# The Effect of Spatial Correlation on the Performance of Uplink and Downlink Single-Carrier Massive MIMO Systems

We present the analysis of a single-carrier massive MIMO system for the frequency selective Gaussian multi-user channel, in both uplink and downlink directions. We develop expressions for the achievable sum-rate when there is spatial correlation among antennas at the base station. We show that although the Channel Matched Filter Precoder (CMFP) performs the best in a spatially uncorrelated downlink channel, in a spatially correlated channel, CMFP does not perform as expected. We suggest better precoding schemes which lead to a better achievable sum-rate. Considering the same groups of users and antennas, we develop a model for the uplink channel. Using two equalizer structures, we show the results of system performance in terms of achievable rate at the base station.

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## Authors

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• ### Capacity Scaling of Massive MIMO in Strong Spatial Correlation Regimes

This paper investigates the capacity scaling of multicell massive MIMO s...
12/21/2018 ∙ by Junyoung Nam, et al. ∙ 0

• ### Transmit Correlation Diversity: Generalization, New Techniques, and Improved Bounds

When the users in a MIMO broadcast channel experience different spatial ...
04/20/2021 ∙ by Fan Zhang, et al. ∙ 0

• ### Uplink Massive MIMO for Channels with Spatial Correlation

A massive MIMO system entails a large number of base station antennas M ...
07/12/2018 ∙ by Ansuman Adhikary, et al. ∙ 0

• ### Communications and Radar Coexistence in the Massive MIMO Regime: Uplink Analysis

This paper considers the uplink of a massive MIMO communication system u...
02/25/2019 ∙ by Carmen D'Andrea, et al. ∙ 0

• ### Full Downlink Channel Reconstruction using Incomplete Uplink Channel Measurements in Massive MIMO networks

While more and more antennas are integrated into single mobile user equi...
02/11/2019 ∙ by Aleksei Fedorov, et al. ∙ 0

• ### Optimal SNR Analysis for Single-user RIS Systems

In this paper, we present an analysis of the optimal uplink SNR of a SIM...
08/21/2020 ∙ by Ikram Singh, et al. ∙ 0

• ### Quantum Annealing for Large MIMO Downlink Vector Perturbation Precoding

In a multi-user system with multiple antennas at the base station, preco...
02/24/2021 ∙ by Srikar Kasi, et al. ∙ 0

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## 1 Introduction

The demands for high-rate wireless communications have been increasing in the past few decades [1]. Due to this fact, there is a great amount of research on the development of new and efficient schemes to obtain higher rates of information for an increasing number of wireless channel users. Massive Multiple-Input Multiple-Output (Massive MIMO) is one of the approaches to achieve a higher rate in a wireless system. As a corollary of increasing the number of transmitters at the base station, the transmit power can be designed to be significantly small, since interference decreases at the same rate of the desired signal power, leading to a power-efficient system [2]

. Also, the channel vectors (or matrices) defined between transmitters and receivers become asymptotically orthogonal, allowing multiple users to use the same bandwidth without interfering with each other and achieving a high spectral efficiency in the system

[3]. However, in order to be able to use this method in an efficient way, better understanding of air interfaces, employment of precoders or equalizers in time or frequency domain, the optimum modulation technique, and limitations due to hardware impairments are needed.

In most studies of massive MIMO, the base station is considered to have the perfect knowledge of the channel, since massive MIMO relies on spatial multiplexing which requires good channel knowledge for the base station [4]

. The base station can estimate the channel response to an individual user using the pilots sent by the terminals in the uplink. This process is more challenging in the downlink. The base station can send out pilot waveforms and users’ terminals will be able to estimate the channel responses and feed them back to the base station. This solution was popular in conventional MIMO systems such as Long Term Evolution (LTE), however, it is not feasible in massive MIMO operation in an environment with high mobility

[4]. Although, operating in time-domain-multiplexing (TDD) mode and employing the reciprocity of the uplink and the downlink is a general solution for massive MIMO, on the other hand, frequency-devision-multiplexing (FDD) mode can be a possible solution in specific cases [4].

It is critical to choose a suitable modulation scheme to cope with the massive MIMO channel. Orthogonal Frequency Division Multiplexing (OFDM) is one of the well-known modulation methods which has been studied in literature, see, e.g., [5]

. In a system with OFDM, discrete Fourier transform and its inverse are important tools to deal with the frequency selectivity of the channel in the air interface. Discrete Fourier transforms also make the allocation of time or frequency resources feasible. A massive MIMO channel with OFDM modulation is considered the main technology in the downlink transmission in LTE

[6], since OFDM modulation can convert a frequency selective channel into multiple sets of flat fading channels and the benefits of MIMO systems can be easily obtained.

