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Channel estimation via gradient pursuit for mmWave massive MIMO systems with onebit ADCs
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 289 (2019)
Abstract
In millimeter wave (mmWave) massive multipleinput multipleoutput (MIMO) systems, 1 bit analogtodigital converters (ADCs) are employed to reduce the impractically high power consumption, which is incurred by the wide bandwidth and large arrays. In practice, the mmWave band consists of a small number of paths, thereby rendering sparse virtual channels. Then, the resulting maximum a posteriori (MAP) channel estimation problem is a sparsityconstrained optimization problem, which is NPhard to solve. In this paper, iterative approximate MAP channel estimators for mmWave massive MIMO systems with 1 bit ADCs are proposed, which are based on the gradient support pursuit (GraSP) and gradient hard thresholding pursuit (GraHTP) algorithms. The GraSP and GraHTP algorithms iteratively pursue the gradient of the objective function to approximately optimize convex objective functions with sparsity constraints, which are the generalizations of the compressive sampling matching pursuit (CoSaMP) and hard thresholding pursuit (HTP) algorithms, respectively, in compressive sensing (CS). However, the performance of the GraSP and GraHTP algorithms is not guaranteed when the objective function is illconditioned, which may be incurred by the highly coherent sensing matrix. In this paper, the band maximum selecting (BMS) hard thresholding technique is proposed to modify the GraSP and GraHTP algorithms, namely, the BMSGraSP and BMSGraHTP algorithms, respectively. The BMSGraSP and BMSGraHTP algorithms pursue the gradient of the objective function based on the band maximum criterion instead of the naive hard thresholding. In addition, a fast Fourier transformbased (FFTbased) fast implementation is developed to reduce the complexity. The BMSGraSP and BMSGraHTP algorithms are shown to be both accurate and efficient, whose performance is verified through extensive simulations.
Introduction
In millimeter wave (mmWave) massive multipleinput multipleoutput (MIMO) systems, the wide bandwidth of the mmWave band in the range of 30–300 GHz offers a high data rate, which guarantees a significant performance gain [1–4]. However, the power consumption of analogtodigital converters (ADCs) is scaled quadratically with the sampling rate and exponentially with the ADC resolution, which renders highresolution ADCs impractical for mmWave systems [5]. To reduce the power consumption, lowresolution ADCs were suggested as a possible solution, which recently gained popularity [6–9]. Coarsely quantizing the received signal using lowresolution ADCs results in an irreversible loss of information, which might cause a significant performance degradation. In this paper, we consider the extreme scenario of using 1 bit ADCs for mmWave systems.
In practice, the mmWave band consists of a small number of propagation paths, which results in sparse virtual channels. In the channel estimation point of view, sparse channels are favorable because the required complexity and measurements can be reduced. Sparsityconstrained channel distributions, however, cannot be described in closed forms, which makes it difficult to exploit Bayesian channel estimation. In [10, 11], channel estimators for massive MIMO systems with 1 bit ADCs were proposed, which account for the effect of the coarse quantization. The near maximum likelihood (nML) channel estimator [10] selects the maximizer of the likelihood function subject to the L^{2}norm constraint as the estimate of the channel, which is solved using the projected gradient descent method [12]. However, the channel sparsity was not considered in [10]. In [11], the Bussgang linear minimum mean squared error (BLMMSE) channel estimator was derived by linearizing 1 bit ADCs based on the Bussgang decomposition [13]. The BLMMSE channel estimator is an LMMSE channel estimator for massive MIMO systems with 1 bit ADCs, whose assumption is that the channel is Gaussian. Therefore, the sparsity of the channel is not taken into account in [11] either.
To take the channel sparsity into account, iterative approximate MMSE estimators for mmWave massive MIMO systems with 1 bit ADCs were proposed in [14, 15]. The generalized expectation consistent signal recovery (GECSR) algorithm in [14] is an iterative approximate MMSE estimator based on the turbo principle [16], which can be applied to any nonlinear function of the linearly mapped signal to be estimated. Furthermore, the only constraint on the distribution of the signal to be estimated is that its elements must be independent and identically distributed (i.i.d.) random variables. Therefore, the GECSR algorithm can be used as an approximate MMSE channel estimator for any ADC resolutions ranging from 1 bit to highresolution ADCs. However, the inverse of the sensing matrix is required at each iteration, which is impractical in massive MIMO systems in the complexity point of view.
The generalized approximate message passingbased (GAMPbased) channel estimator for mmWave massive MIMO systems with lowresolution ADCs was proposed in [15], which is another iterative approximate MMSE channel estimator. In contrast to the GECSR algorithm, only matrixvector multiplications are required at each iteration, which is favorable in the complexity point of view. As in the GECSR algorithm, the GAMPbased algorithm can be applied to any ADC resolutions and any channel distributions as long as the elements of channel are i.i.d. random variable. The performance of the GECSR and GAMP algorithms, however, cannot be guaranteed when the sensing matrix is illconditioned, which frequently occurs in the mmWave band. To prevent the sensing matrix from becoming illconditioned, the GAMPbased channel estimator constructs the virtual channel representation using discrete Fourier transform (DFT) matrices, whose columns are orthogonal. However, such virtual channel representation results in a large gridding error, which leads to performance degradation.
Our goal is to propose an iterative approximate maximum a posteriori (MAP) channel estimator for mmWave massive MIMO systems with 1 bit ADCs. Due to the sparse nature, the MAP channel estimation problem is converted to a sparsityconstrained optimization problem, which is NPhard to solve [17]. To approximately solve such problems iteratively, the gradient support pursuit (GraSP) and gradient hard thresholding pursuit (GraHTP) algorithms were proposed in [17] and [18], respectively. The GraSP and GraHTP algorithms pursue the gradient of the objective function at each iteration by hard thresholding. These algorithms are the generalizations of the compressive sampling matching pursuit (CoSaMP) [19] and hard thresholding pursuit (HTP) [20] algorithms, respectively, in compressive sensing (CS).
With highly coherent sensing matrix, however, the GraSP and GraHTP algorithms do not perform appropriately since the objective function becomes illconditioned. To remedy such break down, we exploit the band maximum selecting (BMS) hard thresholding technique, which is then applied to the GraSP and GraHTP algorithms to propose the BMSGraSP and BMSGraHTP algorithms, respectively. The proposed BMSbased algorithms perform hard thresholding for the gradient of the objective function based on the proposed band maximum criterion, which tests whether an index is the ground truth index or the byproduct of another index. To reduce the complexity of the BMSbased algorithms, a fast Fourier transformbased (FFTbased) fast implementation of the objective function and gradient is proposed. The BMSbased algorithms are shown to be both accurate and efficient, which is verified through extensive simulations.
The rest of this paper is organized as follows. In Section 2, mmWave massive MIMO systems with 1 bit ADCs are described. In Section 3, the MAP channel estimation framework is formulated. In Section 4, the BMS hard thresholding technique is proposed, which is applied to the GraSP and GraHTP algorithms. In addition, an FFTbased fast implementation is proposed. In Section 5, the results and discussion are presented, and the conclusions are followed in Section 6.
Notation:a, a, and A denote a scalar, vector, and matrix, respectively. ∥a∥_{0},∥a∥_{1}, and ∥a∥ represent the L^{0}, L^{1}, and L^{2}norm of a, respectively. ∥A∥_{F} is the Frobenius norm of A. The transpose, conjugate transpose, and conjugate of A are denoted as A^{T},A^{H}, and \(\overline {\mathbf {A}}\), respectively. The elementwise vector multiplication and division of a and b are denoted as a⊙b and a⊘b, respectively. The sum of all of the elements of a is denoted as sum(a). The vectorization of A is denoted as vec(A), which is formed by stacking all of the columns of A. The unvectorization of a is denoted as unvec(a), which is the inverse of vec(A). The Kronecker product of A and B is denoted as A⊗B. The support of a is denoted as supp(a), which is the set of indices formed by collecting all of the indices of the nonzero elements of a. The best sterm approximation of a is denoted as a_{s}, which is formed by leaving only the s largest (in absolute value) elements of a and replacing the other elements with 0. Similarly, the vector obtained by leaving only the elements of a indexed by the set \(\mathcal {A}\) and replacing the other elements with 0 is denoted as \(\mathbf {a}_{\mathcal {A}}\). The absolute value of a scalar a and cardinality of a set \(\mathcal {A}\) are denoted as a and \(\mathcal {A}\), respectively. The set difference between the sets \(\mathcal {A}\) and \(\mathcal {B}\), namely, \(\mathcal {A}\cap \mathcal {B}^{\mathrm {c}}\), is denoted as \(\mathcal {A}\setminus \mathcal {B}\). ϕ(a) and Φ(a) are elementwise standard normal PDF and CDF functions of a, whose ith elements are \(\frac {1}{\sqrt {2\pi }}e^{\frac {a_{i}^{2}}{2}}\) and \(\int _{\infty }^{a_{i}}\frac {1}{\sqrt {2\pi }}e^{\frac {x^{2}}{2}}dx\), respectively. The m×1 zero vector and m×m identity matrix are denoted as 0_{m} and I_{m}, respectively.
mmWave massive MIMO systems with 1 bit ADCs
System model
As shown in Fig. 1, consider a mmWave massive MIMO system with uniform linear arrays (ULAs) at the transmitter and receiver. The Nantenna transmitter transmits a training sequence of length T to the Mantenna receiver. Therefore, the received signal \(\mathbf {Y}=[\mathbf {y}[1]\quad \mathbf {y}[2]\quad \cdots \quad \mathbf {y}[T]]\in \mathbb {C}^{M\times T}\) is
which is the collection of the tth received signal \(\mathbf {y}[t]\in \mathbb {C}^{M}\) over t∈{1,…,T}. In the mmWave band, the channel \(\mathbf {H}\in \mathbb {C}^{M\times N}\) consists of a small number of paths, whose parameters are the path gains, angleofarrivals (AoAs), and angleofdepartures (AoDs) [21]. Therefore, H is
where L is the number of paths, \(\alpha _{\ell }\in \mathbb {C}\) is the path gain of the ℓth path, and θ_{RX,ℓ}∈[−π/2,π/2] and θ_{TX,ℓ}∈[−π/2,π/2] are the AoA and AoD of the ℓth path, respectively. The steering vectors \(\mathbf {a}_{\text {RX}}(\theta _{\text {RX}, \ell })\in \mathbb {C}^{M}\) and \(\mathbf {a}_{\text {TX}}(\theta _{\text {TX}, \ell })\in \mathbb {C}^{N}\) are
where the interelement spacing is halfwavelength. The training sequence \(\mathbf {S}=[\mathbf {s}[1]\quad \mathbf {s}[2]\quad \cdots \quad \mathbf {s}[T]]\in \mathbb {C}^{N\times T}\) is the collection of the tth training sequence \(\mathbf {s}[t]\in \mathbb {C}^{N}\) over t∈{1,…,T}, whose power constraint is ∥s[t]∥^{2}=N. The additive white Gaussian noise (AWGN) \(\mathbf {N}=[\mathbf {n}[1]\quad \mathbf {n}[2]\quad \cdots \quad \mathbf {n}[T]]\in \mathbb {C}^{M\times T}\) is the collection of the tth AWGN \(\mathbf {n}[t]\in \mathbb {C}^{M}\) over t∈{1,…,T}, which is distributed as \(\text {vec}(\mathbf {N})\sim \mathcal {C}\mathcal {N}(\mathbf {0}_{MT}, \mathbf {I}_{MT})\). The signaltonoise ratio (SNR) is defined as ρ.
At the receiver, the real and imaginary parts of the received signal are quantized by 1 bit ADCs. The quantized received signal \(\hat {\mathbf {Y}}\) is
where Q(·) is the 1 bit quantization function, whose threshold is 0. Therefore, Q(Y) is
where sign(·) is the elementwise sign function. The goal is to estimate H by estimating \(\{\alpha _{\ell }\}_{\ell =1}^{L}, \{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\), and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) from \(\hat {\mathbf {Y}}\).
Virtual channel representation
In the mmWave channel model in (2), \(\{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\) and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) are hidden in \(\{\mathbf {a}_{\text {RX}}(\theta _{\text {RX}, \ell })\}_{\ell =1}^{L}\) and \(\{\mathbf {a}_{\text {TX}}(\theta _{\text {TX}, \ell })\}_{\ell =1}^{L}\), respectively. The nonlinear mapping of \(\{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\) and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) to Y renders a nonlinear channel estimation problem. To convert the nonlinear channel estimation problem to a linear channel estimation problem, we adopt the virtual channel representation [22].
The virtual channel representation of H is
where the dictionary pair \(\mathbf {A}_{\text {RX}}\in \mathbb {C}^{M\times B_{\text {RX}}}\) and \(\mathbf {A}_{\text {TX}}\in \mathbb {C}^{N\times B_{\text {TX}}}\) is the collection of B_{RX}≥M steering vectors and B_{TX}≥N steering vectors, respectively. Therefore, A_{RX} and A_{TX} are
