Phase impairment estimation for mmwave mimo systems

Phase Impairment Estimation for mmWave MIMO Systems Phase Impairment Estimation for mmWave MIMO Systems Dinh Ngoc Nguyen Fundamental Faculty Telecommunication University Khanh Hoa, Vietnam Email nguye[.]

2021 8th NAFOSTED Conference on Information and Computer Science (NICS) Phase Impairment Estimation for mmWave MIMO Systems Dinh-Ngoc Nguyen Kien Trung Truong Fundamental Faculty Telecommunication University Khanh Hoa, Vietnam Email: nguyendinhngoc@tcu.edu.vn Undergraduate Faculty Fulbright University Vietnam Ho Chi Minh city, Vietnam Email: kien.truong@fulbright.edu.vn Abstract—Most millimeter-wave (mmWave) multiple-input multiple-output (MIMO) systems are negatively affected by phase noise and carrier frequency offset due to non-ideal transceiver hardware components If not compensated, the phase impairments significantly reduce the data rates of such systems Much prior work either ignored these impairments or proposed highcomplexity estimation algorithms In this paper, we consider a mmWave MIMO system model that takes into account many practical hardware impairments Our main contributions are a problem formulation of phase impairments and a low-complexity estimation method to solve the problem Numerical results are provided to evaluate the performance of the proposed algorithm In particular, it works quite well in the wide range of phase noise variances Index Terms—Hybrid beamforming, Non-ideal hardware, Phase noise estimation, Millimeter wave MIMO, Least Squares I I NTRODUCTION Recently, due to high speed transmission, reliability of connections, minimum signal latency, the fifth generation (5G) of mobile network is the most important technology More specifically, millimeter wave band being studied for the next generation radio systems is a potential technology Operating at high frequency makes it possible to use multiple antenna at transceiver thus increase wireless data traffic and data rate [1], [2] However, the path loss in free space depends on the frequency becomes serious issues need attention In practical and theoretical results showed that transmitters and receivers are equipped with multiple antennas combined with beamforming technique has improved diversity gain and array gain, this helps to reduce transmission losses in the high frequency [1] The existing fully digital beamforming for lower frequency MIMO systems requires each antenna connected to the radio frequency (RF) Clearly, this architecture not only makes the transceiver become complex but also increase power consumption As a result, this architecture cannot be directly applied to mmWave MIMO systems [1] To overcome this problem a solution proposed that beamforming are implemented not only in analog domain but also in digital domain This technique is called hybrid beamforming [3] In practical systems, to reduce hardware costs, radio frequency chains are often made up of inexpensive hardware components This leads to the received signals will be impact of non-ideal transceiver hardware components Phase noise 978-1-6654-1001-4/21/$31.00 ©2021 IEEE is one of the most important hardware impairments resulted from non-ideal oscillators of the transceiver during the up and down conversion between baseband and bandpass signal [4] Phase noise is multiplied by output signal at oscillators Due to the high oscillation frequency of mmWave transceivers and difficult estimation, reducing phase noise is a significant challenge In addition, using low resolution of ADC/DACs has many challenges such as quantization noise and affecting on the other blocks of receivers [5], [6] Recently, the effect of non-ideal transceiver hardware on the performance of MIMO systems at low frequency has obtained a lot of attentions In [7], the authors analyzed how phase noise effect on the performance of MIMO systems operating at low frequency In this work, the authors proposed a method to simultaneously estimate channel gains and phase noise by using an Extended Kalman Filter (EKF) Bjornson et al [8] discussed how non-ideal transmitter hardware impacts massive MIMO systems and derived expressions of corresponding achievable data rates Myers et al in [4], [9] investigated the combining of phase noise and channel estimation technique for fully digital beamforming mmWave systems with few bit quantization In [9], the authors proposed lifting techniques to solve estimation problem for narrowband channel systems while [4] proposed message passing technique for wideband channel estimation Rodrıguez-Fernandez et al [10] analyzed the combining channel estimation and synchronization for mmWave MIMO hybrid analog and digital beamforming mmWave MIMO systems with assumption that phase noise and quantization noise can be ignored In [11], the authors investigated the impacts of non-ideal hardware components on mmWave massive MIMO systems with fully array connected architecture One of their main observations is that phase noise significantly degrades the achievable rates of such systems Assuming the number of bit resolution ADCs on RF chains are the same, J.Mo et al [12] figured out the trade off between power consumption and achievable data rate of a hybrid beamforming mmWave MIMO with fully array connected architecture The results of simulation showed that ADCs at receivers with 4-5 bits quantization is the best for system performance In [13], the authors has investigated how the quantization noise effect on massive MIMO systems with fully digital beamforming architecture and hybrid analog/digital 543 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) beamforming Assumed the number of bit resolution ADCs on RF chains are the different, in [14] proposed the ADC bitallocation algorithm for mmWave MIMO systems with hybrid beamforming to overcome the effects of quantization noise Note that much prior work focused on either phase noise or carrier frequency offset In addition, most of prior estimation algorithms have relatively high complexity In this paper, we consider an mmWave MIMO system model that takes into account hybrid precoding and combining, and many hardware impairments, such as quantization noise, phase noise, and carrier frequency offset Assuming the available knowledge of other modeling parameters, we formulate an estimation problem of phase impairments due to varying phase noise and constant carrier frequency offset We propose a new low-complexity linear method for phase impairment estimation to solve the formulated problem Numerical results are provided to evaluate the accuracy of the proposed method Our major contribution are summarized as follows: • • We formulate an estimation problem of phase impairments for a quite realistic mmWave MIMO system model We propose a low complexity method to solve the formulated phase impairment estimation problem The main idea is using vectorization received signal to estimate the vector of phase rotation factors affecting the received pilot symbols due to phase noise values at the receiver This paper focuses on phase noise estimation in mmWave MIMO systems Note that, system model in this paper is more general than most of previous studies relevant to systems, because it included phase noise component [8] and quantization noise component [6] in hybrid beamforming mmWave MIMO systems The remainder of this paper is organized as follows Section II introduces the system model and the signal model Section III formulates the phase impairment estimation problem and then proposes a low complexity estimation method Section IV provides numerical results Section V concludes the paper and suggests some directions for future work II S YSTEM MODEL Consider a single-user point-to-point mmWave MIMO system where a transmitter with Nt antennas communicates with a receiver with Nr antennas, as illustrated in Fig [10], [15] Extensions to multi-user mmWave systems are left for future work Assume a discrete-time block-fading channel model where the channel coefficients remain unchanged within the duration of a radio frame and change randomly and independently frame-by-frame Let Kp be the number of training symbol periods in each radio frame Denote k as the training symbol index within a frame, where k ∈ Kp = {1, 2, · · · , K} Without loss of generation, we assume that the K pilot symbols are located at the beginning of the frame [8] To focus on phase impairment effects, we assume that the matrix of channel coefficients H ∈ CNr ×Nt is perfectly known at the receiver and leave the joint channel and phase noise estimation for future work Fig The diagram of the considered mmWave MIMO system model where a transmitter with Nt transmit antennas communicates with a receiver with Nr receive antennas Let s[k] ∈ CNp ×1 be the pilot symbol vector to be transmitted at training symbol period k ∈ Kp , where Np is the number of complex-valued modulated pilot symbols to be transmitted, such that E(s[k]sH [k]) = N1p INp We assume that the same set of Mt radio