I Introduction
We investigate the problem of channel estimation in millimeter wave (mmwave) wireless communication networks. Mmwave refers to the wavelength of electromagnetic signals at 30300 GHz frequency bands. At these high frequencies, channel measurement campaigns revealed that wireless communication channels exhibit very limited number of scattering clusters in the angular domain [1, 2, 3]. A cluster refers to a propagation path or continuum of paths that span a small interval of transmit Angles of Departure (AoD) and receive Angles of Arrival (AoA). Moreover, signal attenuation is very significant at mmwave frequencies. This motivates the use of large antenna arrays at the transmitter (TX) and receiver (RX) to provide high antenna gains that compensate for high path losses [4]. Nevertheless, due to the high power consumption of mixed signal components, e.g., Analog to Digital Converters (ADCs) [5], conventional digital transceiver architectures that employ a complete RF chain per antenna is not practical. Hence, alternate architectures have been proposed for mmwave radios with the objective of maintaining a close performance to channel capacity. Among the proposed solutions are the use of i) hybrid analog/digital beamforming [6, 7, 8] and ii) fully digital beamforming with low resolution ADCs [9, 10, 11].
For all proposed solutions, channel estimation remains one of the most critical determinants of performance in communication. Due to the large number of antennas at TX and RX, estimation of the full channel gain matrix may require a large number of measurements, proportional to the product of the number of transmit and receive antennas. This imposes a great burden on the estimation process. To address this issue, various methods have been used, the most prevalent among them, is compressed sensing [12, 13, 6, 14, 11]
, which leverages channel sparsity. Performance of compressed sensing based approaches is heavily dependent on the design of system (sensing) matrices. For instance, while random sensing matrices are known to perform well, in practice, sensing matrices involve the design of transmit and receive beamforming vectors and the choice of dictionary matrices
^{1}^{1}1A dictionary matrix is used to express the channel in a sparse form.. Hence, purely random matrices have not been used in practice [15]. On the other hand, no design that involves deterministic sensing matrices has been considered for sparse channel estimation.Despite the efforts, we do not have a full understanding of the dependence of channel estimation performance on the channel parameters and number of measurements. In an effort to understand this relationship, the study in [16] proposed a multiuser mmwave downlink framework based on compressed sensing in which the authors evaluate the achievable rate performance against the number of measurements.
In this work, we follow a different approach. We propose a systematic method in which we use sequences of error correction codes chosen in a way to control the channel estimation performance. To demonstrate our approach, consider the following simple example. Let a point to point communication channel be such that, there exists 3 possible receive AoA directions, only one of which may have a strong path to TX. We need to obtain the correct AoA at RX, if it exists. Instead of exhaustively searching all 3 possible AoA directions, we alternatively measure signals from combined directions. For instance, by combining directions 1&2 in one measurement and 2&3 in the next measurement, we can find the AoA in just two measurements. Specifically, four different scenarios might occur, namely, i) only the , or ii) only the measurement contains a strong path, iii) both and measurements contain a strong path, and finally, iv) neither measurement reveals a strong path. Interpretation of those cases is: AoA is in i) direction 1, ii) direction 3, iii) direction 2, and iv) none exists. Therefore, only 2 measurements are sufficient for beam detection instead of 3 that are needed for exhaustive search.
We will generalize this idea to develop a systematic method for beam detection, inspired by linear block coding. Specifically, we show that linear block error correcting codes (LBC) possess favorable properties that fit in with the desirable behavior of sparse channel estimation. As a result, we are able to i) provide rigorous criteria for solving the channel estimation problem, ii) significantly decrease the number of required measurements, and iii) utilize a fairly simple and energyefficient transceiver architecture. We design the system using LBCs that leverage the fact that transmission errors are typically sparse in transmitted data streams, and hence, only a few number of erroneous bits need to be corrected per transmitted codeword. Similarly, mmwave channels are also sparse, i.e., only a small number of AoAs/AoDs carry strong signals. LBCs can correct sparse transmission errors by identifying their location in a transmitted sequence (followed by flipping them). We are inspired by LBC’s ability to locate erroneous bits and exploit it to identify the AoAs/AoDs that carry strong signals (and their path gains) among all possible AoA/AoD values. To this end, we exploit hard decision decoding of LBCs, in which the receiver obtains an error syndrome that maps to one of the correctable error patterns. An obtained error pattern determines the positions where errors have occurred. Likewise, for channel estimation, the receiver will be designed to do a sequence of measurements that would result in a channel syndrome. The resultant channel syndrome shall identify the positions (and values) of nonzero angular channel components.