whose gridding AoAs \(\{\omega _{\text {RX}, i}\}_{i=1}^{B_{\text {RX}}}\) and AoDs \(\{\omega _{\text {TX}, j}\}_{j=1}^{B_{\text {TX}}}\) are selected so as to form overcomplete DFT matrices. The gridding AoAs and AoDs are the B_{RX} and B_{TX} points from [−π/2,π/2], respectively, to discretize the AoAs and AoDs because the ground truth AoAs and AoDs are unknown. To make a dictionary pair of the overcomplete DFT matrix form, the gridding AoAs and AoDs are given as ω_{RX,i}= sin−1(−1+2/B_{RX}·(i−1)) and ω_{RX,j}= sin−1(−1+2/B_{TX}·(j−1)), respectively. We prefer overcomplete DFT matrices because they are relatively wellconditioned, and DFT matrices are friendly to the FFTbased implementation, which will be discussed in Section 4. The virtual channel \(\mathbf {X}^{*}\in \mathbb {C}^{B_{\text {RX}}\times B_{\text {TX}}}\) is the collection of \(\{\alpha _{\ell }\}_{\ell =1}^{L}\), whose (i,j)th element is α_{ℓ} whenever (ω_{RX,i},ω_{TX,j}) is the nearest to (θ_{RX,ℓ},θ_{TX,ℓ}) but 0 otherwise. In general, the error between H and \(\mathbf {A}_{\text {RX}}\mathbf {X}^{*}\mathbf {A}_{\text {TX}}^{\mathrm {H}}\) is inversely proportional to B_{RX} and B_{TX}. To approximate H using (7) with negligible error, the dictionary pair must be dense, namely, B_{RX}≫M and B_{TX}≫N.
Before we proceed, we provide a supplementary explanation on the approximation in (7). In this work, we attempt to estimate the Lsparse X^{∗} in (7) because the sparse assumption on X^{∗} is favorable when the goal is to formulate the channel estimation problem as a sparsityconstrained problem. The cost of assuming that X^{∗} is Lsparse is, as evident, the approximation error shown in (7). Alternatively, the approximation error can be perfectly removed by considering X^{∗} satisfying \(\mathbf {H}=\mathbf {A}_{\text {RX}}\mathbf {X}^{*}\mathbf {A}_{\text {TX}}^{\mathrm {H}}\), i.e., equality instead of approximation. One wellknown X^{∗} satisfying the equality is the minimum Frobenius norm solution, i.e., \(\mathbf {X}^{*}=\mathbf {A}_{\text {RX}}^{\mathrm {H}}(\mathbf {A}_{\text {RX}}\mathbf {A}_{\text {RX}}^{\mathrm {H}})^{1}\mathbf {H}(\mathbf {A}_{\text {TX}}\mathbf {A}_{\text {TX}}^{\mathrm {H}})^{1}\mathbf {A}_{\text {TX}}\). Such X^{∗}, however, has no evident structure to exploit in channel estimation, which is the reason why we assume that X^{∗} is Lsparse at the cost of the approximation error in (7).
In practice, the arrays at the transmitter and receiver are typically large to compensate the path loss in the mmWave band, whereas the number of lineofsight (LOS) and near LOS paths is small [23]. Therefore, X^{∗} is sparse when the dictionary pair is dense because only L elements among B_{RX}B_{TX} elements are nonzero where L≪MN≪B_{RX}B_{TX}. In the sequel, we use the shorthand notation B=B_{RX}B_{TX}.
To facilitate the channel estimation framework, we vectorize (1) and (5) in conjunction with (7). First, note that
where the gridding error \(\mathbf {E}\in \mathbb {C}^{M\times T}\) represents the mismatch in (7).^{Footnote 1} Then, the vectorized received signal \(\mathbf {y}=\text {vec}(\mathbf {Y})\in \mathbb {C}^{MT}\) is
where
The vectorized quantized received signal \(\hat {\mathbf {y}}=\text {vec}(\hat {\mathbf {Y}})\in \mathbb {C}^{MT}\) is
The goal is to estimate Lsparse x^{∗} from \(\hat {\mathbf {Y}}\).
Problem formulation
In this section, we formulate the channel estimation problem using the MAP criterion. To facilitate the MAP channel estimation framework, the real counterparts of the complex forms in (16) are introduced. Then, the likelihood function of x^{∗} is derived.
The real counterparts are the collections of the real and imaginary parts of the complex forms. Therefore, the real counterparts \(\hat {\mathbf {y}}_{\mathrm {R}}\in \mathbb {R}^{2MT}, \mathbf {A}_{\mathrm {R}}\in \mathbb {R}^{2MT\times 2B}\), and \(\mathbf {x}_{\mathrm {R}}^{*}\in \mathbb {R}^{2B}\) are
which are the collections of the real and imaginary parts of \(\hat {\mathbf {y}}, \mathbf {A}\), and x^{∗}, respectively. In the sequel, we use the complex forms and the real counterparts interchangeably. For example, x^{∗} and \(\mathbf {x}_{\mathrm {R}}^{*}\) refer to the same entity.
Before we formulate the likelihood function of x^{∗}, note that e is hard to analyze. However, e is negligible when the dictionary pair is dense. Therefore, we formulate the likelihood function of x^{∗} without e. The price of such oversimplification is negligible when B_{RX}≫M and B_{TX}≫N, which is to be shown in Section 5 where e≠0_{MT}. To derive the likelihood function of x^{∗}, note that
given x^{∗}. Then, from (20) in conjunction with (16), the loglikelihood function f(x) is [10]
If the distribution of x^{∗} is known, the MAP estimate of x^{∗} is
where g_{MAP}(x) is the logarithm of the PDF of x^{∗}. In practice, however, g_{MAP}(x) is unknown. Therefore, we formulate the MAP channel estimation framework based on \(\{\alpha _{\ell }\}_{\ell =1}^{L}, \{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\), and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) where we assume the followings:
 1.
\(\alpha _{\ell }\sim \mathcal {C}\mathcal {N}(0, 1)\) for all ℓ
 2.
θ_{RX,ℓ}∼unif([−π/2,π/2]) for all ℓ
 3.
θ_{TX,ℓ}∼unif([−π/2,π/2]) for all ℓ
 4.
\(\{\alpha _{\ell }\}_{\ell =1}^{L}, \{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\), and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) are independent.
Then, the MAP estimate of x^{∗} considering the channel sparsity is
where g(x)=−∥x_{R}∥^{2} is the logarithm of the PDF of \(\mathcal {C}\mathcal {N}(\mathbf {0}_{B}, \mathbf {I}_{B})\) ignoring the constant factor. However, note that only the optimization problems (22) and (23) are equivalent in the sense that their solutions are the same, not g_{MAP}(x) and g(x). In the ML channel estimation framework, the ML estimate of x^{∗} is
In the sequel, we focus on solving (23) because (23) reduces to (24) when g(x)=0. In addition, we denote the objective function and the gradient in (23) as h(x) and ∇h(x), respectively. Therefore,
where the differentiation is with respect to x.
Channel estimation via gradient pursuit
In this section, we propose the BMSGraSP and BMSGraHTP algorithms to solve (23), which are the variants of the GraSP [17] and GraHTP [18] algorithms, respectively. Then, an FFTbased fast implementation is proposed. In addition, we investigate the limit of the BMSGraSP and BMSGraHTP algorithms in the high SNR regime in 1 bit ADCs.
Proposed BMSGraSP and BMSGraHTP algorithms
