I Introduction
Millimeter (mm) and TeraHertz (THz) frequencies offer a huge increase in bandwidth in comparison to conventional subGHz frequencies and are thus strong candidate communication bands to successfully deliver the exponentially rising wireless data traffic [1]. These higher frequencies also enable implementation of massive antenna arrays on small form factors, making massive multipleinputmultipleoutput (MIMO) systems practically viable. However, hardware nonidealities like phasenoise (PhN), which is the random fluctuation in the instantaneous frequency of the local oscillators, also tend to increase with the carrier frequency [2], thus degrading the system performance. While low levels of PhN cause a symbollevel channel aging effect, also known as common phase error (CPE) [3, 4], higher levels of PhN additionally induce symbol distortion. This distortion manifests as intercarrierinterference (ICI) in multicarrier systems, such as those operating with orthogonal frequency division multiplexing (OFDM) [5, 6, 7], and can severely limit the signaltointerferenceplusnoise ratio (SINR) gains offered by the massive antenna arrays, at mmwave/THz frequencies.^{1}^{1}1While use of multicarrier systems at THz is uncertain, 3GPP standards have already adopted OFDM at mmwave frequencies [8]. Consequently, the impact of PhN on multicarrier massive antenna transceivers has received significant attention in the recent works [9, 3, 10, 11, 12, 13, 14].
Several PhN compensation techniques for OFDM have been explored in the literature and standardization activities. One class of approaches focus on estimating PhN with minimal pilot overhead, at the expense of a high computational cost and latency. For example, several works estimate PhN and data iteratively using a decision feedback estimation process [6, 7, 15, 16, 17, 18]. At higher frequencies, where the bandwidth and/or the number of data subcarriers may be large, this approach may lead to a high computational burden and may also increase system latency. Another class of works have explored noniterative, pilotaided estimation approaches [19, 20, 21, 22, 23, 24, 25, 26, 27], which may be more suitable for mmwave and THz OFDM systems due to their speed and low computational burden.^{2}^{2}2Some of these works also involve an optional decisiondirected feedback loop for better performance but with higher complexity. For example, [19] explores the mitigation of CPE due to PhN using pilot subcarriers spread across the system bandwidth, and a similar strategy was adopted by the 3GPP Rel 15 new radio (NR) standard [8]; [20] explores the use of time domain pilots for PhN estimation and [21] additionally identifies the dominant PhN basis for estimation; [23, 22]
explore pilot aided joint CPE and channel estimation, with linear interpolation between the pilots; and
[24] explores a pilot aided Bayesian estimation of PhN via Kalman smoothing. However such estimation techniques either require prior channel equalization, suffer from intersymbolinterference (ISI) and/or may suffer from significant intercarrier interference (ICI) from the data signals. In contrast to the above schemes, the technique in [25, 26, 27] involves transmitting a single high power sinusoidal pilot/reference signal (RS), separated from the data subcarriers by a guardband, and using the received subcarriers corresponding to the RS to estimate and compensate for both CPE and ICI in the frequency domain. While the concentration of RS in frequency obviates the need for prior channel estimates, thus preventing estimation error propagation, the guardband prevents ICI from data subcarriers and the frequency domain estimation prevents ISI during PhN estimation. Recently, the use of such a sinusoidal RS for joint analog channel estimation and PhN compensation in multiantenna analog beamforming systems has also been explored [13, 14]. While this PhN estimation technique has also been verified to work experimentally [26], a rigorous theoretical analysis of the same is lacking in prior literature due to the intractable form of PhN spectral coefficients. Such an analysis can, for example, help determine the throughput maximizing values of many system parameters, such as the power allocation to the RS and data, the required guardband width etc. Furthermore, the use of a strong sinusoidal RS in this technique may also cause a violation of spectral mask regulations.With the motivation of addressing the aforementioned gaps, in this paper we propose a generalization of the above PhN compensation technique, where the transmitter transmits an arbitrary bandlimited RS signal, that occupies adjacent subcarriers separated from the data subcarriers by null subcarriers, and the receiver uses the received RS to estimate the dominant spectral components of PhN in the frequency domain. In addition to decoupling the PhN estimation from channel estimation, such a bandlimited RS reduces the null subcarrier overhead and prevents an increase in the peaktoaverage power ratio of the transmit signal, unlike a wideband RS. By presenting a novel, asymptotically tight approximation to the PhN spectral coefficients, we present a detailed theoretical performance analysis of this PhN compensation technique for an arbitrary RS, and correspondingly find the throughputoptimal power allocation and number of null subcarriers.^{3}^{3}3Throughout the paper, we use ‘throughputoptimal’ to refer to solutions that maximize a tight approximation to the system capacity (including the RS overhead). We also formulate the throughputoptimal RS design problem, and show that signals with good aperiodic autocorrelation properties in the frequency domain, e.g. frequency domain Zero aperiodic Correlation Zone Sequences (ZCZS) [28, 29, 30] and frequencydomain Barker sequences [31, 32, 33], are good candidates. Furthermore, we study the impact of oscillator phaselocked loop (PLL) design on PhN estimation, and show that the PLL design objective can be very different for CPE only cancellation and CPE+ISI cancellation. A preliminary observation in this direction was also made recently in [34]. The proposed technique is also consistent with the PhN estimation framework in 3GPP NR Rel 15, where such an RS is referred to as the phasetracking reference signal (PTRS) [35, 36, 8]. However, the current 3GPP PTRS has a different timefrequency pattern and only estimates and compensates for the CPE. The contributions of this paper are as follows:

We propose a generalized RS aided PhN compensation technique for multiantenna OFDM systems.

We characterize the achievable throughput for a multiantenna hybrid beamforming receiver with the new PhN compensation technique.

We formulate the throughputoptimal RS design problem and provide optimal solutions.

We also analytically characterize the throughputoptimal RS bandwidth, power allocation and the numbers of nulled subcarriers and estimated PhN principal components for the technique.

We compare of the performance of the scheme to several prior works under practically relevant and 3GPP compliant simulations.

We also study the interplay between the oscillator PLL bandwidth and the performance of PhN compensation.
The system model is discussed in Section II; the PhN estimation and compensation is discussed in Section III; the signal and noise components of the demodulated outputs are characterized in Section IV; the system throughput is studied in Section V; the optimal RS design is presented in Section VI; other oscillator PhN models are discussed in Section VII; simulation results are provided in Section VIII; and the conclusions are summarized in Section IX.
Notation:
scalars are represented by lightcase letters; vectors by boldcase letters; and sets by calligraphic letters. Additionally,
, represents the expectation operator, is the complex conjugate of a complex scalar , is the Hermitian transpose of a complex vector , represents the Dirac delta function, is the modulo Kronecker delta function with if and otherwise and / refer to the real/imaginary component, respectively.Ii General Assumptions and System model
We consider the downlink of a singlecell system, with one basestation/transmitter (TX) having antennas and multiple userequipments/receivers (RXs). While the TX may have arbitrary architecture, each RX is assumed to have a subarrayed hybrid beamforming architecture [37, 38], with antennas connected to each downconversion chain via phaseshifters, as illustrated in Fig. 1. Furthermore, we assume that the TX allocates one upconversion chain and one spatially orthogonal datastream to each RX subarray. Neglecting the interuser and interdatastream interference, we shall restrict the analysis to one representative subarray of one RX. For this RX subarray, the TX transmits OFDM symbols with subcarriers, indexed as where . Data is transmitted on the lower and higher subcarriers, i.e. on indices , where . The RS for PhN tracking is transmitted on subcarriers , where , while the remaining subcarriers are nulled to act as a guardband between the reference and data signals. This transmit structure is illustrated in Fig. 2. Under these conditions, the complex equivalent transmit signal of the th OFDM symbol to the representative RX subarray can be expressed as:
(1) 
for , where and are the symbol duration and the cyclic prefix duration, respectively, is the unitnorm TX beamforming vector, is the component of the RS on the th subcarrier, is the data signal on the th subcarrier, is the carrier frequency, represents the frequency offset of the th subcarrier and represents the PhN process of the TX oscillator. Here we define the complex equivalent signal such that the actual (real) transmit signal is given by . For the data subcarriers (), we assume the use of independent, zeromean data streams with equal power allocation and for RS we define . The transmit power constraint is then given by , where is the total OFDM symbol energy (excluding the cyclic prefix).
