An Exploration of the
Heterogeneous Unsourced MAC

In the low-energy cluster, every message $\mathbf{w}_{1i}$ is converted into a binary vector and partitioned into $J$ sub-blocks, where the $j^{\mathrm{th}}$ sub-block consists of $B_{1j}$ bits, so that $\sum_{j\in[J]}B_{1j}=B_{1}=\lfloor NR_{1}\rfloor$. This results in a collection of information fragments $\{\mathbf{w}_{1ij}\}_{j\in[J]}$ for message $\mathbf{w}_{1i}$. On the other hand, users in the high-energy cluster first split their bits into two groups: one for the top layer and one for the bottom layer. Denote these two sets of bits by $\mathbf{\overline{\overline{w}}}_{2i}$, $\mathbf{\overline{w}}_{2i}$ for the top and bottom portions, respectively. Likewise, let ${\overline{\overline{B}}}_{2}$, $\overline{B}_{2}$ be the total numbers of bits assigned to these two layers; and ${\overline{\overline{R}}}_{2}$, $\overline{R}_{2}$ be the corresponding rates. Subsequently, the bottom bits are fragmented exactly as in the low-energy cluster to form the set $\{\mathbf{\overline{w}}_{2ij}\}$. The top bits $\mathbf{\overline{\overline{w}}}_{2i}$ are also partitioned into $J$ fragments, but this time ${\overline{\overline{B}}}_{2j}$ is the size of the $j^{\mathrm{th}}$ fragment (not necessarily the same partitioning as in the bottom layer). Again, we must have $\sum_{j\in[J]}{\overline{\overline{B}}}_{2j}={\overline{\overline{B}}}_{2}$ and, additionally, $\overline{B}_{2}+{\overline{\overline{B}}}_{2}=\lfloor NR_{2}\rfloor$.

The role of the tree decoder is to enable the stitching of message fragments at the decoder. In CCS, this is accomplished by appending parity bits to the $j^{\mathrm{th}}$ fragment based on the preceding information bits. Every device in the low-energy cluster takes the message fragment $\mathbf{w}_{1i(j+1)}$ and encodes it into a vector $\mathbf{v}_{1i(j+1)}$ using a systematic random linear code, together with the message fragment, $\mathbf{w}_{1i(j+1)}$

where $\mathbf{G}_{1j}$ produces $T-B_{1(j+1)}$ random parity bits from all the previous segments, and $\otimes$ indicates modulo-2 matrix multiplication. Thus, $\mathbf{v}_{1i(j+1)}$ is viewed as a binary vector. In this scheme, $B_{11}=T$; that is the first fragment does not contain any parity bits. In (7), $T\geq B_{1j}$, so that effectively we have $T-B_{1j}$ random linear parity constraints embedded in this block to help stitch together fragments of information bits belonging to $\mathbf{w}_{1ij}$ when decoding codeword $\mathbf{v}_{1i}$.

A user in the high-energy cluster performs a similar encoding process. Redundancy for bottom bits is added paralleling the encoding in the low energy cluster, yielding

where $\mathbf{G}_{1j}$ is the same binary matrix that appears in (7). The top bits are encoded according to the information bits in each cluster, i.e.,

where $\mathbf{G}_{2j}$ is, again, a random parity generating matrix.

After tree encoding is complete, each encoded block has size $T$. These blocks are then encoded using two set of inner CS codes: (i) one for the segments of the low-energy cluster and the bottom segments of the high-energy cluster; and (ii) one for the top segments of the high-energy users. To apply the inner encoding, we convert the binary string $\mathbf{v}_{1ij}$ into the one-norm binary vector $\mathbf{m}_{1i}\in\{0,1\}^{2^{T}}$ in which a single one is placed at the location corresponding to the integer value of $\mathbf{v}_{1ij}$. This is the emblematic index representation used in CCS. Blocks $\mathbf{\overline{v}}_{2ij}$ and $\mathbf{\overline{\overline{v}}}_{2ij}$ are converted to their index representations in a similar manner.