Note that h(x) in (23) is concave because f(x) and g(x) are the sums of the logarithms of Φ(·) and ϕ(·), respectively, which are logconcave [24]. However, (23) is not a convex optimization problem because the sparsity constraint is not convex. Furthermore, solving (23) is NPhard because of its combinatorial complexity. To approximately optimize convex objective functions with sparsity constraints iteratively by pursuing the gradient of the objective function, the GraSP and GraHTP algorithms were proposed in [17] and [18], respectively.
To solve (23), the GraSP and GraHTP algorithms roughly proceed as follows at each iteration when x is the current estimate of x^{∗} where the iteration index is omitted for simplicity. First, the best Lterm approximation of ∇h(x) is computed, which is
where T_{L}(·) is the Lterm hard thresholding function. Here, T_{L}(·) leaves only the L largest elements (in absolute value) of ∇h(x), and sets all the other remaining elements to 0. Then, after the estimate of supp(x^{∗}) is updated by selecting
i.e., \(\mathcal {I}\) is the set of indices formed by collecting the L indices of ∇h(x) corresponding to its L largest elements (in absolute value), the estimate of x^{∗} is updated by solving the following optimization problem
which can be solved using convex optimization because the support constraint is convex [24]. The GraSP and GraHTP algorithms are the generalizations of the CoSaMP [19] and HTP [20] algorithms, respectively. This follows because the gradient of the squared error is the scaled proxy of the residual.
To solve (23) using the GraSP and GraHTP algorithms, h(x) is required either to have a stable restricted Hessian [17] or to be strongly convex and smooth [18]. These conditions are the generalizations of the restricted isometry property (RIP) in CS [25], which means that h(x) is likely to satisfy these conditions when A is either a restricted isometry, wellconditioned, or incoherent. In practice, however, A is highly coherent because the dictionary pair is typically dense to reduce the mismatch in (7).
To illustrate how the GraSP and GraHTP algorithms fail to solve (23) when A is highly coherent, consider the real counterpart of ∇h(x). The real counterpart \(\nabla h(\mathbf {x}_{\mathrm {R}})\in \mathbb {R}^{2B}\) is
which follows from \(\nabla \log \Phi (\mathbf {a}_{\mathrm {R}}^{\mathrm {T}}\mathbf {x}_{\mathrm {R}})=\lambda (\mathbf {a}_{\mathrm {R}}^{\mathrm {T}}\mathbf {x}_{\mathrm {R}})\mathbf {a}_{\mathrm {R}}\) and ∇∥x_{R}∥^{2}=2x_{R} where λ(·)=ϕ(·)⊘Φ(·) is the inverse Mills ratio function^{Footnote 2}. Then, the following observation holds from directly computing ∇h(x_{i}), whose real and imaginary parts are the ith and (i+B)th elements of ∇h(x_{R}), respectively.
Observation 1
∇h(x_{i})=∇h(x_{j}) if a_{i}=a_{j} and x_{i}=x_{j}.
However, Observation 1 is meaningless because a_{i}≠a_{j} unless i=j. To establish a meaningful observation, consider the coherence between a_{i} and a_{j}, which reflects the proximity between a_{i} and a_{j} according to [26,27]
Then, using the ηcoherence band, which is [26]
where η∈(0,1), we establish the following conjecture when η is sufficiently large.
Conjecture 1
∇h(x_{i})≈∇h(x_{j}) if j∈B_{η}(i) and x_{i}=x_{j}.
At this point, we use Conjecture 1 to illustrate how the GraSP and GraHTP algorithms fail to estimate supp(x^{∗}) from (28) by naive hard thresholding when A is highly coherent. To proceed, consider the following example, which assumes that x^{∗} and \(\hat {\mathbf {Y}}\) are realized with x representing the current estimate of x^{∗} so as to satisfy
 1)
\(i=\underset {k\in \{1, 2, \dots, B\}}{\text {argmax}}\ x_{k}^{*}\)
 2)
\(i=\underset {k\in \{1, 2, \dots, B\}}{\text {argmax}}\ \nabla h(x_{k})\)
 3)
\(\mathcal {J}\cap \text {supp}(\mathbf {x}^{*})=\emptyset \)
where i is the index corresponding to the largest element of the ground truth^{Footnote 3} virtual channel x^{∗}, and
is the byproduct of i. Here, \(\mathcal {J}\) is called the byproduct of i because
which follows from Conjecture 1, holds despite \(x_{j}^{*}=0\) for all \(j\in \mathcal {J}\). In other words, the byproduct of i refers to the fact that ∇h(x_{i}) and ∇h(x_{j}) are indistinguishable for all \(j\in \mathcal {J}\) according to (34), but the elements of x^{∗} indexed by \(\mathcal {J}\) are 0 according to 3). Therefore, when we attempt to estimate supp(x^{∗}) by hard thresholding ∇h(x), the indices in \(\mathcal {J}\) will likely be erroneously selected as the estimate of supp(x^{∗}) because ∇h(x_{j}) and the maximum element of ∇h(x), which is ∇h(x_{i}) according to 2), are indistinguishable for all \(j\in \mathcal {J}\).
To illustrate how (28) cannot estimate supp(x^{∗}) when A is highly coherent, consider another example where ∇h(x) and T_{L}(∇h(x)) are shown in Figs. 2 and 3, respectively. In this example, supp(x^{∗}) is widely spread, whereas most of supp(T_{L}(∇h(x))) are in the coherence band of the index of the maximum element of ∇h(x). This shows that hard thresholding ∇h(x) is not sufficient to distinguish whether an index is the ground truth index or the byproduct of another index. To solve this problem, we propose the BMS hard thresholding technique.
The BMS hard thresholding function T_{BMS,L}(·) is an Lterm hard thresholding function, which is proposed based on Conjecture 1. The BMS hard thresholding technique is presented in Algorithm 1. Line 3 selects the index of the maximum element of ∇h(x) from the unchecked index set as the current index. Line 4 constructs the byproduct testing set. Line 5 checks whether the current index is greater than the byproduct testing set. In this paper, Line 5 is referred to as the band maximum criterion. If the band maximum criterion is satisfied, the current index is selected as the estimate of supp(x^{∗}) in Line 6. Otherwise, the current index is not selected as the estimate of supp(x^{∗}) because the current index is likely to be the byproduct of another index rather than the ground truth index. Line 8 updates the unchecked index set.
Note that Algorithm 1 is a hard thresholding technique applied to ∇h(x). If the BMS hard thresholding technique is applied to x+κ∇h(x) where κ is the step size, ∇h(x) is replaced by x+κ∇h(x) in the input, output, and Lines 3, 5, and 10 of Algorithm 1. This can be derived using the same logic based on Conjecture 1. Now, we propose the BMSGraSP and BMSGraHTP algorithms to solve (23).