As an alternative approach to OFDM, single-carrier (SC) modulation has been introduced and was first investigated in a massive MIMO channel in [1]. Single-carrier transmission conventionally employs adaptive equalization techniques, however, in [1], the modulation uses the precoding technique to transmit symbols over the channel. In using single-carrier transmission, the main question is defining the equalization (or precoding) technique in either time or frequency domain. Although the frequency domain equalization techniques are usually preferred over the time domain ones because of the smaller complexity, see, e.g., [7], cyclic prefix and discrete Fourier transform modules are non-separable parts of the equalization techniques in the frequency domain. At the cost of increased complexity, time domain equalizers can achieve reasonable performance without requiring cyclic prefix or any discrete Fourier transforms. It is noteworthy to mention that single-carrier modulation combined with frequency domain equalization is a technique similar to OFDM which was proposed to combat intersymbol interference (ISI) without the Peak-to-Average Power Ratio (PAPR) growth of OFDM [8].

The work in this paper is focused on single-carrier transmission for massive MIMO applications. This is motivated by a number of recent studies such as [1]. This work showed that a Channel Matched Filter Precoder (CMFP) is optimum for such systems in channels uncorrelated in space. However, unlike [1], we wish to determine the performance of such systems for correlated channels, such as in massive MIMO where the presence of correlation in the channel is expected to be high due to space limitations for the antenna elements. Also, we expand our studies to the uplink channel, where equalizers are deployed at the base station side. Using the equalizers in the uplink and precoders in the downlink, we investigate the performance of the massive MIMO channel under the influence of correlation patterns. We would like to emphasize that unlike a conventional channel matched filter, CMFP is placed at the transmitter side [1].

## 2 Downlink Channel

In the downlink model, a frequency-selective multi-user MIMO (MU-MIMO) channel with base station antennas and single-antenna users is considered. We model the channel between the -th transmit antenna and the -th user as a finite impulse response (FIR) filter with taps. The taps model different delay components. The -th tap can be written as where and correspond to the slow-varying and fast-varying components of the channel respectively, and where

has a complex Gaussian distribution with zero mean and unit variance

[1]. When the antenna elements or delay components are not correlated, the entries of the matrix consisting of the fast-varying components of the channel, , are independent and identically distributed (i.i.d.). Further, these entries are fixed while the symbols are being transmitted. Define and as the vector of received signals at each user and the vector of transmitted signals from each antenna at the base station at time , respectively. Assume that the noise vector is additive white Gaussian noise (AWGN) consisting of i.i.d. components and complex Gaussian distribution. We assume that this distribution has zero mean and unit variance. The variable is the information symbol to be transmitted to the -th user at time . The vector of information symbols is defined as . This vector is considered to have i.i.d. components. We also need to define , and . The -th element of is .

In this paper, we will model the channel whose antenna elements have spatial correlation. To this end, we introduce the matrix , taking into account all antenna elements at the base station.The effect of this matrix on the channel will be to modify the channel realization from to . A channel whose antenna elements are not spatially correlated will have . In Section 5, we will make use of two commonly used spatial correlation models from the literature. Both of these models will have symmetric matrices.

The received signal vector can be written as

 y[i]=L−1∑l=0^HHlx[i−l]+% n[i], (1)

where the channel state information matrix is defined as and the channel power delay profile (PDP) for each user is normalized such that [1]

 L−1∑l=0dl[k]=1,k=1,…,K. (2)

Also, is the transmitted signal vector and it depends on the precoder used in the channel.

### 2.1 Channel Matched Filter Precoder (CMFP)

In wireless communication, the precoding scheme has a significant role. By using precoding techniques, one can reduce the PAPR and increase the signal-to-noise ratio (SNR) at the receiver [1]. The CMFP response matrix is the Hermitian of the channel state information matrix. Therefore, the transmit symbol vector can be written as

 x[i]=√ρfMKL−1∑l=0^Hls[i+l], (3)

in which is the total average power transmitted by the base station antennas. We define the super channel matrix as the multiplication of the precoder matrix and the channel information matrix . Note that one can obtain , where is a vector with all its elements equal to except the -th one which is . Using the channel composite matrix and placing (3) in (1), we can rewrite the received signal of the -th user at time as separate terms of the desired signal and the effective noise in the form

 yk[i] =√ρfMK(L−1∑l=0E{F(l,l)[k,k]})sk[i]DesiredSignalTerm (4)

By an inspection of (1), (3), and (4), the effective noise term can be calculated as

 +√ρfMKL−1∑b=1−Lb≠0L2∑l=L1F(l,l−b)[k,k]sk[i−b] +√ρfMKK∑q=1q≠kL−1∑b=1−LL2∑l=L1F(l,l−b)[k,q]sq[i−b] +nk[i]. (5)

In (5), we defined and . In (5), the first term can be identified as the additional interference (IF), the second term as the intersymbol interference (ISI), and the third term as multiuser interference (MUI). The final term is AWGN.

