A Stacked Autoencoder-based Decode-and-Forward Relay Networks with I/Q Imbalance

Abstract

We propose a stacked autoencoder (AE) that stacks a novel bit-wise denoising AE and a bit-wise AE for decode and forward (DF) relay network impacted by the I/Q imbalance (IQI) at all the nodes. Within the stacked AE framework, we propose block-coded modulation (BCM) and diferential-BCM (d-BCM) designs depending on the availability of the channel state information (CSI) knowledge. Moreover, IQI estimation increases feedback overhead, thus, we design the stacked AE without utilizing the IQI parameters information that can generalize well on varying levels of IQI and signal-to-noise ratio, completely removing the IQI estimation overhead. By extensive evaluation, we show that the proposed stacked AE framework can remove the deteriorating impact of IQI performing similar to ideal relay networks without IQI.

Keywords

• eol>Autoencoder
• block coded modulation
• decode-and-forward
• deep learning
• I/Q imbalance
• relay networks

1. Introduction

symbol-wise AE needs to perform bit-labeling separately by solving a 2! combinatorial problem, bit-wise AE performs automatic bit-labeling possibly in a Gray-coded format [ 5 ], [ 6 ].

A handful works have analyzed AE for DF relaying networks in [ 7, 8, 9, 10 ]. While [ 7, 8, 9 ] consider a symbol-wise AE, [ 10 ] consider bit-wise AE for cooperative non-orthogonal multiple access. All works [ 7 ]–[ 10 ] consider a separate AE for each phase, where the hard decision decoding (HDD) is performed on the soft probabilistic output of the first phase’s AE before passing it as input to the second phase’s AE. Directly, the chances of incorrectly decoding the soft outputs lying close to the hard decision threshold increases, thus biggest disadvantage of conventional DF relay networks, i.e., the problem of error propagation, still remains unsolved in AE works. Further, [ 7 ] proposed a better two-step training policy compared to an iterative two-step training policy in [ 9 ]. Only [ 7, 9 ] considered a symbol-wise AE-based d-BCM design.

In practice, the DF relay networks are compromised by the hardware impairments, e.g., inphase (I) and quadrature-phase (Q) imbalance (IQI), deteriorating the network performance [ 11, 12, 13 ]. All prior works [ 7 ]–[ 10 ] consider an ideal case of I/Q matching, where the signal-tointerference-ratio (SIR) becomes infinity , while [ 11 ] show that even small IQI can deteriorate the SIR. Any IQI compensation algorithm [ 11 ], [ 12 ] requires the IQI parameters estimation [ 13 ], increasing the feedback overhead. However, none of prior signal processing [ 11 ]–[ 13 ] nor AE [ 3 ]–[ 10 ] works have performed BCM/d-BCM without estimating IQI parameters. We summarize our comparison in Table 1. The major contributions of this work are as follows: • We propose stacked AE-based BCM and d-BCM designs for the DF relay network with IQI at all the nodes. We propose a bit-wise AE for the first phase and a novel denoising bit-wise AE for the second phase, where we directly utilize the soft probabilistic outputs as the input of denoising AE. Further, we propose a two-step training, where we propose new training for denoising AE using the input of the first phase’s AE. Thus, even though the AE in first-phase produces erroneous soft-outputs, the denoising AE can denoise these outputs, while encoding-decoding the signal. Thereby, denoising AE helps in correctly decoding the bits close to hard decision threshold, reducing the error propagation. • We propose BCM and d-BCM that remove the necessity of IQI estimation, reducing the feedback overhead. We focus on generalizability, the trained stacked AE can generalize well on any testing IQI and signal-to-noise-ratio (SNR). Under a low SIR regime, we show that stacked AE completely removes the IQI, performing similarly to ideal relay networks. (a) System model. (b) Impact of IQI on the SIR.

2. System Model

Now, we detail the DF relay network with IQI at all the nodes, as shown in Fig. 1a. Each node has a single antenna and the direct link is absent because of large scale shadowing and path-loss. The efective transmission rate = /2 [bits/channel reuse], where bits are transmitted in 2 phases using transmissions. For explanation, we keep = 1.

