Massive MIMO As an Extreme Learning Machine

Massive multiple-input multiple-output (MIMO) systems, where the base station (BS) is equipped with a large number of antennas, is a promising technology for 5G and future generation wireless communications [1]. However, the requirement for a large number of radio frequency (RF) chains leads to high power consumption. To address this challenge, various techniques have been proposed to reduce the number of RF chains [2] and low-resolution analog-to-digital converters (ADCs) have been considered [3], [4], [5]. In addition to low-resolution ADCs at the BS, hardware impairments of the user equipment may also need to be addressed. For example, the use of cheap power amplifiers (PAs) can lead to severe nonlinear distortions to transmitted signals [6, 7]. Previous work has shown that hardware impairments have to be handled properly to avoid system performance degradation [3], [4], [5], [7]. Existing results consider impairments on either the BS or the user side, however they do not consider impairments on both sides.

An extreme learning machine (ELM) is a single-hidden layer feed-forward neural network (NN). In an ELM, the input weights and biases are randomly initialized and fixed, so that the only parameters to be learned are its output weights. Learning reduces to solving a least squares (LS) problem, making ELM fast in learning [8]. ELM receivers have been designed in our previous works [9, 10] to handle LED nonlinearity and/or cross-LED interference. These works show that ELM is very effective in dealing with nonlinear distortions and delivers much better performance than conventional polynomial based techniques [9, 10]. ELM has also been used for channel estimation and data detection in OFDM systems [11, 12].

In this work, we address the hardware impairments at both the BS and the user side in a massive MIMO system using the concept of an ELM. Although deep NNs have attracted much attention for MIMO receiver design (such as [4, 13]), we focus on ELM (with a single-hidden layer) due to its fast learning and adaptive capabilities. In particular, no back propagation is needed and the size of training samples required is relatively small. We consider the uplink of a massive MIMO system with low-resolution ADCs, where the transmitted signals of users suffer from PA nonlinear distortions. We show that the massive MIMO system forms a natural ELM. Specifically, the transmit antennas of users are regarded as the input nodes of the ELM, the massive number of antennas at the BS serve as the hidden nodes of the ELM so that the channel matrix functions as the ELM input weight matrix, and the low-resolution ADCs act as the activation function. We add random biases to the received signals before the ADC and learn the output weights of the ELM from training signals. As the ELM output weight optimization is simply an LS problem, an adaptive receiver is designed to deal with time-varying channels. Simulation results demonstrate the promising performance of the ELM based system in handling hardware impairments compared to conventional techniques.

Notations: Boldface lower-case and upper-case letters denote vectors and matrices, respectively. The superscripts $(\cdot)^{T}$ and $(\cdot)^{H}$ represent the transpose and conjugate transpose operations. We use $|x|$ and $||\bm{x}||$ to denote the amplitude of $x$ and the norm of $\bm{x}$, and $\Re\{\cdot\}$, $\Im\{\cdot\}$ represent the real and imaginary parts of a complex number, respectively.

Consider the uplink transmission of a massive MIMO system with $K$ active users. Each user has a single antenna and the BS is equipped with $N$ antennas, where $N$ can be much larger than $K$. We consider two hardware impairments: PA nonlinear distortion at the transmitter (user) side and low-resolution ADCs at the receiver (BS) side.

The PA nonlinear distortion can be characterized by the amplitude to amplitude conversion $A(|x|)$ and amplitude to phase conversion $\Phi(|x|)$ [14]:

where $x$ is the signal input to the PA, and $\alpha_{a}$, $\epsilon_{a}$, $\epsilon_{\phi}$ and $\beta_{\phi}$ are parameters. The distorted signal due to the PA nonlinearity is expressed as

where $\text{angle}(x)$ denotes the angle of the complex signal $x$.

The received signal at time instant $m$ is modeled as

where $\bm{H}$ is an $N\times K$ MIMO channel matrix, $\bm{s}(m)=f(\bm{x}(m))$ with $\bm{x}(m)=[x_{1}(m),x_{2}(m),\ldots,x_{K}(m)]^{T}$ being the transmitted signals of the $K$ users at time instant $m$, and $\bm{n}(m)$ denotes an additive white Gaussian noise vector. Conventionally the receiver signal is quantized directly, however in this work, a random bias is added to the received signal, which is then quantized, i.e.,

where $\bm{b^{\text{Re}}}$ and $\bm{b^{\text{Im}}}$ represent the biases for the real part and imaginary part of $\bm{y}(m)$, respectively, and $Q(\cdot)$ denotes an element-wise quantization operation. For a uniform mid-rise quantizer, we have

where $c$ denotes the signal to be quantized, $\lfloor\cdot\rfloor$ is the floor function and $\Delta$ is the quantization step size. Due to the limited number of bits used for the ADCs, the input signal $c$ is clipped when its amplitude exceeds a threshold. The aim of the receiver is to recover $\bm{x}(m)$ based on $\bm{r}(m)$.

ELM is attractive in that it allows fast adaptive learning as its output weights can be easily updated. This endows ELM with the unique capability of dealing with time-varying massive MIMO channels. By leveraging the online sequential-ELM (OSELM) [16], the ELM based receiver is readily extended to an adaptive setting.

