Deep Neural Network Symbol Detection for Millimeter Wave Communications
Yun Liao Nariman Farsad Nir Shlezinger Yonina C. Eldar Andrea J. Goldsmith
eess.SP cs.LG

	
	Synthetic tracking using ZTF Long Dwell Datasets
    Thomas A. Prince
    July 24, 2019
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	 This paper proposes to use a deep neural network (DNN)-based symbol detector for mmWave systems such that CSI acquisition can be bypassed. In particular, we consider a sliding bidirectional recurrent neural network (BRNN) architecture that is suitable for the long memory length of typical mmWave channels. The performance of the DNN detector is evaluated in comparison to that of the Viterbi detector. The results show that the performance of the DNN detector is close to that of the optimal Viterbi detector with perfect CSI, and that it outperforms the Viterbi algorithm with CSI estimation error. Further experiments show that the DNN detector is robust to a wide range of noise levels and varying channel conditions, and that a pretrained detector can be reliably applied to  different mmWave channel realizations with minimal overhead. 
	 
	

	
	
	
	
	

§ INTRODUCTION

	
	TODO Yun and Nir. I recommend as follows:
	
		
  * Paragraph on motivation for using dnn for symbol detection.
		
  * Paragraph on millimeter wave communications and its challenges (from a high level perspective).
		
  * Paragraph surveying previous works on ml for symbol detection and channel decoding. You can base that on Nariman's work <cit.> and the ViterbiNet paper <cit.>. Emphasize what is missing in the previous works in the context of millimeter wave communications.
		
  * In this work paragraph, where you explain what we propose in this work.
		
  * Paragraph presenting a glimpse into the main results (including numerical results) of this work.
	
	
	Do not have the page limitation on your mind. Once we have a paper written, we can usually compress it into five pages without removing much content.
	
	As one of the important techniques being considered for next-generation wireless communications, communication systems designed to operate in the mmw frequency bands have attracted significant attention within the academic and industrial communities. These mmw frequency bands offer less congestion than the conventional sub-6GHz bands, while providing multi-gigahertz channel bandwidths with commensurately high data rates <cit.>. In order for mmw receivers to reliably decode the transmitted signals, it is critical for them to have accurate knowledge of the underlying channel conditions, namely, to obtain csi. However, for mmw communications, acquiring accurate csi can induce significant overhead as mmw channels tend to have very long memory <cit.>. In particular, this overhead grows dramatically when the communicating nodes are equipped with large antenna arrays, as is often deployed in mmw systems to mitigate the large path loss at these frequencies. 
	
	Recently, dnn have shown great potential in addressing data detection in wireless channels due to their ability to disentangle the complicated relationship between the channel inputs and outputs in a data-driven fashion, i.e., without explicitly using a channel model. For example, in <cit.>, deep learning was used in decoding linear codes. An end-to-end communication system was trained to jointly optimize encoding and decoding in memoryless channels in <cit.>. We note that such an end-to-end approach, in which the channel is treated as an intermediate layer, does not easily extend to mulit-user networks and to channels with memory, as encountered in practical wireless setups. The work <cit.> used dnn to learn the weights in the Viterbi algorithm. The computational complexity required to learn these unknown channel conditions using the DNN proposed in <cit.> grows exponentially with the channel memory, thus limiting the feasibility of this technique for equalization of channels with long memory. In <cit.>, the authors proposed a symbol detector based on rnn, and in particular, on a sliding brnn architecture, for scalar channels, focusing on  molecular  communication with a relatively low data rate. For mmw communications,  dnn have  been utilized in non decoding-related applications. In <cit.>, the authors focused on estimating the key characteristics of mmw channels with machine learning. The work <cit.> used dnns to find hybrid precoding schemes for mmw massive MIMO. In <cit.>, the beam management problem in dense mmw networks was addressed by deep learning methods.
	
	In this work, we consider using dnn for symbol detection in mmw communications, bypassing the need to explicitly maintain a channel model with accurate csi. In particular, our deep symbol detector extends the sliding brnn architecture proposed in <cit.> to mimo channels and adapts it to mmw systems. We demonstrate that using this dnn architecture, it is possible to train a detector that can accurately retrieve the symbols without any prior knowledge of the channel. We also show that the dnn decoder generalizes well to different snr levels and to small variations in the cir. In particular, the sliding brnn detector demonstrates improved resiliency to inaccurate training compared to csi-dependent detectors such as the Viterbi detector with the same level of csi inaccuracy.
	
	
	
 	The rest of this paper is organized as follows: In Section <ref> we discuss the mmw channel model and formulate the symbol detection problem. Section <ref>  presents the proposed dnn-based symbol detector. Section <ref> details numerical training and performance results of the proposed detector, and Section <ref> provides concluding remarks. 
 	
 	
 	
 	Throughout the paper, we use boldface lower-case letters to denote vectors, e.g., x;
 	the ith element of x is written as ( x) _i. 

Sets are denoted by calligraphic letters, e.g., X, and specifically, 
 	Z is the set of integers and 
 	 C is the set of complex numbers.


 	
 	
	
	
	
	
	

§ PROBLEM FORMULATION

	
	
	 In order to formulate the mmw communications setup, we first describe the overall wireless communication system model in Subsection <ref>, and then elaborate on the specific mmw channel model in Subsection <ref>.

	  
	
	
	
	  

 §.§ System Model

	  
	  
	  
	  
	  We consider a single-cell multi-user uplink narrowband mmw network, in which a bs equipped with  antennas serves Ν single-antenna ut as illustrated in Fig. <ref>. 
	  The signal transmitted by the kth ut, k ∈{1,2,…,Ν}≜, propagates through a multipath channel with  taps, represented by the vectors _k[l] ∈C^, where l ∈{0,1,…,-1}. 
	  We use [i] ∈C^ to denote the additive noise at time instance i, modeled as an i.i.d. process of zero-mean proper-complex Gaussian random vectors with covariance matrix σ^2 I_. 
	  Let _k[i] be the signal transmitted by the kth user at time instance i, assumed to be a digitally modulated symbol of alphabet size > 1. The channel output observed by the bs, denoted [i]∈C^, can be written as 
	  
    [i] = ∑_k=1^Ν∑_l=0^-1_k[l]_k[i-l] + [i].

	  
	  Our proposed dnn-based receiver recovers the transmitted symbols {_k[i]} from the channel outputs {[i]}. This receiver, detailed in Section <ref>, is designed to detect the symbols transmitted by a single ut, thus implementing separate decoding, which is a common detection strategy in large-scale mimo communications <cit.>. We leave the implementation of a deep joint detection mechanism, which is known to be superior to separate decoding for such multiple access channels <cit.>, for future investigation.
	  
	  Before presenting the proposed receiver architecture, we first elaborate on the channel model, namely, the model of the vectors {_k[l]} which arises in mmw communications, in the following subsection. 
	  







	  Nir comments: Yun, please note that following modification I added here:
	  
	      
  * Do not forget the presence of interference. There is more than a single ut in each cell. While currently we do not really consider the presence of interference in the simulations at some point we will have to mention it. 
	      
  * Some recommended latex coding habits: 
	      
	          
  * Have all your symbols defined as newcommands, and not in the main body. Otherwise it becomes quite challenging to switch symbols in an advanced part of the work.
	            
  * Avoid using non-conventional operations (like convolution between vectors).
	            
  * Try to define the notations you are using before they first appear. 
	            
  * Add label to each equation and subsection. There is a good chance you will wish to refer to them at some point
	      
	  
	  

	
	
	  

 §.§ Millimeter Wave Channel Model

	  
	  
	  
	  We adopt the NYU omnidirectional geometric mmw channel model proposed in <cit.>. Another widely used mmw channel model is given in <cit.>. The main difference between these two models is in the choice of some of the parameters, so that our proposed design can also be applied under the model of <cit.>. 
	  