In (4), for the desired signal term, the average power is equal to [1]

 Sk=Esk[i]{∣∣∣√ρfMKL−1∑l=0E{F(l,l)[k,k]}sk[i]∣∣∣2}=MρfK. (6)

The correlation matrix and the average power of the desired signal are independent, meaning that the desired signal power is not affected by the spatial correlation among antennas at the base station. The power of , the effective noise in (5), can be calculated as

 Var(n′k[i])=tr(A2)Mρf+1. (7)

In (7), is the trace of the square of the correlation matrix. We will introduce Proposition 1 and Proposition 2 below. These two propositions illustrate how (4) can be used to calculate the average power of the desired signal in (6) and the effective noise power in (7). For a channel without spatial correlation, (7) becomes . Although the average power of the desired signal and the correlation matrix are independent, the effective noise power is affected by the correlation matrix.

We are now interested in characterizing for the spatially correlated channel. When we consider the correlated case, the entries on the main diagonal of remain the same as the spatially uncorrelated case. On the other hand, other elements have nonzero values when there is correlation and for the models in this work, these values are nonnegative. Therefore, for the spatially correlated channel, . For this reason, the effective noise power will increase because of channel correlation. On the other hand, this does not affect the desired signal’s average power. The correlation pattern makes the user information rate decrease, but the capacity remains the same. This implies an increase in correlation results in increased difference between the information rate and the capacity.

Each user’s information rate as given in [1] can be obtained by . By considering the fact that each user’s information rate is almost equal to the other users’, we will have the sum-rate as

 Rsum(ρf,M,K)=Klog2(1+MρfKtr(A2)Mρf+K). (8)

Since the correlation has no effect on the average power of the desired signal, the cooperative sum-capacity is independent of the correlation pattern and it can be written as [1]

 Ccoop(ρf,M,K)≈Klog2(1+MρfK). (9)

Equations (8) and (9) are the sum rate (information rate) and the upper bound for CMFP, respectively, that will be illustrated in Fig. 1 and Fig. 2 in the sequel. By observing the information rate and the sum-capacity, one can conclude that CMFP may not be a good choice when there is sufficiently significant spatial correlation among the base station antennas. When the channel does not have spatial correlation, the effect of AWGN becomes more pronounced over the effective noise power. As a result, in that case, CMFP appears as a good choice for a precoder. When reaches larger values, the information rate begins to saturate until the efective noise power is dominated by the interference terms. Computation results in Section 5 will verify this effect. As correlation increases, saturation happens earlier and quicker. The effect of any correlation among base station antennas is stronger interference terms that have a bigger effect than AWGN.

Proposition 1: The average power of the signal using CMFP is given as in (6).
Proof: In order to obtain the average power of the desired signal, we need the following lemma.
Lemma 1: Let and be -dimensional zero-mean circularly symmetric complex Gaussian vectors with the autocorrelation and cross-correlation matrices given as , , and . Then, the following expectations can be expressed as

 E{xHAy} =tr(ARyx) (10) E{|xHAy|2} = tr(ATRxARy)+tr(ARyx)tr(ATRxy) (11)

in which is a real-valued and symmetric matrix. The proof of this lemma is omitted here due to lack of space.111

One can use the moment generating function to obtain (

10).

