Deep Learning Phase Compression for MIMO CSI Feedback by Exploiting FDD Channel Reciprocity

We now present a new DLN called DualNet-MP. As shown in Fig. 2, DualNet-MP splits each complex CSI matrix into

where $\odot$ represents Hadamard product. Denote the $(m,n)$-th entry of $\mathbf{H}$ as $\mathbf{H}_{m,n}=|\mathbf{H}_{m,n}|e^{j\measuredangle{\mathbf{H}_{m,n}}}$. The magnitude matrix $|\mathbf{H}|$ and consists of entries $|\mathbf{H}_{m,n}|$ and phase matrix $e^{j\measuredangle{\mathbf{H}}}$ consists of entries $e^{j\measuredangle{\mathbf{H}_{m,n}}}$.

Similar to [5], we forward the CSI magnitudes to the magnitude encoder network, including four $7\times 7$ circular convolutional layers with 16, 8, 4, 2 and 1 channels and activation functions. Given the circular characteristic of CSI matrices, we introduce circular convolutional layers to replace the traditional linear ones. Subsequently, a fully connected (FC) layer with $\left\lceil{\text{CR}_{\text{MAG}}Q_{t}N_{b}}\right\rceil$ elements is included for dimension reduction after reshaping. $\text{CR}_{\text{MAG}}$ denotes the magnitude compression ratio. The output of the FC layer is then fed into the quantization module, called the sum-of-sigmoid (SSQ) [5] to generate magnitude codewords for feedback.

At the gNB, a magnitude decoder uses magnitude codewords and available UL CSI magnitudes²²2 UL CSI is estimated at the gNB and assumed to be perfectly estimated. as AMI to jointly recover DL CSI magnitudes. The DL magnitude branch is first optimized by updating the network parameters $\Theta_{\text{MAG}}$

to minimize the MSE of recovered MIMO CSI magnitude

in which $\Theta$ denotes DLN parameters and subscripts en, de, UL, and MAG of the $f(\cdot)$ denote the encoder, decoder, UL, and magnitude branch, respectively.

For CSI recovery of MIMO channels, we are only interested in their wrapped phases (i.e., $\measuredangle{\mathbf{H}}$). There is 1-to-1 relationship between a phase value $\phi$ and $\left(\cos(\phi),\mbox{sign}[\sin(\phi)]\right)$. For these reasons, we propose to form a ”cosine” matrix whose entries are cosines of entries from $\mathbf{H}$ denoted by

Thus $(\mathbf{Cos},\mathbf{A})$ uniquely determines $\measuredangle{\mathbf{H}}$.

Since $\mathbf{Cos}$ matrix is real, we can adopt a phase encoder similar to the magnitude encoder. Let $\text{CR}_{\text{PHA}}$ denotes the phase compression ratio. Each $\mathbf{Cos}$ generates a $\lceil\text{CR}_{\text{PHA}}Q_{t}N_{b}\rceil$-element codeword. Our DLN uses tanh activation function in each circular convolutional layer of the phase encoder to capture the underlying features of significant phases associated with large magnitudes. Upon completion of encoder training, the UE processes each CSI $\mathbf{H}$, and feeds back the CSI magnitude codeword $f_{\text{en,MAG}}(|\mathbf{H}|,\Theta_{\text{en,MAG}})$, the phase codeword $f_{\text{en,PHA}}(\mathbf{Cos},\Theta_{\text{en,PHA}})$ and the sign matrix $\mathbf{A}$ to gNB.

At the gNB receiver, the phase codeword $f_{\text{en,PHA}}(\mathbf{Cos},\Theta_{\text{en,PHA}})$ and the feedback sign matrix $\mathbf{A}$ are sent to the phase decoder with the tanh activation function as the last layer to constrain the entries of DL CSI cosine matrix $\widehat{\mathbf{Cos}}$ within $[-1,1]$. The magnitude codeword and side information are used by the magnitude decoder to obtain an estimated CSI magnitude matrix $|\widehat{\mathbf{H}}|$. Based on the relationship $\sin(\phi)=\mbox{sign}(\sin[\phi])\sqrt{1-\cos^{2}(\phi)}$, we form

Therefore, we can directly generate a preliminary CSI estimate

from locally available $\widehat{\mathbf{Cos}}$, $\mathbf{A}$, $\widehat{|\mathbf{H}|}$. The combining network is trainable and can include two residual blocks containing four circular convolutional layers to refine the DL CSI matrix.

For end-to-end optimization, our training criterion relies on

to optimize the parameters $\Theta_{\text{en,PHA}}$ of phase encoder $f_{\text{en,PHA}}$ and parameters $\Theta_{\text{de,PHA}}$ of phase decoder $f_{\text{en,PHA}}$ to estimate

Using the same loss function (8), we also train the combining network $f_{\text{C}}$ by optimizing parameters $\Theta_{\text{C}}$ to generate

Since the training of the magnitude learning branch can be decoupled, our framework optimizes the entire architecture by minimizing the overall CSI MSE of (8). It is possible, however, to also partially incorporate the MSE of (8) to further refine the magnitude DLN branch by adopting a slower learning rate.

Considering the MSE loss function, it may be intuitive to simply rewrite the loss function as follows:

This means that $|\mathbf{H}|$ and $\measuredangle{\mathbf{H}}$ are used as encoder network input variables whereas their estimates are the decoder network output variables. However, the presence of infinitely many and shallow local minima of sinusoidal functions $\sin(\cdot)$ and $\cos(\cdot)$ often lead to training difficulties [9]. To overcome this problem, the authors in [7] recently proposed a weighted MDPP loss function

which still uses $|\mathbf{H}|$ and $\measuredangle{\mathbf{H}}$ as input and output variables. where $\measuredangle{\mathbf{H}}$ and $\measuredangle{\mathbf{\hat{H}}}$ denote the true and estimated phases, respectively. By weighting the original phase discrepancy with the true CSI magnitude, this new loss function helps capture the underlying features of the critical phases associated with CSI coefficents with dominant magnitudes. However, the loss function is not equivalent to our final goal for minimizing MSE of DL CSI. We now propose a reparamterization of the same MSE loss function during training. Instead of changing the loss function, we can overcome the training problem of directly parameterization in Eq. (17). Instead, recognizing that only the wrapped phases of $\measuredangle{\mathbf{H}}$ are of interest, we replace $\measuredangle{\mathbf{H}}$ with $\mathbf{Cos}$ and $\mathbf{A}$ via the following reparameterization:

where we have used the sign matrix $\mathbf{A}$ feedback to generate

This formulation saves about half the bandwidth by sending the sign matrix $\mathbf{A}$ without encoding matrix $\mathbf{Sin}$.

Moreover, the sparsity of $\mathbf{H}$ means that we only need to feed back partial entries of $\mathbf{A}$ associated with a swath of entries with dominant magnitudes. If we define a reduction ratio $R_{\text{s}}$ to further reduce feedback overhead³³3Usually, the reconstruction performance can remain approximately the same even if the sign ratio $R_{\text{s}}$ is less than $0.25$ due to the sparsity.. The total phase feedback overhead (in bits) is summarized as follows:

where $K_{\text{PHA}}$ denotes the number of encoding bits for each entry of the compressed cosine matrix $f_{\text{en,PHA}}(\mathbf{Cos},\Theta_{\text{en,PHA}})$.

To summarize our training strategy of DualNet-MP, we use Eq. (5) as the loss function during the first training stage. In the second training stage, we used Eqs. (• ‣ IV-A) as the loss function to build an end-to-end learning architecture.

In our experiments⁴⁴4The source codes are online available in https://github.com/max821002/DualNet-MP/, we let the UL and DL bandwidths be 20 MHz and the subcarrier number be $N_{f}$ = 1024. We consider both indoor and outdoor cases. We place the gNB with a height of 20 m at the center of a circular cell coverage with a radius of 20 m for indoor and 200 m for outdoor. The number of gNB antennas is $N_{b}=32$ whereas each UE has a single antenna. A half-wavelength inter-antenna spacing is considered. For each trained model, the number of epochs and batch size were set to 1,000 and 200, respectively. We generate two datasets each containing 100,000 random channels for both indoor and outdoor cases from two different channel models. 60,000 and 20,000 random channels are for training and validation. The remaining 20,000 random channels are test data for performance evaluation.

