With the rapid increase of data rates in wireless communications, bandwidth shortage is getting more critical. Accordingly, there is a growing interest in using millimeter wave (mmWave) for future wireless communications taking advantage of the enormous amount of available spectrum . In mmWave systems, a combination of analog beamforming (operating in passband) [2, 3] and digital beamforming (operating in baseband)  is one of the low-cost solutions to higher data rate transmission, and this combination is commonly called hybrid beamforming -. To implement hybrid beamforming at a transmitter and a receiver simultaneously is certainly intractable. Therefore, our previous works in [9, 10] focus on finding the key parameters of the hybrid beamforming gain to alleviate the problem, and eventually all that matters about the hybrid beamforming performance is the analog beam selection.
The problem of analog beam selection for frequency-selective fading channels can be stated as a sum-power (or energy) maximization across all subcarriers [10, 11]. From Parseval’s theorem, we know that it is equivalent to calculating the energy of the observations for the analog beam selection in the delay (or time) domain. Particularly, the observations in the delay domain can be interpreted as coupling coefficients of a matrix-valued channel impulse response (CIR) and all possible analog beam pairs plus noise. Considering an OFDM system, it is easier to obtain the observations in the frequency domain. However, these signals seriously suffer from the noise in the low SNR regime, and it needs more effort to refine them in the frequency domain than in the delay domain because the significant observations are not sparse in the frequency domain. To this end, this paper presents a low-complexity beam selection method and its performance improvement in the delay domain.
In OFDM systems, the delay-domain convolution operation can be constructed as a matrix multiplication, where one of the inputs (that is, the training sequence) is converted into a circulant matrix. Then, left multiplying the received signal vector by the inverse of the circulant matrix leads to the observations for the analog beam selection. In the system, the lengthof a cyclic prefix (CP) is much less than one OFDM symbol duration with samples but is enough to cover the maximum delay spread , which means that at most observations in one OFDM symbol can be used for the beam selection. Unfortunately, the observations are unreliable in the low SNR regime.
In order to improve the quality of the observations for the beam selection, we generate the training sequence of length with a certain period within one OFDM symbol duration at the transmitter. After deconvolution by a small-size circulant matrix, we have periodic signals of length
plus random noise signals. An arithmetic mean of these signals yields the refined observations, where the effective noise variance is reduced by a factor of. According to one of the transmission numerologies in 3GPP 5G New Radio (NR) , so that the mean absolute error (MAE) between the energy estimate and its true value can be significantly reduced. In addition, if the refined observations are reliable enough to find the delay indices, eventually only a few number of signals corresponding to the estimated delay indices are the significant observations for the analog beam selection.
The following notations are used throughout this paper. is a scalar, is a column vector, and is a matrix. denotes the column vector of ; denotes the entry of . and denote the transpose and Hermitian transpose of respectively. denotes the row vector of . and denote respectively the identity and zero matrices. denotes the circular convolution of sequences and of length .
Ii System Model
A system having a transmitter with an -element uniform linear antenna array (ULA) communicates data streams to a receiver with an -element ULA as shown in Fig. 1. The analog beamforming vectors at the transmitter in matrix are selected from a predefined codebook with the member represented as 
where stands for the candidate of the steering angles at the transmitter, is the distance between two neighboring antennas, and is the wavelength at the carrier frequency. At the receiver, the analog beamforming vectors in matrix are selected from the other codebook defined as , where the members can be generated by the same rule as (1). The analog beamforming matrices and are assumed to be constant within one OFDM symbol duration owing to hardware constraints.
Via a coupling of two analog beamforming matrices and a multipath matrix-valued CIR , where denotes the sample in one OFDM symbol, the sampled received signal vector can be written as
where stands for the average received power containing the transmit power, transmit antenna gain, receive antenna gain, and path loss, is the transmitted signal vector, and is an -dimensional independent and identically distributed (i.i.d.) complex Gaussian random vector, .
mmWave channel models have been widely studied recently [14, 15]. Based on the references, a simplified mmWave CIR matrix can be expressed as the sum of outer products of the array response vectors associated with the normalized-quantized delay (the set of natural numbers contains zero), where (the set of positive real numbers contains zero) is the delay for path and is the sampling rate,
where is the attenuation coefficient for path and . Note that the path loss values influenced by an environment and geometry are mentioned in the average received power in (2). characterizes the CIR for path at sample and we assume that when , where is the CP length. The departure array response vector is a function of angle of departure (AoD), , for path ,
and the arrival array response vector , where , has a similar form as (4).
