Due to its enormous potential for overcoming the spectrum crunch, millimeter wave (mmWave) communications has become a promising technology for future wireless cellular systems . However, mmWave communication is vulnerable to blockages due to the limited scattering effects at mmWave frequencies. Furthermore, in conventional mmWave communication systems, the propagation environment is uncontrollable, and therefore, the quality of service (QoS) is significantly degraded when the line-of-sight (LoS) links are blocked.
Recently, intelligent reflecting surfaces (IRSs) have been incorporated into wireless communication systems, mainly due to their capability of customizing favorable wireless propagation environments . Equipped with a large number of low-cost passive reflective elements, e.g., dipoles and phase shifters, IRSs enable the adaptation of wireless propagation environments with limited power consumption . This property of IRSs can be exploited in mmWave systems . Specifically, when the direct LoS links between the transceivers are blocked, the IRSs can reflect the incident signals to provide an effective virtual LoS link for mmWave communications. With well-designed reflecting IRS elements, the communication performance can be further enhanced via programmable and reconfigurable signal reﬂections [5, 3].
Nevertheless, the introduction of IRSs brings new challenges, among which the acquisition of channel state information (CSI) may be the most demanding task. In particular, in addition to the conventional direct channel between the base station (BS) and the user equipment (UE), two IRS-assisted channels need to be estimated, i.e., the BS-IRS channel and IRS-UE channel. Furthermore, since radio frequency (RF) chains are not available at the passive IRSs, it is not possible to estimate the two IRS-assisted channels directly by regarding the IRS as a conventional RF chain-driven transceiver. Therefore, the classical channel estimation techniques are not applicable in the newly-emerged IRS-assisted communications systems .
Recently, several works have investigated channel estimation in IRS-assisted wireless systems [10, 5, 6, 8, 7, 9]. The authors of  characterized the minimum pilot sequence length for channel estimation in IRS-assisted multi-user multiple-input single-output (MISO) systems based on the least square (LS) criterion. A two-timescale estimation scheme was proposed in , where the high-dimensional BS-IRS channel and the low-dimensional IRS-UE channel are estimated in a large timescale and a small timescale, respectively. To further reduce the pilot overhead, by exploiting the sparsity of the channels, compressive sensing techniques were utilized in [7, 9, 8] to solve the estimation problem. However, the algorithms proposed in these existing works are only applicable in wireless systems with single-antenna users. Multiple-input multiple-output (MIMO) systems were studied in , where a channel estimation algorithm for IRS-assisted systems was developed based on parallel factor decomposition (PARAFAC). While this approach, designed for sub-6 GHz bands, is also applicable in mmWave MIMO systems, a significant performance loss is expected as the unique channel characteristics of mmWave MIMO systems are not considered, e.g., the sparsity of mmWave channels.
In this paper, we propose a novel channel estimation algorithm for IRS-assisted point-to-point mmWave MIMO systems. By exploiting the sparsity of mmWave channels, we formulate the channel estimation problem as a non-convex optimization problem with fixed-rank constraints. Then, we apply the alternating minimization principle to divide the original problem into two subproblems, which target the estimation of the BS-IRS channel and the IRS-UE channel, respectively. Finally, manifold optimization (MO) is employed to address the non-convex rank constraint and the subproblems are solved iteratively. The developed algorithm guarantees monotonic convergence to a locally optimal solution. Simulations results clearly illustrate the performance improvement of the proposed MO-based estimation (MO-EST) algorithm over two benchmark schemes including the state-of-the-art PARAFAC approach in . We also demonstrate the robustness of the proposed MO-EST algorithm with respect to different channel sparsity levels.
Notations: In this paper, the imaginary unit of a complex number is denoted by . The set of nonnegative integers is denoted by . denotes the set of all
complex-valued matrices. Matrices and vectors are denoted by boldface capital and lower-case letters, respectively. The-th element of vector is denoted by . denotes the identity matrix. , , , , , , and denote the conjugate, transpose, conjugate transpose, rank, trace, vectorization, and Frobenius norm of a matrix, respectively. The Khatri-Rao matrix product is represented by . and denote the real part of a complex number and expectation, respectively. is a diagonal matrix with the entries of on its main diagonal.
denotes the circularly symmetric complex Gaussian distribution with zero mean and covariance matrix.
