# On Compressed Sensing of Binary Signals for the Unsourced Random Access Channel
## 3 Simulation Results  
### Performance in Compressed Sensing of Binary Signals  
We start by demonstrating the performance of Glauber dynamics in the compressed sensing of binary signals setup of Section 2.  
We run many random recovery experiments, to recover \(\mathbf{x}\in\{0,1\}^{M}\) from \(\mathbf{y}=A\mathbf{x}+\sigma\mathbf{z}\in\mathbb{R}^{n}\). In all the experiments, we use \(M=2^{I}\), \(J=14\), \(n=2^{11}\) and sparsity values \(k\in\{50,100,200,300\}\). These parameters are representative of a typical setup for unsourced random access (see Section 3.2). For each \(k\), we vary the energy per transmitted bit, \(E_{b}/N_{0}=\frac{E_{m}}{2\sigma^{2}\cdot\hat{J}}\) (here, \(E_{m}\) is the average energy per transmitting user--the energy of a column of \(A\)) and plot the corresponding bit error rate.  
We plot the performance under the following schemes:  
1. The scheme of [PERSON] et al. [7]: \(A\) based on BCH codes, and NNLS decoder. To obtain a binary estimator from the NNLS solution, we simply assign every entry to its closest binary value (that is, according to whether it is smaller or greater than \(1/2\)).
2. \(A\) given by a sparse LDPC matrix, with parameters \(\
u=16\) (consequently, \(s=128\)), under the following decoding algorithms: 1. NNLS. 2. Glauber dynamics with initialization at \(\mathbf{x}^{(0)}=\mathbf{0}\). 3. Glauber dynamics, with \(\mathbf{x}^{(0)}\) initialized at the NNLS solution. When using Glauber dynamics, we always let it run for \(T=10M\lg_{2}M\)\(=10\,MJ\) iterations.
3. \(A\) is a dense random i.i.d. Gaussian matrix of mean \(0\) and variance \(1/n\), with Approximate Message Passing (AMP) decoder (thus, \(E_{m}=1\); of course, in the experiments, the noise level \(\sigma\) is normalized according to the appropriate choice of \(E_{b}/N_{0}\)). The denoiser used in AMP is the optimal denoiser for the i.i.d. Bernoulli source, essentially as proposed by [PERSON] et al. [70]. AMP is a state-of-the-art algorithm for compressed sensing of binary signals, and is our main benchmark. For convenience, the exact implementation details of AMP are given in Appendix B.  
Our results are summarized in Figure 1. We see that when the sparsity is moderate (up to \(k=200\)), our proposed scheme attains essentially state-of-the-art performance. However, when \(k\) is large (\(k=300\)), performance falls short of AMP: if initialized at zero, Glauber dynamics consistently gets stuck in a local maximum, far away from the true signal; on the other hand, if one initializes Glauber dynamics with the NNLS solution, the combined scheme eventually attains performance which is substantially better than off-the-self compressed sensing solvers.  
In Figure 2, we plot the evolution, across consecutive iterations, of both the BER and the \"energy\" \(E(\mathbf{x}^{(t)})=-\frac{1}{2\sigma^{2}}\|\mathbf{y}-A\mathbf{x}^{(t)}\|^{2}+\lambda\| \mathbf{x}^{(t)}\|_{1}\) along a single run of Glauber dynamics (initialized at \(\mathbf{x}^{(0)}=0\)). We use \(k=100\) and \(E_{b}/N_{0}=1\,\mathrm{dB}\). Note that the iterations are given in units of \(MJ=M\lg_{2}M\) (meaning, it is \(t/M\)). Ignoring stochastic fluctuations, we see that Glauber dynamics essentially monotonically minimizes the energy (the error, however, is not monotonically decreasing).