Contributions of this work can be summarized as follows:

We set an analogy between beam discovery and channel coding to utilize lowcomplexity decoding techniques for efficient beam discovery.

We provide rigorous criteria for setting the number of channel measurements based on the size of the channel and its sparsity level.

We show that the number of measurements required for beam discovery is linked to the rate of a used linear block code. Hence, maximizing the rate of the underlying code is equivalent to minimizing the number of measurements.

We develop a simple receiver architecture that enables us to measure signals arriving from multiple directions.
Related Work: The main objective of mmwave channel estimation is to find a mechanism that can reliably estimate the channel using as few measurements as possible. For instance, in [6], a compressed sensing based algorithm to estimate singlepath channels is proposed and an upper bound on its estimation error is derived. Further, the authors propose a multipath channel estimation algorithm based on that of singlepath channels. The proposed algorithms in [6] use an adaptive approach with a hierarchical codebook^{2}^{2}2A codebook refers to the set of all possible beamforming vectors. of increasing resolution. Similarly, the work in [13] proposes an adaptive compressive sensing channel estimation algorithm that accounts for offthegrid AoAs and AoDs by using continuous basis pursuit [17] dictionaries. Such adaptive algorithms divide the estimation process into stages and demand frequent feed back to the TX after each stage. Hence, while the number of required measurements are shown to decrease, these methods may add a considerable overhead.
Other works like [18, 19] and [20] have proposed channel estimation algorithms using overlapped beam patterns. For instance, the algorithm in [20] can estimate multipath channel components by sequentially estimating each path gain using an algorithm designed to estimate singlepath channels followed by recursively removing the estimated paths’ effect from subsequent measurements. Similar to [6, 13] adaptive beams with increasing resolution that require feedback to TX are used to refine the AoA/AoD estimates. On the other hand, the beam alignment algorithms proposed in [18, 19]
assume a multipath mmwave channel. These algorithms, with a high probability, can find the best beam alignment in a logarithmic number of measurements (with respect to the total number of available AoA directions). Nonetheless, despite the possible existence of multiple paths, those algorithms are designed to find one path to TX.
Exploiting the results of previous beam alignment operations could be used to reduce the overhead of subsequent alignments. For instance, assuming that successive beam alignments are statistically correlated, the authors in [21] use this contextual information to improve beamforming delay via MultiArmed Bandit based models.
Most research efforts in the field of mmwave channel estimation use the magnitude and phase information of the acquired channel measurements. Nevertheless, if a carrier frequency offset (CFO) error occurs in the transceiver hardware, the phase information might be unreliable. Hence, the work in [18, 19, 22] tackle this problem by ignoring the phase information. Similar to [18, 19], the solution in [22] can only obtain one (dominant) path between TX and RX using a compressed sensing based technique. The CFO problem is tackled in [23] by considering it as a variable to be estimated.
While the power consumption problem of mmwave systems is commonly alleviated using analog or hybrid beamforming transceivers, an alternative solution is to use low resolution ADCs in fully digital architectures. Owing to the fact that low resolution ADCs operate at much lower power than their high resolution counterparts, the work in [9, 10, 11, 24] employ low resolution (singlebit) ADCs in digital transceivers. The work in [14, 25] study the channel estimation problem using such architectures. Moreover, other solutions include integrated mmwave and sub6 GHz systems [26] to provide reliable and energy efficient communication systems.
Notations
A vector and a matrix are denoted by and , respectively, while denotes a scalar or a complex number depending on the context. The transpose, conjugate transpose and frobenius norm of are given by , and , respectively. The sets of real and complex numbers are and . The identity matrix is . A set is denoted by , while is its cardinality. Finally, is the indicator function.