Ii System Model
As shown in Fig. 1, we consider the downlink of an IRS-assisted point-to-point mmWave MIMO communication system. BS and UE are equipped with uniform planar arrays (UPAs) consisting of and antennas, respectively. In addition, a passive IRS that employs phase shifters is deployed in the network to facilitate mmWave communications.
Ii-a Channel Estimation Protocol
The channel estimation protocol adopted in this paper is shown in Fig. 2. Specifically, the time available for estimation is divided into blocks, and each block consists of pilot symbol durations. The reflection coefficient vectors of the IRS may be different in different blocks but are constant within one block [7, 10]. The pilots received at the UE in consecutive time slots of the -th block, denoted by , are compactly written as
where the BS-UE, BS-IRS, and IRS-UE channel matrices are denoted by , , and , respectively. denotes the received Gaussian noise with , . is the training reflection coefficient vector in the -th block. Since the IRS is implemented by phase shifters , the reflecting elements can only change the phases of the received signals, i.e., . Furthermore, we assume that the pilot sequences transmitted by the BS are orthogonal to each other, namely, . Thus, after removing the pilot symbols at the UE, we have
Remark 1: By switching off all IRS elements, the direct BS-UE channel can be estimated via traditional algorithms, e.g., . Therefore, in this paper, we assume that is known and focus on the estimation of and , which is the main challenge in channel estimation for IRS-assisted systems. Thus, the relevant part of is given by
We further concatenate the signals received in all sub-frames as , which leads to
where and .
Ii-B MmWave Channel Model
Before formulating the estimation problem, we introduce the channel model for mmWave propagation. The mmWave propagation environment is well characterized by the Saleh-Valenzuela model , which is given by
where , (), and () denote the complex gain, azimuth (elevation) angle of arrival (AoA), and azimuth (elevation) angle of departure (AoD) of the -th path of the IRS-UE channel. Similarly, , (), and () denote the complex gain, azimuth (elevation) AoA, and azimuth (elevation) AoD of the -th path of the BS-IRS channel. In addition, and denote the receive and transmit array response vectors, respectively. The array response vector of a half-wavelength spaced UPA with elements is given as follows
where and are the antenna element indices in the 2-dimensional plane. An important property of mmWave channels is presented in the following lemma.
Suppose , then we have
Proof: Please refer to Appendix A.
Remark 2: In Section III, the numbers of paths and are assumed to be known at the BS, and hence, the achieved performance is an upper bound for the scenario where and are not available or cannot be accurately estimated. In practice, the numbers of paths can be estimated via low-complexity compressive sensing methods, e.g., the orthogonal matching pursuit (OMP) method in . In Section IV, we consider the case where and are not exactly known to evaluate the robustness of the proposed algorithm with respect to a mismatched number of paths.
Ii-C Problem Formulation
According to and can be obtained based on the LS criterion. By taking the sparsity of the channels into account and leveraging Lemma 1, we formulate the channel estimation problem in IRS-assisted mmWave MIMO systems as follows
where and denote the estimates of and , respectively. Due to the fixed-rank constraints, problem (8) is a highly non-convex problem and a globally optimal solution would entail a very high computational complexity. Besides, the coupling of the two optimization variables in the objective function further complicates the problem. Thus, in the following, we propose an efficient algorithm that achieves a locally optimal solution of problem (8).
Iii Proposed MO-EST Algorithm
To tackle the coupling of the optimization variables in (8), we first decouple the two variables by applying the alternating minimization principle [12, 3]. Specifically, we first fix and minimize with respect to the single variable . The corresponding subproblem is given by
To address the non-convex fixed-rank constraint, we apply the MO technique to solve problem (9). Different from traditional compressive sensing methods, e.g., the OMP and basis pursuit (BP) methods, the proposed MO-based algorithm guarantees convergence to a locally optimal solution of problem (9).
