1 Introduction↩︎

The rapid proliferation of IoT devices and edge computing applications is driving the need for seamless integration of communication and computation in wireless networks, forming the basis for the ICC paradigm [1], [2]. This approach aims to reduce latency and resource overhead by enabling simultaneous data transmission and the OTAC [3]–[6] of nomographic functions over shared channels, particularly in multiple-access scenarios where interference poses significant challenges [7].

Traditional methods treat the communication and computation operations separately, leading to inefficiencies, but recent advancements in ICC have shown that joint designs can significantly enhance performance [8]. However, when discrete data symbols must be communicated alongside computing signals, mutual interference between the two tasks degrades performance, necessitating innovative interference management techniques [9].

DPC, a foundational concept in information theory first proposed in [10], addresses similar issues by allowing a transmitter to communicate reliably in the presence of known interference through precoding, effectively “writing on dirty paper” to achieve the same channel capacity as if the interference were absent.

While many ways to implement DPC have been proposed including approaches leveraging Tomlinson-Harashima precoding [11], nested lattice codes have emerged as a particularly effective method due to their structured nature and ability to approach the theoretical limits of DPC [12]–[16]. In addition, DPC has also been successfully applied in various multifunctional communication system aspects, such as in the ISAC paradigm [17]–[19].

Inspired by these developments, we introduce in this paper Computing on Dirty Paper, a novel ICC approach that leverages DPC to pre-cancel computing symbols at the users in a SIMO uplink setting, enabling the simultaneous transmission of discrete data and the computation of nomographic functions over the same multiple access channel. In particular, we demonstrate that typical DPC schemes such as the one proposed in [13] can be incorporated at the TX, and combined with a classic LMMSE estimator at the RX, to enable ICC with no impact of the OTAC operation over the BER of the communication stream. A simulation-based assessment of the method in a SIMO setup is performed, which validates our approach, showing substantial improvements over SotA benchmarks in terms of both functionalities.

Notation: Throughout the manuscript, vectors and matrices are represented by lowercase and uppercase boldface letters, respectively; \(\mathbf{I}_M\) denotes an identity matrix of size \(M\) and \(\mathbf{1}_M\) denotes a column vector composed of \(M\) ones; the Euclidean norm and the absolute value of a scalar are respectively given by \(\|\cdot\|_2\) and \(|\cdot|\); the transpose and hermitian operations follow the conventional form \((\cdot)^{\mathsf{T}}\) and \((\cdot)^{\mathsf{H}}\), respectively; \(\Re{\{\cdot\}}\), \(\Im{\{\cdot\}}\) and \(\mathrm{min}(\cdot)\) represents the real part, imaginary part and the minimum operator, respectively. Finally, \(\sim \mathcal{CN}(\mu,\sigma^2)\) denotes the complex Gaussian distribution with mean \(\mu\) and variance \(\sigma^2\), where \(\sim\) denotes “is distributed as”.

2 System and Signal Models↩︎

Consider a typical SIMO uplink setup composed of \(K\) single-antenna UE/EDs and one BS/AP equipped with \(N\) antennas, as illustrated in Fig. 1. Throughout this article we refer to the UEs/EDs as users, unless otherwise specified, and refer to the BS/AP as the RX.

2.1 Uplink ICC Signal Model↩︎

Under the assumption of perfect symbol synchronization amongst users, the received signal \(\boldsymbol{y}^{(t)} \in \mathbb{C}^{N\times 1}\) at the BS/AP subjected to fading and noise at a \(t\)-th time slot is given by \[\label{eq:received95signal}\boldsymbol{y}^{(t)} = \sum_{k=1}^K \boldsymbol{h}_k^{(t)} {x}_k^{(t)} + \boldsymbol{w}^{(t)},\qquad{(1)}\] where \({x}_k^{(t)} \in \mathbb{C}\) is a multifunctional transmit signal from user \(k\), incorporating communication and computing signals; \(\boldsymbol{w}^{(t)} \in \mathbb{C}^{N\times1}\sim \mathcal{CN}(0,\sigma^2_w\mathbf{I}_N)\) is the AWGN vector, and \(\boldsymbol{h}_k^{(t)} \in \mathbb{C}^{N \times 1}\) is the SIMO channel vector of user \(k\) to the RX following the uncorrelated block Rayleigh fading model typically assumed in the OTAC literature [4], [20], such that the \((n,k)\)-th elements \(h_{n,k}\sim \mathcal{CN}(0,1)\) of the channel matrix \(\boldsymbol{H}^{(t)}\) are assumed to be iid circularly symmetric complex Gaussian random variable with zero mean and unit variance, and sufficient coherence time.