The channel to the representative RX subarray is assumed to have multipath components (MPCs), and the corresponding
channel impulse response matrix and its Fourier transform, respectively, are given as
[39]:(2a)  
(2b) 
where is the complex amplitude, is the delay and are the TX and RX subarray response vectors, respectively, of the th MPC. As an illustration, the th RX subarray response vector for a uniform planar subarray with horizontal and vertical elements () is given by , where:
(3) 
for and , , are the azimuth and elevation angles of arrival for the th MPC, are the horizontal and vertical antenna spacings and is the carrier wavelength. The expression for can be obtained similarly. We assume the TX and RX to have sufficient prior channel knowledge to implement the analog beamformers and , respectively. Depending on the analog beamformer design, such knowledge may include either instantaneous [40, 41, 42] or average channel parameters [43, 44, 45, 46, 47]. Without loss of generality, let the channel MPCs with nonnegligible power along the TXRX analog beams be indexed as , where . Due to the large antenna array and associated narrow analog beams at the TX and RX, we assume the effective channel to have a small delay spread (i.e., ) and a large coherence bandwidth. Since the TX can afford an accurate oscillator, we also neglect variation of the TX PhN within this small delay spread, i.e., . Additionally we shall also assume that the nondata subcarriers () lie within a coherence bandwidth of the effective channel i.e. .
The RX subarray has a low noise amplifier followed by a bandpass filter (BPF) at each antenna, that leaves the desired signal undistorted but suppresses the outofband noise. The filtered signals at each antenna are then phaseshifted by an analog beamformer, combined and downconverted via an RX oscillator and sampled by an analogtodigital converter (ADC) at samples/sec, as depicted in Fig. 1. Assuming perfect timing synchronization at the RX, the sampled received baseband signal for the th OFDM symbol can be expressed as:
(4) 
for , where the parts of are the outputs corresponding to the inphase and quadraturephase components of the RX oscillator, is the unit norm RX beamformer, , is the sampled PhN process of the RX oscillator, is the postbeamforming effective additive Gaussian noise process with independent and identically distributed samples and is the noise power spectral density. Conventional OFDM demodulation is then performed on the combined baseband signal (4). The demodulated subcarriers are used for PhN estimation, while subcarriers are used for data demodulation, as discussed in the next section. While prior estimates of the effective channel are not required for the proposed PhN estimation algorithm in Section IIIA, for the purpose of SINR and throughput analysis in Section V we shall assume can be estimated perfectly at the RX (after PhN compensation).
The PhN of a freerunning voltage controlled oscillator (VCO) is quite high and is accurately modeled as a Wiener process [5, 7, 16]. To reduce the resulting PhN, VCOs are therefore usually driven by a PLL, for which several PhN models have been proposed [48, 49]. However, for the convenience of analysis, we shall first model the PhN at the TX and RX as Wiener processes. The extension of the results to PLL based PhN models shall be explored later in Section VII. For the Wiener model, is a nonstationary Gaussian process which satisfies , where
is zeromean Gaussian with variance
, and is independent for each . We assume the RX to have apriori knowledge of . Note that in this PhN model, we implicitly assume the mean TX and RX oscillator frequencies to be equal. This assumption shall be relaxed later in Section VIII to demonstrate the oscillator frequency offset suppressing capability of the proposed scheme.Iii Phase noise estimation and compensation
In this section, we first analyze statistics of the channel noise and phase noise, and then discuss the PhN estimation and compensation approach. Note that the sampled channel noise and the sampled PhN for can be expressed using their normalized Discrete Fourier Transform (nDFT) coefficients as:
(5a)  
(5b) 
where and are the corresponding nDFT coefficients. Here nDFT is an unorthodox definition for Discrete Fourier Transform, where the normalization by is performed while finding instead of in (5). These nDFT coefficients satisfy the following lemmas:
Lemma 1.
The nDFT coefficients of are periodic with period and satisfy:
(6a)  
(6b)  
(6c)  
for arbitrary integers , and the approximation in (6c) is tight for . 
Proof.
Lemma 2.
Conditioned upon validity of (6c), satisfy:
(7) 
Proof.
See Appendix B ∎
Lemma 3.
The nDFT coefficients of , are periodic with period and are jointly circularly symmetric Gaussian with:
(8) 
for arbitrary integers (see [14] for proof).
The accuracy of the approximation in (6c) is depicted for a typical Wiener PhN process in Fig. 3. As is evident from Fig. 3, (6c) is reasonably accurate for . While there is a mismatch in the value of for the even subcarriers with , this mismatch vanishes quickly with increasing as also shown in Fig. 3 for . Even for , we observe that is around dB lower than , and thus can reasonably be approximated as . Due to the accuracy of these approximations, we shall henceforth assume validity of (6c) in the rest of the analysis.^{4}^{4}4The results presented in the paper do not depend on the exact expression for . Thus, if the accuracy of in (6c) is insufficient for a specific scenario, it can be replaced by the exact value from (6b).
