Wireless Personal Communications

, Volume 100, Issue 4, pp 1645–1660 | Cite as

Precoding for Non-orthogonal Multiple Access with User Equipment Pairing

  • Chi-Min Li
  • Chen-Feng Shen


Due to the tremendous throughput requirement, current spectrum allocation scheme of 4G long term evolution-advanced system can’t satisfy the need for future communications applications. Hence, researches have proposed a non-orthogonal multiple access (NOMA) technique as a candidate for 5G access system recently. This paper proposes two precoding matrix selection methods for the NOMA system and also provides an appropriate user equipment (UE) pairing method for NOMA. Combining the UE pairing with the proposed precoding selection method, results show that system capacity for NOMA system can be further improved.


NOMA MIMO system Precoding matrix UE pairing 

1 Introduction

Long term evolution-advanced (LTE-A) is currently the most widely deployed fourth-generation (4G) wireless communication system. LTE-A adopts the orthogonal frequency division multiple access (OFDMA) to improve the spectrum efficiency for the conventional wide-band code division multiple access system (WCDMA) [1]. However, as the tremendous throughput demand required for the future wireless communication applications, the spectrum allocation scheme of the LTE-A system can no longer satisfy the need. Therefore, researches have proposed a novel non-orthogonal multiple access (NOMA) technique [2, 3, 4] as the fundamental access method for future 5G system and it has become the working item for the 3rd generation partnership project (3GPP).

Different from the OFDMA system that each individual user occupies a different frequency spectrum, in the NOMA transmission, it allows multiple users to use the same frequency spectrum at the same time to transmit their signals. The advantage of the NOMA is to have better spectrum efficiency than the OFDMA system.

In literatures, many multiple access (MA) techniques have been proposed to share the same spectrum among multiple users, such as the conventional WCDMA and its related variants: the multi-carrier CDMA (MC-CDMA) and multi-carrier direct sequence CDMA (MC-DS-CDMA). Currently, space code multiple access (SCMA) has also been proposed to increase the spectrum efficiency [5, 6]. For NOMA system, the inter-user interferences (IUI) can be eased via the proper power allocation and the successive interference cancellation (SIC) techniques (Fig. 1), and these code-based MA methods separate different signals via using the orthogonality of the adopted spreading codes.
Fig. 1

Illustration of NOMA system [3]

Besides, in LTE-A system, transmitter (Tx) can adopt the precoding matrix to ease the channel fading and has the advantage of beamforming. Receiver (Rx) acquires the channel state information (CSI) via the reference signal (RS) and feedbacks the suitable precoding matrix index (PMI) based on some criteria, such as to maximize the mutual information (MI) or minimize the bit error rate (BER) [7]. Therefore, it is an intuitive and reasonable extension for the future 5G system to adopt the precoding matrix during transmission since that the precoding technique has been shown can efficiently avoid the interferences and increase the signal to noise ratio (SNR) at Rx. Hence, what is the performance of the precoding matrix if applied to the NOMA system and how to select a suitable precoding matrix become interesting problems for the NOMA system. In [8], authors described a precoding matrix selection method to maximize the SNR or MI. This method has excellent capacity performance. Nevertheless, this method requires large amount of calculations that make it impractical in real implementations. In [9], authors reported singular value decomposition (SVD) precoding matrix selection method based on the maximization of the trace of the equivalent channel matrix. Besides, also in [9], a low complexity precoding matrix selection method was proposed by using the QR decomposition of the channel matrix. NOMA communication can increase system capacity by reusing the same frequency spectrum resource among multiple user equipment (UE). In practical situations, there are often several UEs co-exist nearby. Therefore, in this paper, how to choose the appropriate UE pairs for the NOMA communication is also addressed.

This paper is organized as follow, Sect. 2 formulates the considered problem and illustrates the proposed two precoding selection methods. Besides, an appropriate UE pairing method for the NOMA communication is also provided. In Sect. 3, computer simulations and the indoor channel measurements were conducted to evaluate the performances of the proposed schemes and the literature methods. Finally, some conclusions for this paper are given in Sect. 4.

