## 6 Experimental Results
### G.726: Adaptive Differential Pulse Code Modulation  
Adaptive differential pulse code modulation (ADPCM) is a time-domain waveform-following speech codec, and the International Standard for ADPCM is ITU-T G.726. G.726 has the four operational bit rates of 16, 24, 32, and 40 Kbps, corresponding to 2, 3, 4, and 5 bits/sample scalar quantization of a prediction residual. ADPCM is the speech coding technique that was first associated with the log likelihood ratio; so, it is an important codec to investigate initially. Figure 1 shows the \(D\) values as a function of frame number, with the perceptually and statistically significant thresholds superimposed where possible, for the bit rates of 40, 32, 24, and 16 Kbps. