Quantization Precoding in Massive MIMO Systems

Abstract

Example embodiments of the present disclosure relate to a transmitter, method, apparatus, and computer-readable storage medium for quantized precoding in a massive multiple-input multiple-output (MIMO) system. In an example embodiment, a transmitter determines a normalization factor for precoding multiple signals, which are intended for multiple receivers. A first device precodes multiple expected signals using a normalization factor by iteratively performing an action. The action includes determining a decreasing direction of a difference between multiple expected signals and corresponding signals received at multiple receivers by using a normalization factor, in which a decreasing rate of the difference is higher than a threshold rate in the decreasing direction, and determining multiple precoded signals based on the decreasing direction. Then, the first device quantizes the multiple precoded signals for transmission by the transmitter to multiple receivers.

Description

Technical Field

[0001] Example embodiments of the present disclosure relate generally to the field of communications, and more particularly to a transmitter, method, apparatus, and computer-readable storage medium for quantization precoding in a massive multiple-input multiple-output (MIMO) system. Background Art

[0002] Massive Multiple-Input Multiple-Output (MIMO) technology is widely considered to be the core technology in the fifth generation (5G) wireless communication system. This technology can significantly improve spectrum efficiency and reliability by providing a base station (BS) with hundreds of antennas to serve dozens of user equipment (UE) in the same time-frequency resources. However, the large number of antennas and radio frequency (RF) chains at the BS will lead to excessive hardware cost and power consumption, which hinders the commercial deployment of massive MIMO systems. In this case, the practical implementation of massive MIMO technology requires the use of low-cost and energy-efficient hardware components at the BS.

[0003] In large-scale MIMO systems, high-resolution digital-to-analog converters (DACs) and / or analog-to-digital converters (ADCs) can account for most of the power consumption and hardware costs. However, the power consumption of the DACs and / or ADCs can increase exponentially with resolution (in bits) and linearly with linewidth. Low-resolution DACs and / or ADCs can significantly reduce power consumption, system cost, and raw baseband data rates. In addition, the resolution of the DAC can also affect the amount of data that needs to be exchanged between the baseband processing unit and the radio unit. Lower DAC resolution can reduce the amount of data used for exchange, and thus can alleviate the data rate bottleneck in the fronthaul. Therefore, using low-resolution DACs and / or ADCs in antennas and RF chains can be an effective implementation of large-scale MIMO systems.

[0004] Quantization precoding is a key technology for low-resolution DAC. Quantization precoding can bring many benefits to massive MIMO systems. However, traditional quantization precoding algorithms can hardly achieve better performance of signal detection at UE, or have higher computational complexity especially for high-order modulation. Summary of the invention

[0005] Generally speaking, example embodiments of the present disclosure provide a transmitter, method, apparatus, and computer-readable storage medium for quantization precoding in a massive MIMO system.

[0006] In a first aspect, a transmitter is provided, the transmitter comprising at least one processor and at least one memory, the at least one memory containing computer program code. The at least one memory and the computer program code are configured to, together with the at least one processor, cause the transmitter to determine a normalization factor for precoding multiple signals, the multiple signals being intended for multiple receivers. The first device is caused to precode multiple expected signals using the normalization factor by iteratively performing actions. The actions include determining a decreasing direction of a difference between multiple expected signals and corresponding signals received at multiple receivers by using the normalization factor, in which a decreasing rate of the difference is higher than a threshold rate in the decreasing direction, and determining multiple precoded signals based on the decreasing direction. The first device is also caused to quantize multiple precoded signals for transmission by the transmitter to multiple receivers.

[0007] In a second aspect, a method is provided. In the method, a transmitter determines a normalization factor for precoding a plurality of signals intended for a plurality of receivers. A first device precodes the plurality of intended signals using the normalization factor by iteratively performing an action. The action includes determining a decreasing direction of a difference between the plurality of intended signals and corresponding signals received at a plurality of receivers by using the normalization factor, in the decreasing direction, a decreasing rate of the difference being higher than a threshold rate, and determining a plurality of precoded signals based on the decreasing direction. Then, the first device quantizes the plurality of precoded signals for transmission by the transmitter to the plurality of receivers.

[0008] In a third aspect, there is provided an apparatus comprising means for performing the method according to the second aspect.

[0009] In a fourth aspect, a computer readable storage medium is provided, comprising program instructions stored thereon. The instructions, when executed by a processor of a device, cause the device to perform the method according to the second aspect.

[0010] It should be understood that the invention summary is not intended to identify the key or essential features of the exemplary embodiments of the present disclosure, nor is it intended to be used to limit the scope of the present disclosure. Other features of the present disclosure will become readily understood through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Some example embodiments will now be described with reference to the accompanying drawings, in which:

[0012] Figure 1 illustrates example distortions with different bits of quantization;

[0013] Figure 2 An example system model for quantization precoding in a massive MIMO system is illustrated;

[0014] Figure 3 illustrates example curves of quantization functions with different bits;

[0015] Figure 4 illustrates an example environment in which example embodiments of the present disclosure may be implemented;

[0016] Figure 5 illustrates example methods according to some example embodiments of the present disclosure;

[0017] Figure 6 illustrates an example precoding process according to some example embodiments of the present disclosure;

[0018] Figure 7 illustrates example curves of different activation functions according to some example embodiments of the present disclosure;

[0019] Figure 8 illustrates example curves of the real or imaginary part of an approximate quantization function according to some example embodiments of the present disclosure;

[0020] Fig. 9 illustrates an example process of batch gradient descent (BGD) based quantization precoding according to some example embodiments of the present disclosure;

[0021] Figure 10(a) to Figure 10(c) illustrates example bit error rate (BER) performance for different precoders in one scenario (referred to as scenario 1) for the 1-bit case;

[0022] Figure 11(a) to Figure 11(f) illustrates example BER performance for different precoders in an additional scenario (referred to as scenario 2) in the 1-bit case;

[0023] Figure 12(a) to Figure 12(c) illustrates example BER performance of different precoders in scenario 1 for the 2-bit case;

[0024] Figure 13(a) to Figure 13(f) illustrates example BER performance of different precoders in scenario 2 for the 2-bit case;

[0025] Figure 14(a) to Figure 14(c) illustrates example BER performance of different precoders in scenario 1 for the 3-bit case;

[0026] Figure 15(a) to Figure 15(f) illustrates example BER performance of different precoders in scenario 2 for the 3-bit case;

[0027] Fig.16 A simplified block diagram of a device suitable for implementing an example embodiment of the present disclosure is illustrated.

[0028] Throughout the drawings, the same or similar reference numerals refer to the same or similar elements. DETAILED DESCRIPTION

[0029] The principle of the present disclosure will now be described with reference to some example embodiments. It should be understood that these example embodiments are described only to illustrate and help those skilled in the art understand and realize the purpose of the present disclosure, and do not imply any limitation on the scope of the present disclosure. The disclosure described herein can be implemented in various ways different from the following ways.

[0030] In the following description and claims, unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs.

[0031] As used herein, the term "transmitter" refers to a device capable of transmitting a signal. As used herein, the term "receiver" refers to a device capable of receiving a signal. A transmitter or receiver may be implemented by any suitable device or as part of a device, including, for example, a network device or a terminal device. In some example embodiments, the transmitter and / or receiver may be provided with multiple antennas. These antennas may include physical or logical antennas.

[0032] As used herein, the term "terminal device" or "user equipment" (UE) refers to any terminal device capable of wireless communication with each other or with a base station. Communication may involve sending and / or receiving wireless signals using electromagnetic signals, radio waves, infrared signals, and / or other types of signals suitable for transmitting information over the air. In some example embodiments, the UE may be configured to send and / or receive information without direct human interaction. For example, when triggered by an internal or external event, or in response to a request from the network side, the UE may send information to the base station according to a predetermined schedule.

[0033] Examples of UE include, but are not limited to, smartphones, wireless-enabled tablets, laptop embedded devices (LEEs), laptop mounted devices (LMEs), wireless client equipment (CPEs), sensors, metering devices, personal wearable devices such as watches, and / or vehicles capable of communication. For the purposes of discussion, some example embodiments will be described with reference to UEs as examples of terminal devices, and the terms "terminal device" and "user equipment" (UE) may be used interchangeably in the context of the present disclosure. A UE may also correspond to the mobile terminal (MT) portion of an integrated access and backhaul (IAB) node (also known as a relay node).

[0034] As used herein, the term "network device" refers to a device by which a service can be provided to a terminal device in a communication network. As an example, a network device may include a base station. As used herein, the term "base station" (BS) refers to a network device by which a service can be provided to a terminal device in a communication network. A base station may include any appropriate device by which a terminal device or a UE can access a communication network. Examples of base stations include relays, access points (APs), transmission points (TRPs), node Bs (NodeBs or NBs), evolved NodeBs (eNodeBs or eNBs), new radio (NR) NodeBs (gNBs), remote radio modules (RRUs), radio heads (RHs), remote radio heads (RRHs), low-power nodes such as femto and micro, etc. A relay node may correspond to a distributed unit (DU) portion of an IAB node.