Iii Time-Domain Analog Beam Selection
Iii-a Observations for analog beam selection
In order to acquire the observations for the analog beam selection, we simply assume that all the beam pairs selected from and are trained by a known training sequence. Hypothetically there is no data transmission and reception before the transmitter and receiver select the preferable analog beam pairs. Hence, one can use a training sequence of length in one OFDM symbol, , to train one beam pair. The sampled scalar of the received signals by using the beam pair can therefore be expressed as
where , , the combined noise
still has a Gaussian distribution with mean zero and variancedue to the equal-magnitude elements of .
To implement deconvolution of the received signal and get the observations for the beam selection, we intend to decouple the angle- and delay-domain components in by replacing the channel matrix with (3). Consequently, can be further written as follows:
where is the multiplication of beamforming gains at the transmitter and receiver.
Then, we collect samples in a vector and express the circular convolution as a multiplication by a circulant matrix 
The observations can therefore be obtained by pre-multiplying by , where , given by
where . One can design the training sequence so that has a complex Gaussian distribution with mean zero and a variance of .
Iii-B Problem statement
The observations can be interpreted as coupling coefficients of the channel and the trained beam pairs. If the coupling coefficients are acquired in the frequency domain, our previous work in  introduces how to use them to select the analog beam pairs. Simply speaking, the problem of frequency-domain analog beam selection can be formulated as finding the beam pairs that maximize the sum of the power of the observations across all subcarriers. From Parseval’s theorem, we know that the objective function is equivalent to the sum of the power across all samples in the delay domain. As a result, the delay-domain analog beam selection can be expressed as the following maximization problem:
where , is the energy of the observations
and are the sets including the selected analog beamforming vectors from iteration to . The energy estimate is also the objective function used in frequency-domain analog beam selection problem . However, in the frequency domain, we do not have the information that , , only contain noise signals.
In the beam selection problem stated in (10), the sum of the power of noise-free observations, i.e.,
where , would lead to the optimal solution. Let us write down the corresponding objective function
where the second equality follows from that when and , , are different to each other. Compared with (11), it is clear that in (13) only (rather than ) observations associated with the delay indices have to be taken into account. Therefore, our goal is to reduce the error between the energy estimate and its true value without an additional computational overhead.
Iv Performance Improvement of Analog Beam Selection in Time Domain
Iv-a Performance metric
From the discussion in the previous subsection, we know that there is a higher probability to find the optimal solution when the error betweenand approximates to zero. Therefore, we use the MAE between and as a performance metric to quantify the performance of beam selection, which is stated in Theorem 1. We consider the MAE rather than the mean squared error (MSE) due to that fact that and are energy signals; it is redundant to calculate the squared error between these two values.
Given matrix-valued CIRs , , one has the energy estimates
and the corresponding true values
Then the MAE between and is upper bounded by
See Appendix A. ∎
Iv-B Refine observations by averaging random noise signals
In OFDM systems, the CP length () is designed to cover the maximum or root-mean-square (RMS) delay spread, which means that the number of useful observations in one OFDM symbol is less than or equal to . To improve the quality of the observations, we use a property of circular convolutions introduced as follows. First, simply modifying the transmitted training sequence of length as (assume ) repeated sequence blocks, where the length of each block is . Such periodic training sequence blocks make the circular convolution in (6) become
where the second equality follows from that when , and denotes a circular convolution over the cyclic group of integers modulo . Then, following from (9), we can use a circulant matrix of small size (generated by one training sequence block) to sequentially implement the deconvolution of received periodic blocks. An arithmetic mean of the outputs of the deconvolution leads to a result suffering from less noise effect
where , and the variance of is effectively reduced by a factor of . By using these averaged (or refined) observations, the energy estimate in (11) becomes
Based on the derivation of Theorem 1, the MAE between the estimate and its true value conditioned on the same channel realizations, , , is given by
where has a Gaussian distribution
follows a gamma distribution
Iv-C Further refine observations by using knowledge of multipath delay
In the previous subsection, we present how to enhance the quality of the observations. Without any information of multipath delay, the signals in (20), , with respect to a certain beam pair are regarded as useful observations. Nevertheless, only sparse observations corresponding to the CIRs are exactly useful. Fortunately, we can borrow the idea of the analog beam selection in (10) to find the multipath delay indices because the signals are represented in the discrete delay-angle domain, where and respectively denote the delay- and angle-domain indices. Accordingly, the multipath delay estimation can be stated as the following problem: given , one can calculate the sum of the power of observations across all steering angles as
and solve the constrained maximization problem
where denotes the path index whose received power across all steering angles is greater than or equal to a pre-defined threshold , and is the set containing the selected path indices from iteration to . Here we consider the sum of the power of observations in the angle domain; therefore the threshold can be simply assumed to be . Since mmWave channels are sparse in nature, the multipath delay indices can be estimated by using this characteristic to further improve the performance . Due to the page limit, we do not provide more discussion.
According to the estimated delay indices, only refined observations are used for the analog beam selection problem (assume but does not necessarily include ) and the corresponding objective function is given by
Similarly, conditioned on the same channel realizations, , , we have the MAE between and its true value upper bounded by
and with equality iff . Furthermore, and are given as follows:
Compared with and , although the variance of and the shape parameter of become smaller, is not necessarily less than if the difference between and is too large. It depends on the performance of multipath delay estimation.
V Numerical Results
The system parameters used in the simulations are listed below:
Number of antennas Number of RF chains Number of samples per OFDM symbol CP length Codebook size Number of paths
In addition, the effective noise variance is given by , where (dB) is the SNR. In the codebooks, steering angle candidates are: .
As discussed in Section III-B, the true value of the energy yields the optimal solution of the problem in (10). Let us denote the indices of the optimal beam pairs as , and then use the MAE as a performance metric to evaluate the performance of the proposed and reference methods with respect to the beam pairs . In Fig. 2, the curves labeled as Ref, Refined obs., and Estimated delay are respectively calculated by the following equations:
where the energy estimate in (32a) is equivalent to the sum of the power of observations across all subcarriers, which is the objective function of the frequency-domain analog beam selection problem in .
), and they are dominated by the gamma distributed random variables when the values of shape and scale parameters are large. As a result, (32a) and (32b) can be approximated by
and therefore the difference in MAE between Ref and Refined obs. is given by
In (32c), if we only use refined observations associated with estimated delay indices (the estimation error rate is shown in Fig. 3), the MAE can be reduced by dB compared with Refined obs., see curve Estimated delay. Ideally, if the set containing estimated delay indices is equal to , following from (28), the corresponding MAE is upper bounded by
and the simulation results are shown in curve Explicit delay calculated by
In the low SNR regime, is dominated by the gamma distributed random variable as well. Hence, the difference in MAE between Refined obs. and Explicit delay approximates to
In Fig. 3, it shows the estimation error rate of delay indices in curve Estimated delay in Fig. 2. As mentioned in Section IV-C, since the exact number of paths is not available, we try to find paths whose sum of the received power across all steering angles are greater than or equal to the pre-defined threshold . When , the delay estimation error rate of more than leads to an MAE reduction of approximately dB, compared with Refined obs. On the other hand, when , the delay estimation error rate approximates to zero. However, the gap between Estimated delay and Explicit delay in Fig. 2 is still quite obvious, which means that and therefore not only the useful observations but also a large number of noise signals are reserved. The delay estimation approach can be further enhanced by, for example, modifying the threshold; nevertheless, it is beyond the scope of this paper.
The mmWave channel sparsity in the delay domain is widely acknowledged as a powerful cue for analog beam selection. Different to the conventional methods addressing the feature in the frequency domain, this paper presents a new perspective in the delay domain and shows that the significant observations used for the analog beam selection are also sparse. To improve the quality of the observations, we propose a solution that transmits the periodic training sequence of length equal to a CP length. An arithmetic mean can accordingly reduce the noise variance to refine the observations. Then based on the refined signals represented in the delay-angle domain, the sparse significant observations can be simply captured by finding the maximum term in the sum of the power of the refined signals across angle.
Vii-a Proof of Theorem 1
has a normal distribution with mean zero and variance, , and the third term has a gamma distribution with the shape parameter and scale parameter , . Therefore, the MAE between and (denoted as ) is given by