Ii Motivating Example
To elaborate, we present the following example: consider a point to point communication link between a TX with single antenna () and RX with antennas. Therefore, the vector of channel gains^{3}^{3}3Let all the channels have one single significant tap., , is a vector, and its corresponding angular (virtual) channel, , is a vector of the same size and can be derived using the DFT matrix as [27]
(this is merely a linear transformation that maps the sequence of channel gains into a sequence of gains from different AoAs. This mapping will be presented in more detail in Section
III). Assume a singlepath channel, i.e., the channel has only one cluster with a single path in it. Let the path gain be denoted by . For simplicity assume . Further, let us assume perfect sparsity such that the AoA is along one of the directions defined in the DFT matrix , i.e., the channel path will only contribute to one angular bin. Finally, let us also neglect the channel noise.Based on the channel description above, we get an angular channel vector of the form
(1) 
such that and the number of nonzero elements in is . Any component of can be measured using one of the beam patterns shown in Fig. 1.
Objective: Suppose the transmitter sends pilot symbols of the form . Thus, the received vector of size can be obtained as
(2) 
where is the received vector in the angular domain. So, with change of basis, we can think of as a received sequence with just one nonzero component. To identify the position of this nonzero component, the receiver performs a sequence of channel measurements. Let denote the measurement such that
(3) 
where denotes the receive (rx)combining vector.
Our aim is to design channel measurements (i.e., ’s) such that the correct AoA is identified using the minimum number of measurements.
Proposed Solution: We consider this nonzero component to be an anomaly to a normally allzero bin angular channel. Hence, the goal of identifying its position is analogous to finding the most likely bit error pattern of a bit codeword in a linear block code. Now, we need to identify an error correction code with codewords of length and with bit error correction capability [28]. Hence, we can use the binary Hamming code with parity check matrix of size and given by
(4) 
where represents the component at the intersection of row and column of . Using hard decision decoding of LBCs, error syndrome vectors of length are obtained. Every possible syndrome vector maps to only one correctable error pattern^{4}^{4}4A correctable error pattern of a () Hamming code is any binary vector that contains only one ’1’ (at the error’s position).. Similarly, for channel estimation, several measurements should be performed at RX where each measurement mimics the behavior of a corresponding element in the error syndrome vector. Each measurement boils down to adding signals from a subset of the available directions. Since each measurement can either include the direction of the incoming strong path of gain or no strong paths at all, then the elements of the channel syndrome vector are in .
For every measurement , we design based on the entries of the row of such that: if , then we include the beam pattern that points to direction in . For example, the row of is given by . Hence, should include beam patterns pointing to the set of directions .
Fig. 2 illustrates this operation for . We can see that the resultant beam pattern of combines signals coming from a set of selected directions dictated by the row of . We call the obtained measurement vector, , the channel syndrome which is analogous to error syndromes in hard decision decoding of LBCs. Then, a table that maps every possible channel syndrome to a unique corresponding channel can be constructed. Table I shows this mapping.
Channel Syndrome  Angular Channel 

In this example, we are able to estimate the channel based on only measurements as opposed to , which is the number of measurements with exhaustive search. Important aspects of our proposed method include the choice of codes, the design of precoding and rxcombining measurement vectors, the effect of variable gains and phases of different paths and the occurrence of measurement errors.
Remark (Receiver Architecture).
Note that, to achieve beam patterns similar to the one shown in Fig. 2, the receiver architecture needs to be a bit different from those of classical analog/hybrid beamforming architectures. Specifically, in addition to lownoise amplifiers (LNA) typically placed at the output of each antenna, we will need to add controllable lowpower amplifiers, as well. The resultant architecture is still quite simple (see Fig. 3). That is, besides the lowpower amplifiers, the proposed architecture is similar to those of simple analog beamforming. Moreover, relatively lowresolution ADCs can be used which mitigates the high power consumption problem associated with highresolution ADCs.
Motivation for LBCinspired approach: LBCs are designed to discover and correct a certain maximum number of errors in a codeword of a specified length. This objective is achieved by adding redundant parity check bits to the original information sequence. What makes our devised approach attractive is that the number of measurements needed for channel estimation can be shown to be equal to the number of parity bits of some corresponding code. Hence, we can control the estimation performance via appropriate code selection. In this work, we will propose a method to specify the number of necessary channel measurements as a function of the rate of the underlying code.
