# Integrated Radar and Communications Waveform Design Based on Multi-Symbol OFDM
## 4 Proposed Algorithm and Its Complexity
### Algorithm Summary and Complexity Analysis  
By integrating the above derivation, a conjugate gradient-based iterative algorithm for the joint optimization of the PMEPR and PSLR (CGIA-JOPP) of multi-symbol IRC waveforms is summarized in Algorithm 1. In Algorithm 1, the first is the initialization process. We set the initial reserved symbol vector to \(\mathbf{0}\), calculate the initial gradient vector \(\mathbf{G}^{0}\) according to (34), and obtain the initial descent direction \(\mathbf{d}^{0}\) according to (48). The second is the iterative process, and the last is the convergence criterion. In the iterative process of Algorithm 1, appropriate step size is what CGIA-JOPP pursues. In our algorithm, the optimal step size is obtained by solving the minimization problem (36), and its calculation process is summarized in Algorithm 2. [PERSON]'s downhill method is applied here. Specifically, the downhill factor is determined, the search direction is calculated according to (41) and (46), and then based on the update rule (47), the optimal step size required for each iteration in Algorithm 1 is obtained. It should be noted that in the iterative process of Algorithm 2, the search for the downhill factor \(\lambda\) will stop when \(\left|W^{\prime}\Big{(}\mu^{l}_{i+1}\Big{)}\right|<\left|W^{\prime}\Big{(}\mu^ {l}_{i}\Big{)}\right|\) and \(W\Big{(}\mu^{l}_{i+1}\Big{)}<W\Big{(}\mu^{l}_{i}\Big{)}\) are satisfied to prevent the step size from falling into other local optimal ranges due to the excessively large value of \(\lambda\). After obtaining the optimal step size, the new reserved symbol vector is calculated according to the update rule (35), and the new IRC waveform is obtained, so the new gradient vector and descent direction are calculated again. In this way, the iterations are repeated until the convergence condition is satisfied. Complexity Analysis: The number of real multiplications is used to measure the computational complexity of our algorithm. One complex multiplication counts as four real multiplications. A \(Q\)-point IFFT/FFT pair requires \(\mathcal{O}(Q\log_{2}Q)\) complexity. Since the initialization step in Algorithm 1 occurs only once, its computational complexity is ignored. In Algorithm 1, the rest of the steps except Step 6 are only composed of the Hadamard product of the vector, the \(MQ\)-point IFFT/FFT pair, and the \(Q\)-point IFFT/FFT pair, then the complexity is \(\mathcal{O}(MQ\log_{2}(MQ))\). In Step 6, the complexity is mainly brought by the calculation of \(\hat{\mathbf{C}}\) in (34). It can be seen from (31) that each part of the three-part summation term can be regarded as a combination of FFT and matrix Hadamard product. Assuming that the maximum length of the three-part summation index sets for \(k\) is \(\overline{K}\), then the complexity of Step 6 is \(\mathcal{O}(\overline{K}Q\log_{2}Q)\). Note that the number of reserved subcarriers is also related to the complexity of Algorithm 1, but is ignored because it is too small relative to the \(MQ\)-point IFFT/FFT pair. Therefore, for each iteration, the computational complexity of the proposed algorithm is mainly determined by the \(MQ\)-point IFFT/FFT pair and the complexity from (34), which shows that the proposed algorithm can be effectively implemented by IFFT/FFT pair and Hadamard product. ```
1: Consider \(p\), \(N\), \(Q\), \(R_{N}\), \(M\). Store transmit data \(\mathbf{X}_{m}\): \(X_{n,m}\), \(n\in\mathcal{R}^{c},m=0,1,\ldots,M-1\). Initialization conditions: \(l=0\), \(\mathbf{C}_{m}^{0}\): \(C_{n,m}^{0}=0(n\in\mathcal{R})\), \(\mathbf{G}^{0}=\
abla_{\mathbf{C}^{*}}\widetilde{O}(\mathbf{C}^{0})\) and \(\mathbf{d}^{0}=-\mathbf{G}^{0}\). 2:Repeat
3: Compute step size \(\mu^{l}\) according to Algorithm 2. 4: Update the reserved symbol vector: \(\mathbf{C}^{l+1}=\mathbf{C}^{l}+\mu^{l}\mathbf{d}^{l}\). 5: \(\mathbf{s}^{l+1}=\mathbf{\Phi}(\mathbf{X}+\mathbf{C}^{l+1})\). 6: Compute the conjugate gradient \(\mathbf{G}^{l+1}\) according to (34). 7: Compute the descent direction \(\mathbf{d}^{l+1}\) according to (48). 8:\(l=l+1\).