[0035] As used herein, the term "circuitry" may refer to one, more, or all of the following:

[0036] (a) pure hardware circuit implementation (e.g., implementation in analog and / or digital circuitry only), and

[0037] (b) a combination of hardware circuitry and software, such as (where applicable): (i) a combination of analog and / or digital hardware circuitry and software / firmware and (ii) any portion of hardware processor(s) working in conjunction with software (including digital signal processor(s), software and memory(s)) to enable a device (such as a mobile phone or server) to perform various functions) and

[0038] (c) Hardware circuits and / or processor(s), such as microprocessor(s) or portions of microprocessor(s), require software (e.g., firmware) for operation, but the software may not be present when no software is required for operation.

[0039] The definition of circuitry applies to all uses of this term in this application, including in any claims. As a further example, as used in this application, the term circuitry also covers an implementation of only a hardware circuit or processor (or multiple processors) or a portion of a hardware circuit or processor and its (or its) accompanying software and / or firmware. The term circuitry also covers (for example and if applicable to a particular claim element) a baseband integrated circuit or processor integrated circuit for a mobile device or a similar integrated circuit in a server, cellular base station or other computing or base station.

[0040] As used herein, the singular forms "a", "an", and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. The term "including" and variations thereof are to be understood as open terms meaning "including, but not limited to." The term "based on" is to be understood as "based at least in part on." The terms "one embodiment" and "an embodiment" are to be understood as "at least one embodiment." The term "another embodiment" is to be understood as "at least one other embodiment." Other explicit and implicit definitions may be included below.

[0041] As used herein, the terms "first", "second", etc. may be used herein to describe various elements, which should not be limited by these terms. These terms are only used to distinguish one element from another element. For example, a first element may be referred to as a second element, and similarly, a second element may be referred to as a first element without departing from the scope of the example embodiments. As used herein, the term "and / or" includes any and all combinations of one or more of the listed terms.

[0042] Massive MIMO technology utilizes a large number of antennas at the BS to serve many UEs each with a small number of antennas. This technology can bring significant improvements in spectrum efficiency, energy efficiency, reliability and coverage. However, this technology will suffer from high hardware complexity, system cost and circuit power consumption.

[0043] Using low-cost and energy-efficient hardware components at the BS is very important for practical massive MIMO systems. Low-resolution ADC / DAC can significantly simplify the transmitter, receiver, or transceiver and reduce hardware complexity, system cost, and power consumption. However, low-resolution DAC and / or ADC will cause significant distortion on the signal, such as Figure 1 As shown, this will bring great challenges to precoding design.

[0044] Conventional precoding schemes are designed based on the assumption that the resolution of the DAC is almost infinite. However, this assumption cannot be guaranteed in practical systems. In practice, 10 bits or more may allow conventional precoders to achieve satisfactory performance. When using a low-resolution DAC, such as 1 to 3 bits, quantization introduces significant distortion to the precoded signal, which greatly degrades the performance of conventional precoders. Recently, quantized precoding has attracted increasing attention to ensure performance when using low-resolution DACs in massive MIMO systems.

[0045] Figure 2An example system model 200 of quantized precoding in a massive MIMO system is shown. As shown, a BS 205 equipped with N transmit antennas 210-1, 210-2, ..., 210-N simultaneously serves M active flows on a wireless channel 215 in the same frequency band. In this example, the M active flows are directed to M UEs, each of which is provided with a receive antenna 220-1, 220-2, ... or 220-M. N and M represent any suitable positive integers.

[0046] The signal intended for the UE is represented as a vector s (with a size of M×1), and the precoded signal is represented as a vector x (with a size of N×1). Then, the received signal y can be calculated as:

[0047] y=Hx+n (1)

[0048] Where H (having a size of M×N) represents the channel matrix of the wireless channel 215, and the noise vector n (having a size of M×1) represents the noise. Assume that the entries of H (where the entry in the i-th row and j-th column is denoted as h ij ) are independent circularly symmetric complex Gaussian random variables with unit variance, that is, The entries of the channel matrix can be normalized as It is also assumed that the elements of the noise vector n are independent and identically distributed circularly symmetric complex Gaussian variables, and the variance of each complex term is Right now

[0049] If the i-th (1≤i≤M) row of H is represented by h t , then formula (1) can be rewritten as:

[0050]

[0051] The precoded signal x = (x 1 , x 2 , …x N ) T Corresponding to the signals on N transmitting antennas, and the received signal y=(y 1 ,y 2 ,…y M ) T Corresponding to the received signal of M single-antenna UEs. The channel vector h 1· 、h 2· ,...h M· is the row vector of the channel matrix H, each vector corresponds to the channel vector of a single antenna UE. Noise vector n = (n 1 , n 2 ,…n M ) TCorresponds to the noise on the received signal of M single-antenna UEs.

[0052] As shown, the precoding at BS 205 converts the expected or desired signal s into a precoded signal x such that the elements in y can be correctly decoded at the UE into the appropriate symbols in the desired signal vector s, i.e., if the total signal gain is 1, then y i ≈s i (1≤i≤M). If an infinite resolution DAC is used, the elements in the precoded signal x can come from the entire complex domain For low-resolution DACs, only a limited set of Select elements from i (0≤i≤2 L -1) represents the quantization level, and L represents the number of quantization bits per actual dimension.

[0053] If x contains a complex signal with infinite resolution, then quantization of x discretizes x into a discrete signal. If x contains a simple sinusoidal signal, then L-bit quantization of x produces an L-bit quantized signal, such as Figure 1 shown.

[0054] In the case of 1-bit quantization, there are only two quantization levels for the real or imaginary part of a complex number, namely in is determined by the total transmit power P at BS 205 and the number of antennas N. For summary purposes, the average amplitude is extracted as a normalization factor. Then, the 1-bit quantization level is expressed as

[0055] In the L-bit quantization case where L>1, the quantization level for each actual dimension will be 2 L There are several methods for setting the quantization level. One method is to find a set of quantization levels that minimizes the distortion between the quantized vector and the non-quantized vector. Table 1 shows example quantization levels for 1 to 5 bits.

[0056] Table 1

[0057]

[0058]

[0059] As shown, the quantization level is an integer multiple of the base level ζ. The base level ζ is determined based on the power P and the number of antennas N at the BS 205.

[0060] Once the quantization level is determined, the quantization function can be used to quantize the signal. For simplicity, assume that the quantization base level is 1, i.e. ζ = 1. Then, for the real part x of the signal xR or the imaginary part x I , quantization function is defined.

[0061] For 1-bit quantization, the quantization function is expressed as where sgn(·) denotes the sign function applied element-wise to a vector and defined as follows:

[0062]

[0063] For L-bit quantization, the quantization function can be constructed by a sign function. The quantization function for the real part can be expressed as follows.

[0064]

[0065] The quantization function for the imaginary part is similar. Combined with the quantization base level mentioned above, the L-bit quantization function for the complex signal x is (also called quantizer mapping function) can be expressed as follows.

[0066]

[0067] The real and imaginary parts of the quantization function have the same curve. Example curves for quantization functions with 1 to 4 bits are as follows: Figure 3 as shown in .

[0068] If a low-resolution DAC is used, the signal at the antenna will be a quantized signal with only a few quantization levels, such as Figure 1 As shown. It is required that the received signal at the UE side has almost no distortion even when the signal on the antenna at the BS side is quantized. It would be better if the lower DAC resolution at the BS side has no effect on the UE side. This is a huge challenge for precoding design.

[0069] Currently, there are two types of quantization precoding schemes, one of which is called a linear quantization precoding scheme and the other is called a nonlinear quantization precoding scheme. A linear quantization precoder performs linear precoding and then quantizes the elements of the precoding vector into a finite transmission set If W represents the linear precoding matrix, the precoded signal x will be obtained by the following formula:

[0070]

[0071] in Denotes the quantizer mapping function as described above, which is a non-linear function that describes the quantization operation of the DAC at the BS.

[0072] The conventional linear quantization precoding scheme is a zero-forcing (ZF) quantization (ZFQ) precoding scheme, which performs ZF precoding before the quantization operation. zf The ZF precoding matrix can be calculated as follows:

[0073]

[0074] β zf represents the precoding factor and is usually set to where P represents the total transmitted power as described above, tr(·) represents the matrix trace, and H H represents the conjugate transpose of the channel matrix H. Then, ZFQ precoding can be to achieve.

[0075] Linear quantization precoding schemes such as the ZFQ precoding scheme are designed under the assumption of infinite resolution DACs. These schemes do not work well with low resolution DACs. Compared to nonlinear quantization precoders, linear quantization precoders usually have lower complexity but suffer from poor detection performance due to the nonlinear distortion caused by quantization.