Iii System Model
Consider a pointtopoint millimeterwave wireless communication system with a transmitter (TX) equipped with antennas and a receiver (RX) with antennas placed at fixed locations. Uniform Linear Arrays (ULA) are assumed at both TX and RX where each antenna element is connected to a phase shifter and a variable gain amplifier. A single RF chain at the receiver, with inphase (I) and quadrature (Q) channels, is fed through a linear combiner (see Fig. 3). Only two midtread ADCs, with quantization levels, are utilized, where quantization levels take values from the set .
We adopt a singletap channel model where denotes the channel matrix between TX and RX. Assume that the channel has clusters, where each cluster contains a single path with gain , AoD , and AoA . The channel is assumed to be sparse such that . Let denote the baseband channel gain and is defined as
(5) 
where is the length of path and is the carrier wavelength. The angular cosines of AoD and AoA associated with path are denoted by and , respectively. The transmit and receive spatial signatures along the direction are given by and such that
(6) 
where has a similar definition to , and and are the antenna separations at TX and RX normalized by the wavelength . Let the average path loss be denoted by . Thus, can be written as
(7) 
We define and
as the unitary Discrete Fourier Transform (DFT) matrices whose columns constitute an orthonormal basis for the transmit and receive signal spaces
and , respectively. (and similarly ) is given by(8) 
where and are the normalized lengths of the transmit and receive antenna arrays such that and . Let be the channel matrix in the angular domain [27], where
(9) 
The rows and columns of divide the channel into resolvable RX and TX bins, respectively. Further, we assume a perfect sparsity model in which AoDs , and AoA , are along the directions defined in and [14, 6, 20]. Hence, each channel cluster will only contribute to a single pair of TX and RX bins. Therefore, has a maximum of L nonzero TX and RX bins.
The baseband channel model is given by
(10) 
where is the transmitted signal, is the received signal and is an i.i.d. complex Gaussian noise vector.
Let and be the precoding and rxcombining vectors, respectively. The transmit signal is given by where is the transmitted symbol with average power . After the receiver applies the rxcombining vector , the resultant symbol can be written as
(11) 
Afterwards, is passed forward to the ADCs. There, a quantized version, , of is obtained such that
(12) 
where represents the quantization function. Now, constitutes a single, quantized, unit measurement obtained using specific and vectors such that .
We assume that remains fixed throughout the entire estimation process. The noise component normalized by
is also a complex gaussian random variable such that
. We define the signal to noise ratio (SNR) on a per path basis such that SNR of path is given by(13) 
Note that the actual received SNR depends on all path gains included in a measurement.
Iv Problem Statement
Suppose a maximum number of clusters need to be discovered in the channel where . Under the prefect sparsity assumption, has a maximum of nonzero RX and TX angular bins. Our objective is to identify the angular positions at which channel clusters exist and identify their path gain values using the least possible number of measurements. Let the number of measurements be such that each measurement, , is obtained using the precoder and rxcombiner . Let the number of rxcombiners and precoders be and , respectively. Measurements take the form . Let be a mapping function that takes in the measurements as inputs and returns the estimated channel . For each , we stack the measurements in a single (syndrome) vector such that . Our design variables are the precoding vectors , rxcombining vectors , the number of measurements , the mapping function , and the transmitted symbol power .
In its essence, solving this problem boils down to finding the optimal set of measurements and the mapping function such that can be estimated using the minimum number of measurements. For ease of explanation, we first consider a channel with a single transmit antenna and receive antennas. Therefore, no precoding is needed and the design of measurements is reduced to designing the rxcombining vectors . Recall that in the motivating example in Section II, we dealt with a special case of channels where we sought to find the direction of arrival of a channel with a single path of known gain, . In the general case, we should consider arbitrary path gains and channels with multiple paths.
V Beam Discovery
In this section, we present our proposed solution. As an initial step, we solve a simplified version of the problem where communication channels have a single transmit antenna and multiple receive antennas. Afterwards, we will build on it to provide the solution for general channels with multiple transmit and receive antennas.