2 Problem Formulation and Method Descriptions

2.1 Problem Formulation

In Fig. 2, consider a NOMA system with two UEs in the cellular network. The transmitted symbol of the \(n\)-th UE is precoded with the \({\mathbf{W}}_{{\mathbf{n}}}\) precoding matrix, \(n = 1,2\). Assume the 1st user (UE1) has a better channel condition than the 2nd user (UE2), UE1 will apply the SIC technique to decode its symbols. For the UE2, it decodes the received symbols directly. The power levels allocated to the UE1 and UE2 are denoted as \(P_{1}\) and \(P_{2}\) respectively.
Fig. 2

NOMA system with precoding [2]

Assume there are \(N_{t}\) transmit antennas and \(N_{r}\) receive antennas for both users during transmission, the received signal \({\mathbf{y}}_{{\mathbf{n}}}\) for the \(n\)-th UE can be expressed as
$${\mathbf{y}}_{{\mathbf{n}}} = {\mathbf{h}}_{{\mathbf{n}}} ({\mathbf{W}}_{{\mathbf{1}}} \sqrt {\beta_{1} P} {\mathbf{x}}_{{\mathbf{1}}} + {\mathbf{W}}_{{\mathbf{2}}} \sqrt {\beta_{2} P} {\mathbf{x}}_{{\mathbf{2}}} ) + {\mathbf{w}}_{{\mathbf{n}}}$$
where \({\mathbf{h}}_{{\mathbf{n}}}\) is the \(N_{r} \times N_{t}\) channel matrix, \({\mathbf{W}}_{{\mathbf{n}}}\) is the adopted \(N_{t} \times l\) precoding matrix, \(l\) is the number of spatial layer, \({\mathbf{x}}_{{\mathbf{n}}}\) is the \(l \times 1\) transmitted symbol vector for the \(n\)-th UE, \(\beta_{1}\) and \(\beta_{2}\) are the power ratio allocated to the UE1 and UE2 respectively, i.e., \(P_{1} = \beta_{1} P\), \(P_{2} = \beta_{2} P\), \(P\) is the total transmitted power, and \({\mathbf{w}}_{{\mathbf{n}}}\) is the \(N_{r} \times 1\) Additive White Gaussian Noise vector (AWGN). That is, the received signal at UE1 is
$${\mathbf{y}}_{{\mathbf{1}}} = {\mathbf{h}}_{{\mathbf{1}}} ({\mathbf{W}}_{{\mathbf{1}}} \sqrt {\beta_{1} P} {\mathbf{x}}_{{\mathbf{1}}} + {\mathbf{W}}_{{\mathbf{2}}} \sqrt {\beta_{2} P} {\mathbf{x}}_{{\mathbf{2}}} ) + {\mathbf{w}}_{{\mathbf{1}}}$$
In NOMA system, UE1 utilizes the SIC and applies the first equalizer \({\mathbf{G}}_{{\mathbf{1}}}\) to decode the symbol \({\mathbf{x}}_{{\mathbf{2}}}\). For example, the estimated UE2 symbol \({\hat{\mathbf{x}}}_{{\mathbf{2}}}\) at UE1 is
$${\hat{\mathbf{x}}}_{{\mathbf{2}}}\,{\mathbf{ = G}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{1}}}$$
Then, subtract the received signal from the decoded UE2 symbol as
$${\hat{\mathbf{y}}}_{{\mathbf{1}}} = {\mathbf{y}}_{{\mathbf{1}}} - {\mathbf{h}}_{{\mathbf{1}}} \sqrt {\beta_{2} P} {\hat{\mathbf{x}}}_{{\mathbf{2}}}$$
The final decoded symbol for UE1 is
$${\hat{\mathbf{x}}}_{{\mathbf{1}}}\,{\mathbf{ = G}}_{{\mathbf{2}}} {\hat{\mathbf{y}}}_{{\mathbf{1}}}$$
where \({\mathbf{G}}_{{\mathbf{2}}}\) is the second equalizer adopted for UE1 after the SIC.
For the UE2, the decoded symbol is
$${\hat{\mathbf{X}}}_{{\mathbf{2}}}\,{\mathbf{ = U}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{2}}}$$
where \({\mathbf{U}}_{{\mathbf{1}}}\) is the adopted equalizer at UE2.

The purpose of this paper is to appropriately select the precoding pair \(({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} )\) for Tx at the downlink to have a better SNR or capacity performance in NOMA communication.

2.2 Review of the Literature Precoding Selection Methods

Based on the signal model in Eq. (1), the signal-to-interference-plus-noise ratio (SINR) for the UE1 and UE2 can be expressed as
$$SINR_{l,1} = \frac{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{1} } \right|^{2} }}{{\left| {N_{w} + \sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{2} } \right|^{2} }}$$
$$SINR_{l,2} = \frac{{\left| {\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} } \right|^{2} }}{{\left| {N_{w} + \sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{1}}} } \right|^{2} }}$$
where \(N_{w} = E\{ \left| {w_{n} } \right|^{2} \}\) is the average energy of AWGN, \(E\{ \cdot \}\) is the expectation. In [8], the mutual information (MI) is defined as
$$I = \sum\limits_{l = 1}^{L} {\log_{2} (1 + SINR_{l,1} ) + } \log_{2} (1 + SINR_{l,2} )$$
\(L\) is the number of layer and the two precoding matrices can be selected via maximizing the MI as
$$({\mathbf{W}}_{1} ,{\mathbf{W}}_{2} ) = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{1} ,{\mathbf{W}}_{2} \in {\mathbf{W}}}} (I)$$
where \({\mathbf{W}}\) is the set of the available precoding matrices or the adopted code-book, and max(·) is the maximum operator, i.e., the method is to find a pair \(({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} )\) with the maximum of MI.
In [9], authors provided a method to select the proper precoding matrices based on the SVD. Assume the SVD of the channel matrix can be expressed as
$${\mathbf{H = UDV}}^{{\mathbf{H}}}$$
where \({\mathbf{U}}\) and \({\mathbf{V}}\) are the \(N_{r} \times N_{r}\) and \(N_{t} \times N_{t}\) unitary matrix, \({\mathbf{D}}\) is a diagonal matrix. Define an equivalent channel as \({\mathbf{P}}_{{{\mathbf{W}}_{{\mathbf{n}}} }}\)
$${\mathbf{P}}_{{{\mathbf{W}}_{{\mathbf{n}}} }} {\mathbf{ = U}}^{{\mathbf{H}}} {\mathbf{HW}}_{{\mathbf{n}}} {\mathbf{ = DV}}^{{\mathbf{H}}} {\mathbf{W}}_{{\mathbf{n}}}$$
The SVD method selects precoding matrices as
$${\mathbf{W}}_{{\mathbf{n}}} = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{n}}} \in {\mathbf{W}}}} \left| {\sum\limits_{i = 1}^{{N_{r} }} {{\mathbf{P}}_{{{\mathbf{W}}_{{\mathbf{n}}} }} (i,i)} } \right|\quad (n = 1,2)$$
Another SVD-based method in [10] selects the precoding matrices as
$${\mathbf{W}}_{{\mathbf{n}}} = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{n}}} \in {\mathbf{W}}}} (D_{min} )\quad (n = 1,2)$$
where \({\mathbf{D = U}}^{{\mathbf{H}}} {\mathbf{HW}}_{{\mathbf{n}}} {\mathbf{V}}\) and \(D_{min}\) is the minimum value along the diagonal of \({\mathbf{D}}\). In this paper, the SVD method is the scheme by using Eq. (13) to decide the precoding matrix while the method via Eq. (14) is denoted as SVD-SC.
Also in [10], authors provided the precoding selection based on the max–min selection criteria (MSC) and QR decomposition (QRD). Let the QRD of \({\mathbf{HW}}_{{\mathbf{j}}} {\mathbf{ = QR}}\), the MSC method selects the precoding matrix via
$${\mathbf{W}}_{{\mathbf{n}}} = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{n}}} \in {\mathbf{W}}}} (R_{min} )\quad (n = 1,2)$$
where \(R_{min}\) is the minimum value along the diagonal of matrix \({\mathbf{R}}\), i.e., the method is to find a precoding that maximize the minimum diagonal value of the matrix \({\mathbf{R}}\).

2.3 Proposed Precoding Selection Methods for NOMA

In this paper, two precoding selection methods for NOMA system are proposed. One adopts the MI criteria incorporating the SIC operation. The other is a low complexity method to simplify the calculations via maximizing the equivalent channel gain at Rx.

2.4 Proposed Modify Maximum Mutual Information Selection

In NOMA, an important feature for UE1 is to use the SIC method for decoding its symbols. In [8], the calculations of the SINR consider only the received signals for the UE1 and UE2 directly. However, for the UE1, additional SIC procedure is conducted to decode its symbols. Therefore, the proposed modified MI method will incorporate the SIC procedure to determine the suitable precoding matrices for both users.

Let UE1 adopt the zero forcing (ZF) equalizer with \({\mathbf{G}}_{{\mathbf{1}}} = (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1}\) in Eq. (3), the decoded UE2 symbol \({\hat{\mathbf{x}}}_{{\mathbf{2}}}\) is
$${\hat{\mathbf{x}}}_{{\mathbf{2}}}\,{\mathbf{ = G}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{1}}} = (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1} {\mathbf{y}}_{{\mathbf{1}}}$$
Assume the transmitted symbols for UE1 and UE2 are independent with unit energy, the \(SINR_{1}\) for UE1 after the SIC can be derived as
$$SINR_{1} = \frac{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} {\mathbf{x}}_{{\mathbf{1}}} } \right|^{2} }}{{\left| {\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\hat{\mathbf{x}}}_{{\mathbf{2}}} + N_{w} } \right|^{2} }} = \frac{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{1} } \right|^{2} }}{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{1} + \sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{2} + 2N_{w} } \right|^{2} }}$$
Therefore, the MI is
$$I = \sum\limits_{l = 1}^{L} {\log_{2} (1 + SINR_{l,1} ) + } \log_{2} (1 + SINR_{l,2} )$$
where \(SINR_{l,1} = \frac{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} } \right|^{2} }}{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{1} + \sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{2} + 2N_{w} } \right|^{2} }}\) and \(SINR_{l,2} = \frac{{\left| {\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} } \right|^{2} }}{{\left| {N_{w} + \sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{1} } \right|^{2} }}\).
And the precoding matrices can be selected as
$$({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} ) = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} \in {\mathbf{W}}}} (I)$$
On the other hand, if UE1 adopts the minimum mean square error (MMSE) equalizer to compensate the channel fading, the MMSE equalizer can be expressed as [2]
$${\mathbf{G}}_{{\mathbf{1}}} = \sqrt {\beta_{2} } ({\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} )^{{\mathbf{H}}} (\beta_{2} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\mathbf{(h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\mathbf{)}}^{{\mathbf{H}}} + \beta_{1} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} {\mathbf{(h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} {\mathbf{)}}^{{\mathbf{H}}} + \sigma^{2} {\mathbf{I}})^{ - 1}$$
Similar as the ZF case, the precoding matrices can be determined as
$$({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} ) = \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} \in {\mathbf{W}}}} (I)$$
where I(·) is defined in Eq. (18) with \(SINR_{l,1} = \frac{{\left| {\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} } \right|^{2} }}{{\left| {\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\mathbf{G}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{1}}} + N_{w} } \right|^{2} }}\), and \(SINR_{l,2} = \frac{{\left| {\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} } \right|^{2} }}{{\left| {N_{w} + \sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{1} } \right|^{2} }}\). Comparing Eq. (21) with the conventional MI method, the proposed method includes the effect of SIC to have more reasonable SINR calculations for NOMA system.

2.5 Proposed Low Complexity Precoding Selection

Same as the conventional MI method, the proposed modified maximum MI selection has also the concern of large amount of calculations. Hence, a low complexity precoding selection method is proposed.

Consider the case when UE1 adopts the ZF equalizer with transfer function \({\mathbf{G}}_{{\mathbf{1}}} = (\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} )^{ - 1}\), the output of the equalizer is
$${\mathbf{G}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{1}}}\,{\mathbf{ = x}}_{{\mathbf{1}}} + \sqrt {\beta_{2} P} (\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} )^{ - 1} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\mathbf{x}}_{{\mathbf{2}}} + (\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} )^{ - 1} {\mathbf{w}}_{{\mathbf{1}}}$$
The term \(\sqrt {\beta_{2} P} (\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} )^{ - 1} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{2}}} {\mathbf{x}}_{{\mathbf{2}}}\) is the interference from the UE2 and \((\sqrt {\beta_{1} P} {\mathbf{h}}_{{\mathbf{1}}} {\mathbf{W}}_{{\mathbf{1}}} )^{ - 1} {\mathbf{w}}_{{\mathbf{1}}}\) is from the noise. For the UE2, the adopted ZF equalizer is \({\mathbf{U}}_{{\mathbf{1}}} = (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1}\) and the corresponding output is
$${\mathbf{U}}_{{\mathbf{1}}} {\mathbf{y}}_{{\mathbf{2}}} {\mathbf{ = x}}_{{\mathbf{2}}} + \sqrt {\beta_{1} P} (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{1}}} {\mathbf{x}}_{{\mathbf{1}}} + (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1} {\mathbf{w}}_{{\mathbf{2}}}$$

The term \(\sqrt {\beta_{1} P} (\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{1}}} {\mathbf{x}}_{{\mathbf{1}}}\) is the interference from the UE1 and the \((\sqrt {\beta_{2} P} {\mathbf{h}}_{{\mathbf{2}}} {\mathbf{W}}_{{\mathbf{2}}} )^{ - 1} {\mathbf{w}}_{{\mathbf{2}}}\) is the resulted AWGN.

Since that the performance of the interferences can be adjusted via the power allocation \(P_{1} = \sqrt {\beta_{1} P}\) \(P_{2} = \sqrt {\beta_{2} P}\), \(\beta_{1}\) and \(\beta_{2}\) are the power ratio, \(P\) is the total transmitted power at Tx. That is, with property power allocations, the interferences can be reduced. Therefore, the proposed low complexity method aims to reduce the noise enhancement after equalizer. That is,
$$\begin{aligned} {\mathbf{W}}_{{\mathbf{n}}} & = \mathop {\text{argmin}}\limits_{{{\mathbf{Wn}} \in {\mathbf{W}}}} \left( {\sum\limits_{r = 1}^{{N_{r} }} {\sum\limits_{t = 1}^{{N_{t} }} {\left| {({\mathbf{h}}_{{\mathbf{n}}} (r,t){\mathbf{W}}_{{\mathbf{n}}} (t,l))^{ - 1} } \right|^{2} } } } \right)\quad (n = 1,2) \\ & \approx \mathop {\arg \hbox{max} }\limits_{{{\mathbf{W}}_{{\mathbf{n}}} \in {\mathbf{W}}}} \left( {\sum\limits_{r = 1}^{{N_{r} }} {\sum\limits_{t = 1}^{{N_{t} }} {\left| {{\mathbf{h}}_{{\mathbf{n}}} (r,t){\mathbf{W}}_{{\mathbf{n}}} (t,l)} \right|} } } \right) \\ \end{aligned}$$

In Eq. (24), the minimization of the noise enhancement is approximated as the maximization of the equivalent channel gain. With this relation, the calculation of matrix inverse can be avoided and the precoding matrix for each UE can also be determined separately to reduce the required computational complexity.

2.6 Proposed UE Pairing Method for NOMA

In the current LTE-A OFDM system, different UEs are allocated with different spectrums to transmit symbols. To apply the NOMA concept to the current system, consider the scenario that the available spectrum has been fully occupied by N UEs. For example, in Fig. 3, there are five UEs occupied the whole spectrum (N = 5) and system has to decide which user is suitable to conduct the NOMA with another additional (N + 1)-th user (UE6 in this example).
Fig. 3

UE pairing with NOMA

The proposed UE Pairing with precoding selection procedure can be illustrated as:
  • Step 1: For the existing N users (\(UE_{1}\) ~ \(UE_{N}\)), using the precoding matrix selection to decide a suitable pair of the matrices \(({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{2} )\) with the \(UE_{N + 1}\). There will be N pairs of \(({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} )\).