[0076] A 1-bit quantization precoding scheme as a special case of quantization precoding has been studied to alleviate the performance degradation when a 1-bit DAC is used to minimize the energy consumption of the DAC. So far, most nonlinear quantization precoders are specially designed for the 1-bit quantization precoding case and are difficult to generalize.

[0077] The optimization of L-bit nonlinear quantization precoding can be expressed as follows:

[0078] Subject to β>0 (8)

[0079] where β represents the normalization factor (also called the precoding factor), which is generally a real number, and ||·|| 2 represents the 2-norm of the vector. Since the optimization problem is solved for a given signal vector s, the precoding factor β also depends on s. This is different from linear quantization precoding where β depends only on the channel matrix H.

[0080] For the case of a 1-bit DAC, an algorithm called squared-infinity norm Douglas-Rachford splitting (SQUID) is used to approximately solve the above optimization problem. The SQUID algorithm appears to find a near-optimal solution to the 1-bit quantization precoding problem with quadrature phase shift keying (QPSK) modulation. This algorithm can achieve comparable performance to semidefinite relaxation (SDR), an algorithm with higher computational complexity.

[0081] For low-order modulation such as QPSK, the UE does not need to scale the received signal by the scaling factor when decoding the signal. For high-order modulation such as 16-quadrature amplitude modulation (16QAM) or 64QAM, the scaling factor is necessary. In this case, the optimization problem is modified as follows:

[0082] Subject to β>0 (9)

[0083] where T represents the number of time slots, and ||·|| F represents the Frobenius norm of the matrix. In formula (9), both S and X are matrices with T columns corresponding to T time slots. A single value β is selected for a transmitted symbol block s with a length T that does not exceed the channel coherence time. Before applying the SQUID algorithm, formula (9) is rewritten as:

[0084] Subject to β>0 (10) where and represents the vectorized version of the signal matrices S and X, and Represents the identity matrix I T and the Kronecker product of the channel matrix H. Based on formula (10), a single value β can be calculated for the expected or desired signal matrix S. This vectorization allows the UE to estimate β through pilot transmission or blind estimation techniques. However, the dimensionality of the matrix-vector multiplication will increase, and therefore the computational complexity will increase a lot.

[0085] Another algorithm for solving the optimization problem is called biConvex 1-bit PrecOding (C1PO). This algorithm relies on biconvex relaxation (BCR) and can converge to a near-optimal solution faster. To avoid matrix inversion, a low-complexity version of C1PO, called C2PO, was proposed. It seems that the C2PO algorithm can achieve performance close to that of the SQUID algorithm, but with lower computational complexity.

[0086] Currently, a gradient descent (GD) based precoding scheme is proposed for massive MIMO systems. This scheme combines the precoding based on stochastic gradient descent (SGD) and scaling clipping (also known as the SGD-Clip scheme) to achieve 1-bit precoding using low-order modulation such as QPSK. The SGD-Clip scheme seems to achieve similar performance to the SQUID algorithm for QPSK modulation.

[0087] In the case of using low-resolution DAC, a general framework for quantization precoding, called iterative discrete estimation (IDE), is proposed, which is based on the alternating direction method of multipliers (ADMM) framework. To avoid matrix inversion, a low-complexity version of IDE, called IDE2, is proposed. IDE2 can provide a good trade-off between complexity and error rate performance, and is therefore more suitable for massive MIMO systems than IDE.

[0088] The optimization of 1-bit quantization precoding as a special case of L-bit quantization precoding can be simplified. However, 1-bit precoders generally have poor performance for high-order modulations such as 64QAM and 256QAM. On the contrary, L-bit precoders (with 2 bits or more) can achieve better performance than 1-bit precoders, especially for high-order modulation. Therefore, for large-scale MIMO systems with low-resolution DACs, a general L-bit quantization precoder will be required. However, L-bit quantization precoding is much more complex than 1-bit precoding. Achieving a balance between performance and computational complexity will be a huge challenge. Low-complexity schemes often have poor detection performance, while schemes with high detection performance generally have high complexity.

[0089] The example embodiments of the present disclosure provide an effective quantized precoding scheme. With this precoding scheme, a normalization factor (or precoding factor) for precoding the original signal is first obtained at a transmitter that can be configured with multiple physical or logical transmit antennas. The normalization factor can be set to an empirical value. In some example embodiments, the normalization factor can be determined based on the power of multiple signals intended for multiple receivers. Each receiver in the receiver can be configured with at least one receive antenna.

[0090] Once the normalization factor is determined, the precoding scheme focuses on the precoding of the expected or desired signal. Through the precoding scheme, the following actions are performed iteratively. First, the normalization factor is used to determine the descent direction (e.g., gradient). The descent direction indicates the direction in which the difference between the corresponding received signal at the receiver and the multiple expected signals is reduced very quickly. For example, the rate of decline of the difference in the descent direction is higher than the threshold rate. Therefore, multiple precoding signals are determined based on the descent direction. After some iterations, the precoding signal will converge to the optimal solution. Then, the precoding signal is quantized for being sent to multiple receivers by the transmitter.

[0091] According to the precoding scheme of some embodiments of the present disclosure, the normalization or precoding factor does not need to depend on the signal to be transmitted and can be stable during a period of time such as different time slots. On the one hand, this can reduce the computational complexity of precoding at the transmitter. On the other hand, the estimation of the normalization factor can be simplified at the receiver, and thus the detection performance of the original signal can be improved at the receiver.

[0092] Compared with the traditional precoding method, the proposed precoding scheme has lower complexity and better detection performance. The detection performance and complexity are well balanced. In addition, for high-order modulation, it is not necessary to vectorize the signal matrix, thus avoiding the matrix-vector multiplication to a higher dimension. In addition, the quantization precoding scheme is not specified for 1-bit precoding, but can be used for general L-bit quantization precoding. This scheme is a general quantization precoding scheme for large-scale MIMO systems with low-resolution DACs.

[0093] For example, the proposed precoding scheme has the following advantageous benefits:

[0094] High performance: The precoding scheme outperforms traditional linear and nonlinear quantization precoders in terms of detection performance.

[0095] • Low complexity: The proposed precoding scheme has low complexity.

[0096] High versatility: The precoding scheme is a general quantization precoding scheme, which is not specified for 1-bit precoding, but can be used for general L-bit quantization precoding or limited alphabet precoding cases.

[0097] Low dimensionality: The precoding scheme avoids vectorizing the signal vector and the channel matrix into higher dimensions for high-order modulation, thus greatly reducing the computational effort by keeping the matrix-vector multiplications in low dimensions.

[0098] High flexibility: Various gradient descent (GD) algorithms such as batch gradient descent (BGD), stochastic gradient descent (SGD), or mini-batch gradient descent (mini-batchGD) can be used to obtain satisfactory performance.

[0099] Benefits from the field of artificial intelligence (AI): BGD algorithms are basic algorithms in the field of machine learning (ML), and the rapid development of these algorithms in the field of AI can promote the implementation of BGD-based quantization precoding schemes.

[0100] Therefore, the precoding scheme according to example embodiments of the present disclosure is very attractive for practical massive MIMO systems especially with low-resolution DACs.

[0101] Figure 4 An example environment 400 is shown in which example embodiments of the present disclosure may be implemented.

[0102] Environment 400 includes a transmitter 410 provided with a plurality of transmit antennas 415-1, 415-2, ..., 415-N. These transmit antennas may include physical or logical antennas. Environment 400 also includes a plurality of receivers. In this example, each of the plurality of receivers is provided with a receive antenna 420-1, 420-2, ..., 420-M. N and M are any suitable positive integers. In some example embodiments, N≥M.

[0103] The transmitter and receiver form part of a MIMO or massive MIMO system. It should be understood that each receiver is configured with a receiving antenna for illustrative purposes only and does not imply any limitation. In some example embodiments, a receiver may be provided with multiple receiving antennas. It should also be understood that environment 400 may include any appropriate number of transmitters and receivers suitable for implementing embodiments of the present disclosure.

[0104] The transmitter and receiver may be implemented by any suitable device or as part of a device. In some example embodiments, the transmitter may be implemented at a network device such as a base station, and the receiver may be implemented at a terminal device such as a UE. It is also possible that the transmitter is implemented at the terminal device and the receiver is implemented at the network device. In some example embodiments, environment 400 may be part of a relay communication network. In this example, the transmitter may be implemented at the network device and the receiver may be implemented at the relay, or vice versa. In some other embodiments, both the transmitter and the receiver may be implemented at the terminal device in a device-to-device (D2D) communication, which may alternatively be referred to as a side link or vehicle-to-everything (V2X).