Va Beam Detection using LBCinspired approach
To identify the exact number of measurements and their corresponding design, we follow a decodinglike approach of LBC^{5}^{5}5In channel coding, the convention is to use row vectors. Thus, let and be and binary row vectors that represent an information sequence and its corresponding codeword of an LBC, respectively. Also let be a received sequence corrupted by error pattern . To decode , we calculate an error syndrome vector , of size , such that , where is the parity check matrix of the used LBC. Then, a most likely error pattern can be uniquely identified by using a lookup table called the standard array. Finally, the decoded codeword is obtained using . A decoding error occurs if the number of errors, identified using ’s in , is beyond the error correction capability of the used code, denoted by . Note that in this context, all vectors, matrices and math operations are over GF(2).. First, we need to find an LBC, , that has an error correction capability such that i) the maximum number of clusters in the channel, , is equal to and ii) the length of its codewords is equal to the number of antennas ( is also the number of resolvable directions). The code has a parity check matrix which represents the link between channel decoding and beam detection problems. Binary codes deal with data and error sequences defined over the finite field , i.e., addition and multiplication operations are defined over with binary inputs and outputs, i.e., ’s and ’s. However, mmwave channel parameters are defined over the complex numbers field . Therefore, to account for arbitrary path gains, we should be able to extend this concept to .
Although is defined over , we interpret its and entries as real numbers. Then, similar to channel decoding, we seek to obtain a channel syndrome, , such that . This matrix multiplication can be realized using channel measurements such that each measurement gives one component in . Measurements make up the components of the channel syndrome vector . Then, we need to find a mapping function that takes in the channel syndrome vector as an input and returns the estimated channel . The position of each nonzero component in identifies a path’s AoA, and its value identifies its baseband path gain. Finally, for this to work, we need to show that such channel measurements provide onetoone mapping to the channel. In other words, must be a sufficient statistic for estimating the channel. In Section VC, we will show that our design results in the sufficient statistic we seek to achieve.
Remark (Difference between and ).
Both and refer to vectors of measurement symbols, however, is considered to be the noise corrupted and quantized version of . Specifically, such that is the measurement noise vector. While is what we expect to observe, our design of measurements focuses on finding ; an errorfree symbol. Of course, errors degrade beam discovery performance. Thus, in Section VI, we will deal with the effect of measurement errors separately and present a solution that increases reliability of beam discovery. The separate treatment of measurement errors simplifies the design and provides a clear understanding of the nature of our solution.
Remark (Number of Measurements).
The solution we obtain is dependent on channel parameters, namely, the number of antennas and the sparsity level of the channel. That is, at a fixed sparsity level, i.e., fixed number of clusters , a larger number of antennas necessitates more channel measurements. In other other words, the high resolution realized by large comes at a price of an increased number of measurements. Similarly, at fixed , more channel clusters involve more measurements for correct channel estimation.
VB Measurements Design
Recall that each component in represents a resolvable angular direction at the receiver. Let each resolvable direction be given an identification number (#i). Also let #i denote the beam pattern pointing to #i, i.e., a signal coming from #i can be individually measured using #i (similar to beam patterns in Fig. 1).
Now, we seek to obtain using careful design of ’s , i.e.,
(14) 
To achieve this, each rxcombining vector is designed as a multiarmed beam, i.e., composed of several subbeams similar to the beam pattern in Fig. 2. The subbeams included in each are identified by the row of the matrix . That is, only if , the intersection of the row and column, is , do we include #j as a subbeam in (also refer to our discussion in Section II).
The design of rxcombining vectors is a crucial aspect of this work. As an initial step towards obtaining proper rxcombining vectors, we consider designing ’s using linear summation of all analog beamformers that correspond to beam#j’s . Let , such that is the spatial signature of #j. Then, can be designed as
(15) 
VC Sufficient Statistic
We will show in this section that each channel syndrome can only be mapped to a single measurable channel. A measurable channel in this context refers to channels with nonzero components such that , where is the error correction capability of the underlying code and is its CWs length. Let be the set of all measurable channels:
(16) 
Since each measurement combines signals coming from multiple directions, each element in the channel syndrome vector is a linear combination of a subset of the available paths. In other words, each measurement has the possibility that one or more paths are included in it. This setting is rather challenging. To understand why, consider a channel that has two paths with gains . Suppose that and are of equal magnitudes but are outofphase (i.e., phase shift ). Hence, if signals coming from both paths are combined in a single measurement, the resultant value is which is similar to the result we get if no paths exist in the measured directions. Also each channel measurement can be a result of endless possibilities for the combined path gain values. So, a natural question to ask is: does this ambiguity cause measurement errors? The direct answer to this question is: No. In the sequel we will show that the resulting channel syndrome, i.e., the combination of all channel measurements, is sufficient to correctly estimate the channel.
First, recall our discussion in Footnote 5. Then, consider all singlebit error patterns of a code , with maximum number of correctable errors , such that
(17) 
where is the component of . Also let be the corresponding error syndrome of . Recall that . Hence, we can see that is exactly the row of , i.e., column of . Let denote the set of correctable error patterns of the code
(18) 
Now, we can write any correctable error pattern as a linear combination of all singlebit error patterns over the finite field such that
(19) 
and its corresponding error syndrome is
(20) 
Lemma 1 @notefont().
For an error pattern with number of bit errors identical to , its syndrome is a linear combination of linearly independent vectors .
Proof.
We are going to prove this lemma by contradiction. First, assume that is a linear combination of linearly dependent vectors over . Therefore, there exists another error syndrome composed of only linear combination of independent vectors such that . Therefore, there exists another error patter with number of errors strictly less than such that its syndrome . Since has a number of errors less than , then it is a correctable error pattern, and since all error syndromes of correctable error patterns are different, then should be . Hence, we arrive at a contradiction. ∎
It is also easy to see that if and are two different correctable error patterns, then their error syndromes and are composed of a linear combination of different sets of singlebit error syndromes .
Lemma 2 @notefont().
Any dimensional linearly independent vectors over , are also linearly independent over .
Proof.
Let be a set of dimensional vectors defined over . The vectors can be made the columns of an matrix . Since all ’s are linearly independent over , then
is a leftinvertible matrix. Therefore, there exists a nonzero (modulo
) minor of . Now, suppose the entries in are interpreted as real numbers. Therefore, , now taken over , has an submatrix whose determinant is nonzero, which proves that it is invertible. Therefore, the vectors ’s, i.e., columns of , are linearly independent over which, using the same argument, can also be shown to be linearly independent over . ∎Suppose that entries of and are interpreted as real numbers, then we can write the channel as
(21) 
where and . Therefore, each channel syndrome is a linear combination of independent vectors in (columns of ). Therefore, all possible measurable channels yield unique channel syndromes which implies that they are sufficient for the channel estimation problem.
VD Mapping Function
Now that we have shown that each measurable channel can be mapped to a unique channel syndrome, we need to find this mapping function, i.e., , where denotes the estimated channel. Next, we propose two different approaches to find .
VD1 Lookup Table Method
Again, we resolve to a technique used in hard decision decoding where a lookup table is constructed that maps every error syndrome to a corresponding error pattern. Likewise, we construct a lookup table that indicates which channel corresponds to an obtained channel syndrome.
Since we employ ADCs with finite resolution, only a finite number of realizable syndromes, , exist (and a finite number of corresponding channels). Therefore, a lookup table method is feasible. We construct the table by, first, generating all possible sparse angular channels. Then, we find the corresponding channel syndromes using , where . Recall that the actual, noisecorrupted, received channel syndrome is . Therefore, might not exactly match one of the channel syndrome vectors in the lookup table. Hence, we instead search for the table entry that has the closest distance to , and pick its corresponding channel as the estimated channel . We define the distance between the two complex dimensional vectors , to be the norm as follows:
(22) 
By obtaining , we not only identify the AoA at the Rx, but we also obtain the magnitude and phase information associated with every strong path to the TX.
Remark (Size of lookup table).
