$$
\begin{array}{l} \mathbb {E} _ {\widehat {\mathbf {A}}, \mathbf {h}} \left[ \mathbf {W} _ {2} ^ {* (l)} \left(\mathbf {H} ^ {(l - 1)} \widehat {\mathbf {A}} _ {t} \mathbf {e} _ {1, i}\right) \left(\mathbf {H} ^ {(l - 1)} \widehat {\mathbf {A}} _ {t} \mathbf {e} _ {1, i} - \frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) ^ {\top} \mathbf {W} _ {2} ^ {* (l) \top} \right] \\ = \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\widehat {\mathbf {A}}, \mathbf {h}} \left[ \left(\mathbf {H} ^ {(l - 1)} \widehat {\mathbf {A}} _ {t} \mathbf {e} _ {1, i}\right) \left(\mathbf {H} ^ {(l - 1)} \widehat {\mathbf {A}} _ {t} \mathbf {e} _ {1, i}\right) ^ {\top} \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ - \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\widehat {\mathbf {A}}, \mathbf {h}} \left[ \left(\mathbf {H} ^ {(l - 1)} \widehat {\mathbf {A}} _ {t} \mathbf {e} _ {1, i}\right) \left(\frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) ^ {\top} \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ = \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\mathbf {h}} \left[ \left(\frac {p \sum_ {j = 1} ^ {n} \mathbf {h} _ {1 , j} ^ {(l - 1)} + q \sum_ {j = 1} ^ {n} \mathbf {h} _ {2 , j} ^ {(l - 1)}}{n (p + q)}\right) \left(\frac {p \sum_ {j = 1} ^ {n} \mathbf {h} _ {1 , j} ^ {(l - 1)} + q \sum_ {j = 1} ^ {n} \mathbf {h} _ {2 , j} ^ {(l - 1)}}{n (p + q)}\right) ^ {\top} \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ - \mathbf {W} _ {2} ^ {* (l)} \left[ \left(\frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) \left(\frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) ^ {\top} \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ = \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\mathbf {h}} \left[ \left(\frac {p ^ {2} \sum_ {j = 1} ^ {n} \mathbf {h} _ {1 , j} ^ {(l - 1)} \mathbf {h} _ {1 , j} ^ {(l - 1) \top} + q ^ {2} \sum_ {j = 1} ^ {n} \mathbf {h} _ {2 , j} ^ {(l - 1)} \mathbf {h} _ {2 , j} ^ {(l - 1) \top}}{n ^ {2} (p + q) ^ {2}}\right) \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ + \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\mathbf {h}} \left[ \left(\frac {p ^ {2} \sum_ {j = 1} ^ {n} \sum_ {j ^ {\prime} = 1 , \neq j} ^ {n} \mathbf {h} _ {1 , j} ^ {(l - 1)} \mathbf {h} _ {1 , j ^ {\prime}} ^ {(l - 1) \top} + q ^ {2} \sum_ {j = 1} ^ {n} \sum_ {j ^ {\prime} = 1 , \neq j} ^ {n} \mathbf {h} _ {2 , j} ^ {(l - 1)} \mathbf {h} _ {2 , j ^ {\prime}} ^ {(l - 1) \top}}{n ^ {2} (p + q) ^ {2}}\right) \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ + \mathbf {W} _ {2} ^ {* (l)} \mathbb {E} _ {\mathbf {h}} \left[ \left(\frac {p q \sum_ {j = 1} ^ {n} \sum_ {j ^ {\prime} = 1} ^ {n} \mathbf {h} _ {1 , j} ^ {(l - 1)} \mathbf {h} _ {2 , j ^ {\prime}} ^ {(l - 1) \top} + p q \sum_ {j = 1} ^ {n} \sum_ {j ^ {\prime} = 1} ^ {n} \mathbf {h} _ {2 , j} ^ {(l - 1)} \mathbf {h} _ {1 , j ^ {\prime}} ^ {(l - 1) \top}}{n ^ {2} (p + q) ^ {2}}\right) \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ - \mathbf {W} _ {2} ^ {* (l)} \left[ \left(\frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) \left(\frac {p \boldsymbol {\mu} _ {1} ^ {(l - 1)} + q \boldsymbol {\mu} _ {2} ^ {(l - 1)}}{p + q}\right) ^ {\top} \right] \mathbf {W} _ {2} ^ {* (l) \top} \\ \end{array}
$$