frequency (RF) transmit chains are used at the transmitter over all of radio frames We also assume that both pilot transmission and data transmission rely on a hybrid precoding method Denote FBB ∈ CMt ×Np as the precoder in the digital domain and FRF ∈ CNt ×Mt as that in the analog domain Denote F = FRF FBB ∈ CNt ×Np for notation convenience To focus on the hardware impairment at the receiver, we assume that the digital-to-analog converters at the transmitter have relatively high resolutions so that quantization noise is negligible [6], [12] We also assume that the phase noise effects at the transmitter is negligible [4], [9] These assumptions are reasonable for downlink transmission where the base station is often composed of high-quality electronic components The transmitted training signal at symbol k ∈ K can be written as xp [k] = Fs[k] (1)
with the covariance matrix of Rx [k] = E x[k]xH [k] = H Np FF We assume that a common common local oscillator (CLO) is used for all the receive RF chains [8] This means that all the receive RF chains are affected by the same random phase noise θ[k], for all n = 1, 2, · · · , Mr Similar to [8], we adopt the discrete-time Wiener phase noise model such that for all k = 2, 3, · · · , Kp θ[k] =θ[k − 1] + ψ[k] (2) where θ[k−1] is the phase noise value affecting the RF receive chains in the previous symbol and ψ[k] is an independent phase innovation For tractable analysis, we assume that the phase innovation values at the RF receive chains are i.i.d white real Gaussian random variables, i.e ψ[k] ∼ CN (0, σψ2 ) for all k ∈ Kp \ {1} and n = 1, 2, · · · , Mr The phase noise variance may be computed as σψ2 = 4π fc cn Ts where fc is the carrier frequency, cn is a constant that characterizes the quality of the LO and Ts is the symbol time [7], [8], [11] We also assume that the received signal is affected by a constant carrier frequency offset (CFO), which is denoted as ∆f This 544 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) CFO results from the differences between the local oscillator at the transmitter and that at the receiver The receiver applies a hybrid combining method to the signal received at the antennas Denote WRF ∈ CNr ×Mr and WBB ∈ CMr ×Np as the analog combining matrix and the digital (baseband) combining matrix, respectively, where Mr is the number of RF receive chains at the receiver For notation convenience, denote W = WRF WBB ∈ CNr ×Np as the equivalent hybrid analog-digital combining matrix To focus on phase noise estimation, it is assumed that both WRF and WBB have been known, thus W has also been known, for all k ∈ Kp The received training signal in the analog domain, i.e at the output of the RF receive chains, is given by H H yp,A [k] =ejβ[k] WRF Hxp [k] + WRF np [k] (3) where β[k] = θ[k] + 2π∆f k, ∀k ∈ Kp represents that phase impairment at training symbol k ∈ Kp and np [k] ∼ CN (0Nr , σn2 INr ) is the vector of additive white Gaussian noise (AWGN) The covariance matrix of yp,A [k] is denoted as RyA [k] and is given by   H Ryp,A [k] =E yp,A [k]yp,A [k] (4) H WH HFFH HH WRF + σ WRF WRF (5) = Np RF The received signal in the analog domain yp,A [k] is converted into the digital domain by a number of analog-to-digital converter (ADC)s Let Q() denote the input-output relationship of the ADC signal process We assume that the ADCs work with the same resolution √ level of b bits and hence the same distortion factor ξ = π 2−2b Define ξ¯ = − ξ For further tractability, we assume that additive quantization noise model (AQNM) and the gain of the automatic gain control have been set appropriately [12] The pre-processed received baseband training signal is ¯ p,A [k] + eA [k] yp,BB [k] =ξy (6) where eA [k] ∈ CMr ×1 is the additive quantization noise vector with the probability distribution CN (0, ReA [k]) The covariance matrix of eA [k] is computed as [14] ¯ ReA [k] =E(eA [k]eH A [k]) = ξ ξdiag (RyA [k]) In this section, we develop a linear least squares (LS) estimation method of the CLO-based phase noise For most practical oscillators, the phase noise variances are very small Therefore, the LS estimator is expected to accurately obtain the mmWave MIMO channel and phase noise parameters [7] Define z = [ejβ[1] , ejβ[2] , · · · , ejβ[Kp ] ]T ∈ CKp ×1 as the vector of phase rotation factors affecting the received pilot symbols due to phase noise values at the receiver Note jβ[k]that the amplitude of each entry of z is equal to one e = 1, ∀k ∈ Kp We will refer to these as the unitamplitude