# Speckle Noise Detection and Removal for Laser Speech Measurement Systems
## 3 Automatic Noise Detection and Removal
### Decorrelation  
A LP model predicts the future value of a signal from a linear combination of its past values. LP models are used for several applications. The correlation structure of a signal can be modelled using a linear predictor by taking the amplitude of the signal at time \(m\) (\(x(m)\)), and then using a linearly-weighted combination of \(P\) past samples (\(x(m-1)\), \(x(m-2),\ldots\)\(x(m-P)\)):  
\[\widetilde{x}(m)=\sum\limits_{k=1}^{P}a_{P}(k)*x(m-k) \tag{1}\]  
where the integer variable \(m\) is the discrete time index, \(\widetilde{x}(m)\) is the prediction of \(x(m)\) and \(a_{P}(k)\) are predictor coefficients. The linear predictor coefficients can be calculated by using the Levinson-Durbin algorithm [26].  
The linear prediction model for a signal with an error estimation, \(e(m)\), can be expressed as:  
\[e(m)=x(m)-\widetilde{x}(m) \tag{2}\]  
Assuming that a clean speech signal \((x)\) is corrupted by a random additive speckle noise \((n)\). The detected signal is given by:  
\[s=x+n \tag{3}\]  
From Equations (1) and (2), we can rewrite the noisy signal model:  
\[s(m)=\sum\limits_{k=1}^{P}a_{P}(k)*x(m-k)+e(m)+n(m) \tag{4}\]  
In practice, as there is no clean speech signal \(x\), we use the noisy speech \(s(m)\) to calculate an estimate \(\hat{a}_{N}\) of the predictor coefficient vector \(a_{N}\). This can then be used to invert and transform the noisy signal \(s(m)\) to the noisy excitation signal \(v(m)\) as:  
\[\begin{split} v(m)=s(m)-\sum\limits_{k=1}^{P}\hat{a}_{P}(k)*s(m- k)&=x(m)+n(m)-\sum\limits_{k=1}^{P}[a_{P}(k)-\hat{a}_{P}(k)]*[x(m-k)+n(m -k)]\\ &=e(m)+n(m)+\sum\limits_{k=1}^{N}\widetilde{a}_{P}(k)*x(m-k)+ \sum\limits_{k=1}^{P}\hat{a}_{P}(k)*n(m-k)\end{split} \tag{5}\]  
where \(\widetilde{a}_{P}(k)\) is the error in the predictor coefficient estimate. According to [PERSON]'s [27] analysis of extracted signals, there are four basic terms that contribute to the noise in an excitation sequence:  
(a) the error estimation \(e(m)\);  
(b) the speckle disturbance \(n(m)\), which is usually the dominant term;(c) the effect of the past noise samples, run over into the present time by the action of the inverse filtering;  
(d) the increase in the variance of the excitation signal, caused by the error in the parameter vector estimate, and expressed as: \(\sum_{k=1}^{P}\hat{a}_{P}(k)*x(m-k)\).  
As \(n(m)\) is usually the dominant term in a noisy excitation signal \(v(m)\), when a detected speech signal is converted to an excitation signal, the relative energy of the noise in the signal is increased. In other words, the NSR is increased. Before decorrelation, the NSR of a noisy signal is given by:  
\[\frac{\textit{power of speckle noise}}{\textit{power of signal}}=\frac{E \big{[}n^{2}(m)\big{]}}{E\big{[}x^{2}(m)\big{]}} \tag{6}\]  
where \(E\) is the expectation operator. After applying the inverse filter, the NSR is expressed as:  
\[\frac{\textit{power of speckle noise}}{\textit{power of excited signal}}=\frac{E \big{[}n^{2}(m)\big{]}}{E\big{[}v^{2}(m)\big{]}} \tag{7}\]  
The overall gain of the NSR can be obtained by:  
\[gain=\frac{E\big{[}x^{2}(m)\big{]}}{E\big{[}v^{2}(m)\big{]}} \tag{8}\]  
This simple analysis demonstrates that an improvement in speckle noise detectability depends on the characteristics of the power amplification of the linear predictor model and the associated resonances. Figure 5 shows a comparison between a raw measured signal and a decorrelated signal. It can be seen that the speech signal is largely removed and the speckle noise in the signal is highlighted.  
Figure 5: Comparison of (**a**) raw-detected signal and (**b**) extracted-signal by decorrelation.