[0105] As shown, the transmitter and the receiver may communicate via a wireless channel 425. The communication may follow any appropriate communication standard or protocol already in existence or to be developed in the future, such as Universal Mobile Telecommunications System (UMTS), Long Term Evolution (LTE), Advanced LTE (LTE-A), Fifth Generation (5G) New Radio (NR), Wireless Fidelity (Wi-Fi), and Worldwide Interoperability for Microwave Access (WiMAX) standards, and may employ any appropriate communication technology, including, for example, Multiple Input Multiple Output (MIMO), Orthogonal Frequency Division Multiplexing (OFDM), Time Division Multiplexing (TDM), Frequency Division Multiplexing (FDM), Code Division Multiplexing (CDM), Bluetooth, ZigBee, Machine Type Communication (MTC), Enhanced Machine Type Communication (eMTC), Enhanced Mobile Broadband (eMBB), Massive Machine Type Communication (mMTC), Ultra-Reliable Low Latency Communication (URLLC), Carrier Aggregation (CA), Dual Connectivity (DC), and New Radio Unlicensed (NR-U) technology. In some example embodiments, MIMO techniques, such as single-user MIMO (SU-MIMO) and multi-user MIMO (MU-MIMO) techniques, may be used for communications in environment 400 .

[0106] In various example embodiments, transmitter 410 performs quantization precoding. As shown, transmitter 410 includes a precoder 430 for precoding a plurality of signals represented by s into a plurality of precoded signals represented by x. Transmitter 410 also includes a quantizer 435 to quantize the precoded signals x. The quantized precoded signals are then processed at DAC 440 before being transmitted by transmit antennas 415-1, 415-2, ..., 415-N to a plurality of receive antennas 420-1, 420-2, ..., 420-M.

[0107] In this example, as shown in the figure, the number of expected signals M (i.e., the number of elements in the signal vector s) is equal to the number of receiving antennas. It is also possible that the number of expected signals M is less than the number of receiving antennas. For example, if the receiver implemented by the UE has two receiving antennas, the transmitter 410 implemented by the BS can send a stream to the receiver. Therefore, the receiver can combine the two received signals of the two receiving antennas to recover the expected signal stream.

[0108] In this case, the channel matrix H will also be an equivalent channel matrix so that s and Hx have the same dimensions. This equivalent channel matrix will be the coefficients between the combined signal of a particular receiver (which combines the received signals from its multiple antennas) and the transmit antennas. Such a receiver that can be implemented by a multi-antenna UE to receive one stream can be equivalent to a virtual single-antenna UE.

[0109] Figure 5 An example method 500 according to some example embodiments of the present disclosure is shown. The method 500 may be performed by Figure 4 For the purpose of discussion, reference will be made to the transmitter 410. Figure 4 Method 500 is described.

[0110] In block 505, transmitter 410 determines a normalization factor, denoted by β, for use in precoding a plurality of signals (denoted by s) intended for a plurality of receivers provided with a plurality of receive antennas 420-1, 420-2, ..., 420-M. Normalization factor β may be determined in any suitable manner. As discussed above, a linear precoder may use such a normalization or precoding factor. In some example embodiments, the precoding factor β may be determined based on the precoding factor β used for the zero forcing precoder. zf to determine the normalization factor β.

[0111] For example, the precoded signal vector with infinite resolution is represented as x inf , and the expected total transmit power is P, that is, x inf The quantized signal vector x is quantized with L bit resolution q , whose constraints are If the sample x inf If it conforms to the Gaussian distribution, then according to Bussgang's theorem:

[0112]

[0113] Where d represents the distortion term, and the Bussgang constant or quantization constant τ can be calculated as follows.

[0114]

[0115] The value of τ is determined by the quantization resolution, and its values ​​for 1 to 5 bit quantization are listed in Table 2.

[0116] Table 2

[0117] Bit 1 2 3 4 5 τ 0.79788456 0.93869799 0.98110158 0.99421180 0.99825082

[0118] As shown in Table 2, for 1 to 2 bit quantization, the value of τ is significantly less than 1. When the resolution is greater than 3 bits, the value of τ will be closer to 1.

[0119] When the precoding factor β of the precoder used for ZF quantization zf Used to convert x inf When the average power of is normalized to P, the ZF precoded signal vector x inf =W zf s. Then, the linear quantized signal vector x q It can be expressed as follows.

[0120]

[0121] Regardless of the distortion term d, τ -1 β zf can be regarded as the precoding factor of the ZF quantized precoder. Therefore, the normalization factor β for the nonlinear quantized precoder can be determined as τ -1 and β zf (or similar to β zf ). As mentioned above, β zf (or similar to β zf ) usually depends on the channel matrix H, and τ -1 Does not depend on the channel matrix H or the transmitted signal.

[0122] In order to reduce the influence of the channel matrix H on the precoding factor (e.g. β zf ), in some example embodiments, the normalization factor β may depend on the transmit power of the transmitter, rather than the channel matrix H, and the quantization constant τ. For example, the normalization factor β may be calculated as follows:

[0123]

[0124] in represents the average symbol energy of the expected signal s. Therefore, β can be calculated with less computation.

[0125] In some example embodiments, the signal-to-noise ratio (SNR) may also be considered in determining the normalization factor β to further improve precoding efficiency. For example, if the SNR is expressed as in Denotes the variance of the noise, then the normalization factor β will be calculated as follows.

[0126]

[0127] After determining the normalization factor β, in block 510, the transmitter 410 precodes the plurality of desired signals s using the normalization factor β.

[0128] For example, formula (8) describing the quantization precoding problem can be approximated as:

[0129] Subject to β>0 (16)

[0130] In this problem, both β and x are unknown. It is very complicated to find both the optimal precoding factor and the optimal precoding signal simultaneously, thus making the computational complexity very high. Given the normalization factor β, the optimization problem becomes finding the optimal solution x, which thus becomes less complicated.

[0131] The BGD algorithm is widely used in the field of machine learning (ML) due to its low complexity. The BGD algorithm can be used for optimization problems, especially convex optimization problems. In some example embodiments, once the normalization factor β is determined, the BGD algorithm can be used to precode the expected signal s to find the optimal solution x, and the algorithm has low complexity and is very easy to implement.

[0132] The following will refer to Figure 6 Discussing an example precoding process. Assume that the expected signal vector s and the channel matrix H are known. Then, using an initial solution x such as a zero vector, the optimization process can be iteratively implemented by the following steps.

[0133] First, in block 605, the transmitter 410 uses the normalization factor β to determine the decreasing direction of the difference between the plurality of expected signals s and the corresponding signals received by the plurality of receivers, for example, at the plurality of receiving antennas 420-1, 420-1, ..., 420-M. In the decreasing direction, the decreasing rate of the difference is higher than the threshold rate. That is, the difference decreases very quickly. During one number of iterations, the threshold rate may be the fastest decreasing rate (or a faster decreasing rate).

[0134] The descent direction can be determined by setting the derivation of the cost function to zero. For example, the descent direction can be represented by a gradient vector represented by g. In an example embodiment using the BGD algorithm, based on formula (16), the descent direction can be determined by setting Set to zero to derive the gradient vector g:

[0135]

[0136] Then, in block 610, the precoded signal x is determined along a descending direction represented by the gradient vector g. For example, x=x+ηg, where η represents an update step size, such as a learning rate.

[0137] In some example embodiments, the accumulated descent direction is used to determine the precoded signal x. The accumulated descent direction is also called momentum, denoted by m, which represents the trend of gradient descent. The momentum m (initialized to a zero vector) is updated as follows:

[0138] m=αm+g (18)

[0139] Where α represents the predetermined accumulation rate.

[0140] The accumulation rate α may be an empirical value. In some example embodiments, the accumulation rate α is calculated as follows.

[0141]

[0142] Where exp(·) represents the exponential function.

[0143] The precoded signal x is then updated to x=x+ηm.

[0144] The update step size or learning rate η can be set to an automatically calculated value as follows:

[0145]

[0146] In some example embodiments, when calculating the update step size, channel quality information about the wireless channel 425 may be considered. For example, the update step size may be calculated as follows:

[0147]

[0148] The calculation of the update step size η is very flexible. The automatically calculated update step size or learning rate during iterations can promote convergence to the optimal solution x, thereby also reducing the overall complexity.

[0149] In some example embodiments, the function is used to determine the precoded signal x to also reduce distortion caused by quantization. Any suitable activation function in the ML region may be applied. In some example embodiments, the activation function may include a function having a value such as Figure 7 The curve shown is a monotonically increasing, oddly symmetric, and saturating function.

[0150] The sign function 705 can be used to determine the precoded signal x, as shown in formula (4). As candidates for replacing the sign function 705, there are many activation functions in the field of machine learning that have similar curves to the sign function. For example, Figure 7 The soft sign function 710 , inverse square root unit (ISRU) function 715 , tanh function 720 , soft clipping function 725 , and clipping function 730 shown all have similar curves to the sign function 705 .

[0151] In some example embodiments, the soft clipping function 725 or the clipping function 730 is used as an activation function. The soft clipping function 725 can be expressed as follows:

[0152]

[0153] Where p and q are internal parameters of the soft clipping function 725. Parameter p controls the similarity between the soft clipping function 725 and the clipping function 730, and parameter q controls the slope. When p becomes large enough, the soft clipping function 725 will become the clipping function 730, which is represented as follows.

[0154] clip(a;q)=min(max(-1,qa),1) (23)

[0155] Where q is an internal parameter of the clipping function 730 that controls the slope. When q→∞, the clipping function 730 will become the sign function 705, that is, clip(a;∞)≡sgn(a). The use of the clipping function 730 can also simplify the computational complexity.

[0156] When the activation function is applied, the precoded signal x can be updated as:

[0157]

[0158] in Represents an approximate quantization function based on the activation function.

[0159] In case of using the clipping function 730 as the activation function, an approximate quantization function for the real part of the precoded signal x may be constructed as follows.

[0160]

[0161] The approximate quantization function for the imaginary part is similar to formula (25).

[0162] Then, combined with the quantization base level mentioned above, the final approximate quantization function will be expressed as follows.

[0163]

[0164] Figure 8 An example curve of the real part or imaginary part of the approximate quantization function according to some example embodiments of the present disclosure is shown. The real part and the imaginary part have the same curve. As shown in the figure, Figure 3 Compared to the quantization function curve shown in , the slope of the line between two flat steps is not ∞ but q. q may be set to an empirical value. In some example embodiments, q=1.25.

[0165] In addition to the clipping function 730, other functions having a curve similar to the sign function may also be used, such as the soft sign function 710, the inverse square root unit (ISRU) function 715, the tanh function 720, and the soft clipping function 725. With appropriate parameter settings, satisfactory detection performance can be achieved.

[0166] In some example embodiments, the normalization factor β may be updated or refined after multiple iterations to improve convergence. Figure 6 After determining the descent direction (denoted by g) and determining that the plurality of precoded signals x are iteratively performed a predetermined number of times, at block 615, a normalization factor β may be updated based on the precoded signal x.

[0167] For example, the normalization factor β can be updated as follows:

[0168]

[0169] in Represents the real part of a complex number.

[0170] Channel noise can also be considered in the update. For example, the normalization factor η can be updated as follows:

[0171]

[0172] in represents the variance of the noise.

[0173] By iteratively executing Figure 6 The precoded signal x will eventually converge to the optimal solution. Figure 6 The GD-based quantization precoding scheme shown is a general framework for general quantization precoding with high flexibility and scalability. The scheme can be applied to various finite alphabet precoding problems.

[0174] In order to further expand the application of the quantization precoding scheme, in some exemplary embodiments, multiple precoded signals x are also regularized. For example, the quantization precoding problem described in formula (8) can be approximately converted to the following problem.

[0175] Subject to β>0 (29)

[0176] constraint is used in the above formula. Compared with formula (16), formula (29) has a regularization term When regularization is used, the gradient is calculated differently, as follows:

[0177]

[0178] in represents the SNR. The other procedures can be the same as the quantization precoding procedure discussed above without regularization.

[0179] In addition to the BGD algorithm used to search for the optimal algorithm, other GD algorithms such as SGD and mini-batch GD may also be applicable. If other GD algorithms are used, the learning rate may be recalculated to adapt to the specific GD algorithm.

[0180] Take the SGD algorithm as an example. Compared with the BGD algorithm, the SGD algorithm has an inner loop (called an epoch) that iterates by the rows of the channel matrix. In the kth iteration of the inner loop, the gradient will be:

[0181]

[0182] The optimal learning rate for the kth iteration will be:

[0183]

[0184] Then, the solution can be updated by:

[0185] x=x+η k g k (33)

[0186] For each epoch (i.e., one inner iteration loop for SGD), the estimated solution is obtained by Approximately quantized once. If the SGD algorithm is used to search for the optimal solution, the determination of batch gradient descent (or BGD epoch) should be replaced by the SGD epoch mentioned above.

[0187] Still viewing Figure 5 In block 515, the transmitter 410 quantizes the plurality of precoded signals x for transmission by the transmitter 410 to a plurality of receivers, for example, via a plurality of transmit antennas 415-1, 415-2, ..., 415-N and a plurality of receive antennas 420-1, 420-2, ..., 420-M. For example, the precoded signal x may be quantized as follows:

[0188]

[0189] in represents an L-bit quantization function.

[0190] Fig. 9 An example process 900 of a BGD-based quantization precoding algorithm according to some example embodiments of the present disclosure is shown.

[0191] In box 905, parameters such as s, H, β and ρ are input, and some parameters such as x, m and α are initialized. At box 910, the update step size (or learning rate) η is calculated as η = f η (s, H, ρ). At block 915, the descent direction (or gradient) g is calculated as g=f g (s, H, x, β). At block 920, the cumulative descent direction (or momentum) m is calculated as m=f M (m, g, α). At block 925, the precoded signal x is updated to At block 930, it is checked whether the end condition is met. If the end condition is met, process 900 proceeds to block 930, where the precoded signal x is output. If the end condition is not met, process 900 returns to block 915 to calculate the gradient g again. As long as the end condition is not met, blocks 915, 920, and 925 will be iteratively performed.

[0192] Any suitable end condition may be used. In some example embodiments, a threshold number of iterations may be used. If the number of iterations reaches the threshold number of iterations, then the end condition is determined to be satisfied. In some other example embodiments, the end condition may be associated with a convergence level, which may be indicated by a difference between the expected signal s and the received signals at the plurality of receive antennas 420-1, 420-2, ..., 420-M. For example, if the difference is below a threshold difference, then it may be determined that the end condition is satisfied.

[0193] As referenced above Figures 4 to 8 All operations and features described are equally applicable to process 900 and have similar effects. For the purpose of simplicity, details will be omitted.

[0194] Now, an example BGD-based quantization precoding algorithm will be discussed.

[0195] enter:

[0196] Signal vector s.

[0197] Channel matrix H

[0198] Average symbol energy E s .

[0199] The total signal power P.

[0200] SNR

[0201] Bit L.

[0202] Output:

[0203] Precoded signal vector x.

[0204] initialization:

[0205] Set the maximum number of iterations n, where 30 or greater is fine.

[0206] The learning rate η for optimizing BGD is set using formula (21).

[0207] Use equation (19) to set the empirical accumulation rate α for momentum calculation.

[0208] Initialize the precoded signal vector x and momentum m to zero vectors.

[0209] Set the Bussgang constant τ value according to bit L and Table 2.

[0210] The normalization factor β is set using formula (15).

[0211] The precoding factor refinement interval J is set empirically, such as J=25 (where 10 to 30 is sufficient).

[0212] Set the approximate quantization level ζ according to bit L and Table 1.

[0213] The approximate quantization function is set using formula (26), where the parameter q=1.25.

[0214] Main routine:

[0215] For t = 1 to n

[0216] #Step 1: Optimized batch gradient descent (approximate quantization is integrated)

[0217] The gradient g is calculated using formula (17).

[0218] Update the momentum m using formula (18).

[0219] Update the precoded signal vector x using formula (24).

[0220] #Step 2: Normalization factor refinement

[0221] If (t+1)%J=0, that is, t+1 in the set {J, 2J, 4J, ...}:

[0222] Then use formula (28) to update the normalization factor β.

[0223] #End of loop

[0224] The precoded signal vector x is quantized by changing the elements of x to the nearest quantization level using formula (34).

[0225] Output precoded signal vector x

[0226] #End the quantization precoding algorithm based on optimized BGD

[0227] For each iteration of the optimized BGD (OBGD) algorithm, the computational complexity is only O(2NM). It takes about 25 to 50 iterations to reach convergence, and the total complexity is much lower than that of traditional precoders. For example, for 1-bit quantization, the computational complexity of the SQUID precoder with two loops is O(2(k 1 N 3 +k 2 N 2 )), where k 1 and k 2 is the number of iterations of the two loops. In each iteration, the computational complexity of the C2PO precoder is O(N 2 ). However, it is difficult to extend the C2PO precoder to L-bit quantization precoding.

[0228] For the general L-bit quantization case, the complexity of the IDE2 precoder is O(2NM) for each iteration, which is the same as the OBGD-based quantization precoder. However, the IDE2 precoder usually requires 50 to 100 iterations to have satisfactory performance, and the total complexity will be greater than the proposed scheme.

[0229] In addition, for high-order modulation, such as 16QAM precoder, 64QAM precoder and 256QAM precoder, SQUID precoder, C2PO precoder and IDE2 precoder need to vectorize the signal vector block and thus expand the channel matrix, which will also increase the computational complexity for each iteration by T. 2 (T represents the number of time slots). In contrast, the OBGD-based quantization precoder does not need to vectorize the signal blocks or expand the channel matrix. Signal vectors in different time slots can be processed independently and in parallel. Therefore, the OBGD-based quantization precoder will be much more practical than the traditional quantization precoder.