Consider the average power of the desired signal in (4) as

 Sk =ρfMKEsk[i]{∣∣∣L−1∑l=0E{F(l,l)[k,k]}sk[i]∣∣∣2} =ρfMKEsk[i]{L−1∑l′=0sHk[i]E{F(l′,l′)[k,k]} (12)

We claim that the first multiplier can be written as

 L−1∑l′=0sHk[i]E {F(l′,l′)[k,k]}= L−1∑l′=0sHk[i]E{eTkD1/2l′HHl′AHl′D1/2l′ek}. (13)

Let’s focus on the term . If we expand this term, it will be

 E{F(l,l)[k,k]}=E{eTkD1/2lHHlAHl% D1/2lek}. (14)

Note that using , we can rewrite the above equation as

 E{F(l,l)[k,k]}=√dl[k]E{hHl[k]Ahl[k]}√dl[k], (15)

where . Note that and . By using Lemma 1, one can say which is equal to . Therefore

 Sk=ρfMKEsk[i]{∣∣sk[i]∣∣2}(ML−1∑l=0dl[k])2=MρfK. (16)

Based on Lemma 1, a set of expectations necessary to calculate the noise power in (7) is provided in the Appendix.

Proposition 2: The effective noise power using CMFP is given as in (7).
Proof: By an inspection of (5), we notice that different terms are independent from each other. Therefore, by using the expectations (58) and (61), we can write the variance of the effective noise as

 Var {n′k[i]}= +ρfMKVar{L−1∑b=1−Lb≠0L2∑l=L1F(l,l−b)[k,k]sk[i−b]} +ρfMKVar{K∑q=1q≠kL−1∑b=1−LL2∑l=L1F(l,l−b)[k,q]sq[i−b]} +1. (17)

Note that the means of IF, ISI, and MUI terms are zero. Considering the fact that and the information symbols are independent from all other terms, after carefully reindexing (17) and removing a number of redundant terms, we can rewrite the effective noise variance as

 Var{n′k[i]}= ρfMKK∑q=1L−1∑b=1L−1∑l=bE{F(l,l−b)[q,k]F(l−b,l)[k,q]} +ρfMKK∑q=1L−1∑b=1L−1∑l=bE{F(l−b,l)[q,k]F(l,l−b)[k,q]} −ρfMKL−1∑l=0[E{F(l,l)[k,k]}]2+1. (18)

Then, if we replace the expectations inside (18) with their corresponding expressions in (55)-(61) in the Appendix, the effective noise variance can be obtained as

 Var{n′k[i]})= tr(A2)ρfMKK∑q=1L−1∑b=1L−1∑l=b(dl−b[q]dl[k]+dl[q]dl−b[k]) +tr(A2)ρfMKK∑q=1L−1∑l=0dl[k]dl[q]+1. (19)

One can recognize that (19) can be expressed as

 (20)

Then, by using (2), (20) can be simplified as (7).

### 2.2 Conventional Precoders

The Zero-Forcing Precoder (ZFP) and Regularized Zero-Forcing Precoder (RZFP) are two well-known precoders in the massive MIMO field of study, see, e.g., [2]. A common theme among the precoders we will discuss in this paper is the fact that they are defined in the frequency domain and are translated into the time domain. Computations indicated that precoders defined in the time domain, such as those in [9], do not perform as well as precoders in the frequency domain. The two precoders will be given as functions of the channel state matrix as the following

• ZFP: It forces the system to eliminate the interference and is given by

 WZFPν=aZFPW^Hν(^HHν^Hν)−1, (21)

where is the -point () Fourier transform of the channel state matrix and is a normalization factor for this precoder.

• RZFP: This precoder maximizes the power of the desired signal compared to the power of the noise and interference at the receiver. It is given by

 WRZFPν=aRZFPW^Hν(^HHν^Hν+βWIK)−1, (22)

where is also a power normalization factor and is a system parameter which depends on the SNRs and the path losses of the users.

Using these new precoders, one can generate a new model for the transmit signal vector. Using a general notation of to indicate the precoder model, the transmit signal vector can be obtained as

 x[f]=√ρfWνs[f], (23)

where and are the Fourier transforms of the information symbols vector and the transmit signal vector, respectively. Note that in the definition of the precoders, we already consider the normalization factor. Considering the inverse Fourier transform, the vector of the transmit signals can be obtained in the time domain as the following

 x[i]=√ρfN−1∑m=0Wms[i−m], (24)

where is the inverse Fourier transform of the precoder matrix. Note that (24) represents the cyclic convolution and all the indices of equation defining the transmit signal are taken modulo (where ). Considering that (1) still holds, the vector of the received signals at the users’ site can be written as

 y[i]=√ρfN−1∑m=0L−1∑l=0^HHlWms[i−l−m]+n[i]. (25)

By defining the new super channel matrix as and considering the fact that (24) represents circular convolution, one can rewrite (25) as

 y[i]=√ρf0∑m=1−NL−1∑l=0F(l,m)s[i−l−m]+n[i]. (26)

By changing the variable to and considering the fact that , we can rewrite (26) as

 y[i]=√ρfK∑q=1L−1∑b=1−NL3∑l=L1F(l,b−l)[:,q]sq[i−b]+n[i] (27)

where is a column vector and where we defined . Note that the desired signal at the -th user and time is given by

 gk[i]=√ρfL−1∑l=0E{F(l,−l)[k,k]}sk[i]. (28)

Using the equations of the desired and received signal, one can express the system model in terms of desired signal and effective noise of the channel as

 yk[i]=gk[i]+n′k[i], (29)

where represents the effective noise and can be written as

 n′k[i] =√ρfL−1∑l=0(F(l,−l)[k,k]−E{F(l,−l)[k,k]})sk[i] +√ρfL−1∑b=1−Nb≠0L3∑l=L1F(l,b−l)[k,k]sk[i−b] +√ρfK∑q=1q≠kL−1∑b=1−NL3∑l=L1F(l,b−l)[k,q]sq[i−b] +nk[i], (30)

which again includes IF, ISI, MUI, and AWGN terms, respectively. The system model introduced in this section is used to run computations. Note that since the power of the AWGN is considered to be , one can define the capacity of the system as

 Csum=K∑k=1log2(1+Sk), (31)

where one can obtain the power of the desired signal for the -th user as it was calculated in Section 2.1 by . The achievable sum-rate of the system can be obtained as

 Rsum=K∑k=1log2(1+SkVar(n′k[i])). (32)

Equations (31) and (32) are the upper bound and the sum-rate (information rate) for the precoders, respectively. They will also apply to the case of equalizers for the uplink channel as will be discussed in the next section. These equations will be employed in Fig. 3 through Fig. 15.

## 3 Uplink Channel

Just like the downlink model, a frequency-selective multi-user MIMO (MU-MIMO) channel with base station antennas and single-antenna users is considered. Perfect knowledge of channel state information is considered at the users’ terminal in the uplink transmission. Since the correlation pattern is also considered in the uplink channel, the channel state information matrix can be modeled as . Considering the uplink, users are supposed to transmit through the channel (to be able to control the power of the transmit signal from users’ terminal, has been added to the signal). However, using equalizers in the frequency domain at the base station requires employment of cyclic prefix techniques. In this work, the conventional cyclic prefix technique, where the last samples of a

-sample transmission block are added to the beginning of the block, is preferred over the zero-padding or the known symbol padding techniques due to its circular convolution property

[10].

In the same manner as it was introduced earlier in Section 2, one can obtain the capacity and the achievable sum-rate of the uplink channel with an equalizer using (31) and (32). respectively. Note that is the power of the desired signal for the -th user at the base station and is the effective noise of the system for the -th user at the base station at time considering the uplink channel.

### 3.1 Channel Matched Filter Equalizer (CMFE)

The cyclic prefix is designed in the way that the length of the added symbols are larger than the length of the channel (i.e., ). We can write the received signal vector before applying the proper equalizer as

 r[i]=L−1∑l=0^Hlx[(i−l)modT]+n[i], (33)

where denotes the transmitted signal vector and consists of the transmitted symbols (e.g., ) and the added symbols as the cyclic prefix. In this section, the channel matched filter is considered as the equalizer in the system. For that reason we designate it as CMFE, as opposed to CMFP employed in the downlink channel. The following shows how CMFE will affect the unprocessed output signal of the channel.

 y[i]=1√MKL−1∑l=0^HHlr(i+l), (34)

where is the vector of the received signal after the equalizer block. Substituting (33) in (34), one can obtain

 y[i] =√ρfMKL−1∑l=0L−1∑l′=0^HHl^Hl′s[(i−l′+l)modT] +1√MKL−1∑l=0^HHln[i+l]. (35)

We can rewrite the received signal for each individual user at the base station as the following

 yk[i]=gk[i]+n′k[i], (36)

where is the desired signal of the -th user at the base station and can be written as

 gk[i]=√ρfMK(L−1∑l=0F(l,l)[k,k])sk[imodT], (37)

where is the super channel matrix. The second is the effective noise of the system and, similar to the downlink channel, it can be written as

 n′k[i]= √ρfMKL−1∑b=1−Lb≠0L2∑l=L1F(l,l−b)[k,k]sk[(i−b)modT] +√ρfMKK∑q=1q≠kL−1∑b=1−LL2∑l=L1F(l,l−b)[k,q]sq[(i−b)modT] +1√MKL−1∑l=0eTk^HHln[i+l]. (38)

As it can be seen, the effective noise term in the uplink is very similar to that of the downlink, except the difference of using CMFE makes in AWGN noise and perfect knowledge of CSI eliminates the IF term.