The BMSGraSP and BMSGraHTP algorithms are the variants of the GraSP and GraHTP algorithms, respectively. The difference between the BMSbased and nonBMSbased algorithms is that the hard thresholding function is T_{BMS,L}(·) instead of T_{L}(·). The BMSGraSP and BMSGraHTP algorithms are presented in Algorithms 2 and 3, respectively. Lines 3, 4, and 5 of Algorithms 2 and 3 roughly proceed based on the same logic. Line 3 computes the gradient of the objective function. Line 4 selects \(\mathcal {I}\) from the support of the hard thresholded gradient of the objective function. Line 5 maximizes the objective function subject to the support constraint. This can be solved using convex optimization because the objective function and support constraint are concave and convex, respectively. In addition, b is hard thresholded in Line 6 of Algorithm 2 because b is at most 3Lsparse. A natural halting condition of Algorithms 2 and 3 is to halt when the current and previous \(\text {supp}(\tilde {\mathbf {x}})\) are the same. The readers who are interested in a more indepth analyses of the GraSP and GraHTP algorithms are referred to [17] and [18], respectively.
Remark 1
Instead of hard thresholding b, we can solve
which is a convex optimization problem, to obtain \(\tilde {\mathbf {x}}\) in Line 6 of Algorithm 2. This is the debiasing variant of Algorithm 2 [17]. The advantage of the debiasing variant of Algorithm 2 is a more accurate estimate of x^{∗}. However, the complexity is increased, which is incurred by solving (35).
Remark 2
In this paper, we assume that only h(x) and ∇h(x) are required at each iteration to solve (23) using Algorithms 2 and 3, which can be accomplished when the first order method is used to solve convex optimization problems in Line 5 of Algorithms 2 and 3. An example of such first order method is the gradient descent method with the backtracking line search [24].
Fast implementation via FFT
In practice, the complexity of Algorithms 2 and 3 is demanding because h(x) and ∇h(x) are required at each iteration, which are highdimensional functions defined on \(\mathbb {C}^{B}\) where B≫MN. In recent works on channel estimation and data detection in the mmWave band [14,15,28], the FFTbased implementation is widely used because H can be approximated by (7) using overcomplete DFT matrices. In this paper, an FFTbased fast implementation of h(x) and ∇h(x) is proposed, which is motivated by [14,15,28].
To facilitate the analysis, we convert the summations in h(x) and ∇h(x_{R}) to matrixvector multiplications by algebraically manipulating (21) and (30). Then, we obtain
where we see that the bottlenecks of h(x) and ∇h(x) come from the matrixvector multiplications involving A_{R} and \(\mathbf {A}_{\mathrm {R}}^{\mathrm {T}}\) resulting from the large size of A. For example, the size of A is 5120×65536 in Section 5 where M=N=64,B_{RX}=B_{TX}=256, and T=80.
To develop an FFTbased fast implementation of the matrixvector multiplications involving A_{R} and \(\mathbf {A}_{\mathrm {R}}^{\mathrm {T}}\), define \(\mathbf {c}_{\mathrm {R}}\in \mathbb {R}^{2MT}\) as \(\mathbf {c}_{\mathrm {R}}=\lambda (\sqrt {2\rho }\hat {\mathbf {y}}_{\mathrm {R}}\odot \mathbf {A}_{\mathrm {R}}\mathbf {x}_{\mathrm {R}})\odot \sqrt {2\rho }\hat {\mathbf {y}}_{\mathrm {R}}\) from (37) with \(\mathbf {c}\in \mathbb {C}^{MT}\) being the complex form of c_{R}. From the fact that
we now attempt to compute Ax and A^{H}c via the FFT. Then, Ax and A^{H}c are unvectorized according to
where \(\mathbf {X}=\text {unvec}(\mathbf {x})\in \mathbb {C}^{B_{\text {RX}}\times B_{\text {TX}}}\) and \(\mathbf {C}=\text {unvec}(\mathbf {c})\in \mathbb {C}^{M\times T}\). If the matrix multiplication involving S can be implemented using the FFT, e.g., ZadoffChu (ZC) [29] or DFT [11] training sequence, (40) and (41) can be implemented using the FFT because A_{RX} and A_{TX} are overcomplete DFT matrices. For example, each column of A_{TX}X^{H} in (40) can be computed using the B_{TX}point FFT with pruned outputs, i.e., retaining only N outputs, because we constructed A_{TX} as an overcomplete DFT matrix.
In particular, the matrix multiplications involving A_{TX},S^{H}, and A_{RX} in (40) can be implemented with B_{TX}point FFT with pruned outputs repeated B_{RX} times, Tpoint IFFT with pruned inputs repeated B_{RX} times, and B_{RX}point FFT with pruned outputs repeated T times, respectively.^{Footnote 4} Using the same logic, the matrix multiplications involving \(\mathbf {S}, \mathbf {A}_{\text {TX}}^{\mathrm {H}}\), and \(\mathbf {A}_{\text {RX}}^{\mathrm {H}}\) in (41) can be implemented using Tpoint FFT with pruned outputs repeated M times, B_{TX}point IFFT with pruned inputs repeated M times, and B_{RX}point IFFT with pruned inputs repeated B_{TX} times, respectively. Therefore, the complexity of the FFTbased implementation of (40) and (41) is O(B_{RX}B_{TX} logB_{TX}+B_{RX}T logT+TB_{RX} logB_{RX}) and O(MT logT+MB_{TX} logB_{TX}+B_{TX}B_{RX} logB_{RX}), respectively.
To illustrate the efficiency of the FFTbased implementation of (40) and (41), M/N,M/B_{RX},M/B_{TX}, and M/T are assumed to be fixed. Then, the complexity of the FFTbased implementation of Ax and A^{H}c is O(M^{2} logM), whereas the complexity of directly computing Ax and A^{H}c is O(M^{4}). Therefore, the complexity of Algorithms 2 and 3 is reduced when h(x) and ∇h(x) are implemented using the FFT operations.
Remark 3
Line 5 of Algorithms 2 and 3 is equivalent to solving
where
and \(\mathbf {A}_{\mathcal {I}}\in \mathbb {C}^{MT\times \mathcal {I}}\) is the collection of a_{i} with \(i\in \mathcal {I}\). Therefore, only \(h_{\mathcal {I}}(\mathbf {x}_{\mathcal {I}})\) and \(\nabla h_{\mathcal {I}}(\mathbf {x}_{\mathcal {I}})\) are required in Line 5 of Algorithms 2 and 3, which are lowdimensional functions defined on \(\mathbb {C}^{\mathcal {I}}\) where \(\mathcal {I}=O(L)\). If \(h_{\mathcal {I}}(\mathbf {x}_{\mathcal {I}})\) and \(\nabla h_{\mathcal {I}}(\mathbf {x}_{\mathcal {I}})\) are computed based on the same logic in (40) and (41) but A replaced by \(\mathbf {A}_{\mathcal {I}}\), the complexity of Algorithms 2 and 3 is reduced further because the size of the FFT is reduced in Line 5.
Results and discussion
In this section, we evaluate the performance of Algorithms 2 and 3 from different aspects in terms of the accuracy, achievable rate, and complexity. Throughout this section, we consider a mmWave massive MIMO system with 1 bit ADCs, whose parameters are M=N=64 and T=80. The rest vary from simulation to simulation, which consist of B_{RX},B_{TX}, and L. In addition, we consider S, whose rows are the circular shifts of the ZC training sequence of length T as in [15,33]. Furthermore, H is either random or deterministic. If H is random, \(\alpha _{\ell }\sim \mathcal {C}\mathcal {N}(0, 1), \theta _{\text {RX}, \ell }\sim \text {unif}([\pi /2, \pi /2])\), and θ_{TX,ℓ}∼unif([−π/2,π/2]) are independent. Otherwise, we consider different H from simulation to simulation.
The MSEs of \(\{\alpha _{\ell }\}_{\ell =1}^{L}, \{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\), and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) are