[0230] According to the simulation results, the OBGD-based quantization precoding scheme outperforms the conventional precoding scheme. The comparison between the performance of the OBGD-based quantization precoding scheme and the conventional precoding scheme is performed based on simulations in various scenarios for different quantization precoders with 10,000 samples. For comparison, the detection performance of the ZF precoder (marked by ZF-inf) with infinite resolution is proposed.

[0231] Two scenarios were considered in this simulation. In scenario 1, it is assumed that the symbol level normalization or precoding factor is known to the receiver. This scenario is not realistic but provides a benchmark for detection performance. In scenario 2, the transmitter implemented by the base station sends a pilot signal in the first time slot (T=10 time slots for each signal block) which is used to calculate the signal gain at the receiver. This is a realistic scenario that can provide a more reasonable estimate of the detection performance of different precoding schemes.

[0232] The simulation parameters are set as follows:

[0233] ο The average power is set to 1, i.e. P = 1.0;

[0234] ο Modulation: BPSK, QPSK, 16QAM, 64QAM and 256QAM;

[0235] ο The iterations for SQUID, C2PO, IDE2 and OBGD are 200, 50, 100 and 50.

[0236] In the case of one-bit quantization, the performance of ZFQ precoder, SQUID precoder, C2PO precoder, IDE2 precoder and OBGD precoder are compared for different scenarios and different numbers of antennas. In scenario 1, it is assumed that the receiver knows the normalization factor (which is impossible in practice). In this ideal scenario, the ideality of the precoder is evaluated and Figure 10(a) to Figure 10(c) The uncoded BER for different precoders is plotted in .

[0237] In Figure 10(a), the system load factor (defined as N / M) is 2, and BPSK modulation is used. Among all the quantization precoders, the OBGD precoder achieves the best BER performance, especially when the SNR is greater than 10dB. In Figure 10(b), the system load factor is N / M=4, and QPSK modulation is used. In this simulation, the OBGD precoder also achieves the best BER performance. In Figure 10(c), the system load factor is N / M=8, and 16QAM modulation (high-order modulation) is used. In this simulation, both the OBGD precoder and the IDE2 precoder achieve the best BER performance.

[0238] In scenario 2, both non-vectorized and vectorized data are used to evaluate the BER performance for different precoders. In general, vectorization will greatly increase the computational complexity because the vectorized data usually has high dimensions. In Figure 11(a) and (b), the system load factor N / M=2, and BPSK modulation is adopted. In this simulation, the OBGD precoder achieves the best BER performance, and the OBGD precoder with and without vectorization achieves similar BER performance. For the SQUID precoder, the performance gap between the vectorized version and the non-vectorized version is relatively larger than that of other precoders.

[0239] In Figure 11 (c) and (d), the system load factor N / M = 4, and QPSK modulation is adopted. In this simulation, the OBGD precoder with and without vectorization achieves the best BER performance, while the IDE2 precoder with and without vectorization achieves slightly worse BER performance.

[0240] In Figure 11(e) and (f), the system load factor N / M = 8, and 16QAM modulation is used. In this simulation, the OBGD precoder still achieves the best BER performance, especially for non-vectored data. The IDE2 precoder achieves poor BER performance for non-vectored data, but achieves similar BER performance as the OBGD precoder for vectored data. Similarly, the SQUID precoder and C2PO precoder with vectorization achieve better performance than the SQUID precoder and C2PO precoder without vectorization. It is obvious that the IDE2 precoder, SQUID precoder, and C2PO precoder have a great dependence on vectorization for high-order modulation.

[0241] Vectoring for a specific signal block can help keep the signals in the block with the same precoding factor, so that the receiver implemented by the UE can use the normalization factor of the pilot signal to calculate the data signal. But for BPSK modulation and QPSK modulation, such calculation is not required on the UE side. Therefore, for low-order modulations, such as BPSK and QPSK, the dependence on vectoring is irrelevant for all precoders. For high-order modulations like 16QAM, the normalization factor is essential for the UE to calculate the data signal. Therefore, except for the OBGD precoder, which has little dependence on vectoring, the dependence on vectoring is very large for most nonlinear precoders.

[0242] In the case of two-bit quantization, the performance of ZFQ precoder, IDE2 precoder and OBGD precoder is compared for different scenarios and different numbers of antennas. SQUID precoder and C2PO precoder are not tested here because they are dedicated to the case of 1-bit quantization.

[0243] In scenario 1 (ideal scenario), the BER performance for three configurations is evaluated. In Figure 12(a), the system load factor N / M = 2, and QPSK modulation is used. In this simulation, the OBGD precoder achieves the best BER performance (closest to ZF-inf), the IDE2 precoder and the ZFQ precoder achieve the second best and worst performance, respectively. In Figures 12(b) and (c), the results are similar, with the OBGD precoder achieving the best BER performance, the IDE2 precoder achieving the second best BER performance, and the ZFQ precoder achieving the worst BER performance.

[0244] In scenario 2 (real scenario), the BER performance for the same three configurations is evaluated. For the first configuration with QPSK modulation in Figures 13(a) and 13(b), the receiver does not need precoding factors to calculate the data signal. Therefore, the performance between vectored and non-vectored is similar. For both vectored and non-vectored data, the OBGD precoder achieves better BER performance than the IDE2 precoder and the ZFQ precoder.

[0245] In Figure 13(c) and Figure 13(d), the system load factor N / M = 4, and 16QAM modulation is used. In this simulation, the IDE2 precoder is highly dependent on vectoring, and the gap between the BER performance with and without vectoring is very large. For the OBGD precoder, the performance gap between non-vectoring and vectoring is very small, which means that the OBGD precoder has little dependence on vectoring.

[0246] In Figures 13(e) and 13(f), the system load factor N / M=8, and 64QAM modulation is used. In this simulation, without vectorization, the IDE2 precoder achieves the worst BER performance, even worse than the linearly quantized ZFQ precoder. Although the IDE2 precoder with vectorization achieves better BER performance, it is only slightly worse than the OBGD precoder. In contrast, the OBGD precoder with vectorization and the OBGD precoder without vectorization achieve similar BER performance, which is the best among all quantized precoders. Therefore, the OBGD precoder has little dependence on vectorization.

[0247] In the case of three-bit quantization, the performance of ZFQ precoder, IDE2 precoder and OBGD precoder is compared for different scenarios and different numbers of antennas. For 3-bit quantization, the distortion caused by quantization is much smaller than that of 1-bit and 2-bit cases. Therefore, 16QAM modulation, 64QAM modulation and 256QAM modulation are used in the case of 3-bit quantization.

[0248] In scenario 1 (ideal scenario), the BER performance for three configurations is evaluated. In Figure 14(a), the system load factor is 2 and 16QAM modulation is used. Among all quantization precoders, the OBGD precoder achieves the best BER performance, which is much better than the IDE2 precoder and the ZFQ precoder. In Figure 14(b), the system load factor is 4 and 64QAM modulation is used. The OBGD precoder and the IDE2 precoder achieve similar BER performance, which is much better than the ZFQ encoder. In Figure 14(c), the system load factor is 8 and 256QAM modulation is used. The OBGD precoder achieves the best BER performance, and the IDE2 precoder achieves slightly worse performance.

[0249] In scenario 2 (actual scenario), the BER performance of the same three configurations in scenario 1 is evaluated. In Figures 15(a) and 15(b), the system load factor is 2 and 16QAM modulation is used. The OBGD precoder without vectorization and the OBGD precoder with vectorization achieve similar BER performance, which are the best among all quantization precoders. In contrast, the IDE2 precoder with vectorization achieves the second best BER performance, while the IDE2 precoder without vectorization achieves the worst BER performance. Therefore, the IDE2 precoder is highly dependent on vectorization.

[0250] In Figure 15(c) and Figure 15(d), the system load factor is 4 and 64QAM modulation is used. The OBGD precoder still achieves the best BER performance, and vectorization has almost no effect on the OBGD precoder. For the IDE2 precoder, vectorization is necessary for satisfactory BER performance. Without vectorization, the IDE2 precoder can only achieve performance similar to that of the ZFQ precoder, which is much worse than the OBGD precoder. With vectorization, the IDE2 precoder can achieve much better performance than the ZFQ precoder, which is slightly worse than the OBGD precoder.

[0251] In Figures 15(e) and 15(f), the system load factor is 8 and 256QAM modulation is used. In this simulation, the OBGD precoder still achieves the best BER performance regardless of whether vectoring is applied. For the IDE2 precoder, vectoring has a great impact on the BER performance. With vectoring, the IDE2 precoder can achieve satisfactory performance that is much better than the ZFQ precoder but slightly worse than the OBGD precoder. Without vectoring, the IDE2 precoder achieves the worst BER performance, which is slightly worse than the ZFQ precoder.

