Infant Cry Classification Using Modified Group Delay Cepstral Coefficients

Abstract

Classification of pathological vs. normal infant cries is used to infer the infant’s health conditions. Such an approach can be beneficial in many situations and even to save infants’ lives. In this paper, we propose a novel classification system based on the Modified Group Delay Cepstral Coefficients (MGDCC), for classifying infant cries. We investigate generalizability of proposed MGDCC features. The Convolutional Neural Network (CNN) was used as a pattern classifier in this study. Proposed MGDCC features are found to perform better than widely used spectral features, such as Mel Frequency Cepstral Coefficients (MFCC), Linear Frequency Cepstral Coefficients (LFCC), and Group Delay Cepstral Coefficients (GDCC). Experiments are performed on two datasets namely, Baby Chillanto (D1) dataset, and DA-IICT Infant Cry (D2) corpus and for various experimental evaluation factors, such as noise robustness under signal degradation conditions, cross-database scenario, and analysis of latency period. We obtained 2.25% increase accuracy as compared to existing optimal accuracy for proposed task. Better performance of MGDCC is may be due to its capability to implicitly capture time dependencies in the sequence of audio samples via fourier transform phase information.

Appendix: Noise Robustness of MGDCC

Let clean $_{\text {signal }}(x)$ be a clean signal, degraded by adding uncorrelated, additive noise (x) with 0 mean and $\sigma ^2$ variance. Then, the noisy $_{\text {signal }}(x)$ can be represented as,

$$\begin{aligned} \operatorname {noisy}_{\text {signal }}(x)=\text { clean }_{\text {signal }}(x)+\operatorname {noise}(x) . \end{aligned}$$

(7)

Obtaining the power spectrum, and taking the Fourier transform, we get,

$$\begin{aligned} P_{\text {noisy }}\left( e^{j \omega _0}\right) =P_{\text {clean }}\left( e^{j \omega _0}\right) +P_{\text {noise }}\left( e^{j \omega _0}\right) . \end{aligned}$$

(8)

Two frequency regions, which are mutually exclusive (higher and lower SNR), can be obtained from Eq. (8). For the scenario of lower signal-to-noise ratio (SNR), we examine frequencies $\omega _0$ satisfying $P_{\text {clean }}\left( e^{j \omega _0}\right) \ll \sigma ^2\left( \omega _0\right)$, while for higher SNR, we focus on frequencies $\omega _0$, where $P_{\text {clean }}\left( e^{j \omega _0}\right) \gg \sigma ^2\left( \omega _0\right)$ [29]. For low SNR, we have:

$$\begin{aligned} P_{\text {noisy }}\left( e^{j \omega _0}\right) =\sigma ^2\left( \omega _0\right) \left( 1+\frac{P_{\text {clean }}\left( e^{j \omega _0}\right) }{\sigma ^2\left( \omega _0\right) }\right) . \end{aligned}$$

(9)

Solving Eq. (9), and neglecting higher order terms, we get:

$$\begin{aligned} \ln \left( P_{\text {noisy }}\left( e^{j \omega _0}\right) \right) \approx \ln \left( \sigma ^2\left( \omega _0\right) \right) +\frac{1}{\sigma ^2\left( \omega _0\right) }\left[ d_0+\sum _{x=1}^{+\infty } d_x \cos \left( \frac{2 \pi }{\omega _0} \omega _0 x\right) \right] . \end{aligned}$$

(10)

Equation (10) can be further solved and GDF can be obtained as mentioned in [29]:

$$\begin{aligned} \tau \left( e^{j \omega _0}\right) \approx \frac{1}{\sigma ^2\left( \omega _0\right) } \sum _{x=1}^{+\infty } x d_x \cos \left( \omega _0 x\right) . \end{aligned}$$

(11)

Similarly for higher SNR, we have:

$$\begin{aligned} P_{\text {noisy }}\left( e^{j \omega _0}\right) = P_{\text {clean }}\left( e^{j\omega _0}\right) \left( 1 + \frac{\sigma ^2\left( \omega _0\right) }{P_{\text {clean }}\left( \omega _0\right) }\right) . \end{aligned}$$

(12)

Taking the logarithm on both sides of Eq. (12) and using the Taylor series expansion results in expanded term as:

$$\begin{aligned} \ln \left( P_{\text {noisy }}\left( e^{j \omega _0}\right) \right) \approx \frac{d_0}{2}+\frac{\sigma ^2\left( \omega _0\right) e_0}{2}+\sum _{x=1}^{+\infty }\left( d_x+\sigma ^2\left( \omega _0\right) e_x\right) \cos \left( \omega _0 x\right) . \end{aligned}$$

(13)

Equation (13) can be solved to GDF and the term obtained can be represented as [29]:

$$\begin{aligned} \tau \left( e^{j \omega _0}\right) \approx \sum _{x=1}^{+\infty } x\left( d_x+\sigma ^2\left( \omega _0\right) e_x\right) \cos \left( \omega _0 x\right) \end{aligned}$$

(14)

The respective GDF for these cases (Eq. (11), and Eq. (14)) summarized and represented as follows [29]:

$$\begin{aligned} \tau _{G D F}\left( e^{j \omega _0}\right) \approx \left\{ \begin{array}{l} \frac{1}{\sigma ^2\left( \omega _0\right) } \sum \nolimits _{x=1}^{+\infty } x d_x \cos \left( \omega _0 x\right) , \text { for lower SNR}, \\ \sum \nolimits _{x=1}^{+\infty } x\left( d_x+\sigma ^2\left( \omega _0\right) e_x\right) \cos \left( \omega _0 x\right) , \text { for higher SNR}, \end{array}\right. \end{aligned}$$

(15)

The Fourier series coefficients of $\ln \left( P_{\text {noisy }}\left( e^{j \omega _0}\right) \right)$ and $\frac{1}{P_{c l e~\hbox {a} n}\left( e^{j \omega _0}\right) }$ are denoted by $d_x$’s and $e_x$’s, respectively. Equation (15) reveals that in the lower SNR scenario, the GDF is inversely proportional to the noise power, suggesting that the GDF effectively preserves peaks and valleys amidst additive noise. Conversely, for higher SNR values, the GDF is proportional to noise power, although the noise power is lower than the signal power. These findings imply that the GDF tracks the signal spectrum rather than the noise spectrum.