <cit.>, the solution to  (𝒫_3) is derived by the following iterative steps:

    (𝒫_3A): 𝐙_(n) = min_𝐙1/2‖(1+α)𝐙-𝐖_opt+Λ_(n-1)
        -α𝐖_RF(n-1)Δ_(n-1)𝐖_BB(n-1)‖_F^2, 
    
    
       (𝒫_3B): 𝐖_RF(n) = min_𝐖_RF1_𝒲^N_R× L_R{𝐖_RF}+α/2×
       𝐙_(n)+Λ_(n-1)/α- 𝐖_RFΔ_(n-1)𝐖_BB(n-1)_F^2, 
    
    
       (𝒫_3C): Δ_(n) = min_Δ‖𝐲_c-Ψvec(Δ)‖_2^2 +γ P(Δ), 
        subject to Δ∈𝒟,
    
    
       (𝒫_3D): 𝐖_BB(n) = min_𝐖_BBα/2‖𝐙_(n)+Λ_(n-1)/α
        -𝐖_RF(n)Δ_(n)𝐖_BB‖_F^2, 
    
    
       Λ_(n) = Λ_(n-1) + α(𝐙_(n)-𝐖_RF_(n)Δ_(n)𝐖_BB_(n)),

where n denotes the iteration index, 𝐲_c=vec(Z_(n)+Λ_(n-1)/α) and Ψ=𝐖_BB_(n-1)⊗𝐖_RF_(n) (⊗ is the Khatri-Rao product). 


We solve the optimization problems (𝒫_3A)-(𝒫_3D) and the solutions are provided in Algorithm 1. The algorithm provides the complete procedure to obtain the optimal analog combiner matrix 𝐖_RF, the optimal bit resolution matrix Δ and the optimal baseband (or digital) combiner matrix 𝐖_BB. It starts by initializing the entries of the matrices 𝐙, 𝐖_RF, Δ, 𝐖_BB with random values and the entries of the Lagrange multiplier matrix Λ with zeros. For iteration index n, 𝐙_(n), 𝐖_RF_(n), Δ_(n) and 𝐖_BB_(n) are updated at each iteration step using the solutions provided in Steps 4, 7, 8, 10 and 11 of Algorithm 1. In Step 7, Π_𝒲 is the operator that projects the solution onto the set 𝒲. This is computed by solving the following optimization problem <cit.>:


    (𝒫_4):    min_𝐀_𝒲𝐀_𝒲-𝐀_F^2, subject to 𝐀_𝒲∈𝒲,


where 𝐀 is an arbitrary matrix and 𝐀_𝒲 is its projection onto the set 𝒲. The solution to (𝒫_4) is given by the phase of the complex elements of 𝐀. Thus, for 𝐀_𝒲  = Π_𝒲{𝐀} we have


    𝐀_𝒲(x,y) = 
    0,     𝐀(x,y) = 0 
    𝐀(x,y)/|𝐀(x,y)|,     𝐀(x,y) ≠ 0 
    ,

where 𝐀_𝒲(x,y) and 𝐀(x,y) are the elements at the xth row-yth column of matrices 𝐀_𝒲 and 𝐀, respectively. Furthermore, as shown in Step 8, the minimization problem in (𝒫_3C) is solved by implementing CVX <cit.>. A termination criterion related to the maximum permitted number of iterations of the ADMM sequence (N_max) is considered. Upon convergence, the number of bits for each ADC is obtained by using (<ref>) and quantized to the nearest integer value. 



  §.§.§ Computational complexity analysis of Algorithm 1

In Algorithm 1, mainly Step 8 involves multiplication by Ψ whose dimensions are L_RN_R× N_s L_R. In general, the solution of (𝒫_3C) can be upper-bounded by 𝒪((L_R^2N_RN_s)^3) which can be improved significantly by exploiting the structure of Ψ.


















































 


















































































§ SIMULATION RESULTS

In this section, we evaluate the performance of the proposed ADMM technique using computer simulation results. 

The results have been averaged over 1,000 Monte-Carlo realizations.


  §.§.§ System setup

We set the following parameters, unless specified otherwise, to obtain the desired results: N_T = 32, N_R = 16, L_R = 4, N_s = 4, N_cl=2, N_ray=4, N_max=40, m=1, M=8, α=1 and σ^2_α, i = 1. The azimuth angles of departure and arrival are computed with uniformly distributed mean angles; each cluster follows a Laplacian distribution about the mean angle. The antenna elements in the ULA are spaced by distance d = λ/2. The signal-to-noise ratio (SNR) is given by the inverse of the noise variance, i.e., 1/σ_n^2. The transmit vector 𝐱 is composed of the normalized i.i.d. Gaussian symbols. The values used for the terms in the power model in (<ref>) of Section III are P_ADC = 100 mW, P_CP = 10 W, P_R = 100 mW and P_PS = 10 mW. Note that to measure the spectral efficiency (SE) performance, we compute the ratio R/B bits/s/Hz  where B represents the bandwidth, and for the simulations we set B=1 Hz. For simulations, the precoder matrix 𝐅 is considered equal to the optimal fully digital precoder matrix <cit.>, i.e., the product of  1/√(N_s) and first N_s columns of the right singular matrix 𝐕_H. 