	  The channel model given in <cit.> has a sparse time cluster structure: a mmw channel typically exhibits no more than 6 time clusters, with relatively little delay/angle spreading within each cluster. To formulate the model, let  be the number of time clusters. The c-th time cluster contains  subpaths with gains {a_m,c,n^2}, phases {ϕ_m,c,n}, and delays {τ_m,c,n}, m = 1,2,…, M_c, n = 1,2,…,. For each n∈{1,2,…,}, the nth entry of the resulting channel vector [l] can be written as
	  
    ([l])_n = ∑_c = 1^∑_m = 1^ a_m,c,n e^j ϕ_m,c,n p(lT_s - τ_m,c,n),

	  where p(τ) represents the pulse shaping function evaluated at time τ, and T_s is the time interval between two symbols. 
	  
	 Is B the bandwidth? Is T_s the sampling period? Do they have to be related via  T_s=1/B or does (<ref>) holds for any channel model? And how come the gains / phases / delays do not depend on the antenna index n (which you originally denoted n_r?  
	 The resulting channel tends to exhibit a very long memory, typically several hundred taps. This makes maximum likelihood based algorithms, as well the Viterbi detection algorithm, prohibitively complicated, and, since the csi acquisition is more likely to be imprecise in this case when limited channel estimation overhead is allowed, these detection algorithms are more prone to errors. In this sense, an efficient data-driven symbol detection scheme, as proposed in the following section, is highly desirable.
	  
	  Here we should detail the millimeter wave channel models, introducing the mathematical notations.
	  Explain that this is a multi-user uplink setup, and illustrate that we propose to utilize a dnn for recovering the message of each user separately, leaving a joint deep detector for future investigation. 
	   It would be great to have a figure illustrating the considered input-output relationship. Do not forget to discuss the unique characteristics of such channels. Conclude with the need to implement efficient data-driven symbol detectors for the considered channel, paving the way for the next section.
	   

	 
	 
	
	
	
	
	

§ DEEP SYMBOL DETECTION

	
	
	In this section, we present the architecture of the proposed dnn detector for recovering uplink messages.
	
	
	
	
	

 §.§ DNN Architecture

	
	

	 First elaborate on the model, emphasize in what sense is it different from <cit.>. Adding a diagram of the dnn architecture can be helpful.
	 (We may want to move this paragraph to the Introduction section or the discussion subsection III.B. I feel it might be better to discuss the reason why we use sliding BRNN before describing the detailed architecture).
	 Our proposed dnn architecture is based on rnn, and specifically on an extension of the sliding brnn strategy considered in <cit.>. The motivation for utilizing this structure stems from the fact that rnn are known to be capable of exploiting temporal correlation among time sequences. However, conventional rnn only take the current input and the state, which represents the history, to make predictions. This can be far from optimal in mmw symbol detection where the first tap of the cir may not be dominant compared to the subsequent taps, especially in non line of sight scenarios. A brnn, which takes information from both the history and the future into account in making predictions, helps to improve the performance in this case. The brnn scheme, however, also has some drawbacks: the detection algorithm needs to wait until the entire signal is received. To overcome this issue, we adopt the sliding brnn architecture, proposed in <cit.> for scalar molecular channels, for symbol detection in multiple-antenna mmw communications.
	 
	 
	 
	 The detection scheme for recovering the symbols of a single ut is illustrated in Fig. <ref>. When the first  symbols arrive at the bs, the brnn takes the N_r length- sequences from all receive antennas and produces a soft prediction on the symbols (x_1 … x_) sent by the ut. When the next symbol arrives, the brnn slides one symbol ahead, and returns the soft prediction on (x_2, …, x_+1). The procedure is repeated until the entire signal is received. Let  𝒥_k = {j|k ∈ [j, j+-1]} be the set of all valid starting positions of the sliding brnn such that the prediction range covers x_k. Let p_k^(j) be the estimated pmf for x_k when the starting position of the brnn is j.  Note that each p_k^(j) is an × 1 vector. These initial pmf are used to obtain the final pmf of x_k, which is estimated by averaging over 𝒥_k, i.e.,
	 
    p_k = 1/|𝒥_k|∑_j ∈𝒥_kp_k^(j).