Iii-a Preliminaries of MO
First, we note that the feasible set of problem (9) can be represented as a fixed-rank manifold
which is a smooth complex Riemannian manifold. The Riemannian optimization method for the real-valued fixed-rank manifold has been studied in . By extending the definitions of the fixed-rank manifold to the complex domain, we introduce the key operations that are necessary for the Riemannian optimization method for .
1) Inner product: By endowing the complex space with the Euclidean metric, the inner product between two points is defined as
2) Tangent space: For a point on the manifold, its tangent space , which is composed of all the vectors that tangentially pass through , is given by 
where and denote the semi-unitary matrices containing the first left and right singular vectors of , respectively. In addition, and lie in the null spaces of and , respectively, i.e., .
3) Orthogonal projection: The orthogonal projection of a point onto the tangent space of , , is given by
where , , , and .
4) Retraction: Retraction is a mapping from the tangent space to the manifold itself. Particularly, for a point
, the retraction operation can be formulated via a truncated singular value decomposition (SVD)
where , , and are the ordered singular values, left singular vectors, and right singular vectors of , respectively.
Iii-B Conjugate Gradient Method on
With the basic definitions of the key operations on at hand, we can formulate the counterpart of the classic conjugate gradient (CG) algorithm in the Euclidean space on the manifold [14, 12]. The main idea is illustrated in Fig. 3. In each iteration, we first find a local minimizer in the tangent space, and then project the obtained point back to the manifold. For problem (9), the update rule of the CG method is given by 
where denotes the Armijo backtracking step size in the -th iteration [15, Eq. (59)], is the estimate in the -th iteration, and is the updated local minimizer in the -th iteration. In addition, is the conjugate direction in the -th iteration, given by
where the first term is the negative Riemannian gradient representing the steepest descent direction of the objective function in the tangent space , and represents the chosen Polak-Ribiere parameter [15, p. 42]. Since the conjugate direction in the previous iteration does not lie in , the sum operation in (16) can not be performed directly. To this end, we introduce the vector transport operation to project to the current tangent space . According to (13), the vector transport for is given by
Therefore, the remaining task to determine the conjugate direction in (16) is to derive the Riemannian gradient. Since is embedded in , the Riemannian gradient is obtained by projecting the Euclidean gradient onto the tangent space , i.e.,
The Euclidean gradient of with respect to is given by
After updating the local minimizer in the -th iteration according to (15), we need to map this minimizer back to to obtain the estimate in the -th iteration, which is achieved by the retraction operation shown in (14), i.e.,
The proposed generalized CG method for the fixed-rank manifold, referred to as the CG-MO algorithm, is summarized in Algorithm 1, where is the convergence threshold.
Iii-C Estimation of
In this subsection, we present the optimization of for given . First, we establish the following equality
where . The subproblem that optimizes for given is then formulated as follows
Thus, the CG-MO algorithm is also applicable to solving problem (22). The main modification compared to the optimization of is the replacement of the Euclidean gradient in (19) by the Euclidean gradient of with respect to , which is given by
Finally, the overall estimation scheme is referred to as the MO-EST algorithm and summarized in Algorithm 2. With the proposed algorithm, the objective values achieved by the sequence form a non-increasing sequence that converges to a stationary value, and any limit point of the sequence is a stationary point of problem (9) .
