## 2 Log Likelihood Spectral Distance Measure  
A key problem in signal processing, starting in the 1970s, has been to determine an expression for the difference between the spectra of two different discrete time signals. One distance measure that was proposed by [PERSON] [1] and found to be effective for some speech recognition applications and for comparing the linear prediction coefficients calculated from two different signals is the log likelihood ratio defined by [2; 3; 4; 5]:  
\[d=\log\left[\frac{\mathit{AVA}^{T}}{BVB^{T}}\right], \tag{1}\]  
where \(A=[1,a_{1},a_{2},\ldots,a_{n}]\) and \(B=[1,b_{1},b_{2},\ldots,b_{n}]\) are the augmented linear prediction coefficient vectors of the original signal, \(x(n)\), and processed signal, \(\hat{x}(n)\), respectively, and \(V\) is the Toeplitz autocorrelation matrix [14; 15] of the processed signal, with diagonal components \(r(|i-j|)=\sum_{m=1}^{N-|i-j|}\hat{x}(m)\hat{x}(m+|i-j|),i-j=0,1,\ldots,n\). \(N\) is the number of samples used for the analysis window, and \(n\) is the predictor order [5]. We thus see that \(BVB^{T}\) is the minimum mean squared prediction error for predicting the processed signal, and that \(\mathit{AVA}^{T}\) is the mean squared prediction error obtained when predicting the processed signal with the coefficients calculated based on the original signal [2; 3; 4; 5; 14; 15]. [PERSON] [2] employed the ratio \(\mathit{AVA}^{T}/BVB^{T}\) to determine when to transmit linear prediction coefficients in a variable rate speech coder based on Linear Predictive Coding (LPC). A filtering interpretation corresponding to \(\mathit{AVA}^{T}\) and \(BVB^{T}\) is also instructive [3; 5]. Defining \(A(z)=1+\sum_{i=1}^{n}a_{i}z^{-i}\) and \(B(z)=1+\sum_{i=1}^{n}b_{i}z^{-i}\), we consider \(\mathit{AVA}^{T}\) as the mean squared prediction error when passing the signal \(\hat{x}(n)\) through \(A(z)\), which, upon letting \(z=e^{i\omega}\), can be expressed as \(\mathit{AVA}^{T}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|A(e^{i\omega})\right|^{2 }\hat{X}(e^{i\omega})\right|^{2}d\omega\); also, \(BVB^{T}\) is the mean squared prediction error when passing the signal \(\hat{x}(n)\) through \(B(z)\), which can be expressed as \(BVB^{T}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|B(e^{i\omega})\right|^{2}\hat{X} (e^{i\omega})\right|^{2}d\omega\). Here, \(\hat{X}(e^{i\omega})\) denotes the discrete time Fourier transform of the sequence \(\hat{x}(n)\). We can thus write the log likelihood ratio in (1) in terms of the spectral densities of autoregressive (AR) processes as [3; 5]:  
\[d=\log\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{\left|A(e^{i\omega})\right|^ {2}}{\left|B(e^{i\omega})\right|^{2}}d\omega \tag{2}\]  
where the spectral density \(\left|\hat{X}(e^{i\omega})\right|^{2}\) divides out. The quantity \(d\geq 0\) has been compared to different thresholds in order to classify how good of a spectral match is being obtained, particularly for speech signals. Two thresholds often quoted in the literature are \(d=0.3\), called the statistically significant threshold, and what has been called a perceptually significant threshold, \(d\) = 0.9. The value \(d\) > 0.3 indicates that \"there is less than a 2% chance\" that the set of coefficients \(\{b_{k}\}\) are from an unprocessed signal with coefficients \(\{a_{k}\}\), and thus \(d\) = 0.3 is called the statistically significant threshold [4]. [PERSON] [1,4] found that for the difference in coefficients to be significant for speech recognition applications, the threshold should be 3 times the statistically significant threshold or \(d\) > 0.9. These values were based on somewhat limited experiments, and it has been noted that these specific threshold values are not set in stone, but it was generally accepted that \(d\) > 0.9 is a poor spectral fit [3,4]. Therefore, for speech coding applications, \(d\) > 0.9, is considered an indicator that the codec being evaluated is performing poorly.