2.1. Modelling the I/Q Imbalance (IQI)

We can model the IQI efects at the complex local oscillator (LO) signals, operating with angular frequency , at transmitter (Tx) and receiver (Rx) sides as [ 11 ]: () = 1 + 2− , () = 1− + 2 (1) Let { , } and { , } represent the efective amplitude and phase imbalances of the Tx and Rx sides, respectively. Using (1) we can obtain the IQI parameters at Tx and Rx sides as 1 = (1 + )/2, 2 = (1 − − )/2, 1 = (1 + − )/2, 2 = (1 − )/2. In the ideal without IQI scenario, the IQI parameters at Tx and Rx sides becomes = = 1 (or 1 = 1 = 1) and = = 0∘ (or 2 = 2 = 0), respectively.

2.2. Signal Transmission–Reception

We detail the steps for signal transmission–reception between the Tx node Γ = {, } and Rx node ϒ = {, }, where Γ ̸= ϒ, in two phases of DF relay network, as below:

Firstly, the Tx node Γ maps intended bits sΓ ∈ {0, 1} to a complex symbol Γ ∈ C, such that E{|Γ|2} = 1. The up-converted signal in presence of Tx IQI becomes Γ, as where (· )⋆ denotes conjugate operation. Secondly, Tx node Γ transmits signal to Rx node ϒ, as ϒ = √ΓℎΓϒΓ + ϒ, where Γ is Γ’s transmit power, ℎΓϒ ∼ (0, 1) is fading channel between Γ, ϒ, and ϒ ∼ (0, ϒ2) is AWGN at ϒ. Thirdly, the received signal at ϒ with the Rx side IQI, becomes Γ = 1Γ + 2⋆⋆Γ ϒ = 1ϒ + 2ϒ⋆ (2) (3) ϒ = √︀Γ (11ℎΓϒ +22ℎ⋆Γϒ) Γ+√︀Γ (12⋆ℎΓϒ +21⋆ℎ⋆Γϒ) ⋆Γ+1ϒ +2⋆ϒ ⏟ Desired signal,⏞Λ(Γ,ϒ)Γ ⏟ Self-interference sig⏞nal, Ω(Γ,ϒ)⋆Γ ⏟Noise, ˜ϒ⏞(Γ,ϒ) Thus, IQI leads to signal distortion, Λ(Γ, ϒ)Γ, and causes self-interference, Ω(Γ, ϒ)⋆Γ. Fourthly, we apply traditional zero-forcing (ZF)-based IQI compensation at the Rx node as follows (4) (5) [︃ϒϒ⋆ ]︃ = [︂ Λ(Γ, ϒ) Ω(Γ, ϒ)⋆ Ω(Γ, ϒ) ]︂ [︂ Γ]︂ +︂[ 1 Λ(Γ, ϒ)⋆ ⋆Γ 2⋆ 2]︂ [︂ ϒ]︂ 1⋆ ⋆ϒ y = A(Γ, ϒ)xΓ + B(Γ, ϒ)nϒ

ϒ We perform ZF-based IQI compensation to get ^ϒ, as [^ϒ, ^ϒ⋆ ] = (A(Γ, ϒ))− 1 × yϒ. Please note in ZF-based IQI compensation, we know IQI parameters, but in its absence, we = ϒ. Fifthly, we perform maximum likelihood decoding (MLD) as ^sΓ = only have ^ϒ arg min∈ ||^ϒ − √ΓℎΓϒ||2, where denotes all possible symbols and ^sΓ is decoded bits.

2.3. Impact of IQI on DF Relay Networks

Considering there are no noise terms ϒ = 0, the SIR for each phase can be given as SIR (in dB) = E{|Λ(Γ, ϒ)Γ|2} = |1|2|1|2 + |2|2|2|2

E{|Ω(Γ, ϒ)⋆Γ|2} |1|2|2|2 + |2|2|1|2 In Fig. 1b, we analyze the impact of IQI on SIR. In the ideal without IQI scenario, SIR becomes infinity , whereas, even a small phase/amplitude ofset (IQI) can deteriorate SIR significantly.

3. Proposed Stacked AE-based DF Relay Networks with IQI

In this section, we propose a stacked AE-based DF relay network with IQI. We consider each phase in the DF relay network as a separate AE-based transmission because the direct link is absent and the relay node operates in DF mode. For the first phase, we consider a bit-wise AE with its NN encoder at the source node S and its NN decoder at the relay node R. Now, for the ifrst time, we introduce the bit-wise denoising AE, defined as below.

Definition 1. A bit-wise denoising AE is a bit-wise AE with the diference that the input at the NN encoder is the soft probabilistic values lying between [ 0, 1 ] instead of bits {0, 1}. In the second phase, we employ the bit-wise denoising AE with its NN encoder at the relay node R and its NN decoder at the destination node D because the NN decoder (of bit-wise AE) at the relay node R produces soft outputs, which can be directly fed as an input to the denoising AE. Thus, we remove the HDD on the soft outputs of the bit-wise AE in the first phase as [ 7 ]–[ 10 ], which sufers from error propagation because the wrongly decoded bits are fed for re-transmission. Further, the probability of erroneous bit decoding is highest for soft outputs close to HDD threshold due to the ambiguity. Instead, by directly utilizing soft outputs as input to the bit-wise denoising AE, we can remove noise from input and decode soft probabilities close to HDD threshold correctly. Note stacked AE mimics the operations of conventional DF mode that employs decoding and re-encoding, with additional denoising of decoded signal. Thus, processing requirements of stacked AE remains same as conventional DF relay network.

3.1. Designing of the Bit-wise AE for Phase 1

In this work, we utilize dense layers in the NN architectures, where any th dense layer in a NN can be represented as (x) = (Wx + b) where is number of neurons, x ∈ R is input, W ∈ R × +1 is weight matrix between the th and ( + 1)th dense layers, b ∈ R is bias vector, and is activation function. We denote (· ) / as the weight and bias terms of the Tx or Rx at the (· ) node with constituent or , respectively.

The source node S takes bits s ∈ {0, 1} as input and maps to complex symbols x ∈ C using the mapping function, given as

(s, x) = PN ( (...1(s)...)) where PN is the power normalization layer that mandates ||x||22 = . Then, symbol-bysymbol transmission takes place using (2)–(3) with Γ, ϒ = , to obtain symbols y ∈ C at NN decoder of relay node, that obtains soft probabilistic outputs ˜ (s|y) ∈ [ 0, 1 ], using the de-mapping function, given as (y, ˜

(s|y)) = (...1(LL(y))...) where LL denotes the Lambda layer with no trainable NN parameters. (6) (7)

Comments

Training dataset creation parameters

NN

Settings Step decay for learning rate (LR) SNR /0

Phase ofset Amplitude ofset

Optimizer Weight initializer Batch size Initial LR

Drop

Step size Minimum LR

Values

= {3, 8, 13, 23, 33, 43, 53, 63} dB = {25∘ , 35∘ , 40∘ , 45∘ } = {0.4, 0.5, 0.6, 0.7}

Adam

Glorot = 6000 0 = 0.002 = 0.5 = 25 min = 10− 5

3.2. Designing of the Bit-wise Denoising AE for Phase 2

The relay node R takes the soft probabilistic outputs ˜ maps to complex symbols x ∈ C using mapping function, given as (s|y) ∈ [ 0, 1 ] as input and (˜ (s|y), x) = PN( (...1(˜ (s|y))...)) (8) where PN ensures ||x||22 = . Then, symbol-by-symbol transmission takes place using (2)–(3) with Γ, ϒ = , to obtain symbols y ∈ C at the NN decoder of destination node, that obtains soft probabilistic outputs ˜ (|y) ∈ [ 0, 1 ], for all , using de-mapping function, (where LL denotes the Lambda layer), given as (y, ˜ (s|y)) = (...1(LL(y))...) (9)

3.3. Proposed AE-based BCM and d-BCM Designs

We propose stacked AE-based BCM and d-BCM, where for generalizability, we employ same NN architecture for BCM/d-BCM, except Lambda layer LL in the NN decoders is designed as • BCM – Herein, we assume the CSI knowledge and perform channel equalization in

Lambda layer with ℎΓϒ. • d-BCM – Herein, we assume absence of CSI knowledge and employ a radio transformer network (RTN), widely employed to estimate the CSI knowledge [ 3 ]. However, we propose an RTN that also helps in removing the IQI from the received signal at the Rx node ϒ. Remark 1. Unlike the conventional networks performing ZF-based IQI compensation (in Sec. 2.2) using IQI parameters, we do not utilize the IQI parameter information. Thus, removing the feedback overhead for IQI estimation.