2.2 SotA Transmit Signal Design↩︎

Under the SotA ICC paradigm [8], [21], a trivial way to model the transmit signal is to decompose it as a sum of communication and computing components, \(i.e.,\) \[\label{eq:transmit95sig95decomposition} x_k^{(t)} \triangleq d_k^{(t)} + \psi_k(s_k^{(t)}),\tag{1}\] where \(d_k^{(t)} \in \mathcal{D}\) and \(s_k^{(t)} \in \mathcal{S}\) denote the modulated symbol for communication and computing for user \(k\), respectively, with \(\mathcal{D}\) representing an arbitrarily scaled discrete constellation of cardinality \(D\), \(e.g.\) QAM; \(\mathcal{S}\) representing a discrete set following [22] for the computing symbols; while \(\psi_k(\cdot)\) denotes a general pre-processing function for OTAC to be elaborated in the following subsection.

The SotA RX design [8] typically consists of a two-stage approach, where the RX first estimates the individual communication symbols \(\{d_k^{(t)}\}_{k=1}^K\) per time slot via a typical LMMSE or a lower complexity GaBP technique under the assumption that the computing signal is noise, followed by the evaluation of a target function \(f(\boldsymbol{s}^{(t)})\) via a combiner.

For future convenience, the received signal can now be reformulated in terms of matrices as \[\boldsymbol{y}^{(t)} = \boldsymbol{H}^{(t)} \boldsymbol{x}^{(t)} + \boldsymbol{w}^{(t)} =\boldsymbol{H}^{(t)} \big(\boldsymbol{d}^{(t)} + \boldsymbol{s}^{(t)}\big) + \boldsymbol{w}^{(t)}, \label{eq:received95signal95matrix}\tag{2}\] where the complex channel matrix \(\boldsymbol{H}^{(t)} \triangleq [\boldsymbol{h}_1^{(t)},\dots, \boldsymbol{h}_K^{(t)}] \in \mathbb{C}^{N\times K}\), the concatenated transmit signal \(\boldsymbol{x}^{(t)} \triangleq [x_1^{(t)},\dots, x_K^{(t)}]^{\mathsf{T}}\in \mathbb{C}^{K\times1}\), the data signal vector \(\boldsymbol{d}^{(t)} \triangleq [d_1^{(t)},\dots, d_K^{(t)}]^{\mathsf{T}}\in \mathcal{D}^{K \times 1} \subset \mathbb{C}^{K\times1}\) and the computing signal vector \(\boldsymbol{s}^{(t)} \triangleq [\psi_1(s_1^{(t)}),\dots, \psi_K(s_K^{(t)})]^{\mathsf{T}}\in \mathcal{S}^{K \times 1} \subset \mathbb{C}^{K\times1}\) are explicitly defined.

In addition, concatenating the received signals over \(T\) time slots and assuming that the channel \(\boldsymbol{H}^{(t)}\) (hereafter denoted as \(\boldsymbol{H}\) for brevity) remains unchanged¹, we may express the received signal matrix as \[\boldsymbol{Y} = \boldsymbol{H} \boldsymbol{X} + \boldsymbol{W} = \boldsymbol{H} (\boldsymbol{D} + \boldsymbol{S}) + \boldsymbol{W}, \label{eq:received95signal95matrix95T}\tag{3}\] where \(\boldsymbol{Y} \triangleq [\boldsymbol{y}^{(1)},\dots, \boldsymbol{y}^{(T)}] \in \mathbb{C}^{N\times T}\), \(\boldsymbol{X} \triangleq [\boldsymbol{x}^{(1)},\dots, \boldsymbol{x}^{(T)}] \in \mathbb{C}^{K\times T}\), \(\boldsymbol{D} \triangleq [\boldsymbol{d}^{(1)},\dots, \boldsymbol{d}^{(T)}] \in \mathcal{D}^{K\times T} \subset \mathbb{C}^{K\times T}\), \(\boldsymbol{S} \triangleq [\boldsymbol{s}^{(1)},\dots, \boldsymbol{s}^{(T)}] \in \mathcal{S}^{K \times T} \subset \mathbb{C}^{K\times T}\) and \(\boldsymbol{W} \triangleq [\boldsymbol{w}^{(1)},\dots, \boldsymbol{w}^{(T)}] \in \mathbb{C}^{N\times T}\) are the corresponding matrices.

Finally, for the sake of derivation of the proposed scheme, we assume that the computing symbols \(\boldsymbol{s}^{(t)}\) across a block of \(T\) time slots are identical (repeatedly transmitted), \(i.e.,\) \(s_k^{(t)} = s_k, \forall k\), and \(\boldsymbol{s}^{(t)} = \boldsymbol{s}, \forall t\), and thus \(\boldsymbol{S} = [\boldsymbol{s},\dots, \boldsymbol{s}] \in \mathcal{S}^{K \times T} \subset \mathbb{C}^{K\times T}\).

2.3 Description of the AirComp Operation↩︎

The OTAC operation consists of the evaluation of a target function \(f(\boldsymbol{s})\) at the RX, which can be described as [4] \[\label{eq:target95function95def} f(\boldsymbol{s}) = \phi\bigg(\sum_{k=1}^K \psi_k(s_k)\bigg),\qquad{(2)}\] where \(\phi\) represents the OTAC post-processing function for a general nomographic expression.

While many examples of nomographic functions are often considered in the OTAC literature [3], [23], we choose the arithmetic sum operaton for ease of exposition, given by \[\label{eq:target95function95def95SUM} f(\boldsymbol{s}) = \phi\bigg(\sum_{k=1}^K \psi_k(s_k)\bigg) = \sum_{k=1}^K s_k,\qquad{(3)}\] where the corresponding pre- and post-processing functions are defined as \(\psi_k(s_k) \triangleq s_k\) and \(\phi\big(\sum_{k=1}^K \psi_k(s_k)\big) \triangleq \sum_{k=1}^K \psi_k(s_k)\).

3 Computing on Dirty Paper↩︎

3.1 Fundamentals on Dirty Paper Coding↩︎

DPC, first introduced by Costa in [10], is a powerful coding technique that allows the transmission of information over a noisy channel in the presence of known interference at the TXs, without being affected by the latter. In more detail, the TX – the arbitrary UEs or EDs in our case – needs to transmit a signal \(d\) subject to interference \(s\). Then, the RX receives signal \(y = d + s + w\), where \(w\) is random noise. When \(s\) is unknown to both the TX and RX, it can only be incorporated into the noise as previously proposed in [8] for the ICC paradigm. However, when \(s\) is known to the TX, the RX can completely remove the interference \(s\) by leveraging DPC, even though it does not know \(s\). In this paper, we introduce a novel scheme for the DPC operation in the context of ICC, termed Computing on Dirty Paper.

3.2 Fundamentals on Nested Lattice Codes↩︎

For the design of the lattice, consider a standard Gaussian integer lattice \(\mathbb{Z}[j] = \{ a + bj \mid a, b \in \mathbb{Z} \}\) with generator matrix defined to be \(\mathbf{I}_2\), as illustration in Fig. 2, which is simple² and effective for complex signals (equivalent to a square lattice in \(\mathbb{R}^2\)) in a similar fashion to [13] as follows:

Coarse lattice: Let \(\Lambda_c = \Delta \mathbb{Z}[j] \subset \mathbb{C}\), where \(\Delta > 0\) sets the modulo region; \(i.e.,\) the “coarse lattice cube” bounded by sides of length \([-\Delta/2, \Delta/2]\). This choice ensures that the transmit power is bounded during the modulo operation.

Fine lattice: Let \(\Lambda_f = \delta \mathbb{Z}[j]\), with \(\delta = \Delta / \sqrt{M}\) and \(M = 2^{2R}\) where \(M\) represents the (integer) number of constellation points inside each coarse lattice cell and rate \(R\) denotes the number of bits per complex symbol for \(d_k\).

Cosets: Each data symbol \(d_k\) now corresponds to a coset of \(\Lambda_f\) where \[\label{eq:coset95definition}v_k + \Lambda_c, v_k \in \Lambda_f \cap \mathcal{V}(\Lambda_c),\tag{4}\] with \(\mathcal{V}(\Lambda_c)\) denoting the Voronoi region of the coarse lattice \(\Lambda_c\), which is a square in the complex plane (of area \(\Delta^2\)), with distortion bounded by \(\Delta / \sqrt{2}\) in Euclidean norm for nearest-neighbor quantization.

Note that the transmit constellation is the set of all coset representatives inside a single Voronoi cell, and not the entire coset. For a more compresensive overview of nested lattice codes, the reader is referred to [24].