 


  §.§.§ Convergence of the proposed ADMM solution

Fig. 2 shows the convergence of the ADMM solution as proposed in Algorithm 1 to obtain the optimal bit resolution at each ADC and corresponding optimal combiner matrices. The proposed solution converges rapidly at around 20 iterations and mean square error (MSE), 𝐖_opt - 𝐖_RF(N_max)Δ_(N_max)𝐖_BB(N_max)_F^2, goes as low as -20 dB. A lower number of RX antennas shows lower MSE as expected, since fewer parameters are required to be estimated.




 
 
 

 
 
 
   



  §.§.§ Benchmark techniques




  §.§.§ Digital combining with full-bit resolution

We consider the conventional fully digital beamforming architecture, where the number of RF chains at the RX is equal to the number of RX antennas, i.e., L_R = N_R. The fully digital combining solution may be provided by SVD and waterfilling <cit.>. In terms of the resolution sampling, we consider full-bit resolution, i.e., M = 8-bit, which represents the optimum from the achievable SE perspective.



  §.§.§ A/D Hybrid combining with 1-bit and 8-bit resolutions

We also consider a A/D hybrid combining architecture with L_R < N_R, for two cases of bit resolution: a) 1-bit resolution which usually shows reasonable EE performance, and b) 8-bit resolution which usually shows high SE results. 



  §.§.§ Brute force with A/D hybrid combining

We also implement an exhaustive search approach as an upper bound for EE maximization called brute force (BF), based on <cit.>, which clearly shows the energy-rate performance trade-offs in the simulations. It makes a search over the number of RF chains L_R and all the available bit resolutions, i.e., b = 1,..., M. It then finds the best EE out of all the possible cases and chooses the corresponding optimal resolution for each ADC. This method provides the best possible EE performance, but it is computationally intractable for L_R>4.

Fig. 3 shows the performance of the proposed ADMM solution compared with existing benchmark techniques with respect to (w.r.t.) SNR at N_R=16. The proposed ADMM solution achieves high EE which has performance close to the BF approach and better than the 8-bit hybrid, 1-bit hybrid and full-bit digital baselines. For example, at SNR = 20 dB, the proposed ADMM solution outperforms 1-bit hybrid, 8-bit hybrid and full-bit digital baselines by about 0.45 bits/Joule, 1.375 bits/Joule and 1.44 bits/Joule, respectively. It also exhibits better SE than 1-bit hybrid and has similar performance to the 8-bit hybrid baseline. 

There is an energy-rate trade-off between the proposed solution and the BF approach as we can achieve better rate with lower EE and vice-versa. Moreover, the proposed solution has lower complexity than the BF approach because the BF involves a search over all the possible bit resolutions while the proposed solution directly optimizes the number of bits to obtain an optimal number of bits at each ADC. We constrain the number of RF chains L_R=4 for the BF approach due to the high complexity order which is 𝒪(M^L_R). Also note that the proposed approach enables the selection of different resolutions for different ADCs and thus, it offers a better trade-off for EE versus SE than existing approaches which are based on a fixed ADC resolution. 
 
Figs. 4 and 5 show the performance results w.r.t. the number of RX and TX antennas at 30 dB SNR. The proposed ADMM solution again achieves high EE and performs close to the BF approach and better than the 8-bit hybrid, 1-bit hybrid and full-bit digital baselines. For example, at N_R = 20, the proposed ADMM solution outperforms 1-bit hybrid, 8-bit hybrid and full-bit digital baselines by about 0.85 bits/Joule, 1.75 bits/Joule and 1.875 bits/Joule, respectively. Also, for N_T = 20, the proposed solution outperforms 1-bit hybrid, 8-bit hybrid and full-bit digital baselines by about 1.0 bits/Joule, 1.5 bits/Joule and 1.625 bits/Joule, respectively. The proposed solution also exhibits better SE than 1-bit hybrid and has similar performance to the 8-bit hybrid baseline. Both the figures follow the energy-rate trade-off with the BF approach.

Furthermore, we investigate the performance over the trade-off parameter γ introduced in (𝒫_2). Fig. 6 shows the bar plot of average of the optimal number of bits selected by the proposed solution for each ADC versus γ. The average optimal number decreases with the increase in γ, for example, it is 4 for γ = 0.001, 3 for γ = 0.01 and 2 for γ = 0.1. Fig. 6 also shows that the power consumption in the proposed case is considerably low and decreases with the increase in the trade-off parameter γ unlike digital 8-bit, several fixed bit hybrid baselines and the BF approach. Fig. 7 shows the EE and SE plots for several solutions w.r.t. γ. It can be observed that the proposed solution achieves higher EE than the fixed bit allocation solutions and achieves comparable EE and SE results to the BF approach. These curves also show that adjusting γ allows the system to vary the energy-rate trade-off.



§ CONCLUSION

This paper proposes an energy efficient mmWave A/D hybrid MIMO system which can vary the ADC bit resolution at the RX. This method uses the decomposition of the A/D hybrid combiner matrix into three parts representing the analog combiner matrix, the bit resolution matrix and the digital combiner matrix. These three matrices are optimized by the novel ADMM solution which outperforms the EE of the full-bit digital, 1-bit hybrid combining and 8-bit hybrid combining baselines. There is an energy-rate trade-off with the BF approach which yields the upper bound for EE maximization. The proposed approach enables the selection of the optimal resolution for each ADC and thus, it offers better trade-off for data rate versus EE than existing approaches based on fixed ADC resolution. In future work, we will jointly optimize the DAC and ADC bit resolution and hybrid precoder and combiner matrices at the TX and the RX.





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