The two CS code differ as follows: the former has entries from $\{+\sqrt{P_{1}},-\sqrt{P_{1}}\}$, while the latter features entries from $\{+\sqrt{P_{2}-P_{1}},-\sqrt{P_{2}-P_{1}}\}$. This difference in support results in a superposition constellation; the top bits are effectively transmitted at a higher power than the bottom bits, based on our assumption $P_{2}>2P_{1}$. Accordingly the CSS signal corresponding to section $\mathbf{v}_{1ij}$ is

where $\mathbf{A}_{1}\in\{+\sqrt{P_{1}},-\sqrt{P_{1}}\}^{Q\times 2^{T}}$ is a matrix formed by picking $Q=\lfloor NR/J\rfloor$ rows uniformly at random (excluding the row of all ones) from a Hadamard matrix of dimension $2^{T}\times 2^{T}$ and re-scaling them to meet the power constraint. Similarly, after tree encoding, the bottom bits are CS encoded,

using the same sensing matrix. The top bits are processed using a different signal dictionary,

The rows of matrix $\mathbf{A}_{2}\in\{+\sqrt{P_{2}-P_{1}},-\sqrt{P_{2}-P_{1}}\}^{Q\times 2^{T}}$ are also scaled versions of randomly selected rows from a Hadamard matrix (excluding the row of all ones).

Finally, the channel inputs are obtained by concatenating the partial signals. For the low-energy users, we get

and, for the high energy users, we have

ver the ensemble of random coding parameters, we obtain expected transmit power

as mandated by the constraint associated with (1). The parameters of the scheme are summarized in Table. I.

As the transmitted channel inputs are composed of coded segments of the same length, we observe that the CCS construction naturally maps to the CS interpretation in (6). To see this, let $\mathbf{m}_{1j}$ be the sum of coded fragments $\mathbf{m}_{1ij}$ for $i\in\mathcal{A}_{1}$, then let $\mathbf{m}_{1}$ be the concatenation of the fragments $\mathbf{m}_{1j}$ as in (13). Define $\mathbf{\overline{m}}_{2j}$, $\mathbf{\overline{m}}_{2}$ and $\mathbf{\overline{\overline{m}}}_{2j}$, $\mathbf{\overline{\overline{m}}}_{2}$ in a similar manner. Let $\boldsymbol{\Phi}_{1}$ be the tensor product between $\mathbf{I}_{J}$ and $\mathbf{A}_{1}$; likewise, let $\boldsymbol{\Phi}_{2}$ be that between $\mathbf{I}_{J}$ and $\mathbf{A}_{2}$. Then, we can express the received signal as

The signal aggregate obtained by adding $\mathbf{m}_{1}$ and $\mathbf{\overline{m}}_{2}$ has a sparsity of $J(M_{1}+M_{2})$, whereas $\mathbf{\overline{\overline{m}}}_{2}$ is $JM_{2}$ sparse.

Upon getting observation $\mathbf{y}$, the receiver separates it into sections $\{\mathbf{y}_{j}\}_{j\in[J]}$, each block corresponding to the summation of the sections $\mathbf{A}_{1}\mathbf{m}_{1j}$, $\mathbf{A}_{1}\mathbf{\overline{m}}_{2j}$, and $\mathbf{A}_{2}\mathbf{\overline{\overline{m}}}_{2j}$, as in (15). The receiver begins by decoding $\mathbf{\overline{\overline{m}}}_{2j}$ using the CSS algorithm with coding matrix $\mathbf{A}_{2}$. During this phase of the decoding process, it treats the remaining terms in $\mathbf{y}_{j}$ as additional noise. Parameters are selected to make sure that this portion of the decoding process is successful with high probability. Furthermore, for ease of exposition, we assume that $\mathbf{\overline{\overline{m}}}_{2j}$ is correctly recovered, although in reality, an error at this stage will produce some interference at the subsequent stage.

Once $\mathbf{\overline{\overline{m}}}_{2j}$ is recovered, the decoder computes a residual, or effective observation, for every section,

In the spirit of successive interference cancellation, these sections, $\left\{\tilde{\mathbf{y}}_{j}\right\}_{j\in[J]}$ are then passed to the CSS algorithm with coding matrix $\mathbf{A}_{1}$ and sparsity level $M_{1}+M_{2}$.

The recovery algorithm we adopt for individual sections is an AMP-based CS solver [maleki2010optimally]. Such composite algorithms iterate through two equations:

with initial conditions $\mathbf{m}^{(0)}={\bf 0}$ and $\mathbf{z}^{(0)}=\mathbf{y}$. The function $\boldsymbol{\eta}_{t}(\cdot)$ in (17) is the denoiser, which can take the form of a posterior mean estimate [Giuseppe] or a standard soft thresholding operator [daubechies2004iterative, beck2009fast]. Equation (16) can be interpreted as the computation of a residual enhanced enhanced with an Onsager correction [bayati2011dynamics, donoho2013information].

After converting the terms $\mathbf{m}$ into $\mathbf{v}$, the tree decoder uses the random parity bits to stitch together the information bits from each of the users, thus reconstructing the transmitted terms $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$.¹¹1In actuality, one also needs some random parity bits to stitch together the top and the bottom bits from the high-energy users. For simplicity this coding step is not presented here, as it is analogous to the step in Sec. III-B.

The encoding and decoding process is also conceptually represented in Fig. 1. Successive steps in the scheme are represented in vertical sections, proceeding from left to right. Separated horizontal section represent the processing of three sets of bits: the bits from cluster 1, the bottom bits from cluster 2, and the top bits from cluster 2. In the figure “be2i” indicates the conversion from binary string of length $l$ to the index in the $2^{l}$ binary vector. Also, “concat” indicates the concatenation of the segment as in (13).

In this section, we elaborate on the baseline settings for our simulation campaign. The performance associated with the original CSS code in each fragment is plotted in Fig. 2 for the settings in Table II.

Let us consider the performance of the classic (coordinated) RA scheme in which each active user transmits using time-sharing for a portion $1/(M_{1}+M_{2})$ of the time. In this case, the largest attainable rate (ignoring the block-length effects) is $R_{1}=0.067$: note the CCS code here attains reliable decoding up to $R_{l}=0.07$ but shows a $P_{\rm UP}$ saturation at $R=0.08$.

To attain the desired block-length, the various fragment is stitched together using a tree code expressed as $\mathbf{B}_{1}=[0,5,5,5,5,5,5,5,5,5,9]$ where the $j$ element of $\mathbf{B}_{1}$ is $B_{1j}$ in (7). With this choice choice of tree-coding parameters, we achieve an error rate below $1\%$ at the tree decoder.

Next, we wish to compare the performance of the scheme in Sec. III with the simpler scheme relying time-division multiple access (TDMA) as follows. Given a block-length $N$, transmission takes place in a low-rate phase of duration $\lambda N$ and a high-rate phase of duration $(1-\lambda)N$. In the low-rate phase, all users send at the baseline rate in Sec. IV-A at power $P_{1}^{\prime}=P_{1}/\lambda$ while, in the high-phase rate, the high-rate users transmit at rate the baseline rate for $M_{2}$ users at power $P_{2}^{\prime}$ obtained as

Let the rate attained with this scheme in the two clusters as $(R_{1}^{\prime},R_{2}^{\prime})$. We can compare the performance of this TDMA approach with the approach in Sec. III by letting the rate in the low-energy user be $R_{1}^{\prime}$ and see what rate for the high-energy user is attainable. We can then interpret ${\overline{\overline{R}}}_{2}$ and $R_{2}^{\prime}$ as the rates that the two approaches afford the high-energy users for a given degradation of the baseline performance.

Thorough numerical experiments, we have noted that the decoding procedure in Sec. III-E actually greatly outperforms the performance predicted by SIC assuming that the effecting noise $𝚽{}_1(𝐦{}_1+\overline{𝐦}{}_2𝐳isnormaldistributed,i.e.IAN.OneinherentfeatureoftheCCScodesusedinSec.\text{<a href="\#S3" title="III A Coded Compressed Sensing Scheme ‣ An Exploration of the Heterogeneous Unsourced MAC" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">III</span></a>}isthattherandomlygeneratedcodewordsin𝐀{}_1/𝐀{}_{2}areuniformlydistributedinthecodespace.Thismeansthat,whenR{}_{1}andR{}_{2}aresufficientlylow,thecodewordsin𝐀{}_{1}and𝐀{}_{2}areperpendicularwithhighprobability.WeindeedobservethatthedecodingprocedureinSec.\text{<a href="\#S3" title="III A Coded Compressed Sensing Scheme ‣ An Exploration of the Heterogeneous Unsourced MAC" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">III</span></a>}isinherentlyabletoexploitthecodewordperpendicularityintheprojectionstep.Wearecurrentlyunabletopreciselycharacterizethiseffect;nonetheless,wecannumericallyinvestigatethisphenomenaasinFig.\text{<span class="ltx\_ref ltx\_missing\_label ltx\_font\_italic ltx\_ref\_self">LABEL:fig:SIC\_CDMA</span>}.Hereweplotthelargest\overline{\overline{R}}{}_2=R{}_2-R{}_{1}attainablefordifferentnumberofusers,togetherwiththeIANprediction.\text{Figure 44Figure 418202224262830323410-1.410-1.210-1P1 [dB]RR2-R1 for M1=15R2-R1 for M1=10IAN prediction for R2-R14Decodingimprovementwithrespecttotheinterferenceasnoise(IAN)predictiondescribedinSec}$