3.4. Training of the Proposed Stacked AE

denoted as ℒΓϒ for clarity, solving the multi-label binary classification problem, as Both AEs are optimized by minimizing binary cross-entropy loss, ℒΓϒ(s, ˜ ϒ(s|yϒ)), ℒΓϒ = ∑︁ − (1 − ) log2(˜ ϒ (|yϒ)) − log2(1 − ˜ ϒ (|yϒ)) =1 (10) Note that minimization of (10) only takes place during training (ofline phase), once the stacked AE is trained we can deploy the trained NNs (testing phase). The NN architectures for encoder, decoder, and RTN are generalizable for both the AEs, as shown in Fig. 2. We assume = = and = = . We create a training dataset (using simulations) with 2+2 blocks of data for each combinations of /0, phase ofsets and amplitude ofsets from the sets [, , ] detailed in Table 2. Using this training set we train both the AEs individually by estimating the expected loss in (10) with mini-batch training, using the hyper-parameter settings detailed in Table 2. Specifically, we employ Adam optimizer and Glorot initializer for weight initialization. Dependence on diferent weight initializations is left for future work. We utilize the step-decay method to update the learning rate and reduce overfitting [ 14 ]. The distinct advantages are: • We create a single training dataset such that a single trained stacked AE framework can generalize well for varying levels of testing /0 and IQI. • For the bit-wise denoising AE we utilize the input bits at source node and soft outputs of NN decoder for minimizing the loss in (10), with Γ, ϒ = , , as shown in Fig. 2. Thus, NN decoder of bit-wise denoising AE learns the distribution ˜ (s|y), learning the soft outputs for input bits at source node. If end-to-end training between the input-output of the bit-wise denoising AE have been performed, then the NN decoder of would have learnt the distribution ˜ (˜ (s|y)|y), learning the soft outputs of the NN decoder of bit-wise AE in first phase, propagating the errors made in first phase. During deployment, we can monitor the decoding performance, if it falls below a threshold due to varying environmental conditions, we can re-train the NN using transfer learning [ 14 ].

4. Performance Evaluation

In this section, we evaluate the proposed stacked AE under Rayleigh block fading channels with = 8/7 [bits/channel reuse], where the channel remains constant for = 7 symbols and then changes randomly. For the conventional scenarios, we utilize QPSK (with CSI)/d-QPSK (without CSI) with (7, 4) Hamming codes and consider the following as benchmarks – (1) MLD without any IQI compensation (MLD: No IQIC), (2) MLD with ZF-based IQI compensation (MLD: ZF IQIC), and (3) MLD in ideal relay network without IQI (Ideal MLD).

In Fig. 3a, we compare the proposed staked AE with state-of-the-art AE works in [ 7 ], [ 9 ] for an ideal relay network without IQI because no prior works consider IQI. Also, we can’t compare with [ 10 ] because it considers a NOMA scenario. Proposed stacked AE-based d-BCM design outperforms [ 7 ], [ 9 ] by 2.5, 3.5 dB, showing the merits of proposed stacked AE framework. 10-1 R E B

Proposed AE 2.5 dB and 10-2 3.5 dB better [ 7 ] [ 9 ]

Proposed 0 5 Eb/N100[dB] 15 20 ER10-3 B 10-4 10-5 IMMdLLeaDDl::MZNFLoDIIQQ(nIICCo IQI)

Proposed AE: No IQIC 10-60 10 20 30 40

Eb/N0 [dB] 10-1 10-2 ER10-3 B 10-4 10-5 IMdLeDal:MNLoDIQ(nICo IQI)

Proposed AE: No IQIC 10-60 10 Eb/N0 [dB] 20 30 40 (a) d-BCM - Stacked AE ver- (b) BCM design (with CSI) for (c) d-BCM design (without CSI) sus [ 7 ], [ 9 ]. , = 45∘ , 0.7. for , = 30∘ , 0.8.

In Fig. 3b, 3c, we analyze the stacked AE-based BCM and d-BCM designs for DF relay networks with varying IQI levels. In Fig. 3b, we analyze the BCM design for SIR < 3 dB. In Fig. 3c, we analyze the d-BCM design for SIR < 6 dB. We can see that the MLD with ZF-based IQI compensation (MLD: ZF IQIC) is always able to decode the signals because of the presence of IQI parameters, while MLD without any IQI compensation (MLD: No IQIC) is unable to decode the signals because of absence of IQI parameters. Also, the proposed stacked AE is always able to decode the signal, even without utilizing the IQI parameters information. In fact, stacked AE performs similar to MLD for an ideal relay network without IQI (Ideal MLD: No IQI), indicating stacked AE completely removes the impact of IQI, without utilizing the IQI parameters information (reducing feedback overhead), even under low SIR regimes, due to: • Bit-wise AE in the first phase forms 2 codewords in 2-dimensional space with kurtosis as 1, indicating that spherical codes are formed, which are optimal for small block lengths. Also, it maximizes the minimum Euclidean distance between codewords to 1.5 and 1.2 for BCM and d-BCM designs compared to 1.4 and 0.76 in QPSK and d-QPSK, respectively. • Bit-wise denoising AE in the second phase takes soft probabilistic outputs as input, thus it learns almost a slightly diferent codeword for diferent soft outputs, helping in removing the noise in the input soft probabilistic outputs while decoding the signal at NN decoder.