	 
	 Fig. <ref> shows the architecture of each brnn adopted in this work. In particular, lstm units with normalized outputs are used as the basic units in the brnn. The input y^(i) = (y_1^(i), y_2^(i), …, y_N_r^(i)) to each lstm in the first layer is a N_r × 1 vector containing the received signal at time step i at all receive antennas. Then, three layers of the bidirectional lstms are stacked. The design of the output layer depends on the modulation scheme. When a modulation scheme with >2 symbols is used by the ut, the output layer has output size  with softmax activation to estimate the pmf of the symbols. When a binary constellation, i.e., BPSK, is adopted, we simplify the output layer by using the sigmoid activation function with a scalar output, which represents the estimate of the probability that the symbol is the BSPK symbol +1.
	 
	 
	
	Note that in this work, we do not include channel coding schemes that help recover the original message. The focus of this considered receiver architecture is the reliable detection of the received symbols over mmw channels. Since the detector  estimates the pmf of the symbol sequence, it is straightforward to add channel coding at the transmitter and then feed the estimated pmf to the associated soft channel decoding algorithms in the receiver.
	
	
	
	
	
	

 §.§ Discussion

	
	
	Here you need to discuss the expected pros and cons of the proposed receiver. 
	Explain why its architecture is suitable for millimeter wave bs. 
	Explain in what sense is it different from the deep detector of <cit.>. 
	Explain that we propose to use a dedicated network per each user thus implementing separate decoding, which is a common approach in massive MIMO communications (you can cite here <cit.> and <cit.>. You can also say that we consider the implementation of joint-decoding scheme as a topic for future investigation. 
	
	Maybe the first paragraph of III.A is exactly what you expect here?
	The proposed sliding brnn detector has several main advantages over model-based detectors for mmw systems. First, being a data-driven detector, it requires no prior csi at the bs, thus avoiding the difficult task of accurately estimating the exact mmw cir. Since the detector learns the channel conditions from the training data, there is no need to do channel estimation separately. As we show in the numerical results in Section <ref>, the sliding brnn detector is quite robust to inaccurate training. In particular, it is demonstrated that a sliding brnn detector trained using samples acquired from a set of channel realizations generated according to (<ref>) generalizes well to a different mmw channel realization.
	
	Furthermore, the sliding detector structure with a moderate window length also allows for near real-time detection. 
	In particular, the proposed detector is capable of recovering the symbol transmitted at time index i once the channel output y[i+-1] is received, i.e., a decoding delay of merely -1 samples. To see this, we note that after receiving -1 consecutive channel outputs, the detector starts to provide the pmf estimations up to the current symbol as soon as the th related channel output is obtained.   This allows the detector to recover the symbols  without having to wait for the entire block of channel outputs to be received.
	
	
   
   Finally, a notable advantage of the sliding brnn architecture compared to model-based approaches stems from the fact that, once trained, the decoding can be carried out significantly faster than iterative model-based approaches, and that this delay hardly increases as the number of antennas  grows. This gain is illustrated Table <ref>, which shows the average running time of detecting one 200-bit message using the sliding brnn detector and the Viterbi detector measured on an AMD Phenom(tm) II X6 1045T 2.70 GHz Processor. This makes the sliding brnn a promising decoding approach for mmw receivers, which typically utilize massive mimo arrays.
   
   Yun - I do not understand the following discussion. Why does increasing the number of antennas affect Viterbi? its complexity should be determined by the memory length and the input alphabet, not by the dimensionality of the output. What Viterbi does is compute the likelihood function p_X|Y(X|Y) for each possible x, the dimensionality of Y does not affect the complexity here. Can you please explain where does this gain come from? I also recommend that if you use O notations you either prove it or cite a reference. Additionally, I do not think that the SBRNN has a concrete advantage over Viterbi in real-time decoding, since Viterbi also does not need the entire block for detection but only has a delay which is equal to the memory size. So I recommend not to mention this as an advantage of our detector.
   
   
   
	Let H be the hidden size of each gru cell. The computational complexity of the sliding brnn detector to detect a message of length  is 𝒪(( + H + )H ). Also, it turns out that the sliding brnn detector is able to maintain a satisfying performance with a small constant window size  and a moderate constant H even when  increases. Therefore, the complexity of the sliding brnn to detect a message only scales linearly with the number of receive antennas and the message length, which is independent of the memory length of the channel. In comparison, the computational complexity of the Viterbi detector with beam search is 𝒪( X), where X is the number of most likely paths to maintain in the trellis search. Besides, to represent the states in the trellis, the space complexity of the Viterbi detector also scales up with the channel memory length. Table <ref> shows the average running time of detecting one 200-bit message using the sliding brnn detector and the Viterbi detector measured on AMD Phenom(tm) II X6 1045T 2.70 GHz Processor. Clearly, the sliding brnn is significantly more computational efficient than the Viterbi detector. The efficiency gap is even larger when the number of receive antenna increases.
	
