## 5 Results
### CS Recovery Performance  
We apply the optimized Kronecker-product-based frames in CS model to measure sparse signals, where the orthogonal matching pursuit (OMP) [49] is deployed to recover the sparse signals. For a sparse signal \(\mathbf{x}\), we select the support set \(\mathcal{K}\subset\{1,\cdots,N\}\) randomly, where \(|\mathcal{K}|=s\ll N\). The non-zero elements are drawn from the complex Gaussian distribution with zero mean and variance \(1\). Then, \(\mathbf{x}\) is normalized to \(\|\mathbf{x}\|_{2}=1\). The linear measurement \(\mathbf{y}\in\mathbb{C}^{m\times 1}\) is given as  
\[\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n}, \tag{20}\]  
where \(\mathbf{A}\in\mathbb{C}^{m\times N}\) is a measurement matrix. The additive white Gaussian noise (AWGN) vector is \(\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma_{N}^{2}\mathbf{I})\). The signal-to-noise ratio (SNR) is defined as \(\mathbb{E}(\|\mathbf{A}\mathbf{x}\|_{2}^{2})/(m\sigma_{N}^{2})\), and we fix the SNR to 15 dB. We assume that the OMP has the prior knowledge of the sparsity \(s\). We recover \(I_{s}=10\),000 signals for sparse recovery performance. We denote the \(i\)th original and recovered signal by \(\mathbf{x}^{(i)}\) and \(\hat{\mathbf{x}}^{(i)}\), respectively. We also denote the \(i\)th original and recovered support set by \(\mathcal{K}^{(i)}\) and \(\hat{\mathcal{K}}^{(i)}\), respectively. The mean squared error (MSE) is defined as \(I_{s}^{-1}\sum_{i=1}^{L_{s}}\left\|\mathbf{x}^{(i)}-\mathbf{\dot{x}}^{(i)}\right\|_ {2}^{2}\), and the success rate of support set recovery is defined as \(|\mathcal{K}\cap\hat{\mathcal{K}}|/|\mathcal{K}|\). Figure 2: The mutual coherence of various frames over frame length \(m\). (**a**) \(N=64\), (**b**) \(N=256\). In simulations, the CS measurement matrix \(\mathbf{A}\) is from the optimized harmonic, optimized Hadamard-based, random Kronecker-product-based, and optimized Kronecker-product-based frames, respectively, where \(N=256\) and \(m=64\). For random Kronecker-product-based and optimized Kronecker-product-based frames, we set \(q=16\). In Figure 4, we show the sparse recovery performance for the four frames in CS. If the sparsity \(s\) is small, e.g., \(s\leq 19\), the optimized harmonic, optimized, and random Kronecker-product-based frames show almost the same performances of success rate and MSE, respectively. Meanwhile, if \(s\geq 20\), the success rate and the MSE of random Kronecker-product-based frames becomes worse than those of the optimized Kronecker-product-based and optimized harmonic frames due to their high coherence. Figure 4 reveals that the optimized Hadamard-based frames show the worst performance of success rate and MSE. From the distribution of inner product magnitudes of all possible frame vector pairs, we observed that the optimized Hadamard-based frame has discrete magnitudes of inner products, and many pairs have large magnitudes of inner products, although the largest one is smaller than that of the random Kronecker-product-based frame. Thus, many frame vectors of large inner products are likely to be involved in CS measurement process, which will degrade the CS recovery performance of optimized Hadamard-based frame. Moreover, if \(s\geq 20\), we observed the optimized Kronecker-product-based frame with \(q=16\) achieves better performance of success rate and MSE than the optimized harmonic frame. To investigate this observation, we compare the distributions of the  
Figure 3: The mutual coherence of various frames over frame dimension \(N\). (a) \(m=32\), (b) \(m=64\). average magnitudes of inner products for randomly selected 30-frame vectors in the optimized harmonic and the optimized Kronecker-product-based frames, respectively. We performed 10,000 random trials of selecting the frame vectors and averaged the inner product magnitudes to sketch the histogram.