# Optimal Reference Signal Design for Phase Noise Compensation in Multi-carrier Massive MIMO Systems

Millimeter-wave and Terahertz frequencies, while promising high throughput and abundant spectrum, are highly susceptible to hardware non-idealities like phase-noise, which degrade the system performance and make transceiver implementation difficult. In this paper, a novel reference-signal (RS) aided low-complexity and low-latency technique is proposed to mitigate phase-noise in high-frequency multi-carrier massive multiple-input-multiple-output systems. Unlike in existing methods, the proposed RS is transmitted in each symbol and occupies adjacent sub-carriers separated from the data by null sub-carriers. The receiver uses the received RS to estimate and compensate for the dominant spectral components of the phase-noise at each symbol. While the null sub-carriers reduce the interference between the RS and data, the frequency compactness of the RS decouples phase-noise estimation from channel equalization, reducing error propagation. A detailed theoretical analysis of the technique is presented and correspondingly, throughput-optimal designs for the RS sequence, RS bandwidth, power allocation and the number of nulled sub-carriers and estimated spectral components, are derived. A hitherto unexplored interplay between oscillator phase-locked loop design and the performance of phase-noise compensation is also studied. Simulations, performed under 3GPP compliant settings, suggest that the proposed scheme, while achieving better performance than several existing solutions, also effectively compensates for oscillator frequency offsets.

## Authors

• 3 publications
07/19/2018

### On the Phase Tracking Reference Signal (PT-RS) Design for 5G New Radio (NR)

The volume of mobile data traffic has been driven to an unprecedented hi...
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### A Rate Splitting Strategy for Mitigating Intra-Cell Pilot Contamination in Massive MIMO

The spectral efficiency (SE) of Massive MIMO (MaMIMO) systems is affecte...
07/19/2019

### Rate Splitting with Finite Constellations: The Benefits of Interference Exploitation vs Suppression

Rate-Splitting (RS) has been proposed recently to enhance the performanc...
11/28/2018

### Data Detection in Single User Massive MIMO Using Re-Transmissions

Single user massive multiple input multiple output (MIMO) can be used to...
03/29/2018

### Massive MIMO in Sub-6 GHz and mmWave: Physical, Practical, and Use-Case Differences

The use of base stations (BSs) and access points (APs) with a large numb...
02/25/2020

### Amplitude and Phase Estimation for Absolute Calibration of Massive MIMO Front-Ends

Massive multiple-input multiple-output (MIMO) promises significantly hig...
03/30/2016

### FPGA Impementation of Erasure-Only Reed Solomon Decoders for Hybrid-ARQ Systems

This paper presents the usage of the Reed Solomon Codes as the Forward E...
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## I Introduction

Millimeter (mm) and TeraHertz (THz) frequencies offer a huge increase in bandwidth in comparison to conventional sub-GHz 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 multiple-input-multiple-output (MIMO) systems practically viable. However, hardware non-idealities like phase-noise (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 symbol-level channel aging effect, also known as common phase error (CPE) [3, 4], higher levels of PhN additionally induce symbol distortion. This distortion manifests as inter-carrier-interference (ICI) in multi-carrier systems, such as those operating with orthogonal frequency division multiplexing (OFDM) [5, 6, 7], and can severely limit the signal-to-interference-plus-noise ratio (SINR) gains offered by the massive antenna arrays, at mm-wave/THz frequencies.111While use of multi-carrier systems at THz is uncertain, 3GPP standards have already adopted OFDM at mm-wave frequencies [8]. Consequently, the impact of PhN on multi-carrier 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 sub-carriers may be large, this approach may lead to a high computational burden and may also increase system latency. Another class of works have explored non-iterative, pilot-aided estimation approaches [19, 20, 21, 22, 23, 24, 25, 26, 27], which may be more suitable for mm-wave and THz OFDM systems due to their speed and low computational burden.222Some of these works also involve an optional decision-directed feedback loop for better performance but with higher complexity. For example, [19] explores the mitigation of CPE due to PhN using pilot sub-carriers 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 inter-symbol-interference (ISI) and/or may suffer from significant inter-carrier 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 sub-carriers by a guard-band, and using the received sub-carriers 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 guard-band prevents ICI from data sub-carriers 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 multi-antenna 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 guard-band 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 band-limited RS signal, that occupies adjacent sub-carriers separated from the data sub-carriers by null sub-carriers, 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 band-limited RS reduces the null sub-carrier overhead and prevents an increase in the peak-to-average power ratio of the transmit signal, unlike a wide-band 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 throughput-optimal power allocation and number of null sub-carriers.333Throughout the paper, we use ‘throughput-optimal’ to refer to solutions that maximize a tight approximation to the system capacity (including the RS overhead). We also formulate the throughput-optimal RS design problem, and show that signals with good aperiodic auto-correlation properties in the frequency domain, e.g. frequency domain Zero aperiodic Correlation Zone Sequences (ZCZS) [28, 29, 30] and frequency-domain Barker sequences [31, 32, 33], are good candidates. Furthermore, we study the impact of oscillator phase-locked 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 phase-tracking reference signal (PTRS) [35, 36, 8]. However, the current 3GPP PTRS has a different time-frequency pattern and only estimates and compensates for the CPE. The contributions of this paper are as follows:

1. We propose a generalized RS aided PhN compensation technique for multi-antenna OFDM systems.

2. We characterize the achievable throughput for a multi-antenna hybrid beamforming receiver with the new PhN compensation technique.

3. We formulate the throughput-optimal RS design problem and provide optimal solutions.

4. We also analytically characterize the throughput-optimal RS bandwidth, power allocation and the numbers of nulled sub-carriers and estimated PhN principal components for the technique.

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

6. 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 light-case letters; vectors by bold-case 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 single-cell system, with one base-station/transmitter (TX) having antennas and multiple user-equipments/receivers (RXs). While the TX may have arbitrary architecture, each RX is assumed to have a sub-arrayed hybrid beamforming architecture [37, 38], with antennas connected to each down-conversion chain via phase-shifters, as illustrated in Fig. 1. Furthermore, we assume that the TX allocates one up-conversion chain and one spatially orthogonal datastream to each RX sub-array. Neglecting the inter-user and inter-datastream interference, we shall restrict the analysis to one representative sub-array of one RX. For this RX sub-array, the TX transmits OFDM symbols with sub-carriers, indexed as where . Data is transmitted on the lower and higher sub-carriers, i.e. on indices , where . The RS for PhN tracking is transmitted on sub-carriers , where , while the remaining sub-carriers are nulled to act as a guard-band 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 sub-array can be expressed as:

 stx(t) = √2Tst[∑k∈^Gpkej2πfkt+∑k∈K∖Gxkej2πfkt]ej[2πfct+θtx(t)], (1)

for , where and are the symbol duration and the cyclic prefix duration, respectively, is the unit-norm TX beamforming vector, is the component of the RS on the -th subcarrier, is the data signal on the -th sub-carrier, is the carrier frequency, represents the frequency offset of the -th sub-carrier 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 sub-carriers (), we assume the use of independent, zero-mean 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 sub-array is assumed to have multi-path components (MPCs), and the corresponding

channel impulse response matrix and its Fourier transform, respectively, are given as

[39]:

 H(t) = ~L−1∑ℓ=0αℓarx(ℓ)atx(ℓ)†δ(t−τℓ) (2a) H(f) = ~L−1∑ℓ=0αℓarx(ℓ)atx(ℓ)†e−j2π(fc+f)τℓ, (2b)

where is the complex amplitude, is the delay and are the TX and RX sub-array response vectors, respectively, of the -th MPC. As an illustration, the -th RX sub-array response vector for a uniform planar sub-array with horizontal and vertical elements () is given by , where:

 [~arx(ψrxazi,ψrxele)]¯MVh+v=exp{j2πΔHhsin(ψrxazi)sin(ψrxele)λ+j2πΔV(v−1)cos(ψrxele)λ}, (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 non-negligible power along the TX-RX 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 non-data subcarriers () lie within a coherence bandwidth of the effective channel i.e. .