Iv Simulation Results
In this section, we provide simulation results for performance evaluation of the proposed MO-EST algorithm. The signal-to-noise-ratio (SNR) is defined as. Square UPAs are equipped at both the BS and UE. For both and , the same number of paths are assumed, i.e., . According to the channel model in (5), without loss of generality, we let and represent the indices of the LoS components in and . The complex channel gains are distributed as () and () for . The azimuth and elevation AoAs/AoDs, i.e., () and (
), are generated uniformly distributed in, and , , respectively. For the reflecting elements, we set
and use the discrete Fourier transform (DFT) matrix as. The normalized mean square error (NMSE) is adopted as the performance metric. The NMSE is defined as , where and denote the cascaded channel and its estimate, respectively111As the and are coupled in the received signal , there inevitably exist scaling ambiguities between and . Therefore, the NMSE of is adopted as performance metric to avoid the scaling issues .. The convergence threshold in both Algorithm 1 and 2 is set as . To show the effectiveness of the proposed MO-EST algorithm, the PARAFAC algorithm  is adopted as a benchmark. In addition, by dropping the rank constraints, the LS problems in (9) and (22) can be alternately solved in closed form. This approach is also adopted as a benchmark and is referred to as the ALT-LS algorithm.
In Fig. 4, the NMSE is plotted as a function of SNR when . It can be observed that our proposed MO-EST algorithm achieves a significant performance gain of more than
dB compared to the two benchmark schemes. This is mainly because the proposed MO-EST algorithm exploits the sparsity of the involved mmWave channels. In contrast, the ALT-LS algorithm yields the highest NMSE. This phenomenon highlights the importance of incorporating the rank constraints into the alternating optimization algorithm for channel estimation in IRS-assisted mmWave MIMO systems. Furthermore, as more antennas and reflection elements provide more spatial degrees of freedom for channel estimation, the performance of all three algorithms is improved for larger values of, and .
In Fig. 5, we investigate the impact of the number of paths of the estimated channels, i.e., , when , , and . As can be observed, MO-EST outperforms the two benchmark algorithms for all considered values of . The performance gain is especially significant in the high sparsity regime. This is because the performance gain mainly comes from the exploitation of channel sparsity. As the number of paths of the estimated channels increases, the channel sparsity level decreases. Therefore, the performance gap is larger when is small, which is typically the case for mmWave channels where scattering is very limited.
In Fig. 6, we consider the case where the number of paths, , is not perfectly known for channel estimation and test the robustness of the MO-EST algorithm with respect to the resulting uncertainty. The parameters are set as , , , , and the estimated number of paths is denoted as . As can be observed, with the proposed MO-EST algorithm, the lowest NMSE is achieved when , i.e., the number of paths is perfectly known. In contrast, the performance of the two benchmark algorithms does not depend on the number of paths, and therefore the achieved NMSEs are independent of . In addition, for the MO-EST algorithm, the mismatch between the estimated and the true value of leads to a performance loss, which, nevertheless, is limited especially when . In particular, the channel matrix and its estimate can be decomposed by their SVDs, i.e., and , where (), (), and () denote the ordered singular values, left singular vectors, and right singular vectors, respectively. In order to minimize the objective function in (8) based on the LS criterion, the MO-EST algorithm chooses the largest singular values of and the corresponding singular vectors to be close to the true values while making the remaining singular values small. In other words, the solution obtained by the MO-EST algorithm satisfies , , for , and for , which still maintains a satisfactory estimation performance when . Hence, the proposed MO-EST algorithm is robust with respect to imperfect knowledge of the exact number of paths of the estimated channels.
In this paper, we investigated the channel estimation problem for IRS-assisted mmWave MIMO systems. By exploiting the sparsity of the mmWave channel, a manifold optimization-based alternating optimization algorithm, i.e., the MO-EST algorithm, was developed to effectively estimate the BS-IRS and IRS-UE channels. Simulation results showed the achieved performance improvements compared to two existing benchmark schemes, even when the sparsity level of the channels was not accurately unknown. As a next step, it is of great interest to extend this work to multi-user and broadband scenarios.
According to (6), the receive array response vector can be written as
where and . When , it can be shown that matrix is an Vandermonde matrix, whose column vectors are linearly independent. Therefore, the vectors are also linearly independent and matrix satisfies . Similarly, also satisfies . According to (5), can be expressed as
where is also a rank- matrix. We have the following inequalities
and similarly we can prove .
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