  • Step 2: Calculate the \(SINR_{1}\) and \(SINR_{2}\) for each pair, and use Eq. (25) to decide the system capacity \(C_{n}\).

    $$C_{n} = B[\log_{2} (1 + SINR_{1} ) + \log_{2} (1 + SINR_{2} )]\quad n = 1 \ldots N$$
  • Step 3: The \(\hat{n}\)-th user with the largest capacity is the final decision to conduct the NOMA with \(UE_{N + 1}\), i.e.,

    $$\hat{n} = \mathop {\arg }\limits_{n} \hbox{max} \{ C_{n} \}$$
In Step 1, different precoding selection methods can be applied to decide the proper precoding pair of matrices \(({\mathbf{W}}_{{\mathbf{1}}} ,{\mathbf{W}}_{{\mathbf{2}}} )\). Basically, the proposed UE pairing method is to select the \(\hat{n}\)-th user that maximizing the total capacity. It can be shown in the following simulations that system capacity can be further improved for the proposed UE pairing method than the random selection in NOMA communication. The flow chart of the proposed UE pairing with the precoding selection for the NOMA system can be depicted in Fig. 4.
Fig. 4

Flow chart of the proposed UE pairing with precoding selection for NOMA

3 Computer Simulations and Experiments

In this section, the capacity performances of the proposed precoding selection methods are evaluated and compared with the related literature methods [8, 9, 10]. The required computation complexities are also analyzed to have fair comparisons.

3.1 Capacity Performance of Precoding Selection

Assume there are four transmit antennas and one receive antenna at the Tx and Rx respectively. The fading channels are assumed to be independent, identical distributed (i.i.d) complex Gaussian and can be perfect estimated at Rx. Table 1 lists the detail parameters of this simulation. The power ratios for UE1 and UE2 are 0.2 and 0.8 respectively to have a better SIC performance [3].
Table 1

Simulation parameters



Number of Tx antenna


Number of Rx antenna



Rayleigh fading channel


5 kHz

Power Ratio

\(\beta_{1} = 0.2\,\,\beta_{2} = 0.8\)


0–20 (dB)


\(C = B(\log_{2} \left( {1 + SINR_{1} } \right) + \log_{2} (1 + SINR_{2} ))\)

In Fig. 5, the proposed modified MI precoding selection methods with ZF and MMSE are compared with the conventional MI method. Since that the proposed method considers the effect of SIC for NOMA system, the proposed methods have better capacity performances than the compared MI method, especially for the MMSE case. Besides, Fig. 6 depicts the capacity performances for the proposed methods and the compared methods [8, 9, 10]. The proposed modified MI selection (MMSE) still has the best capacity performance among these methods. Note that the proposed low complexity method has almost the same performance as the SVD-based selection, SVD-SC and MSC selection methods. However, the proposed low complexity method has the least required calculations during the precoding determination which is analyzed below.
Fig. 5

Capacity analysis for the proposed modified mutual information precoding selection

Fig. 6

Capacity analysis for the different precoding selections

Table 2 lists the required computation complexity for different precoding selection methods. Based on [9], let \(R\) be the required computations for SVD of a \(N_{r} \times N_{t}\) matrix and can be approximately as \(2N_{r} N_{t}^{2} + 2N_{t}^{3}\). \(K\) is the complexity for the SVD-SC, it requires the \(2 {\text{N}}_{\text{r}} l^{2} + 2l^{3}\) calculations approximately. \(Q\) is the complexity for the MSC method and can be derived as \(2N_{r} l^{2}\) [10]. \(l\) denotes the number of the spatial layer (\(l\) = 4 is the maximum of LTE system). Besides, the inverse calculation for a \(N \times N\) matrix is \(N^{3} + N^{2} + N\) [11]. In Table 2 \(S\) denotes the number of the candidate for the precoding matrices during selection (\(S = 16\) for the LTE system with four antennas). Take the parameters of Table 1 into calculations and let the complexity for the proposed MI method be 100%, results show that the required complexity for the proposed low complexity method is only 1.5% which is the lowest among the compared methods.
Table 2

Complexity analysis

Complexity required


Proposed maximum mutual information selection



\(\begin{aligned} & S^{2} \times \{ 6 \times [N_{r} lN_{t} + N_{r} l(N_{t} - 1)] + 3 \times [N_{r}^{2} l + N_{r}^{2} (l - 1)] \\ & \quad + N_{r}^{3} + N_{r}^{2} (N_{r} - 1) + N_{r}^{2} l + N_{r} l(N_{r} - 1) \\ & \quad + 2 \times [N_{r} lN_{t} + N_{r} l(N_{t} - 1) + N_{r} ]\} \\ \end{aligned}\)