constraints The problem is to estimate the phase noise vector z given the knowledge of the received training symbols y[k] for k ∈ Kp , channel matrix H, precoding matrices, combining matrices and statistical information of phase noise Define Yp = [yp [1], yp [2], · · · , yp [Kp ]] ∈ CNp ×Kp as the processed pilot signal matrix in one frame Define S = [s[1], s[2], · · · , s[Kp ]] ∈ CNp ×Kp as the matrix of transmitted pilot symbols and sv = vec(S) ∈ CNp Kp ×1 as the vectorized version of S Define EA = [eA [1], eA [2], · · · , eA [Kp ]] ∈ CMr ×Kp as the additive quantization noise matrix due to the ADCs during the pilot transmission Define Np = [np [1], np [2], · · · , np [Kp ]] ∈ CNr ×Kp as the AWGN matrix at the receiver As a result, we can rewrite Yp as follows ¯ H HFSdiag(z) + ξW ¯ H Np + WH EA Yp =ξW BB ¯ jβ[k] WH HFs[k] = ξe | {z } desired training signal (8) ef f ective noise The receiver has to estimate the phase impairment values, which include ∆f and θ[k], ∀k ∈ Kp , based on the postprocessed received signal yp [k], ∀k ∈ Kp (9) Define yv ∈ CNp Kp ×1 as the vectorization of Yp By applying the following properties vec(ABC) =
CT ⊙ A diag(B) and vec(AC) = (Ik ⊙ A) vec(C) for any A ∈ Cm×n , B ∈ Cn×n and C ∈ Cn×k where B is a diagonal matrix, we obtain

¯ ¯ yv =ξvec WH HFSdiag(z) + ξvec WpH Np
H + vec WBB EA (10)


H H ¯ ¯ = ξ IKp ⊙ W HFS z + ξ IKp ⊙ W vec(Np ) {z } | {z } | g2 G1 + (IKp ⊗ | (7) The post-processed received baseband signal at symbol k ∈ Kp is   ¯ jβ[k] HFs[k] + ξn ¯ p [k] + WH eA [k] yp [k] = WH ξe BB ¯ H np [k] + WH eA [k] + ξW | {z BB } III P ROPOSED L INEAR P HASE I MPAIRMENT E STIMATION M ETHOD H WBB )vec(EA ) {z g4 (11) } =G1 z + g2 + g3 (12) Due to the above mentioned assumptions on the beamforming matrices, and on the training signal matrix, the receiver knows G1 perfectly while it has only statistical information on g2 and g3 , since they are random vectors Moreover, since all the entries of the random vectors/matrices in Np and EA,p have zero mean, then we have E[g2 ] = E[g3 ] = ˆLS as the linear LS estimate of z Given that the Define z receiver knows G1 perfectly and by ignoring the terms with unknown coefficients in yv = f (z), we formulate a linear LS estimation problem of z as follows [16] 545 (LS) : ˆLS = arg ∥yv − G1 z∥2F z z∈CKp ×1 (13) 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) After some manipulation, we obtain the linear LS estimate as IV N UMERICAL RESULTS −1 H ˆLS =(GH z G1 yv G1 ) This section provides numerical results to investigate the performance of the proposed phase impairment estimation method The proposed method is evaluated by using the mean squared errors (MSE) of the phase impairment β[k], which is defined as follows   Kp X (22) MSE =E  β[k] − βˆLS [k]  Kp (14) Define HLS = WH HFS = [hLS,1 hLS,2 · · · hLS,Kp ] ∈ CNp ×Kp as the baseband equivalent end-to-end channel estimate Given the channel H, HLS and hence hLS,k ∈ CNp ×1 for all k ∈ {1, 2, · · · , Kp } can be computed exactly in advance at the receiver By definition, we have   hLS,1 0Np ×1 · · · 0Np ×1 0Np ×1 hLS,2 · · · 0Np ×1    (15) IKp ⊙ HLS =     0Np ×1 0Np ×1 · · · hLS,Kp It follows that  hLS,1 0Np ×1  G1 = ξ¯   0Np ×1 hLS,2 ··· ···  0Np ×1 0Np ×1     0Np ×1 0Np ×1 ··· hLS,Kp (16) Thus, we have −1 H ˆLS =(GH z G1 yv , G1 )  hH y[1]  LS,1 −2 |hH LS,1 |2 hH LS,2 y[2] −2 |hH LS,2 |2  
−1   ¯ = ξ     hH         y[Kp] (17) (18) k=1 Consider the following key simulation parameters as in [17] There are Nt = 128 transmit antennas, which are connected to RF chains There are Nr = 64 receive antennas, which are connected to RF chains The antenna elements at transceivers form uniformly-linear array (ULA) with the same distance d = λ/2 between adjacent antenna elements The number of pilot streams Np = The carrier frequency is 28 GHz while the bandwidth is 100MHz Monte-Carlo simulation results are obtained based on 100.000 realizations of channel vectors, phase noise, and quantization noise We also consider the mmWave channel model with the number of paths L = and AoAs/AoDs are uniformly distributed in [0, 2π] [18] Furthermore, analog precoding/combining matrices (FRF , WRF ) and digital precoding/combining matrices (FBB , WBB ) are designed using fully-array connected architecture [19] The training symbol matrix is generated by Zadoff-Chu sequences [4] LS,Kp 101 −2 |hH LS,Kp |2 CFO = 10ppm Thus, the LS estimate of the phase noise at the pilot symbol k, for k = 1, 2, · · · , Kp , is given by CFO = 1ppm 100 (19) MSE (dB)