where \((\tilde {\alpha }_{\ell }, \tilde {\theta }_{\text {RX}, \ell }, \tilde {\theta }_{\text {TX}, \ell })\) corresponds to some nonzero element of \(\tilde {\mathbf {X}}=\text {unvec}(\tilde {\mathbf {x}})\in \mathbb {C}^{B_{\text {RX}}\times B_{\text {TX}}}\). The normalized MSE (NMSE) of H is
where \(\tilde {\mathbf {H}}=\mathbf {A}_{\text {RX}}\tilde {\mathbf {X}}\mathbf {A}_{\text {TX}}\). In (45)–(48), the symbol \(\tilde {\hphantom {\mathbf {y}}}\) is used to emphasize that the quantity is an estimate.
Throughout this section, we consider the debiasing variant of Algorithm 2. The halting condition of Algorithms 2 and 3 is to halt when the current and previous \(\text {supp}(\tilde {\mathbf {x}})\) are the same. The gradient descent method is used to solve convex optimization problems, which consist of (35) and Line 5 of Algorithms 2 and 3. The backtracking line search is used to compute the step size in the gradient descent method and κ in Line 3 of Algorithm 3. In addition, η is selected so that Conjecture 1 is satisfied. In this paper, we select the maximum η satisfying
For example, the maximum η satisfying (49) is η=0.6367 when B_{RX}=2M and B_{TX}=2N. The channel estimation criterion of Algorithms 2 and 3 is either MAP or ML, which depends on whether H is random or deterministic. To compare the BMSbased and nonBMSbased algorithms, the performance of the GraSP and GraHTP algorithms is shown as a reference in Figs. 4, 5, 6, and 7. The GraSP and GraHTP algorithms forbid B_{RX}≫M and B_{TX}≫N because the GraSP and GraHTP algorithms diverge when A is highly coherent. Therefore, the parameters are selected as B_{RX}=M and B_{TX}=N when the GraSP and GraHTP algorithms are implemented. Such B_{RX} and B_{TX}, however, are problematic because the mismatch in (7) is inversely proportional to B_{RX} and B_{TX}.
In Figs. 4 and 5, we compare the accuracy of the BMSbased and band excludingbased (BEbased) algorithms at different SNRs using \(\text {MSE}(\{\alpha _{\ell }\}_{\ell =1}^{L}), \text {MSE}(\{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}), \text {MSE}(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L})\), and NMSE(H). The BE hard thresholding technique was proposed in [26], which was applied to the orthogonal matching pursuit (OMP) algorithm [34]. In this paper, we apply the BE hard thresholding technique to the GraSP algorithm, which results in the BEGraSP algorithm. In this example, B_{RX}=B_{TX}=256 for the BMSbased and BEbased algorithms. We assume that L=8 and H is deterministic where \(\alpha _{\ell }=(0.8+0.1(\ell 1))e^{j\frac {\pi }{4}(\ell 1)}\). However, \(\{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L}\) and \(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L}\) vary from simulation to simulation, which are either widely spread (Fig. 4) or closely spread (Fig. 5). In Figs. 4 and 5, the notion of widely and closely spread paths refer to the fact that the minimum 2norm distance between the paths are either relatively far or close, i.e., mini≠j∥(θ_{RX,i}−θ_{RX,j},θ_{TX,i}−θ_{TX,j})∥_{2} of Fig. 4, which is \(\sqrt {(\pi /18)^{2}+(\pi /18)^{2}}\), is greater than that of Fig. 5, which is \(\sqrt {(\pi /36)^{2}+(\pi /36)^{2}}\). The path gains, AoAs, and AoDs are assumed to be deterministic because the CRB is defined for deterministic parameters only [35]. A variant of the CRB for random parameters is known as the Bayesian CRB, but adding the Bayesian CRB to our work is left as a future work because applying the Bayesian CRB to nonlinear measurements, e.g., 1 bit ADCs, is not as straightforward.
According to Figs. 4 and 5, the BMSbased algorithms succeed to estimate both widely spread and closely spread paths, whereas the BEbased algorithms fail to estimate closely spread paths. This follows because the BE hard thresholding technique was derived based on the assumption that supp(x^{∗}) is widely spread. In contrast, the BMS hard thresholding technique is proposed based on Conjecture 1 without any assumption on supp(x^{∗}). This means that when supp(x^{∗}) is closely spread, the BE hard thresholding technique cannot properly estimate supp(x^{∗}) because the BE hard thresholding technique, by its nature, excludes the elements near the maximum element of x^{∗} from its potential candidate. The BMS hard thresholding technique, in contrast, uses the elements near the maximum element of x^{∗} to construct the byproduct testing set only, i.e., Line 4 of Algorithm 1. Therefore, the BMSbased algorithms are superior to the BEbased algorithms when the paths are closely spread. The CramérRao bounds (CRBs) of \(\text {MSE}(\{\alpha _{\ell }\}_{\ell =1}^{L}), \text {MSE}(\{\theta _{\text {RX}, \ell }\}_{\ell =1}^{L})\), and \(\text {MSE}(\{\theta _{\text {TX}, \ell }\}_{\ell =1}^{L})\) are provided, which were derived in [36]. The gaps between the MSEs and their corresponding CRBs can be interpreted as a performance limit incurred by the discretized AoAs and AoDs. To overcome such limit, the AoAs and AoDs must be estimated based on the offgrid method, which is beyond the scope of this paper.
In addition, note that \(\text {MSE}(\{\alpha _{\ell }\}_{\ell =1}^{L})\) and NMSE(H) worsen as the SNR enters the high SNR regime. To illustrate why x^{∗} cannot be estimated in the high SNR regime in 1 bit ADCs, note that
in the high SNR regime with c>0, which means that x^{∗} and cx^{∗} are indistinguishable because the magnitude information is lost by 1 bit ADCs. The degradation of the recovery accuracy in the high SNR regime with 1 bit ADCs is an inevitable phenomenon, as observed from other previous works on lowresolution ADCs [11,14,15,33,37].
In Figs. 6 and 7, we compare the performance of Algorithms 2 and 3, and other estimators when H is random. The Bernoulli GaussianGAMP (BGGAMP) algorithm [15] is an iterative approximate MMSE estimator of x^{∗}, which was derived based on the assumption that \(x_{i}^{*}\) is distributed as \(\mathcal {C}\mathcal {N}(0, 1)\) with probability L/B but 0 otherwise, namely, the BG distribution. The fast iterative shrinkagethresholding algorithm (FISTA) [38] is an iterative MAP estimator of x^{∗}, which was derived based on the assumption that the logarithm of the PDF of x^{∗} is g_{FISTA}(x)=−γ∥x∥_{1} ignoring the constant factor, namely, the Laplace distribution. Therefore, the estimate of x^{∗} is
which is solved using the accelerated proximal gradient descent method [38]. The regularization parameter γ is selected so that the expected sparsity of (51) is 3L for a fair comparison, which was suggested in [17]. In this example, L=4, whereas B_{RX} and B_{TX} vary from algorithm to algorithm. In particular, we select B_{RX}=B_{TX}=256 for Algorithms 2, 3, and the FISTA, whereas B_{RX}=M and B_{TX}=N for the BGGAMP algorithm.
