A FrameTheoretic Scheme for
Robust Millimeter Wave Channel Estimation
Abstract
We propose a new scheme for the robust estimation of the \acmmWave channel. Our approach is based on a sparse formulation of the channel estimation problem coupled with a frame theoretic representation of the sensing dictionary. To clarify, under this approach, the combined effect of transmit precoders and receive beamformers is modeled by a single frame, whose design is optimized to improve the accuracy of the sparse reconstruction problem to which the channel estimation problem is ultimately reduced. The optimized sensing dictionary frame is then decomposed via a Kronecker decomposition back into the precoding and beamforming vectors used by the transmitter and receiver. Simulation results illustrate the significant gain in estimation accuracy obtained over state of the art alternatives. As a bonus, the work offers new insights onto the sparse \acmmWave\acMIMO channel estimation problem by casting the tradeoff between correlation and variation range in terms of frame coherence and tightness.
GEVGEVGeneralized Extreme Values \newacronymEVEVExtreme Values \newacronymCSCSCompressed Sensing \newacronymCSICSIchannel state information \newacronymBPBPbasis pursuit \newacronymOMPOMPorthogonal matching pursuit \newacronymBPDNBPDNbasis pursuit denoising \newacronymRIPRIPrestricted isometry property \newacronymETFETFequiangular tight frame \newacronymUNTFUNTFunitnorm tight frames \newacronymSIDCOSIDCOsequential iterative decorrelation via convex optimization \newacronymHSIDCOHSIDCOhungarian pairing SIDCO \newacronymCSIDCOCSIDCOcomplex SIDCO \newacronymQCSIDCOQCSIDCOquadratic complex SIDCO \newacronymSVDSVDsingular value decomposition \newacronymmmWavemmWavemillimeter wave \newacronymMIMOMIMOmultipleinput multipleoutput \newacronymAoAAoAangles of arrival \newacronymAoDAoDangles of departure \newacronymBSBSbase station \newacronymUEUEuser equipment \newacronymAWGNAWGNadditive white Gaussian noise \newacronymNMSENMSEnormalized mean square error \newacronymPTFPTFParseval tight frame \newacronymPDFPDFprobability density function
I Introduction
The increasing demands in terms of higher rate, more access and lower latency at the physical link, coupled with the lack of spectral resources in conventional cellular systems is recently strongly motivating the development of \acmmWave communications [1].
In principle, the larger bandwidths and shorter wave lengths of \acmmWave systems enable the provision of higher communication rates [2], and respectively, the equipping of larger antennas arrays at transmitters and receivers favors the utilization of \acMIMO architectures [3, 4].
In practice, however, hardware costs and other implementation issues challenge the realization of \acmmWave systems, which therefore must be counteracted via dedicatedly designed signal processing methods [5, 6].
In turn, previous work has demonstrated [7, 8, 9, 10, 11, 12] that the efficacy of signal processing in ameliorating the hardware challenges of \acmmWave systems depends highly on the quality of \acCSI. In fact, although hybrid precoding with partial \acCSI has been well studied [13], the substantial performance losses resulting from imperfect/partial \acCSI only further motivate the quest for better methods for channel acquisition [12, 11, 9, 7, 10, 8].
Retrieving complete and accurate \acmmWave \acCSI is challenging in practice due to the rapid variation and severe pathloss experienced under the high operating frequencies.
In answer to this challenge, a sparse formulation of the \acmmWave \acMIMO channel estimation problem was proposed in [7] which allowed the use of \acCS [14] for the scant channel recovery problem [10], which posteriorly was improved by the introduction of a greedy \acOMP recovery algorithm [15].
Recognizing that the efficacy of \acOMP in noisy systems is limited, as the method fails to exactly fit linear systems, an alternative solver to mitigate this problem has been proposed in our previous work [16] in which the \acBPDN [17] has been leveraged as a more efficient solution. Furthermore, in [16] the sparsity of the problem was enhanced through a reweighted minimization formulation [18], which led to an efficient iterative \acBPDN sparse recovery.
In this paper, we continue this trend and further contribute with a technique for joint channel estimation and training beamformer optimization. The generic optimization of training vectors is performed based on Frame Theory and its applicability to sparse recovery, [19, Ch.9], [20]. In turn, the measurement matrix selection problem is solved by a decoupled, flexible lowcoherence tight frame design, with increased robustness compared to conventional random or optimized training vectors [21].
In the remainder of the paper the following notation is used:

, and represent a matrix, a vector, and a scalar;

and denote the Frobenius, Euclidean and norms;

denote the transpose, complex conjugate transpose and conjugate of matrix ;

is the Kronecker product of and ;

denotes the diagonal matrix with diagonal ;

is a column vector with all columns of stacked;

and denote the sized identity and null matrices.
Ii Problem Formulation
Iia Millimeter Wave Channel Estimation & Compressive Sensing
A downlink \acMIMO \acmmWave system formed of a \acBS with transmit antennas and an \acUE with receive antennas is considered. It is also assumed that the \acBS uses training beamforming vectors to transmit a known training signal , while the \acUE applies combining vectors for each beamforming one in order to estimate the channel .
It follows that the receive signal matrix at the \acUE, denoted by , is given by
(1) 
where the precoding and combining (TX/RX beamforming) matrices are given by and , respectively, denotes circularly symmetric complex \acAWGN, and is the \acmmWave channel matrix.
Following the usual sparse multipath scatter channel model [3, 12, 7, 9, 11, 8, 10], we may rewrite as
(2) 
where is the number of propagation paths, is the complex gain of the th path, and and are the array response vectors respectively at the receiver and transmitter, with corresponding \acAoA and \acAoD denoted by .
The channel matrix described by equation (2) can also be more compactly expressed as
(3) 
with , , and .
For the sake of simplicity, identity signaling is assumed hereafter, such that the training transmit symbol matrix is given by , which in turn implies that equation (1) can be rewritten in a vectorized form as
(4) 
where .
A sparse characterization of equation (4) can be obtained as follows [7, 8]. First, consider expanded versions of the scatter matrices and defined by and where the \acpAoD and \acpAoA and lay on a sufficiently fine quantization grid,
(5) 
with , , and respectively, .
Next, expand also into a sparse matrix , whose only nonzero entries are the elements satisfying
(6) 
for every .
And finally, obtain [16]
(7) 
where the measurement matrix , the sparse dictionary , the sparse vector and the noise vector have been implicitly defined.
IiB Previous Contributions
Under the assumption that the \acAoA and \acAoD angles and are known^{1}^{1}1As literature on \acAoA estimation is vast we refrain from further discussion., and in light of the model expressed by equation (2), the mmWave channel estimation problem amounts to estimating the complex gains . And under the further assumption that the precoding and combining matrices and are given, the vectorized expression of equation (7) enables the mmWave channel estimation problem to be solved as the sparse recovery \acCS optimization problem:
(8a)  
(8b) 
where we have explicitly identified the equivalent sensing matrix for future convenience.
The problem formulated above can be solved via the \acOMP [15] algorithm, e.g. as proposed in [7] and [10]. More recently, we have shown in [16] that the latter can be enhanced by relaxing the problem (8) to the associated norm formulation
(9a)  
(9b) 
which can then be solved via \acBPDN, thus mitigating the noisy recovery limitations encountered by classical \acOMP.
In fact, the problem, can be even more accurately solved if the \acBPDN solver is further combined with the sparsityenhancing iterative reweighing scheme of [18], as also shown in [16, Alg. 1].
Another approach to further improve the performance of mmWave channel estimation that received comparatively less attention so far is to optimize the sensing matrix , given a certain discrete angle dictionary , which is usually fixed by means of hardware/processing requirements.
Deriving a method to optimize given , which in turn reduces to optimizing , is the objective and the main contribution of this article, and the focus of the next section.
Iii Frametheoretical Design of
Precoding & Beamforming Matrices
\ac
CS is a direct application of a larger framework of linear projections, namely Frame Theory [19, Ch. 9], [22, Sect. 7.2]. In a general sense, a frame is defined as a set of vectors over a Hilbert space (reduced to in the current setup) with and
(10) 
where (), , are the finite highest lower and lowest higher frame bounds, respectively [19].
A frame is tight iff , and unitnorm iff . \AcUNTF have both these properties so that , where is known as the redundancy of the frame [19, Ch. 1].
Similar to the \acRIP in \acCS, the bounding property expressed by equation (10) offers a measure of how close a frame is to an orthogonal basis with respect to any projected vector in the spanned space. But another measure of the frame’s similarity to an orthonormal basis is its mutual coherence, defined (for a unitnorm frame) as
(11) 
where is known as the Gram operator, and the lower bound on the righthand side is the Welch bound [23] for .
The performance of pursuit algorithms can be studied under concepts like coherence [24] and the restricted isometry property \acRIP [25]. As a result, there are two general requirements on the design of the measurement matrix :