Then from (4), the received signal on subcarrier can be expressed as:
(9) 
where and we use for , which is reasonable due to the large coherence bandwidth of the effective channel (see Section II). As is evident from (9), the subcarrier output contains contributions from other datasubcarriers, i.e., it suffers from ICI and phase rotation/CPE due to the PhN, which can be suppressed by PhN estimation as discussed next.
Iiia Phase noise estimation
From Lemma 2 and Fig. 3, we observe that the magnitude of decreases quickly with , and therefore the PhN nDFT coefficients for lower frequency indices dominate its behavior and impact. In fact, from (6c), these indices form the principal components of the PhN process. Consequently, we shall only estimate for the dominant principal components: , where . Here is a design parameter whose throughputoptimal value shall be discussed later in Section VI. These coefficients are estimated from the subcarrier outputs . To reduce ICI from the data subcarriers to these subcarrier outputs, we assume that the number of null subcarriers is designed such that . Then from (9), the received signal on subcarrier can be expressed as:
(10)  
where follows by neglecting ICI due to data subcarriers and follows by approximating for . Note that these approximations are tight for sufficiently large . We shall henceforth assume validity of the approximation in (10) for the rest of the analysis. Equation (10) can then be expressed in matrix form as:
(11) 
where are matrices with the th entries being and , respectively, is a matrix with and is a rectangular, banded Toeplitz matrix with for and otherwise.
Assuming to be full rank, the least squares (LS) estimate for can be obtained as:
(12) 
where . Note that the LS estimate in (12) does not require knowledge of and is thus decoupled from channel estimation. This prevents channel estimation errors (with ICI) from crawling into the PhN estimates i.e. error propagation. While linear minimum mean square error (LMMSE) estimation of may lead to less sum square error accumulation than with LS estimation, we do not consider it here for analytical tractability. A comparison of the two estimators is performed via simulations later in Section VIII. As shall be shown, optimizing the choice of enables limiting noise accumulation of the LS estimator, this achieving almost identical performance to the LMMSE estimator.
IiiB Phase noise compensation
To compensate for the phase rotation and ICI, a simple PhN compensation technique is considered, where the th OFDM output (after compensation) can be obtained as:
(13)  
where we use (12) and define . These PhN compensated subcarriers are then used to demodulate the data signals . Using (6a), it can be readily shown that the above technique can completely cancel the PhN in the absence of estimation noise in (12) and for . The demodulated outputs in the more general case are analyzed in the next section. Note that CPEonly compensation [19, 50] is a special case of (13), obtained by picking .
Iv Analysis of the demodulated outputs
We shall split in (13) as where , referred to as the signal component, involves the terms in (13) containing and not containing the channel/estimation noise, , referred to as the interference component, involves the terms containing and not containing the channel/estimation noise, and , referred to as the noise component, containing the remaining terms. These signal, interference and noise components are analyzed in the following subsections. Note that while the mathematical tools from our prior work [14] can be directly extended to analyze for the case of (i.e. sinusoidal RS), the generic case handled here requires novel analysis techniques.
Iva Signal component analysis
From (13), the signal component for can be expressed as:
(14) 
As is evident, the phase rotation due the PhN is suppressed by the compensation technique and the magnitude of signal component increases with . Taking an expectation with respect to the PhN and , the energy of the signal component can be obtained as:
(15) 
where , and the last step follows from Jensen’s inequality and (6c).
IvB Interference component analysis
From (13), the interference component for can be expressed as:
(16) 
The first and second moment of
, averaged over the PhN and can be expressed as:(17)  
where follows from , (6a) and the zero mean assumption for data; also follows from the independent, zero mean assumption for subcarrier data; follows by using the CauchySchwartz inequality for the first term and defining ; follows by using (6a) for both the terms; and follows by using Jensen’s inequality for the second term and (6c) for both terms. Despite being in closed form, equation (IVB) usually yields a loose bound on for , as also observed in [14] for the case of .
Remark IV.1.
A tighter approximation is obtained by replacing in (IVB) by where .
IvC Noise component analysis
The noise component of the received signal on subcarrier can be expressed as:
(18a)  
(18b)  
(18c)  
(18d) 
where is obtained by using change of variables . From Lemma 3 and equations (10)–(12), it can be readily verified that and are circularly symmetric, zeromean Gaussian and mutually independent for . Therefore the first and second moments of the noise signal, averaged over the PhN, channel noise and data signals, can be expressed as:
(19a)  
(19b) 
where:
(20a)  
(20b)  
(20c)  
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