The size of the lookup table scales proportionally with ADC resolution. As the resolution of ADCs increases, the size of increases as well; since every nonzero component of every can take more values. If low resolution ADCs could be tolerated, then the lookup table is a plausible choice for mapping owing to the small size of the lookup table. However, if high resolution ADCs are needed, the lookup table size creates a problem for memorylimited devices especially for large antenna arrays. It also increases the complexity of finding the closest table entry to the measurement . Next, we will propose a different approach for mapping called search method which does not scale with ADC resolution.
VD2 Search Method
Recall that (Eq. (14)), and let the parity check matrix be represented as:
(23) 
where is the column of . Thus, we can write as:
(24) 
where is the component of . Note that is sparse, i.e., we have no more than nonzero components . Let the indices of the nonzero components be , hence, can succinctly be written as:
(25) 
Let
(26) 
Then we can write Eq. (25) in matrix form as:
(27) 
where is a shortened version of that only has dimensions. Also is an matrix of rank , since , and ’s are linearly independent columns of (recall our discussion in Section VC). Therefore, has a left MoorePenrose inverse (pseudo inverse), where of size . Thus, if we have knowledge of , we can then find as:
(28) 
The problem we need to solve is obtaining the matrix . We can solve this problem using an exhaustive search method which can be explained as follows:

Candidate matrices are generated by choosing different combinations of columns of where .

Find . Note that at this step, we obtain a vector identical to if and only if .

Let be such that
(29) Hence, if the correct choice is made, then
else, if , then^{6}^{6}6Since is the left pseudoinverse of , and since is not a square matrix, then
Hence, if , we declare its corresponding matrix the true matrix defined in Eq. (26) which satisfies Eq. (27). Also, we have that . Since, identifying is equivalent to identifying the indexes . Thus, we found the angular channel which is all zeros except  potentially^{7}^{7}7This means that if the number of paths is less than , then some ’s might have zero values as well.  for the components .
The previous discussion dealt with an idealized version of the measurements (i.e., ), however, in practice, we observe as an errorcorrupted version of . Define to be the error vector that captures the effect of both channel noise and quantization error which satisfies
(30) 
Suppose that we know the matrix for which we have
(31) 
then, we can find (compare to Eq. (28)) as follows
to be a noisecorrupted version of .
Now, to find an estimate of , we follow a very similar procedure to the one described before as follows:

Matrices are generated similar to the step before.

Find such that
(33) Unlike the step of the noerror case, will not be identical to the true with probability , since is not identical to with probability ( is the difference between continuous and discrete quantities).

Let be such that
(34) (35) Then find such that
(36) such that is given by
(37) which at is further reduced to
(38)
VE Multiple Transmit and Receive Antennas
So far, we have considered channels with single transmit antennas and shown how to perform beam discovery at RX. To extend our approach to a general setting, we consider channels with antennas at TX, and antennas at RX. Thus, instead of the TX just sending signals omnidirectionally, now it can perform highly directional transmission. Recall that the RX is able to perform channel measurements using multiarmed beams. Similarly, the TX can send signals using multiarmed beams to simultaneously focus on multiple directions using precoding vectors .
The design of precoding vectors can also be obtained using an LBC approach. Similar to the method of designing rxcombining vectors , we look for an LBC, , that has CWs of length and can correct for errors. Let the parity check matrix of be , using which, we will design the precoding vectors . Let beam#i denote the TX beam which points to TX direction dir #i. Then, just as before, we envisage as an array whose columns are associated with resolvable TX directions such that: i) its column corresponds to dir#j, and ii) its row corresponds to the measurement. We note that no actual measurements are performed at TX; we use the word measurement to refer to precoding, consistent with the case of RX. That is, the TX measurement is actually the precoder . Thereby, we design the precoder as a multiarmed TX beam such that, only if , the intersection of the row and columns of , is , do we include subbeam beam#j in . Each TX measurement provides a component in a TX channel syndrome vector . The total number of TX measurements (i.e., precoding vectors), denoted by , is equal to the number of parity check bits of the code . That is, , where is the length of ’s information sequences. To obtain AoDs of strong paths at TX, we define the function as the mapping function between all possible TX channel syndromes and their corresponding angular channels denoted by . Note that, for every dir#i, there exists a corresponding which represents the row of . Also, since the maximum number of clusters is , then, the number of nonzero vectors is .
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