SVD based selection



\(2S \times (N_{r} lN_{t} + N_{r} l(N_{t} - 1) + N_{r}^{2} l + N_{r} l(Nr - 1) + N_{r} + R)\)

Singular value decomposition selection criterion



\(2S \times (N_{t} lN_{t} + N_{r} l(N_{t} - 1) + K)\)

Max–min selection criterion



\(2S \times (N_{t} lN_{t} + N_{t} l(N_{t} - 1) + N_{r} + Q)\)

Proposed low complexity method



\(2S \times (N_{t} lN_{t} + N_{t} l(N_{t} - 1) + N_{r} )\)

From above analysis, we can conclude that the proposed modified MI precoding selection method has better capacity performance that the conventional maximum MI method. And the proposed low complexity method has the similar capacity performance than the literature methods [9, 10]. Nevertheless, it requires the least computations among these methods.

3.2 Performance Analysis of the Proposed UE Pairing

Consider a scenario that there are five UEs has been allocated the whole spectrum and system has to decide which user is suitable to conduct the NOMA with the additional 6-th user (UE6). The parameters in this simulation are the same as Table 1.

Figure 7 is the system capacity analysis for the proposed UE pairing and the random selection (without UE pairing). The proposed MI precoding selection (MMSE) is adopted to decide the precoding matrix pair \(({\mathbf{W}}_{{\mathbf{1}}} {\mathbf{,W}}_{{\mathbf{2}}} )\). It shows that the proposed UE pairing can further improve the system capacity about 30% in average for NOMA system with proper UE pairing selection.
Fig. 7

Capacity analysis for the proposed UE pairing in NOMA

3.3 Measured Experiments

In this paper, indoor wireless channels were also measured to further verify the predictions of the computer simulations. The channels were measured by using the vector network analyzer (VNA). In this measurement, signal can be enhanced by the power amplifier and low noise amplifier at the Tx and Rx (Fig. 8). We considered the same scenario as the computer simulations with four transmit antennas and one receive antenna at the Tx and Rx respectively. Two cases of channel are considered in the experiments, one is the line-of-sight (LOS) case and the other is the non-LOS (NLOS) case. The distances from the UE1 and the UE2 to the Tx are 5 and 10 m respectively. The carrier frequency is at 2.4 GHz and the separation between the transmit antennas is half wavelength. Figure 9 depicts the LOS scenario in this experiment. For the NLOS case, the UE1 and UE2 were moved into the rooms nearby to have the NLOS channels with Tx.
Fig. 8

Channel measurement

Fig. 9

Experiment scenario (LOS case)

Figures 10 and 11 are the capacity performances for different precoding methods in the LOS and NLOS measured channels. Both results are consistent with the simulation predictions that the proposed modified MI has the best performance. Besides, the proposed low complexity precoding method has very similar performance compared with the literature SVD and MSC methods. However, it has the lowest computational complexity advantage analyzed in Table 2.
Fig. 10

Capacity analysis for different precoding selections (LOS)

Fig. 11

Capacity analysis for different precoding selections (NLOS)

Furthermore, Figs. 12 and 13 are the measured capacity results if applying the proposed modified MI precoding method with the UE pairing. In the indoor measurements environment, the proposed UE pairing method has better performance than without UE pairing or random selection in the NOMA system. Based on the experiments, the proposed method can have 20–40% capacity improvement and the measured results are quite consistent with the simulation in Fig. 7.
Fig. 12

Capacity analysis for the proposed UE pairing (LOS)

Fig. 13

Capacity analysis for the proposed UE pairing (NLOS)

4 Conclusions

In this paper, two precoding selection methods for the NOMA system are proposed. One is to maximize the MI after SIC and the other is to enhance the equivalent channel gain at Rx. Besides, a UE pairing method is also provided for the NOMA system.

Based on the results of simulations and the experiments, it shows that the proposed MI precoding selection method has excellent performance in the system capacity. For the proposed low complexity method, it has the least required complexity among the compared methods. Besides, with the help of the proposed UE pairing method, system capacity can be further improved about 30% in average for NOMA system.



Funding was provided by Ministry of Science and Technology, Taiwan (Grant No.: 105-2119-M-019-006-).


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Communications, Navigation and Control Engineering, Room 912National Taiwan Ocean UniversityKeelungTaiwan, ROC

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