Update 2021JUL21.md

2021JUL21.md

@@ -29,6 +29,10 @@ Algorithm 1. EM for the Fusion of GMMs
     ```math
     \gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}
     ```
+    - **M-step.** Re-estimate the parameters using the current responsibilities by setting the derivatives to zero
+    ```math
+    \mu_{k}^{\text{new}} = \frac{1}{N_{k}}(\Sigma^{H}_{k} + \Sigma^{L}_{k} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})(\Sigma_{k}^{H}\mathbf{Ux}^{H}_{n} + \Sigma_{k}^{L}\mathbf{x}_{n}^{L} )
+    ```
 
 ### Simulation results
 #### We considered three scenarios