[0252] Obviously, for higher order modulations such as 16QAM, 64QAM, and 256QAM, IDE2 must vectorize the signal blocks to search for near-optimal solutions in a higher dimensional space. Therefore, the computational complexity increases a lot (up to T) due to the high dimensional matrices and vectors. 2). Therefore, the traditional nonlinear quantization precoder is not practical. The proposed OBGD-based quantization precoder has little dependence on vectorization, and the precoder can achieve excellent performance without vectorization. Therefore, for large-scale MIMO systems with low-resolution DAC, the OBGD-based quantization precoding scheme is much more practical than the traditional quantization precoder. Compared with the traditional nonlinear quantization precoder, the quantization precoding algorithm proposed in the present invention has lower complexity and better detection performance.

[0253] Fig.16 1 is a simplified block diagram of a device 1600 suitable for implementing an example embodiment of the present disclosure. The device 1600 may be implemented at or as part of the transmitter 410, such as Figure 4 as shown in .

[0254] As shown, the device 1600 includes a processor 1610, a memory 1620 coupled to the processor 1610, a communication module 1630 coupled to the processor 1610, and a communication interface (not shown) coupled to the communication module 1630. The memory 1620 stores at least a program 1640. The communication module 1630 is used for bidirectional communication via multiple antennas. The communication interface can represent any interface required for communication.

[0255] Assume that program 1640 includes program instructions that, when executed by associated processor 1610, enable device 1600 to operate in accordance with example embodiments of the present disclosure, as described herein with reference to Figures 4 to 9 The example embodiments herein may be implemented by computer software executable by the processor 1610 of the device 1600, or by hardware, or by a combination of software and hardware. The processor 1610 may be configured to implement various example embodiments of the present disclosure.

[0256] Memory 1620 may be of any type suitable for a local technology network and may be implemented using any appropriate data storage technology, such as non-transitory computer-readable storage media, semiconductor-based memory devices, magnetic memory devices and systems, optical memory devices and systems, fixed memory, and removable memory, as non-limiting examples. Although only one memory 1620 is shown in device 1600, there may be several physically distinct memory modules in device 1600. Processor 1610 may be of any type suitable for a local technology network and may include one or more of a general-purpose computer, a special-purpose computer, a microprocessor, a digital signal processor (DSP), and a processor based on a multi-core processor architecture, as non-limiting examples. Device 1600 may have multiple processors, such as dedicated integrated circuit chips that are time-slave to a clock synchronized with a main processor.

[0257] When the device 1600 acts as the transmitter 410, the processor 1610 and the communication module 1630 may cooperate to implement the above reference Figures 4 to 9 The method 500 and the process 600 and the process 900 described above are Figures 4 to 9 All operations and features described are equally applicable to device 1600 and have similar effects. For the sake of simplicity, the details will be omitted.

[0258] In general, various example embodiments of the present disclosure may be implemented in hardware or dedicated circuits, software, logic, or any combination thereof. Some aspects may be implemented in hardware, while other aspects may be implemented in firmware or software that may be executed by a controller, microprocessor, or other computing device. Although various aspects of the example embodiments of the present disclosure are illustrated and described as block diagrams, flow charts, or using some other graphical representations, it should be understood that the blocks, devices, systems, techniques, or methods described herein may be implemented in hardware, software, firmware, dedicated circuits or logic, general hardware or controllers or other computing devices, or some combination thereof as non-limiting examples.

[0259] The present disclosure also provides at least one computer program product tangibly stored on a non-transitory computer-readable storage medium. The computer program product includes computer executable instructions, such as instructions contained in a program module, which are executed in a device on a target real or virtual processor to implement the above referenced Figures 4 to 9 The method 500 and process 600 and process 900 are described. Typically, a program module includes routines, programs, libraries, objects, categories, components, data structures, etc. that perform specific tasks or implement specific abstract data types. In various example implementations, the functionality of the program modules can be combined or split between program modules as needed. Machine executable instructions for program modules can be executed in local or distributed devices. In distributed devices, program modules can be located in local and remote storage media.

[0260] The program code for executing the method of the present disclosure can be written in any combination of one or more programming languages. These program codes can be provided to a processor or controller of a general-purpose computer, a special-purpose computer or other programmable data processing device so that the program code causes the function / operation specified in the flow chart and / or block diagram to be implemented when executed by the processor or controller. The program code can be executed entirely on the machine, partially on the machine, as an independent software package, partially on the machine and partially on a remote machine or entirely on a remote machine or server.

[0261] In the context of the present disclosure, computer program codes or related data may be carried by any suitable carrier to enable a device, apparatus or processor to perform various processes and operations as described above. Examples of carriers include signals and computer-readable media.

[0262] The computer readable medium may be a computer readable signal medium or a computer readable storage medium. The computer readable medium may include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared or semiconductor systems, devices or apparatuses, or any suitable combination of the foregoing. More specific examples of computer readable storage media would include an electrical connection with one or more wires, a portable computer floppy disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), a digital versatile disk (DVD), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing.

[0263] In addition, although the operations are described in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequence, or that all illustrated operations be performed to obtain the desired result. In some cases, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are contained in the above discussion, these should not be interpreted as limitations on the scope of the present disclosure, but rather as descriptions of features that may be specific to a particular example implementation. Certain features described in the context of a separate example embodiment may also be implemented in combination in a single embodiment. On the contrary, the various features described in the context of a single embodiment may also be implemented in multiple example embodiments individually or in any suitable sub-combination form.

[0264] Although the disclosure has been described in language specific to structural features and / or methodological acts, it should be understood that the disclosure defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.

[0265] Various example embodiments of the technology have been described. In addition to or as an alternative to the above, the following embodiments are described. Features described in any of the following examples may be used with any of the other examples described herein.

[0266] In some aspects, the transmitter includes: at least one processor; and at least one memory containing computer program code; the at least one memory and the computer program code are configured to, together with the at least one processor, cause a first device to: determine a normalization factor for precoding multiple signals, the multiple signals being expected for multiple receivers; precode multiple expected signals using the normalization factor by iteratively performing actions, the actions including: determining a decreasing direction of a difference between the multiple expected signals and corresponding signals received at multiple receivers by using the normalization factor, in the decreasing direction, a decreasing rate of the difference being higher than a threshold rate, and determining multiple precoded signals based on the decreasing direction; and quantizing the multiple precoded signals for transmission by the transmitter to the multiple receivers.

[0267] In some example embodiments, the transmitter is caused to determine the normalization factor by determining the normalization factor for precoding the plurality of original signals based on a transmission power of the transmitter and a quantization constant.

[0268] In some example embodiments, the normalization factor is determined as:

[0269]

[0270] Where β is the normalization factor, E s is the average energy of the multiple original signals, P is the transmit power, N is the number of transmit antennas, M is the number of desired signals and τ is a quantization constant.

[0271] In some example embodiments, the determination of the normalization factor is also based on the signal-to-noise ratio.

[0272] In some example embodiments, the transmitter is further caused to determine a plurality of precoded signals based on the descent direction by: determining a cumulative descent direction based on the descent direction using a predetermined accumulation rate; and updating the plurality of precoded signals based on the cumulative descent direction and an update step size.

[0273] In some example embodiments, the update step size is determined based at least in part on the plurality of expected signals and channel state information regarding a wireless channel between the transmitter and the plurality of receivers.

[0274] In some example embodiments, the update step size is also determined based on the signal-to-noise ratio.

[0275] In some example embodiments, the transmitter is caused to update the multiple precoded signals based on the accumulated descent direction and the update step size by the following operation: based on the accumulated descent direction and the update step size, the multiple precoded signals are updated by monotonically increasing odd symmetric and saturated functions of the real and imaginary parts of the multiple precoded signals, respectively.

[0276] In some example embodiments, the function comprises a clipping function.

[0277] In some example embodiments, the actions further include updating a normalization factor based on the plurality of precoded signals after the determining of the descent direction and the determining of the plurality of precoded signals are iteratively performed a predetermined number of times.

[0278] In some example embodiments, the actions further include regularizing the plurality of precoded signals.

[0279] In some aspects, a method implemented at a transmitter includes: determining a normalization factor for precoding multiple signals, the multiple signals being intended for multiple receivers; precoding the multiple expected signals using the normalization factor by iteratively performing actions, the actions including: determining a decreasing direction of a difference between the multiple expected signals and corresponding signals received at multiple receivers by using the normalization factor, in the decreasing direction, a decreasing rate of the difference is higher than a threshold rate, and determining multiple precoded signals based on the decreasing direction; and quantizing the multiple precoded signals for transmission by the transmitter to the multiple receivers.

[0280] In some example embodiments, determining the normalization factor includes determining the normalization factor for precoding the plurality of original signals based on a transmit power of the transmitter and a quantization constant.

[0281] In some example embodiments, the normalization factor is determined as:

[0282]

[0283] Where β is the normalization factor, E s is the average energy of the multiple original signals, P is the transmit power, N is the number of transmit antennas, M is the number of desired signals and τ is a quantization constant.

[0284] In some example embodiments, the determination of the normalization factor is also based on the signal-to-noise ratio.