   
   
   
	
	
	
	
	

§ NUMERICAL STUDY

	
	
	
	
   
   
   In this section, we numerically evaluate the performance of our dnn-based symbol detector in terms of the detection accuracy and the convergence speed in training. The main characteristics of the simulated environment in this section are listed in Table <ref>. The parameters C, M_c, {a_m,c,n}, {ϕ_m,c,n}, τ_m,c,n in (<ref>) are generated according to the model suggested in <cit.> under the parameter settings of Table  <ref>. For the sliding brnn detector, the hidden size of each lstm unit is 20, and the window length  is set to 30. For the Viterbi detector, we use beam-search <cit.> with 300 survivor paths.
   
   
   
   
   	
	
	
	

 §.§ Error Rate Comparison

	
	
	
	
	In this subsection, the performance of the sliding brnn detector is compared with that of the Viterbi algorithm with either perfect or imperfect csi in terms of ser. In the simulations, BPSK modulation is used. The number of receive antennas is fixed to = 4. For each snr value, the ser is averaged over 5 different independently generated mmw channel realizations, and 2 × 10^5 symbols, divided into blocks of 200 symbols, are detected for each channel realization. 
	
	
	
	We first evaluate the ser performance of  the sliding brnn detector compared to the Viterbi detector with perfect csi. The resulting ser values versus snr are depicted in Fig. <ref>. 
	
	The dashed curve in Fig. <ref> is the error rate of the optimal Viterbi detector in which the detector has perfect csi and waits to receive the entire block with the tail before detecting the symbols. This receiver implements maximum-likelihood detection for the mmw channel. Since the sliding brnn detector gives the detection results as soon as it receives the last symbol in the message, we also evaluate the performance of the Viterbi detector that only considers a finite window of the received signal, whose window size equals the message length.  The performance of the Viterbi detector in this case is shown by the dot dashed curve with the legend “Viterbi-cut”. 
	
	The solid curve with cross markers in Fig. <ref> shows the error rate of the sliding brnn detector that is trained and evaluated at the same snr level under the same channel realization. The training set for each snr contains 4,000 training samples, and each training sample is a tuple of a 200-symbol transmitted block {x[i]} and the corresponding channel outputs {y[i]}. It is clear that the performance of the sliding brnn detector is close to that of the optimal Viterbi detector when the snr is low. Note that the sliding brnn detector only takes the received signal in a limited window for prediction, while the optimal Viterbi detector utilizes the entire received signal. When taking in the same amount of signal, the sliding brnn detector outperforms the Viterbi-cut detector over a large range of snr levels. This demonstrates the ability of the data-driven sliding brnn receiver to reliably detect the transmitted symbols without requiring prior knowledge of the csi.
	
	To represent practical setups in which the exact value of the snr is not known during training, we also depict in Fig. <ref> the ser performance of the sliding brnn receiver when trained using training samples corresponding to several noise levels. The legends of the curves in Fig. <ref> show the range of snr in the training signal streams. In particular, to generate the training set, we randomly pick an snr value uniformly from the given range as the snr for each block. After training, we use the same sliding brnn to detect signal streams under different snr levels. As shown in Fig. <ref>, the performance of the sliding brnn decoders remain within a small gap from the Viterbi detector. 
	It is also interesting that the sliding brnn trained across different snr ranges yield similar performance. Evidently, the trained sliding brnn generalizes well to a wide range of snr. These observations indicate that it suffices to train a sliding brnn under a single snr and use it regardless of the change to the snr, resulting in a robust symbol detection scheme in time-varying snr environments.
	
	
	
	
	Next, we numerically evaluate the generalization capability of the sliding brnn under csi uncertainty. 
	In particular,  we consider the case where the csi estimate is imprecise or the channel varies slightly over time. With this aim, we evaluate the ser of the sliding brnn detector on a mmw channel realization whose cir varies by 2.5% or 1.0% compared to the cir used during training. Specifically, the sliding brnn is trained under one particular channel realization and a 0 dB snr. Then, in evaluation, we randomly distort the amplitude at each tap {a_m,c,n} by 2.5% or 1.0% on average for each message block compared to the channel used during training. For comparison, we use the Viterbi detector with the same level of csi error. The results of this simulation are depicted in Fig. <ref>, where it is observed that the sliding brnn detector consistently outperforms the Viterbi detector with imperfect instantaneous csi, and that the performance of the sliding brnn remains close to the optimal Viterbi detector with perfect csi. The result shows that the sliding brnn detector is able to generalize well to not only the various noise levels, but also to different channel realizations.
	
	
	
	The numerical evaluations presented in Figs. <ref>-<ref> focused on a bs with only 4 receive antennas. In order to demonstrate that the gains of the sliding brnn receiver are also maintained for large antenna arrays, which are commonly utilized in mmw communications, we depict in Fig. <ref>  the ser curves of the sliding brnn detector when the number of receive antennas is =128. While we do not increase the size of the neural network or the window length, the performance of the sliding brnn detector is still close to the Viterbi detector, while the running time is significantly reduced, as discussed in Subsection <ref>. Similarly to the results with fewer receive antennas presented in Fig. <ref>, a single sliding brnn detector trained under a wide range of snr generalizes well to different noise levels. These results demonstrate the great potential of using a data-driven dnn-based detector for mmw communications.
	
	
	
	
	
	

 §.§ Training Size Analysis

	
	
	
	
	
    Next, we numerically study the convergence speed in training the sliding brnn detector.
 
   
   
   
   
   Fig. <ref> shows the convergence speed of the brnn in the massive SIMO case. It could be seen that 4,000 blocks of messages are sufficient to train the network from scratch.
   
   
   

  
   To that aim, we depict in  Fig. <ref> the convergence time of the sliding brnn detector. We consider two different initial values for the network weights: in the first case (filled bars in Fig. <ref>) the detector was already trained under some channel, and needs to be re-trained for a different channel realization. This simulates the practical scenario where the channel changes over time, and the detector needs to be adjusted to track the channel periodically. In the second case (slashed bars in Fig. <ref>) the weights in the sliding brnn are randomly initialized, representing the scenario where the network is trained from scratch. In particular, 9 different channel realizations are generated independently, indexed 0,…,8, and a sliding brnn symbol detector trained on the last realization is utilized as the pretrained model. For each of the other 8 channel realizations, we train the sliding brnn detector either from scratch or from the pretrained model with the same learning rate. Fig. <ref> shows the number of training samples required to achieve at least 90% detection accuracy. The channels are sorted by the number of the required samples for training from a pretrained model to illustrate the distribution of the required number. Observing Fig. <ref>, we note that it takes many fewer samples for the sliding brnn detector to be adapted to a new channel compared to training it from scratch. In particular, for some channel realizations, such as channels with indices 0, 1, and 2 in the figure, it takes less than 150 samples for the detector to adapt to the new channel. This study indicates that a previously trained sliding brnn architecture can be adjusted to time-varying channel conditions with minimal overhead for re-training the network.


	
	
	

	

§ CONCLUSIONS

	
	
	This paper studied a dnn-based symbol detector for uplink mmw communications, which does not require csi and learns the decoding mapping in a data-driven fashion. Our proposed receiver was based on a sliding brnn architecture, which is suitable for the long channel memory and large number of receive antennas commonly used in mmw systems. Numerical evaluations demonstrated that the sliding brnn detector is capable of achieving performance within a small gap from the optimal Viterbi detector, which requires full csi. Moreover, the sliding brnn detector was shown to generalize well when trained using samples taken from an inaccurate model, and in fact it outperformed the Viterbi detector with the same level of csi uncertainty. Finally, we numerically showed that a pretrained dnn detector could be adjusted to another channel realization with minimal overhead. 
 
	
	
	
	
	

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