In Fig. 6, we compare the accuracy of Algorithms 2, 3, and other estimators at different SNRs using NMSE(H). According to Fig. 6, Algorithms 2 and 3 outperform the BGGAMP algorithm and FISTA as the SNR enters the medium SNR regime. The accuracy of the BGGAMP algorithm is disappointing because the mismatch in (7) is inversely proportional to B_{RX} and B_{TX}. However, increasing B_{RX} and B_{TX} is forbidden because the BGGAMP algorithm diverges when A is highly coherent. The accuracy of the FISTA is disappointing because the Laplace distribution does not match the distribution of x^{∗}. Note that (23), which is the basis of Algorithms 2 and 3, is indeed the MAP estimate of x^{∗}, which is in contrast to the FISTA. According to Fig. 6, NMSE(H) worsens as the SNR enters the high SNR regime, which follows from the same reason as in Figs. 4 and 5.
In Fig. 7, we compare the achievable rate lower bound of Algorithms 2, 3, and other estimators at different SNRs when the precoders and combiners are selected based on \(\tilde {\mathbf {H}}\). The achievable rate lower bound shown in Fig. 7 is presented in [15], which was derived based on the Bussgang decomposition [13] in conjunction with the fact that the worstcase noise is Gaussian. According to Fig. 7, Algorithms 2 and 3 outperform the BGGAMP algorithm and FISTA, which is consistent with the result in Fig. 6.
In Fig. 8, we compare the complexity of Algorithms 2, 3, and other estimators at different B_{RX} and B_{TX} when H is random. To analyze the complexity, note that Algorithms 2, 3, and the FISTA require h(x) and ∇h(x) at each iteration, whose bottlenecks are Ax and A^{H}c, respectively, while the BGGAMP algorithm requires Ax and A^{H}c at each iteration. Therefore, the complexity is measured based on the number of complex multiplications performed to compute Ax and A^{H}c, which are implemented based on the FFT. In this example, L=4, whereas SNR is either 0 dB or 10 dB.
In this paper, the complexity of the BGGAMP algorithm is used as a baseline because the BGGAMP algorithm is widely used. The normalized complexity is defined as the number of complex multiplications performed divided by the periteration complexity of the BGGAMP. For example, the normalized complexity of the FISTA with B_{RX}=B_{TX}=256 is 160 when the complexity of the FISTA with B_{RX}=B_{TX}=256 is equivalent to the complexity of the 160iteration BGGAMP algorithm with B_{RX}=B_{TX}=256. In practice, the BGGAMP algorithm converges in 15 iterations when A is incoherent [39]. In this paper, an algorithm is said to be as efficient as the BGGAMP algorithm when the normalized complexity is below the target threshold, which is 15. As a sidenote, our algorithms, namely, the BMSGraSP and BMSGraHTP, requires 2.1710 and 2.0043 iterations in average, respectively, across the entire SNR range.
According to Fig. 8, the complexity of the FISTA is impractical because the objective function of (51) is a highdimensional function defined on \(\mathbb {C}^{B}\) where B≫MN. In contrast, the complexity of Algorithms 2 and 3 is dominated by (42), whose objective function is a lowdimensional function defined on \(\mathbb {C}^{\mathcal {I}}\) where \(\mathcal {I}=O(L)\). The normalized complexity of Algorithms 2 and 3 is below the target threshold when B_{RX}≥192 and B_{TX}≥192. Therefore, we conclude that Algorithms 2 and 3 are as efficient as the BGGAMP algorithm when B_{RX}≫M and B_{TX}≫N.
Conclusions
In the mmWave band, the channel estimation problem is converted to a sparsityconstrained optimization problem, which is NPhard to solve. To approximately solve sparsityconstrained optimization problems, the GraSP and GraHTP algorithms were proposed in CS, which pursue the gradient of the objective function. The GraSP and GraHTP algorithms, however, break down when the objective function is illconditioned, which is incurred by the highly coherent sensing matrix. To remedy such break down, we proposed the BMS hard thresholding technique, which is applied to the GraSP and GraHTP algorithms, namely, the BMSGraSP and BMSGraHTP algorithms, respectively. Instead of directly hard thresholding the gradient of the objective function, the BMSbased algorithms test whether an index is the ground truth index or the byproduct of another index. We also proposed an FFTbased fast implementation of the BMSbased algorithms, whose complexity is reduced from O(M^{4}) to O(M^{2} logM). In the simulation, we compared the performance of the BMSbased, BEbased, BGGAMP, and FISTA algorithms from different aspects in terms of the accuracy, achievable rate, and complexity. The BMSbased algorithms were shown to outperform other estimators, which proved to be both accurate and efficient. Our algorithms, however, addressed only the flat fading scenario, so an interesting future work would be to propose a lowcomplexity channel estimator capable of dealing with the wideband scenario.
Methods/experimental
The aim of this study is to propose an accurate yet efficient channel estimator for mmWave massive MIMO systems with 1 bit ADCs. Our channel estimator was proposed based on theoretical analysis. To be specific, we adopted and modified CS algorithms to exploit the sparse nature of the mmWave virtual channels. In addition, we carefully analyzed the proposed channel estimator to reduce the complexity. To verify the accuracy and complexity of the proposed channel estimator, we conducted extensive (MonteCarlo) simulations.
Notes
 1.
In practice, X^{∗} may be either approximately sparse or exactly sparse to formulate (10). If X^{∗} is approximately sparse, the leakage effect is taken into account so the mismatch in (7) becomes 0, namely, vec(E)=0_{MT}. In contrast, the mismatch in (7) must be taken into account with a nonzero E when X^{∗} is exactly sparse. Fortunately, E is inversely proportional to B_{RX} and B_{TX}. Therefore, we adopt the latter definition of X^{∗} and propose our algorithm ignoring E assuming that B_{RX}≫M and B_{TX}≫N. The performance degradation from E will become less as B_{RX} and B_{TX} become sufficiently large.
 2.
The elementwise vector division in the inverse Mills ratio function is meaningless because the arguments of the inverse Mills ratio function are scalars in (30). The reason we use the elementwise vector division in the inverse Mills ratio function will become clear in (37), whose arguments are vectors.
 3.
We use the term “ground truth” to emphasize that the ground truth x^{∗} is the true virtual channel which actually gives the quantized received signal \(\hat {\mathbf {Y}}\) from (16), whereas x merely represents the point where ∇h(x) is computed to estimate supp(x^{∗}) via hard thresholding.
 4.
Abbreviations
 ADC:

Analogtodigital converter
 AoA:

Angleofarrival
 AoD:

Angleofdeparture
 AWGN:

Additive white Gaussian noise
 BE:

Band excluding
 BGGAMP:

Bernoulli Gaussiangeneralized approximate message passing
 BLMMSE:

Bussgang linear minimum mean squared error
 BMS:

Band maximum selecting
 CoSaMP:

Compressive sampling matching pursuit
 CRB:

CramérRao bound
 CS:

Compressive sensing
 DFT:

Discrete fourier transform
 FFT:

Fast Fourier transform
 FISTA:

Fast iterative shrinkagethresholding algorithm
 GAMP:

Generalized approximate message passing
 GECSR:

Generalized expectation consistent signal recovery
 GraHTP:

Gradient hard thresholding pursuit
 GraSP:

Gradient support pursuit
 HTP:

Hard thresholding pursuit
 i.i.d.:

Independent and identically distributed
 LOS:

Lineofsight
 MAP:

Maximum a posteriori
 MIMO:

Multipleinput multipleoutput
 mmWave:

Millimeter wave
 nML:

Near maximum likelihood
 NMSE:

Normalized mean squared error
 OMP:

Orthogonal matching pursuit
 RIP:

Restricted isometry property
 SNR:

Signaltonoise ratio
 ULA:

Uniform linear array
 ZF:

ZadoffChu
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IK and JC led the research and wrote the manuscript. Both authors read and approved the final manuscript.
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Correspondence to Junil Choi.
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Kim, I., Choi, J. Channel estimation via gradient pursuit for mmWave massive MIMO systems with onebit ADCs. J Wireless Com Network 2019, 289 (2019). https://doi.org/10.1186/s136380191623x
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Keywords
 MmWave
 Massive MIMO
 1 bit ADC
 MAP channel estimation
 GraSP
 GraHTP
 CoSaMP
 HTP
 CS
 BMS hard thresholding technique
 FFT