[0285] In some example embodiments, determining the plurality of precoded signals based on the descent direction comprises: determining a cumulative descent direction based on the descent direction using a predetermined accumulation rate; and updating the plurality of precoded signals based on the cumulative descent direction and an update step size.

[0286] In some example embodiments, the update step size is determined based at least in part on the plurality of expected signals and channel state information regarding a wireless channel between the transmitter and the plurality of receivers.

[0287] In some example embodiments, the update step size is also determined based on the signal-to-noise ratio.

[0288] In some example embodiments, updating the plurality of precoded signals based on the cumulative descent direction and the update step size comprises updating the plurality of precoded signals by monotonically increasing odd symmetric and saturated functions for real and imaginary parts of the plurality of precoded signals, respectively, based on the cumulative descent direction and the update step size.

[0289] In some example embodiments, the function comprises a clipping function.

[0290] In some example embodiments, the actions further include updating a normalization factor based on the plurality of precoded signals after the determining of the descent direction and the determining of the plurality of precoded signals are iteratively performed a predetermined number of times.

[0291] In some example embodiments, the actions further include regularizing the plurality of precoded signals.

[0292] In some aspects, an apparatus includes: a component for determining a normalization factor for precoding multiple signals, the multiple signals being intended for multiple receivers; a component for precoding the multiple expected signals using the normalization factor by iteratively performing actions, the actions including: determining a decreasing direction of a difference between the multiple expected signals and corresponding signals received at multiple receivers by using the normalization factor, in which a decreasing rate of the difference is higher than a threshold rate in the decreasing direction, and determining multiple precoded signals based on the decreasing direction; and quantizing the multiple precoded signals for transmission by a transmitter to the multiple receivers.

[0293] In some example embodiments, means for determining a normalization factor includes means for determining a normalization factor for precoding the plurality of original signals based on a transmit power and a quantization constant.

[0294] In some example embodiments, the normalization factor is determined as:

[0295]

[0296] Where β is the normalization factor, E s is the average energy of the multiple original signals, P is the transmit power, N is the number of transmit antennas, M is the number of desired signals and τ is a quantization constant.

[0297] In some example embodiments, the determination of the normalization factor is also based on the signal-to-noise ratio.

[0298] In some example embodiments, the means for determining the plurality of precoded signals based on the descent direction comprises: means for determining a cumulative descent direction based on the descent direction using a predetermined accumulation rate; and means for updating the plurality of precoded signals based on the cumulative descent direction and an update step size.

[0299] In some example embodiments, the update step size is determined based at least in part on the plurality of expected signals and channel state information regarding a wireless channel between the transmitter and the plurality of receivers.

[0300] In some example embodiments, the update step size is also determined based on the signal-to-noise ratio.

[0301] In some example embodiments, the means for updating the plurality of precoded signals based on the cumulative descent direction and the update step size comprises means for updating the plurality of precoded signals based on the cumulative descent direction and the update step size by monotonically increasing odd symmetric and saturated functions of the real and imaginary parts of the plurality of precoded signals, respectively.

[0302] In some example embodiments, the function comprises a clipping function.

[0303] In some example embodiments, the actions further include updating a normalization factor based on the plurality of precoded signals after the determining of the descent direction and the determining of the plurality of precoded signals are iteratively performed a predetermined number of times.

[0304] In some example embodiments, the actions further include regularizing the plurality of precoded signals.

[0305] In some aspects, a computer-readable storage medium includes program instructions stored thereon, which when executed by a processor of a device causes the device to perform methods according to some example embodiments of the present disclosure.

Claims

1. A transmitter, comprising: at least one processor; as well as at least one memory containing computer program code; The at least one memory and the computer program code are configured to, with the at least one processor, cause the transmitter to: determining a normalization factor for precoding the plurality of signals based on a transmit power of the transmitter, a quantization constant, an average energy of the plurality of signals, a number of transmit antennas, and a number of expected signals, the plurality of signals being intended for a plurality of receivers; Precoding a plurality of desired signals using the normalization factor by iteratively performing actions, the actions comprising: determining, by using the normalization factor, a decreasing direction of the differences between the plurality of expected signals and corresponding signals received at the plurality of receivers, in which decreasing direction a decreasing rate of the differences is higher than a threshold rate, and determining a plurality of precoded signals based on the descending direction; as well as The plurality of precoded signals are quantized for transmission by the transmitter to the plurality of receivers.

2. The transmitter of claim 1 , wherein the normalization factor is determined as: where β is the normalization factor, E s is the average energy of the multiple signals, P is the transmit power, N is the number of transmit antennas, M is the number of the expected signals and τ is the quantization constant.

3. The transmitter of claim 1, wherein the determination of the normalization factor is further based on a signal-to-noise ratio.

4. The transmitter of claim 1 , wherein the transmitter is further configured to determine the plurality of precoded signals based on the descent direction by: determining a cumulative descent direction based on the descent direction using a predetermined cumulative rate; and The plurality of precoded signals are updated based on the accumulated decreasing direction and the updating step size. 5 . The transmitter of claim 4 , wherein the update step size is determined based at least in part on the plurality of expected signals and channel state information about a wireless channel between the transmitter and the plurality of receivers. The transmitter of claim 5 , wherein the update step size is further determined based on a signal-to-noise ratio.

7. The transmitter according to claim 4, wherein the transmitter is caused to update the plurality of precoded signals based on the accumulated decreasing direction and the updating step size by: Based on the cumulative decreasing direction and the updating step size, the plurality of precoded signals are updated by monotonically increasing odd symmetric and saturation functions for real parts and imaginary parts of the plurality of precoded signals, respectively. The transmitter of claim 7 , wherein the function comprises a clipping function.

9. The transmitter of claim 1, wherein the actions further comprise: After the determination of the descent direction and the determination of the plurality of precoding signals are iteratively performed a predetermined number of times, the normalization factor is updated based on the plurality of precoding signals.

10. The transmitter of claim 1, wherein the actions further comprise: The plurality of precoded signals are regularized.

11. A method implemented at a transmitter, comprising: determining a normalization factor for precoding a plurality of signals intended for a plurality of receivers based on a transmit power of the transmitter, a quantization constant, an average energy of the plurality of signals, a number of transmit antennas, and a number of intended signals; Precoding a plurality of desired signals using the normalization factor by iteratively performing actions, the actions comprising: determining, by using the normalization factor, a decreasing direction of the differences between the plurality of expected signals and corresponding signals received at the plurality of receivers, in which decreasing direction a decreasing rate of the differences is higher than a threshold rate, and determining a plurality of precoded signals based on the descending direction; as well as The plurality of precoded signals are quantized for transmission by the transmitter to the plurality of receivers.

12. The method of claim 11, wherein the normalization factor is determined as: where β is the normalization factor, E s is the average energy of the multiple signals, P is the transmit power, N is the number of transmit antennas, M is the number of the expected signals and τ is the quantization constant. The method of claim 11 , wherein the determination of the normalization factor is further based on a signal-to-noise ratio.

14. The method according to claim 11, wherein determining the plurality of precoded signals based on the descent direction comprises: determining a cumulative descent direction based on the descent direction using a predetermined cumulative rate; as well as The plurality of precoded signals are updated based on the accumulated decreasing direction and the updating step size.

15. The method of claim 14, wherein the update step size is determined based at least in part on the plurality of expected signals and channel state information about a wireless channel between the transmitter and the plurality of receivers. The method of claim 15 , wherein the update step size is further determined based on a signal-to-noise ratio.

17. The method according to claim 14, wherein updating the plurality of precoded signals based on the accumulated decreasing direction and the updating step size comprises: Based on the cumulative decreasing direction and the updating step size, the plurality of precoded signals are updated by monotonically increasing odd symmetric and saturation functions for real parts and imaginary parts of the plurality of precoded signals, respectively. The method of claim 17 , wherein the function comprises a clipping function.

19. The method of claim 11, wherein the actions further comprise: After the determination of the descent direction and the determination of the plurality of precoding signals are iteratively performed a predetermined number of times, the normalization factor is updated based on the plurality of precoding signals.

20. The method of claim 11, wherein the actions further comprise: The plurality of precoded signals are regularized.

21. An apparatus for precoding, comprising: means for determining a normalization factor for precoding a plurality of signals intended for a plurality of receivers based on a transmit power of the device, a quantization constant, an average energy of the plurality of signals, a number of transmit antennas, and a number of intended signals; Means for precoding a plurality of desired signals using the normalization factor by iteratively performing actions comprising: determining, by using the normalization factor, a decreasing direction of the differences between the plurality of expected signals and corresponding signals received at the plurality of receivers, in which decreasing direction a decreasing rate of the differences is higher than a threshold rate, and determining a plurality of precoded signals based on the descending direction; as well as Means for quantizing the plurality of precoded signals for transmission by the apparatus to the plurality of receivers.

22. A computer-readable storage medium comprising program instructions stored thereon, which instructions, when executed by a processor of a device, cause the device to perform the method according to any one of claims 11 to 20.

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