In the first (indoor) dataset, we used the COST 2100 [10] simulator and select the scenario IndoorHall at 5GHz to generate indoor channels at 5.1-GHz UL and 5.3-GHz DL with LOS paths. The antenna and band types are set as MIMO VLA omni and wideband, respectively. As for the outdoor dataset, we utilized QuaDRiGa simulator [11] with the scenario features of 3GPP 38.901 UMi. We considered the UMi scenario at 2 and 2.1 GHz of UL and DL bands, respectively, without LOS paths. The number of cluster paths was set as 13. The antenna type is set to omni. By DFT/IDFT operations and truncating described in Eqs. (3) and (4) of the revised manuscript, the resulting $32\times 32$ UL/DL CSI matrices in DA domain are obtained and be fed as the encoder input after proper processes.

The performance metric is the normalized MSE

where the number $D$ and subscript $d$ denote the total number and index of channel realizations, respectively. Instead of evaluating the estimated DL CSI matrix $\widehat{\mathbf{H}}_{\text{DL}}$, we evaluate the estimated SFCSI matrix $\widehat{\mathbf{H}}_{\text{DL}}^{\text{SF}}$ that can be obtained by reversing the Fourier processing and padding zero matrix. Note that this NMSE includes both the errors caused by truncation at the encoder and the overall recovery error. Thus, it is practically more meaningful.

In the following section, we evaluate the performance of CSI recovery by adopting the proposed optimization method and encoder/decoder architecture. Thus, we trained DualNet-MP with the same core network design for magnitude recovery. However, we test different methods to reconstruct the CSI phases for two phase compression ratios of $\text{CR}_{\text{PHA}}=1/8$ and $1/16$ ⁵⁵5All alternate approaches consume $1.2$ and $0.625$ bits/phase entry:

• •

  SMAPE: the network architecture follows DualNet-MP as illustrated in Fig. 2 of the revised manuscript. The sign ratio $R_{\text{s}}$ varies between $[0.25,0.125]$ and we use $K_{\text{PHA}}=8$ bits for both $\text{CR}_{\text{PHA}}=\{1/8,1/16\}$. The loss function for phase reconstruction is given as follows:

  	$\displaystyle\text{MSE}_{\text{SMAPE}}(\widehat{|\mathbf{H}|},\widehat{\mathbf{Cos}},\mathbf{A})$	$\displaystyle=\mathinner{\!\left\lVert\mathbf{H}-\mathbf{\hat{H}}\right\rVert}^{2}_{\text{F}}$
  		$\displaystyle=\mathinner{\!\left\lVert|\mathbf{H}|\odot\mathbf{Cos}-|\hat{\mathbf{H}}|\odot\widehat{\mathbf{Cos}}\right\rVert}^{2}_{\text{F}}$
  		$\displaystyle+\mathinner{\!\left\lVert|\mathbf{H}|\odot\mathbf{Sin}-|\hat{\mathbf{H}}|\odot\widehat{\mathbf{Sin}}\right\rVert}^{2}_{\text{F}}.$

• •

  MDPQ [4]: the design assigns more encoding bits to encode the significant phases according the corresponding magnitudes. It assigns $[0,0,0,3,7]$ and $[0,0,0,0,5]$ bits for $\text{CR}_{\text{PHA}}=[1/8,1/16]$, respectively, to encode the CSI phases corresponding to $[0,0.5,0.7,0.8,0.9]$ of the cumulative distribution of CSI magnitude.

• •

  MSE₀: instead of cosine, CSI phases are fed directly to the phase encoder. Both cosine and sine functions are appended as the final layer of the phase decoder to transform the decoded phase into cosine and sine, respectively, to recover the real and imaginary parts of CSI. We set $K_{\text{PHA}}$ to $8$ bits. The loss function for phase reconstruction is given as follows:

  		$\displaystyle\mbox{Loss}_{\text{0}}=\text{MSE}_{\text{CSI}}(\widehat{|\mathbf{H}|},\measuredangle{\hat{\mathbf{H}}})=\mathinner{\!\left\lVert\mathbf{H}-\mathbf{\hat{H}}\right\rVert}^{2}_{\text{F}}$		(17)
  		$\displaystyle=\mathinner{\!\left\lVert|\mathbf{H}|\odot\text{cos}(\measuredangle{\mathbf{H}})-|\hat{\mathbf{H}}|\odot\text{cos}(\measuredangle{\hat{\mathbf{H}}})\right\rVert}^{2}_{\text{F}}$
  		$\displaystyle+\mathinner{\!\left\lVert|\mathbf{H}|\odot\text{sin}(\measuredangle{\mathbf{H}})-|\hat{\mathbf{H}}|\odot\text{sin}(\measuredangle{\hat{\mathbf{H}}})\right\rVert}^{2}_{\text{F}}.$

• •

  MDPP [7]: we reuse the loss function Eq.(18) with the same network architecture. We set $K_{\text{PHA}}$ to $10$ bits.

  $$\text{MSE}_{\text{MDPP}}=\mathinner{\!\left\lVert|\measuredangle{\mathbf{H}}-\measuredangle{\mathbf{\hat{H}}}|{\odot}|\mathbf{H}|\right\rVert}^{2}_{\text{F}}\\$$

  (18)

To demonstrate the superiority of the proposed SMAPE loss function, we applied different phase reconstruction approaches to DualNet-MP for different phase compression ratios $\text{CR}_{\text{PHA}}$. Figs. 3 (a) and (b) show the NMSE performance of different approaches under indoor and outdoor scenarios, respectively, at different compression ratios. As expected, DaulNet-MP encounters training difficulties when using the simple loss function $Loss_{0}$. By adopting MDPP loss functions, DualNet-MP performs much better than the simple loss function $Loss_{0}$. Although DualNet-MP appears to be better when using MDPQ instead of MDPP, encoding bit-assignment require careful tuning to achieve a satisfactory result. Finally, DualNet-MP based on the proposed SMAPE loss function achieves 4-dB performance improvement in terms of NMSE reduction for $\text{CR}_{\text{PHA}}$=1/8 at outdoor and 7-dB improvement for $\text{CR}_{\text{PHA}}$=1/8 at outdoor.

To investigate the appropriate core layer designs of DualNet-MP in order to efficiently extract the underlying features of CSI phases, we provide a performance evaluation using FC, linear convolutional, and circular convolutional layers, respectively, for the core network. Denoted respectively as DNN, CNN and C-CNN, these networks adopted SSQ [5] and binary-level quantization (BLQ) as the quantization module at the encoder. Denote that the DNN design follows the recent work [7]. We consider the phase compression ratio of $\text{CR}_{\text{PHA}}=1/8$. For SSQ, we assign $K_{\text{PHA}}=8$ bits for each codeword. That is, there are $\text{CR}_{\text{PHA}}Q_{t}N_{b}=128$ 8-bit codewords sent to the gNB. In contrast, there are $K_{\text{PHA}}\text{CR}_{\text{PHA}}Q_{t}N_{b}=1024$ 1-bit codewords when applying BLQ.

Figs. 4.(a) and (b) show the NMSE performance for the considered core layer designs. For both indoor and outdoor scenarios, DualNet-MP demonstrates superiority when adopting SSQ and C-CNN, which can be attributed to two possible reasons. Firstly, unlike BLQ, SSQ is differentiable such that it is easier to train. Secondly, there are many structural and circular features of CSI phases in the angle-delay domain that can be extracted better with the proposed structural changes.

In terms of storage and complexity of the proposed architecture, we note that C-CNN with only 826K parameters is considerably simpler than DNN of 11.6M parameters, using comparable floating point operations. Thus, the newly proposed DualNet-MP architecture combining SSQ and C-CNN provides both performance gain and cost benefits.

We evaluated the numbers of floating point operations (FLOPs) and parameters of DualNet-MP with different phase encoding approaches and core layer designs in Tables I and II, respectively. The comparison shows similarly complexity in terms of FLOPS. Since the complexity and storage issues are both crucial in the design of encoders deployed at UEs, we provide Table III to compare the encoder complexity between the proposed framework and two state-of-the-art alternatives.

Since the FC layers are responsible for most of the parameters in many models, CoCsiNet requires the most parameters as it uses multiple FC layers at the encoder. Compared with CsiNet+, the proposed framework requires about half storage by separately encoding CSI phase and magnitude. To be specific, the FC layers require $(2048\times 256+256)$ and $2(1024\times 128+128)$ parameters at the encoders of CsiNet+ and DualNet-MP, respectively.

Often, convolutional layers contribute most FLOPs in a model due to the considerable number of two-dimensional convolution operations. However, DualNet-MP uses more convolutional layers at the encoder but still requires fewer FLOPs than CoCsiNet. Thus the proposed new architecture exhibits improvement in complexity reduction.