The adaptive receiver consists of two learning phases, an initialization phase and a sequential learning phase. Suppose that we have ${M_{0}}$ training samples for initialization. The output weights for the users can be computed using (14) and (15), which are denoted as $\bm{\beta}_{k}^{\text{Re}}(0)$ and $\bm{\beta}_{k}^{\text{Im}}(0)$ for user $k$. Define

In the subsequent learning phase with new training samples $\{(\bm{s}^{\text{Train}}(m),\bm{x}^{\text{Train}}(m))\}$ arriving one by one, the adaptive receiver recursively updates its weights. Taking $\bm{\beta}_{k}^{\text{Re}}$ as an example, we have

where $\lambda\in[0,1]$ is a forgetting factor to adjust the tracking capability and convergence rate. The above sample-by-sample update can also be extended to a chunk-by-chunk update [17].

Although the adaptive receiver may require a relatively large number of training samples (still smaller compared to deep NNs) for initialization, a small number of training samples are sufficient for tracking. This makes the adaptive receiver very attractive in handling time-varying massive MIMO channels, as demonstrated in Section V.

Assume that the BS is equipped with a uniform linear array (ULA) of $N=256$ antennas, number of users $K$ = 10, and 16-QAM is used. As in [14], the parameter setting for the PA nonlinearity is $\alpha_{a}=1.96$, $\epsilon_{a}=0.99$, $\alpha_{\phi}=2.53$ and $\epsilon_{\phi}=2.82$. Assume 6-bit ADCs. The SNR is defined as $\text{SNR}=P_{s}/\sigma^{2}_{n}$, where $P_{s}$ is the power of the signal (per transmit antenna), and $\sigma^{2}_{n}$ is the power of the noise (per receive antenna). We employ the massive channel model in [18], [19] and [20], which considers both spatial and temporal correlations, and assume that the ULA has a half-wavelength spacing. Table I lists the parameters used for massive MIMO channel generation. Small scale fading is considered and the multiple rays of each user are normalized. The power angular spectrum is modelled using a truncated Laplacian distribution. We consider the channel under a typical mobility scenario, i.e., urban macro [20]. For the ZF receiver with training, the weights are computed with real and imaginary parts separated. For the ELM receiver in [9], 512 hidden nodes are used, and the input weights and the biases (also the biases of the new ELM based receiver) are drawn independently from a uniform distribution [-0.1, 0.1].

Figure 3 shows the symbol error rate (SER) of the new ELM based receiver, the ELM receiver borrowed from [9] and the ZF and MMSE receivers with perfect channel state information and noise power (but note that it is difficult to acquire them when low resolution ADCs are used). Quasi-static channels are assumed without considering the mobility of users, and the training length is 3000. It can be seen from Fig. 3 that the conventional ZF and MMSE receivers deliver poor performance due to the inability to mitigate the PA nonlinearity. The trained and regularized ZF receiver performs better than them as the impact of nonlinearity is considered in training. In contrast, the ELM receiver in [9] and the new ELM based receiver can effectively handle the hardware impairments, and the new ELM based receiver delivers the best performance, with significantly lower complexity compared to the ELM receiver in [9].

The difference between the trained and regularized ZF receiver and the new ELM receiver is that the received signals are biased before quantization in the new receiver. Their huge performance difference indicates the impact of signal biasing. To further examine the impact of signal biasing and quantization, We assume two trained and regularized ZF receivers without quantization: one performs signal biasing but the other one does not. The results are shown in Fig. 4. It is interesting to see that the trained ZF with biasing performs better than the trained ZF without biasing. This indicates that, for a linear receiver, adding biases to the received signals is helpful to deal with the nonlinear distortions. However, the trained ZF receiver with biasing still performs much worse than the new ELM based receiver. This indicates that the low resolution ADCs are helpful in dealing with the nonlinear distortion when they are exploited as the ELM activation function. The ADCs mimic a scaled version of typically used activation functions, e.g., tanh, due to the clipping affect.

Figure 5 shows the SER performance of the adaptive receiver with time-varying channels, where a mobile velocity of 100 km/h is assumed. After the initialization with 3000 training symbols, only 300 training symbols in each subsequent frame are used for receiver update. The forgetting factor $\lambda$ is 0.98, and chunk-by-chunk update is used. As a benchmark, we also show the performance of the receiver, where we assume that 3000 training symbols are available for training in each frame. We can see that the adaptive receiver with training length 300 can achieve almost the same performance as that with training length 3000, indicating that training length 300 is sufficient for the receiver to track the channel.

Note that no explicit channel estimation is needed for the new receiver and the ZF receiver with training, which is in contrast to the conventional ZF and MMSE receivers. Batch training of the new ELM receiver and the ZF receiver requires the same complexity of $\mathcal{O}(N^{3}+MN^{2})$. As for detection, the new ELM receiver has the same complexity as the ZF and MMSE receivers, i.e., $\mathcal{O}(N)$ per user.

We have shown that massive MIMO with low resolution ADCs can be treated as a natural ELM where the massive number of antennas serve as the hidden nodes and the ADCs act as the activation function of the ELM. By adding biases to the received signals and optimizing the ELM output weights, the receiver can effectively handle hardware impairments. An adaptive receiver is also designed and its capability of tracking time-varying channels is demonstrated.