−1 hH LS,k y[k] ˆLS,k = ξ¯ z −2 ∥hH LS,k ∥2 CFO = 5ppm ˆLS,k , k = 1, 2, · · · , Kp This means that the Kp values of z can be computed in parallel, thus reducing the computational complexity significantly Since GH G1 is always invertible, the LS estimation is always feasible Thus, we propose an estimate of the phase impairment value at training symbol k ∈ Kp , which is denoted as βˆLS [k], ∀k ∈ Kp and is computed as the ˆLS,k We also propose the following estimate of the angle of z CFO ∆f ∆fLS Kp −1   X βˆLS [k + 1] − βˆLS [k] = 2π(Kp − 1) (20) k=1 As a result, the following estimate of phase noise θ[k], ∀k ∈ Kp can be obtained as θˆLS [k] = βˆLS [k] − 2π∆fLS k (21) Note that the estimates of phase impairments in the training symbols can be used to interpolate those in the data symbols Nevertheless, the methods for interpolation and the corresponding achievable rate computation are left for future work 10-1 10 -2 The resolution of ADC (bits) Fig MSE as a function of the number of bit ADC, the number of pilot Kp = 128 and phase noise variance is 10−3 rad2 First, Fig presents the MSE in dB as a function of the number of ADC bits for different values of CFO, which are 1ppm, 5ppm and 10ppm when the phase noise variance is fixed at 10−3 rad2 As expected that the MSE value decreases with the resolution of the ADCs Moreover, there are little differences in MSE performance when low-resolution ADCs, i.e 5-bit or less ADCs, are used When higher-resolution ADCs are used, however, the MSE values depend significantlt on the value of CFO, especially in the low CFO regime 546 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) 10 V C ONCLUSION CFO = ppm, number pilot = 128 CFO = ppm, number pilot = 256 CFO = ppm, number pilot = 256 CFO = ppm, number pilot = 128 MSE (dB) 100 10-1 10 -2 10-3 10-4 The resolution of ADC (bits) Fig MSE as a function of bit ADC for the different values of the number pilot and CFO, phase noise variance is 10−3 rad2 Next, Fig presents the MSE versus the number of ADC bits for different numbers of pilots and CFO as the phase noise variance σψ2 r = 10−3 rad2 The simulation results pointed out that when increasing the number of bit ADC (4-5 bits), the MSE decreases and the trade off is the power consumption increase It is due to the fact that quantization noise assumed as an additive noise and depend on the number of bit ADC This result can be anticipated given that low resolution ADC as high distortion noise factor 101 MSE (dB) 100 10-1 10-2 number pilot = 128 number pilot = 256 number pilot = 512 10 -3 0.2 0.4 0.6 phase noise variance (rad 2) 0.8 10 -3 Fig MSE as a function of phase noise variance for the different the number of pilot, 4-bit ADC Finally, in Fig we investigate the MSE versus the phase noise variance In Fig shows the different values of the number of pilot Kp = [128, 256, 512] and 4-bit ADC As expected, when the phase noise variance increases (the quality of oscillator lower), the phase of signal is difficult to estimate so MSE increases In this paper, we consider millimeter-wave MIMO systems with practical assumptions on hardware components, signal processing techniques and available channel state information (CSI) In particular, the considered system model takes into account hybrid precoding and combining, and many hardware impairments, such as quantization noise, phase noise, and carrier frequency offset Due to limited space, we have formulated only the estimation problem of phase impairment, which results from varying phase noise and constant carrier frequency offset A very low complexity linear LS method was proposed to estimate the phase impairment values Numerical results have shown the effect of phase noise variance and the number of bit ADCs on the estimated performance Some directions for future work may be the extension to the minimum mean squared errors (MMSE) estimation method and joint channel and phase noise estimation R EFERENCES [1] R W Heath et al., “An overview 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We propose a new low-complexity linear method for phase impairment estimation. .. estimation problem of phase impairments for a quite realistic mmWave MIMO system model We propose a low complexity method to solve the formulated phase impairment estimation problem The main... the vector of phase rotation factors affecting the received pilot symbols due to phase noise values at the receiver This paper focuses on phase noise estimation in mmWave MIMO systems Note that,