5. Conclusion

In this work, we propose stacked AE-based BCM and d-BCM designs for the DF relay network with IQI at all the nodes. We propose to employ bit-wise AE in the first phase and a novel bit-wise denoising AE in the second phase, with a new training policy for bit-wise denoising AE. The proposed stacked AE generalizes well on any testing IQI and SNR. Under a low SIR regime, we show that stacked AE performs similar to ideal DF relay network without IQI, even without utilizing the IQI parameters, thereby saving bandwidth and computational resources, highly suitable for IoE applications that mandates low latency. Further, stacked AE can be directly re-utilize for low-density parity-check (LDPC) codes as the outer codes, similar to the work [ 15 ]. This work is supported by the COG-MHEAR: Towards cognitively-inspired 5G IoT enabled, multi-modal Hearing Aids (https://cogmhear.org) under Grant EP/T021063/1.

References

[1] K. Singh , A. Gupta , T. Ratnarajah , M.-L. Ku , A general approach toward green resource allocation in relay-assisted multiuser communication networks , IEEE Transactions on Wireless Communications 17 ( 2018 ) 848 - 862 .

[2] A. Gupta , K. Singh , M. Sellathurai , Time-switching eh-based joint relay selection and resource allocation algorithms for multi-user multi-carrier af relay networks , IEEE Transactions on Green Communications and Networking 3 ( 2019 ) 505 - 522 .

[3] T. O'Shea , J. Hoydis , An introduction to deep learning for the physical layer , IEEE Transactions on Cognitive Communications and Networking 3 ( 2017 ) 563 - 575 .

[4] A. Gupta , M. Sellathurai , End-to-end learning-based two-way af relay networks with i/q imbalance , in: 2021 IEEE 22nd International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) , 2021 , pp. 111 - 115 .

[5] S. Cammerer , et. al., Trainable communication systems: Concepts and prototype , IEEE Transactions on Communications 68 ( 2020 ) 5489 - 5503 .

[6] A. Gupta , M. Sellathurai , End-to-end learning-based framework for amplify-and-forward relay networks , IEEE Access 9 ( 2021 ) 81660 - 81677 .

[7] Y. Lu , et. al., Deep autoencoder learning for relay-assisted cooperative communication systems , IEEE Transactions on Communications 68 ( 2020 ) 5471 - 5488 .

[8] Y. Lu , P. Cheng, Z. Chen , W. H. Mow , Y. Li , A learning approach to cooperative communication system design , in: ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2020 , pp. 5240 - 5244 .

[9] A. Gupta , M. Sellathurai , A stacked-autoencoder based end-to-end learning framework for decode-and-forward relay networks , in: ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2020 , pp. 5245 - 5249 .

[10] Y. Lu , et. al., Deep multi-task learning for cooperative noma: System design and principles , IEEE Journal on Selected Areas in Communications 39 ( 2021 ) 61 - 78 .

[11] A. E. Canbilen , S. S. Ikki , E. Basar , S. S. Gultekin , I. Develi , Impact of i/q imbalance on amplify-and-forward relaying: Optimal detector design and error performance , IEEE Transactions on Communications 67 ( 2019 ) 3154 - 3166 .

[12] Y. Gao , et. al., Performance analysis of dual-hop relaying with i/q imbalance and additive hardware impairment , IEEE Transactions on Vehicular Technology 69 ( 2020 ) 4580 - 4584 .

[13] W. Hou , M. Jiang , Enhanced joint channel and iq imbalance parameter estimation for mobile communications , IEEE Communications Letters 17 ( 2013 ) 1392 - 1395 .

[14] I. Goodfellow , Y. Bengio , A. Courville , Deep Learning, MIT Press, 2016 .

[15] E. Balevi , J. G. Andrews , Autoencoder-based error correction coding for one-bit quantization , IEEE Transactions on Communications 68 ( 2020 ) 3440 - 3451 .