In order to determine the power of effective noise in the uplink channel, we need to determine the power of AWGN affected by CMFE, which is as the following

 =1MKE(L−1∑l=0eTk^HHln[i+l]L−1∑l′=0nT[i+l′]^Hl′ek) =1MKE(L−1∑l=0L−1∑l′=0eTk^HHln[i+l]nT[i+l′]^Hl′ek) =1MKEHl(L−1∑l=0L−1∑l′=0eTk^HHlEn[i](n[i+l]nT[i+l′])^% Hl′ek). (39)

By the nature of AWGN, if , then . It will be equal to if . Keeping this in mind, we can rewrite the power of the AWGN affected by CMFE as

 Var(zk[i]) =Var(1√MKL−1∑l=0e% Tk^HHln[i+l]) =1MKEHl(L−1∑l=0eTk^HHl^Hl′ek) =1MKEHl(L−1∑l=0eTkD1/2lHHlAHl% D1/2lek) =tr(A)MK. (40)

Therefore, using the power of AWGN affected by CMFE in (40) and the power of interference terms calculated in [11] neglecting the IF term, one can obtain the power of the effective noise in the uplink channel as

 Var(n′k[i])=tr(A2)M(1−L−1∑l=0d2l[k]K)ρf+tr(A)MK, (41)

where for the correlation patterns we consider in this work.

Based on what is stated in [11], the power of the desired signal can be obtained as . Using (3) and by considering the fact that each user’s capacity information rate is almost equal to the other users’, we will have the capacity sum-rate as

 Csum=Klog2(1+ρfMKtr(A2)L−1∑l=0d2l[k]+ρfMKtr2(A)). (42)

Using (4), one can obtain the achievable sum-rate as

 Rsum=Klog2(1+tr(A2)∑L−1l=0d2l[k]ρf+tr2(A)ρftr(A2)(K−∑L−1l=0d2l[k])ρf+M). (43)

### 3.2 Conventional Equalizers

Clearly, CMFE is a special case of implementing equalizers in the system. Using the general notation of to denote the equalizer equation in the frequency domain, one can obtain the received signal as

 y[f]=Qνr[f], (44)

where and are the Fourier transforms of the unprocessed signal vector and the received signal vector, respectively.

Selecting the equalizer scheme has been the subject of study in MIMO channel as in [9]. Two well-known equalizers are considered in this work:

• Zero-Forcing Equalizer (ZFE): As it was introduced earlier as a precoder, ZFE can bring down the ISI to zero. This equalizer is given by the following

 QZFEν=aZFEQ(GνGHν)−1Gν, (45)

where is the normalization factor, is the Fourier transform of the composite channel matrix, and .

• Minimum Mean Square Error Equalizer (MMSEE): As it is known in the downlink channel as RZFP, MMSEE is able to maximize the power of received signal compared to the noise and interference. This equalizer can be obtained as

 QMMSEEν=aMMSEEQ(GνG% Hν+βQIK)−1Gν, (46)

where is also a normalization factor and is a system parameter just like the one in RZFP, which depends on the SNRs and the path losses of the users. By inspecting the similarities in the definition of MMSEE and RZFP, it can be concluded that MMSEE in the uplink performs the same functionality as RZFP in the downlink.

Considering the inverse Fourier transform of the equalizer matrix , (44) can be rewritten in the time domain as

 y[i]= √ρfN−1∑m=0L−1∑l=0Qm^Hls[(i−l−m)modT] + N−1∑m=0Qmn[i−m]. (47)

Note that using an equalizer in the channel, the super channel state information matrix can be obtained by . Similar to the downlink channel, one can rewrite the vector of the received signal using the desired signal and the effective noise terms

 yk[i]=gk[i]+n′k[i], (48)

where is the desired signal at the -th user and time

 gk[i]=√ρfL−1∑l=0F(−l,l)[k,k]sk[i]. (49)

Note that the channel length is smaller than the frequency range (i.e., ). One can obtain the effective noise of the channel at the -th user as the following

 n′k[i]= √ρfL−1∑b=1−Nb≠0L3∑l=L1F(b−l,l)[k,k]sk[i−b] + √ρfK∑q=1q≠kL−1∑b=1−NL3∑l=L1F(b−l,l)[k,q]sq[i−b] + K∑q=1N−1∑m=0Qm[k]nq[i−m],