3.3 Transmitter Design↩︎

Reintroducing the time-varying component for the data, let each transmit user know its own encoded data symbol \(v_k^{(t)} \in \Lambda_f \cap \mathcal{V}(\Lambda_c)\) and computing symbol picked from the rotated set \(\mathcal{S}_{\text{rot}} \triangleq \mathbf{R} \, \mathcal{S} = \bigl\{ \mathbf{R} \, s_k \,\big|\, s_k \in \Lambda_f \cap \mathcal{V}(\Lambda_c) \bigr\}\) to maximize orthogonality, where \[\mathbf{R} \triangleq \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}.\]

Nested lattice-based DPC encodes the communications symbols \(v_k^{(t)}\) leveraging a modulo operation such that the transmit symbol \(\tilde{v}_k^{(t)}\) is given by \[\label{eq:lattice95DPC95encoding} \tilde{v}_k^{(t)} = \text{modL}(v_k^{(t)} - s_k, \Delta),\tag{5}\] where the \(\text{modL}(a,b)\) operation is defined as³ \[\begin{align} \text{modL}(a,b) \triangleq &&\\ && \Big(\text{mod}\big(\Re\{{a}\}\!+\!\tfrac{b}{2}, b\big)\!-\!\tfrac{b}{2}\Big)\!+\!j \Big( \text{mod}\big(\Im\{{a}\}\!+\!\tfrac{b}{2}, b\big)\!-\!\tfrac{b}{2} \Big),\nonumber \end{align}\] with \(\text{mod}(a,b)\) denoting the standard modulo operation.

Then, the multifunctional transmit signal \(x_k^{(t)}\) across \(T\) time slots (analogous to the SotA expression in equation 1 ) can be expressed as \[\label{eq:lattice95DPC95transmit95signal} x_k^{(t)} = \tilde{v}_k^{(t)} + s_k,\tag{6}\] where we emphasize that the computing symbol \(s_k, \forall k\) remains unchanged across \(T\) time slots.

3.4 Receiver Design↩︎

After transmission of \(T\) slots over a typical Rayleigh fading channel, the received signal in matrix form can be expressed as \[\boldsymbol{Y} = \boldsymbol{H} \boldsymbol{X} + \boldsymbol{W} = \boldsymbol{H} (\tilde{\boldsymbol{V}} + \boldsymbol{S}) \in \mathbb{C}^{N \times T}, \label{eq:received95signal95lattice}\tag{7}\] where similarly to previous definitions, \(\tilde{\boldsymbol{V}} \triangleq [\tilde{\boldsymbol{v}}^{(1)},\dots, \tilde{\boldsymbol{v}}^{(T)}] \in \mathbb{C}^{K\times T}\) with \(\tilde{\boldsymbol{v}}^{(t)} \triangleq [\tilde{v}_1^{(t)},\dots, \tilde{v}_K^{(t)}]^{\mathsf{T}}\in \mathbb{C}^{K\times1}\) and \(\boldsymbol{S} = [\boldsymbol{s},\dots, \boldsymbol{s}] \in \mathcal{S}_{\text{rot}}^{K \times T} \subset \mathbb{C}^{K\times T}\).

Note that different time slots contain different data symbols \(\tilde{v}_k\), retaining the full rate in terms of data transmission identical to typical communications systems, while the computing symbols \(s_k\) remain unchanged across \(T\) time slots to improve the MSE performance of the function evaluation. One can then apply a LMMSE estimator per \(t\)-th time frame at the receiver to obtain \(\hat{\boldsymbol{x}}^{(t)}\), which can be expressed as \[\hat{\boldsymbol{x}}^{(t)} = (\boldsymbol{H}^{\mathsf{H}}\boldsymbol{H} + \sigma_w^2 \mathbf{I}_K)^{-1} \boldsymbol{H}^{\mathsf{H}}\boldsymbol{y}^{(t)}.\]