must be highly incoherent in order to preserve the salient information of sparse vectors;

must satisfy the \acRIP in order to afford robustness to the reconstruction.
We may remark at this stage that the dictionary in equation (7) can in fact be identified as a harmonic frame, sampled out of the discrete Fourier matrix of size , so that is an \acUNTF by construction [22]. Also interesting to notice is the fact that this frame admits a natural Kroneckerdecomposable form , as seen before.
In light of all the above, our goal is to design the optimized measurement matrix as a Kroneckerdecomposable, normalized tight frame with lowcoherence and with a \acRIPcompliant associated sensing matrix , which is addressed in the sequel.
Iiia QCSIDCO: Measurement Matrix as a Lowcoherence Frame
A lowcoherence frame can be generated from a given unitnorm frame by iteratively decorrelating its vectors while constraining the feasibility region to an ball. This scheme, referred to as \acSIDCO, was originally introduced in [20] only for frames in . More recently, a strategy to generalize \acSIDCO to frames in , referred to as \acCSIDCO, was reported by [26]. An explicit and complete mathematical formulation of \acCSIDCO was, however, not given in [26]. A variation of the latter based on an explicit quadratic program is offered below.
Consider an existent unitnorm frame . The strategy of \acCSIDCO [26] is to minimize mutual coherence by iteratively solving the problem
(12a)  
(12b) 
for all vectors, where denotes the pruned existent frame and the search ball radius of vector is given by
(13) 
so that is constrained to the largest ball such that the prospective solution cannot be collinear with other .
In order to circumvent the additional challenge of optimizing in the complex domain, the space is reinterpreted as , with interlaced real and imaginary components.
The generically formulated \acCSIDCO approach described by equation (12) can be explicitly reformulated as the quadratic program
(14a)  
(14d)  
where  
(14e)  
(14f)  
(14g)  
(14h)  
(14i)  
(14j)  
(14k)  
(14l) 
We remark that thanks to the replacement of the linear program adopted in [20, Alg. 1] with the quadratic program given by equation (14), and the direct interlacing from to achieved in our formulation via the matrices and , \acCSIDCO becomes here a simple extension of the original \acSIDCO algorithm, which converges (absolutely) to the local minimum coherence points since the original concept of ball packings is preserved.
For all the above, the explicit quadratic reformulation of \acCSIDCO offered above is original, and can be directly coded on top of optimized standard quadratic solvers, unlike the formulation presented in [26].
We therefore refer to this realization of \acSIDCO as the \acQCSIDCO algorithm.
IiiB Beamformers: Decomposition of QCSIDCO
The frame obtained by the method described above is not strictly tight, unless an \acETF is reached^{2}^{2}2\acpETF are the closest frame analogies to orthogonal bases, attaining tightness and uniform lowest coherence, and therefore also the Welch bound [23]., which rarely happens in practice, since \acpETF exist only for particular dimensions.
Fortunately, tightness of the frame constructed via the \acQCSIDCO frame method can be posteriorly enforced by applying polar decomposition [27, Th. 2], which yields the \acUNTF closest, in Frobeniusnorm sense, \acQCSIDCO frame. For details we refer the reader to [27].
Finally, in order to uniformly distribute the sensing cost of , the polardecomposed \acQCSIDCOdesigned measurement matrix is obtained by normalizing the latter frame ,
(15) 
Returning to our original problem of mmWave channel estimation, however, we remark that the measurement matrix obtained as explained above still needs to be decomposed into precoding and combining beamforming matrices and in order for the channel estimation method to be practically implementable. This can be achieved by solving the problem
(16) 
which reformulated by vectorization becomes
(17) 
where
with
(19) 
Notice that the matrix defined above is a rectangular matrix, such that as a result of this reshaping, the solution of equation (17) can be easily obtained via \acSVD, yielding
(20) 
where and are the dominant left and right singular vectors, and the dominant singular value of .
With knowledge of the vectorized forms , as in equation (20), the precoding and combining matrices and are finally obtained such that , as desired for a practical implementation.
Given all the above, the proposed measurement matrix construction for application in the mmWave channel estimation problems described by equations (8) and (9) can therefore be summarized as follows:

Generate a unitnorm lowcoherence frame through the \acQCSIDCO scheme described by equation (14);

Apply \acSVD and polar decomposition to , [27, Th. 2], thus obtaining an \acUNTF with lowcoherence ;

Obtain the ideal measurement matrix from eq. (15);

Decompose using equation (20), and reshape and accordingly for suboptimal but practical realizations.
Iv Results and Analysis
In this section, the performance of the proposed mmWave channel estimation method employing the beamformers obtained as described above is evaluated numerically.
The simulation scenario is as follows. Both the transmitter and the receiver are assumed to have the same number of antennas , the number of training beamforming vectors are such that , and a sparse channel model as described by equation (2) with and was considered, with the \acBS/\acUE antenna subsystems assumed to be uniformly spaced linear antenna arrays, such that
(21) 
(22) 
with an interantenna spacing of half transmission wavelength , i.e. , and \acAoA/\acAoD uniformly and randomly distributed in the interval .
Let us start our numerical evaluation of the proposed art by studying the impact of the proposed \acQCSIDCO measurement matrix design. To this end, we first compare in Fig. 1 the coherence profile – defined as the distribution of innerproducts for all distinct column pairs – corresponding to measurement matrices obtained with the \acPTF construction approach of [21], against that achieved by the lowcoherence \acfQCSIDCO frames constructed as described in Section III. The empirical realizations have been fitted by families of \acGEV, and respectively, \acEV distributions [28], i.e Fig. 1.
It can be observed that indeed the proposed \acQCSIDCO approach is superior as it both reduces the frame coherence as defined in equation (11), yet also preserves the frame tightness, as defined in equation (10).
Next we compare the performance of the \acCSbased channel estimation algorithms employing the ideal measurement matrices highlighted above. To this end, a grid granularity of was considered, and different methods were used to solve the channel estimation problem. In particular, the classical \acOMP algorithm of [15] was used to solve equation (8), and both the standard \acBPDN algorithm of [17] as well as our previously proposed \acBPDN variation given in [16, Alg. 1] (with maximum iterations and tolerance ) were used to solve equation (9).
The \acNMSE was used as accuracy measure to compare the performance of the different mmWave channel estimation end schemes.
In Fig. 2, it can be seen that employing the measurement matrix proposed in Section III results in an improved estimation accuracy compared to the alternative of employing the stateoftheart \acPTF construction approach of [21], regardless of whether the estimation problem itself is performed via OMP or BPDN, corresponding to equations (8) and (9), respectively.
Finally, in Fig. 3, the performance of mmWave channel estimation schemes based on the improved iterative \acBPDN reweighed sparse estimator proposed in [16] and employing \acPTF and \acQCSIDCO is evaluated.
In this last comparison, however, we let assume different values while maintaining constant. Referring to equations (7) and (17), it can be seen that this impacts on the aspect ratio of the measurement matrices (, number of rows divided by number of columns), such that the results capture the robustness of the beamforming designs. Not only is the superiority of the \acQCSIDCO method of Section III over the \acfPTF approach of [21] once again confirmed, but also it is found that the relative gain obtained is robust against the noise power and the frame’s aspect ratio.
V Conclusion
We discussed in this paper the problem of sparse recovery of \acmmWave\acMIMO channels. In particular, we focused on the design of the training vectors for sparse channel recovery. We reformulated the problem in the context of Frame Theory and we proposed a novel tight lowcoherence generic design associated with the Kronecker product of the transmit and receive beamformers. We also introduced a recovery mechanism for practical realizations based on the beamforming matrices by means of a vectorized \acSVD decomposition. Furthermore, we analyzed the proposed scheme against the state of the art and outlined its design advantages and robust performance gains for the estimation of \acmmWave\acMIMO channels.
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