# On Compressed Sensing of Binary Signals for the Unsourced Random Access Channel
## 4 Conclusions and Additional Discussion  
We propose a scheme for compressed sensing of binary signals, consisting of a sparse sensing matrix, based on [PERSON]'s ensemble of LDPC codes, and a decoder based on MCMC. When used as a building block in grant-based random access, the scheme is demonstrated numerically to attain essentially state-of-the-art performance. To conclude, we mention several points that rise up as follow-up questions to our results. **Belief Propagation. ** One of the most popular algorithms for decoding LDPC codes is Belief Propagation (BP) (see, e.g., [56; 63]). We conducted very limited experiments with sum-product and max-product BP (not reported in this paper); our preliminary findings suggest that our MCMC decoder outperforms BP (in terms of its tolerance to noise), at least in the regime considered in Section 3.1. A possible explanation for this could be that the sensing matrix \(A\) has many small cycles, which severely violates the tree assumption and is common in BP analysis of LDPC codes. A thorough study of BP for compressed sensing with binary signals is left as an interesting direction for future research. **Granless unsourced random access. ** In Section 3.2, we demonstrate that our scheme can attain essentially state-of-the-art performance in grant-based unsourced random access (wherein a compressed sensing problem is solved in the first, scheduling, step). However, most previous works on unsourced random access have considered a different approach, which does not allow for feedback. The idea is to divide transmission into several blocks and perform coding in two steps: (1) an outer code, to allow the decoder to relate (\"stitch\") messages across different blocks to one another; and (2) an inner code, wherein each user codes its message (payload + parity bits) over an AWGN multiple access channel--in this framework, decoding the inner code boils down to solving a compressed sensing problem with a binary signal. An interesting question is whether our proposed scheme can provide any gains if used to construct an inner code in this framework. In [11], the authors proposed to use a certain tree code (outer code) and an i.i.d. Gaussian sensing matrix for  
Figure 3: Total \(E_{b}/N_{0}\) required to achieve end-to-end PUPE \(\leq\) 0.05. We see that, by using a better compressed sensing algorithm for binary signals, significant gains can be achieved over the current state of the art [12]. the inner code, together with a certain AMP decoder, which in decoding iteratively passes information between the inner and outer codes. We tried replacing the AMP decoder with our scheme. Specifically, we considered an iterative procedure that alternates between the following steps: (1) run Glauber dynamics on each block, producing a soft decision rule for the (sparse, per-block) activity pattern; and (2) a tree code inference step, that takes the per-block \"likelihoods\" produced by Glauber dynamics and computes a posterior over the entire activity pattern, by integrating information across all the blocks. The next time we decode the inner code, this posterior is used for the new prior of the signal. Our preliminary experiments indicate that the performance of this combined scheme is rather disappointing and quite far off from state of the art [11]. **Generalizing to non-equal channel gains. ** When discussing random access, we modeled the received signal at the base station as \(\mathbf{y}=A\mathbf{x}+\sigma\mathbf{z}\) where \(\mathbf{x}\in\{0,1\}^{M}\) is the pattern of active users and \(\sigma\mathbf{z}\) is Gaussian noise; namely, the channel between the users and the basestation is an AWGN multiple access channel where all gains are equal. This model is based on the assumption of perfect power and phase control, which is not always realistic, and designing communication schemes for the fading model, where channel gains are not assumed equal, is desired. Generalizing our MCMC decoder to incorporate fading looks somewhat challenging.