Next, a temporary variable \(\hat{\boldsymbol{z}}\) can be calculated via the modulo operation \[\hat{\boldsymbol{z}}^{(t)} = \text{modL}(\hat{\boldsymbol{x}}^{(t)}, \Delta),\] which can then be leveraged to estimate \(\hat{\boldsymbol{v}}^{(t)}\) via \[\hat{\boldsymbol{v}}^{(t)} = \text{modL}\big( \mathcal{Q}_{\Lambda_f}(\hat{\boldsymbol{z}}^{(t)}, \delta), \Delta\big),\] where \(\mathcal{Q}_{\Lambda_f}(\cdot)\) denotes the nearest-neighbor quantizer to the fine lattice \(\Lambda_f\), and is defined as \[\mathcal{Q}_{\Lambda_f}(a, b)\! \triangleq \! b \Big(\text{round}\big(\tfrac{\Re\{{a}\}}{b} - \epsilon\big) + \epsilon \Big) + jb \Big( \text{round}\big(\tfrac{\Im\{{a}\}}{b} - \epsilon\big) + \epsilon \Big),\] with \(\text{round}(\cdot)\) denoting the rounding operation to the nearest integer and \(\epsilon \triangleq \text{mod}\Big(\frac{\sqrt{M} - 1}{2}, b\Big)\).

Finally, a hard decision against the constellation \(\Lambda_f \cap \mathcal{V}(\Lambda_c)\) can be performed to obtain the final estimate of the encoded data symbols \(\hat{\boldsymbol{v}}^{(t)}\).

In order to recover the computing symbols, a novel best-case bound  can be derived in the form of a maximum likelihood estimator which can be expressed as \[\label{eq:OTAC95recever} \hat{\boldsymbol{s}}\!=\!\arg\min_{\boldsymbol{s} \in \mathcal{S}_{\text{rot}}^{K \times 1}} \sum_{t=1}^T \big\| \boldsymbol{y}^{(t)}\!-\!\boldsymbol{H}\big( \mathrm{modL}(\hat{\boldsymbol{v}}^{(t)}\!-\!\boldsymbol{s}, \Delta)\!+\!\boldsymbol{s} \big) \big\|_2^2,\tag{8}\] where each element \(\hat{s}_k\) of \(\hat{\boldsymbol{s}}\) is chosen from the discrete set \(\mathcal{S}_{\text{rot}}\) that has the lowest error across all time instances \(T\), naturally improving the MSE performance as \(T\) increases.

The computational complexity of equation 8 is dominated by the number of \(K\) users, since an exponential scaling is needed to compare with a constellation of size \(M\) such that the dominant complexity is given by \(M^K\) in order to obtain a maximum likelihood estimate.

Finally, the target function can be computed as \[\hat{f}(\boldsymbol{s}) = \sum_{k=1}^K \hat{s}_k.\]

4 Performance Analysis↩︎

To evaluate the performance of the proposed scheme, let us first visualize the complete nested lattice structure that is used, as shown in Fig. 3. Notice that each coarse lattice cell contains \(M = 2^{2R}\) constellation points, where \(R\) is the number of bits per complex symbol for data and this fine lattice (corresponding to a scaled QPSK) is used for data symbol transmission. In addition, to maximize orthogonality between data and computing symbols, the computing symbols are chosen from a rotated version of the fine lattice constellation, as previously described.

In order to bound the transmit power for a fair comparison with the SotA, we set the coarse lattice parameter to be \(\Delta = 2\), which leads to a transmit power of \(E[|x_k|^2] = E_d + E_s = 0.5 + 0.5 = 1\) per user, where \(E_d = E[|d_k|^2] = 0.5\) and \(E_s = E[|s_k|^2] = 0.5\) are the average powers of data and computing symbols, respectively.

The parameters used in the simulations are \(K = 2\) users, \(N = 5\) antennas at the RX, an alternating time slot of \(T = 5,10\), and \(R = 2\) bits/symbol (i.e., for QPSK).

Figure 4 (a) and 4 (b) show the BER and MSE performance of the proposed scheme compared to the SotA ICC method in [8], where the latter uses a two-stage approach with a GaBP detector for data detection followed by a linear combiner for function estimation. As seen, the proposed scheme achieves significant performance gains in both BER and MSE compared to the SotA, with the performance gap increasing with SNR. This is because the proposed scheme effectively eliminates the interference between data and computing symbols via DPC, allowing both tasks to be performed more accurately. In addition, increasing the number of time slots \(T\) improves the MSE performance for the proposed scheme, as more observations are available for function estimation.

5 Conclusion↩︎

We proposed Computing on Dirty Paper, a new scheme for ICC inspired by DPC. By pre-canceling computing symbols at the transmitting users, the scheme enables interference-free communication and computation over multiple-access channels. Analysis under a SIMO ICC system and supporting simulations show that the approach achieves superior BER and MSE performance compared to SotA ICC methods.

References↩︎