The RX sub-array has a low noise amplifier followed by a band-pass filter (BPF) at each antenna, that leaves the desired signal un-distorted but suppresses the out-of-band noise. The filtered signals at each antenna are then phase-shifted by an analog beamformer, combined and down-converted via an RX oscillator and sampled by an analog-to-digital converter (ADC) at samples/sec, as depicted in Fig. 1. Assuming perfect timing synchronization at the RX, the sampled received base-band signal for the -th OFDM symbol can be expressed as:

 srx,BB[n]=L−1∑ℓ=0αℓr†arx(ℓ)atx(ℓ)†t[∑k∈^Gpkej2πfk(nTsK−τℓ)+∑k∈K∖Gxkej2πfk(nTsK−τℓ)]ej[θtx[n]+θrx[n]]+w[n], (4)

for , where the parts of are the outputs corresponding to the in-phase and quadrature-phase components of the RX oscillator, is the unit norm RX beamformer, , is the sampled PhN process of the RX oscillator, is the post-beamforming 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 base-band signal (4). The demodulated sub-carriers are used for PhN estimation, while sub-carriers 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 III-A, 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 free-running 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 non-stationary Gaussian process which satisfies , where

is zero-mean 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:

 w[n] = ∑k∈KWkej2πkn/K (5a) ejθ[n] = ∑k∈KΩkej2πkn/K, (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:

 ∑k∈KΩk1Ω∗k2 = δKk1,k2, (6a) Δk1,k2 ≜ E{Ωk1Ω∗k2} (6b) = 1K2K−1∑˙n,¨n=0e−σ2θ|˙n−¨n|Ts2Ke−j2π[k1˙n−k2¨n]K ≈ δKk1,k2K[1−e−(σ2θTs−j4πk14)eσ2θTs−j4πk12K−1+1−e−(σ2θTs+j4πk14)1−e−σ2θTs+j4πk12K], (6c) for arbitrary integers k1,k2, and the approximation in (6c) is tight for σ2θTs≫1.
###### Proof.

This lemma was first derived by us in [14], and is reproduced in Appendix A for convenience. ∎

###### Lemma 2.

Conditioned upon validity of (6c), satisfy:

 Δk,k≤{Δk−2,k−2% for 2≤k≤K/2Δk+2,k+2for −K≤k≤−2. (7)

See Appendix B

###### Lemma 3.

The nDFT coefficients of , are periodic with period and are jointly circularly symmetric Gaussian with:

 E{Wk1Wk2†} = δKk1,k2N0, (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 sub-carriers 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.444The 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 sub-carrier can be expressed as:

 Yk = 1KK−1∑n=0srx,BB[n]e−j2πkn/K=∑˙k∈^Gβ0p˙kΩk−˙k+∑¯k∈K∖Gβ¯kx¯kΩk−¯k+Wk, (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 sub-carrier output contains contributions from other data-subcarriers, i.e., it suffers from ICI and phase rotation/CPE due to the PhN, which can be suppressed by PhN estimation as discussed next.

### Iii-a 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 throughput-optimal value shall be discussed later in Section VI. These coefficients are estimated from the sub-carrier outputs . To reduce ICI from the data sub-carriers to these sub-carrier outputs, we assume that the number of null sub-carriers is designed such that . Then from (9), the received signal on sub-carrier can be expressed as:

 Yk (1)≈ β0[^g∑˙g=−^gp˙gΩk−˙g]+Wk (10) (2)≈ β0[min{^g+k,^g}∑¨g=max{−^g+k,−^g}pk−¨gΩ¨g]+Wk,

where follows by neglecting ICI due to data sub-carriers 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:

 Y(u)=β0P(u)Ω(u)+W(u), (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:

 ˆβ0Ω(u)=R−1p[P(u)]†Y(u)=β0Ω(u)+^W(u), (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.

### Iii-B 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:

 ^Yk = ∑˙u∈U[ˆβ0Ω(u)]∗˙u+u+1Yk+˙u (13) = ∑˙u∈U[β∗0Ω∗˙u+^W∗˙u][∑˙k∈^Gβ0p˙kΩk+˙u−˙k+∑¯k∈K∖Gβ¯kx¯kΩk+˙u−¯k+Wk+˙u],

where we use (12) and define . These PhN compensated sub-carriers 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 CPE-only 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.

### Iv-a Signal component analysis

From (13), the signal component for can be expressed as:

 ^Sk=∑˙u∈Uβ∗0βkxk|Ω˙u|2. (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:

 E{|^Sk|2} = |β0βk|2EdE{∣∣∑˙u∈U|Ω˙u|2∣∣2}≥|β0βk|2Edμ(0,u)2, (15)

where , and the last step follows from Jensen’s inequality and (6c).

