Algorithm 1: Two-Step Joint Optimization Procedure with Auxiliary ASR Loss.
Require: $m$, batch size. $n$, the number of iterations. ${\alpha }_1$, auxiliary ASR loss weight. ${\alpha }_2$, loss weight ratio between speech enhancement and ASR. ${\theta }_{SE}$, speech enhancement learnable parameters. ${\theta }_{ASR}$, ASR learnable parameters.
1:	for$1,\cdots ,n$do
2:	Mini-batch of $m$ clean speech frames $\left\{s^{\left(1\right)},\cdots ,s^{\left(m\right)}\right\}$ and corresponding acoustic features $\left\{a^{\left(1\right)},\cdots ,a^{\left(m\right)}\right\}$
3:	Mini-batch of $m$ enhanced speech frames $\left\{{\widehat{s}}^{\left(1\right)},\cdots ,{\widehat{s}}^{\left(m\right)}\right\}$ and corresponding acoustic features $\left\{{\widehat{a}}^{\left(1\right)},\cdots ,{\widehat{a}}^{\left(m\right)}\right\}$
4:	Mini-batch of $m$ linguistic units $\left\{{\mathit{y}}^{\left(1\right)},\cdots ,{\mathit{y}}^{\left(m\right)}\right\}$
5:	First-Step Processing: Update the speech enhancement parameters using SI-SNR and auxiliary ASR loss
	${\nabla }_{{\theta }_{SE}}\frac{1}{m}{\displaystyle {\displaystyle \sum }_{i=1}^m}\left({\alpha }_1·{ℒ}_{SI-SNR}\left(s^{\left(i\right)},{\widehat{s}}^{\left(i\right)}\right)+\left(1-{\alpha }_1\right)·{ℒ}_{aux}\left(a^{\left(i\right)},{\widehat{a}}^{\left(i\right)}\right)\right)$
6:	Second-Step Processing: Update both speech enhancement and ASR parameters using SI-SNR and ASR loss
	${\nabla }_{{\theta }_{SE}, {\theta }_{ASR}}\frac{1}{m}{\displaystyle {\displaystyle \sum }_{i=1}^m}\left({\alpha }_2·{ℒ}_{SI-SNR}\left(s^{\left(i\right)},{\widehat{s}}^{\left(i\right)}\right)+\left(1-{\alpha }_2\right)·{ℒ}_{ASR}\left(a^{\left(i\right)},{\mathit{y}}^{\left(i\right)}\right)\right)$
7:	end for