# Robustness and Sensitivity Tuning of the Kalman Filter for Speech Enhancement
## 2 Kalman Filter for Speech Enhancement
### Impact of Bused \(K_{0}(n)\) on KF-Based Speech Enhancement in Real-Life Noise Conditions  
To analyze \(K_{0}(n)\) and its impact on KF-based speech enhancement, we repeated the experiment in Figure 1, except that the utterance sp05 was corrupted with a typical real-life non-stationary noise, _bubble_[24], at 5 dB SNR. A 32 ms rectangular window with 50% overlap ([25], Section 7.2.1) was considered for converting \(y(n)\) into frames, \(y(n,k)\) (as in Equation (28)).  
As shown in Section 2.2, in the oracle case, \(K_{0}(n)\) also shows a smooth transition between 0 and 1 depending on the speech absence and speech presence of noisy speech (Figure 2c). Technically, during speech pauses of \(y(n,k)\), the total a priori prediction error of the AR model, \([\alpha^{2}(n)+\sigma_{w}^{2}]=0\) (e.g., 0-0.15 s or 1.8-2.19 s in Figure 2d). Substituting \([\alpha^{2}(n)+\sigma_{w}^{2}]=0\) in Equation (21)) gives \(K_{0}(n)=0\), which in turn yields \(\hat{s}(n|n)=0\) (Equation (23)), i.e., nothing is passed to the output (e.g., 0-0.15 s or 1.8-2.19 s of Figure 2c,g). Conversely, it was observed that \([\alpha^{2}(n)+\sigma_{w}^{2}]>>\sigma_{w}^{2}\) in speech regions of \(y(n,k)\), for which \(K_{0}(n)\) is approaching 1 (e.g., 0.16-0.33 s or 0.9-1.06 s in Figure 2c). As demonstrated in Section 2.2, a higher \(K_{0}(n)\) enables the KF to produce enhanced speech with less residual background noise as well as less distortion (Figure 2g).  
In the non-oracle case, the biased estimates of \((\{\hat{a}_{i}\},\sigma_{w}^{2})\) and \(\sigma_{v}^{2}\), resulted in \([\hat{\alpha}^{2}(n)+\hat{\sigma}_{w}^{2}]\approx\sigma_{v}^{2}\) (e.g., 0-0.15 s or 1.8-2.19 s in Figure 2e). According to Equation (21), this condition introduces around 0.5 bias in \(\hat{K}_{0}(n)\) (e.g., 0-0.15 s or 1.8-2.19 s in Figure 2c). During speech presence of \(y(n,k)\), it is observed that \(\hat{\sigma}_{v}^{2}>>[\hat{\alpha}^{2}(n)+\hat{\sigma}_{w}^{2}]\) (e.g., 0.16-0.33 s or 0.9-1.06 s of Figure 2e), resulting in an underestimated \(\hat{K}_{0}(n)\) as compared to the oracle case (Figure 2c). The 0.5 biased \(\hat{K}_{0}(n)\) leaks 50% residual noise to \(\hat{s}(n|n)\) particularly in silent regions (Figure 2h). Additionally, the underestimated \(\hat{K}_{0}(n)\) in the speech regions introduce a significant distortion in the enhanced (Figure 2h). In addition, \(J_{2}(n)\) and \(J_{1}(n)\) metrics (Figure 2f) do not comply with the desired characteristics as found in WGN condition (Figure 1c). Therefore, it is inappropriate to apply \(J_{2}(n)\) and \(J_{1}(n)\) metrics in Figure 2f for tuning of the biased \(\hat{K}_{0}(n)\) (Figure 2c) using Equations (31) and (32).  
In the AKF-RMBT method, the speech LPC parameters were computed from the pre-whitened speech to utilize \(J_{2}(n)\) metric for the tuning of biased \(K_{0}(n)\) in colored noise conditions ([22], Figure 5d). As in [20], \(J_{2}(n)\) metric-based tuning of \(K_{0}(n)\) still produces distorted speech. In addition, the noise LPC parameters computed from initial speech pauses keep constant during the processing of all noisy speech frames for an utterance. The whitening filter was also constructed with the constant noise LPCs to pre-whiten each noisy speech frame prior to compute speech LPC parameters. As a result, the tuning of \(K_{0}(n)\)[22] becomes irrelevant in conditions having time-varying amplitudes, such as babble noise.  
Motivated by the shortcomings of [20, 21, 22], we propose a \(J_{2}(n)\) and \(J_{1}(n)\) metric-based tuning of the KF gain, \(K_{0}(n)\), for speech enhancement in real-life noise conditions.