### Iv-B Interference component analysis

From (13), the interference component for can be expressed as:

 ^Ik = ∑˙u∈U∑˙k∈^G|β0|2p˙kΩ∗˙uΩk+˙u−˙k+∑˙u∈U∑¯k∈K∖[G∪{k}]β∗0β¯kx¯kΩ∗˙uΩk+˙u−¯k. (16)

The first and second moment of

, averaged over the PhN and can be expressed as:

 E{^Ik} (1)= 0 E{|^Ik|2} (2)= E∣∣∑˙u∈UΩ∗˙u[∑˙k∈^Gp˙k|β0|2Ωk+˙u−˙k]∣∣2+∑¯k∈K∖[G∪{k}]|β∗0β¯k|2EdE∣∣∑˙u∈UΩ∗˙uΩk+˙u−¯k∣∣2 (17) (3)≤ E{[∑˙u∈U|Ω˙u|2][∑˙u∈U∣∣∑˙k∈^Gp˙k|β0|2Ωk+˙u−˙k∣∣2]} +∑˙u,¨u∈U|β0¯β|2EdE{Ω∗˙u[∑¯k∈K∖{k}Ωk+˙u−¯kΩ∗k+¨u−¯k]Ω¨u} (4)≤ E[∑˙u∈U∣∣∑˙k∈^Gp˙k|β0|2Ωk+˙u−˙k∣∣2]+∑˙u,¨u∈U|β0¯β|2EdE[δK˙u,¨uΩ∗˙uΩ¨u−|Ω˙u|2|Ω¨u|2] (5)≤ ∑˙k∈^G|p˙k|2|β0|4μ(k−˙k,u)+|β0¯β|2Ed[μ(0,u)−μ(0,u)2],

where follows from , (6a) and the zero mean assumption for data; also follows from the independent, zero mean assumption for sub-carrier data; follows by using the Cauchy-Schwartz 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 (IV-B) 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 (IV-B) by where .

This heuristic is obtained by assuming

and to be independently distributed in step of (IV-B), but we skip the proof for brevity. As shall be verified in Section VIII, Remark IV.1 offers a tighter approximation to and hence we shall use instead of in the forthcoming derivations in Section V.

### Iv-C Noise component analysis

The noise component of the received signal on subcarrier can be expressed as:

 ^Zk = ^Z(1)k+^Z(2)k+^Z(3)k+^Z(4)k, where: ^Z(1)k = ∑˙u∈U∑˙k∈^G^W∗˙uβ0p˙kΩk+˙u−˙k(1)=^g+u∑v=−^g−umin{u,^g−v}∑˙u=−min{u,v+^g}^W∗˙uβ0pv+˙uΩk−v (18a) ^Z(2)k = ∑˙u∈U∑¯k∈K∖G^W∗˙uβ¯kx¯kΩk+˙u−¯k (18b) ^Z(3)k = ∑˙u∈Uβ∗0Ω∗˙uWk+˙u (18c) ^Z(4)k = ∑˙u∈U^W∗˙uWk+˙u, (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, zero-mean 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:

 E{^Zk} = 0 (19a) E{|^Zk|2} = 4∑i=1E{|^Z(i)k|2}, (19b)

where:

 E{|^Z(1)k|2} (1)= ^g+u∑v=−^g−uΔk−v,k−vE∣∣∣min{u,^g−v}∑˙u=−min{u,v+^g}^W∗˙uβ0pv+˙u∣∣∣2 (20a) (2)≤ ^g+u∑v=−^g−uηk,^g+umin{u,^g−v}∑˙u,¨u=−min{u,v+^g}|β0|2N0Rinvp(¨u,˙u)pv+˙up∗v+¨u (3)= ηk,^g+u∑˙u,¨u∈U|β0|2N0Rinvp(¨u,˙u)^g−min{˙u,¨u}∑v=−^g−min{˙u,¨u}pv+˙up∗v+¨u (4)= ηk,^g+u∑˙u,¨u∈U|β0|2N0Rinvp(¨u,˙u)Rp(˙u,¨u)=ηk,^g+u|U||β0|2N0 E{|^Z(2)k|2} = (20b) ≤ (5)= Tr{R−1p}N0|¯β|2Ed E{|^Z(3)k|2} (6)= ∑˙u∈U|β0|2Δ˙u,˙uN0=|β0|2μ(0,u)N0 (20c) E{|^Z(4)k|2} (7)= ∑˙u∈URinvp(˙u,