# Iterative Parameter Estimation Algorithms for Dual-Frequency Signal Models
## 2 The Gradient-Based Iterative Parameter Estimation Algorithm
The gradient at \(\mathbf{\theta}=\hat{\mathbf{\theta}}_{k-1}\) is given by  
\[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\] \[\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a} =-\sum_{i=1}^{L}\cos(\hat{\omega}_{1,k-1}t_{i})[y(t_{i})-\hat{a}_ {k-1}\cos(\hat{\omega}_{1,k-1}t_{i})-\hat{b}_{k-1}\cos(\hat{\omega}_{2,k-1}t_ {i})],\] \[\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b} =-\sum_{i=1}^{L}\cos(\hat{\omega}_{2,k-1}t_{i})[y(t_{i})-\hat{a}_ {k-1}\cos(\hat{\omega}_{1,k-1}t_{i})-\hat{b}_{k-1}\cos(\hat{\omega}_{2,k-1}t_ {i})], \tag{8}\] \[\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\hat{ \omega}_{1}} =\sum_{i=1}^{L}\hat{a}_{k-1}t_{i}\sin(\hat{\omega}_{1,k-1}t_{i})[y(t_{i}) -\hat{a}_{k-1}\cos(\hat{\omega}_{1,k-1}t_{i})-\hat{b}_{k-1}\cos(\hat{\omega}_ {2,k-1}t_{i})],\] \[\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\hat{ \omega}_{2}} =\sum_{i=1}^{L}\hat{b}_{k-1}t_{i}\sin(\hat{\omega}_{2,k-1}t_{i})[y(t_{i}) -\hat{a}_{k-1}\cos(\hat{\omega}_{1,k-1}t_{i})-\hat{b}_{k-1}\cos(\hat{\omega}_ {2,k-1}t_{i})].\]  
Using the negative gradient search and minimizing \(J_{1}(\mathbf{\theta})\), introducing an iterative step-size \(\mu_{k}\), we can get the gradient-based iterative (GI) algorithm for dual-frequency signal models:  
\[\hat{\mathbf{\theta}}_{k} =\hat{\mathbf{\theta}}_{k-1}-\mu_{k}\text{grad}[J_{1}(\hat{\mathbf{ \theta}}_{k-1})] \tag{9}\] \[=\hat{\mathbf{\theta}}_{k-1}-\mu_{k}\mathbf{\Phi}^{\top}(\hat{\mathbf{\theta }}_{k-1},L)[Y(L)-\mathbf{F}(\hat{\mathbf{\theta}}_{k-1},L)],\] \[Y(L) =[y(1),y(2),\cdots,y(L)]^{\top},\] (10) \[\hat{\mathbf{\Phi}}_{k} :=\mathbf{\Phi}(\hat{\mathbf{\theta}}_{k-1},L)=[\mathbf{\varphi}(\hat{\mathbf{ \theta}}_{k-1},1),\mathbf{\varphi}(\hat{\mathbf{\theta}}_{k-1},2),\cdots,\mathbf{\varphi}( \hat{\mathbf{\theta}}_{k-1},L)]^{\top},\] (11) \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]0 (12) \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]1 \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]2 (13) \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]3 \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]4 (14) \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]5 (15) \[\text{grad}[J_{1}(\hat{\mathbf{\theta}}_{k-1})] =\bigg{[}\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial a },\frac{\partial J_{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial b},\frac{\partial J _{1}(\hat{\mathbf{\theta}}_{k-1})}{\partial\omega_{1}},\frac{\partial J_{1}(\hat {\mathbf{\theta}}_{k-1})}{\partial\omega_{2}}\bigg{]}^{\top}\in\mathbb{R}^{4},\]6  
The steps of the GI algorithm to compute \(\hat{\mathbf{\theta}}_{k}\) are listed as follows:  
1.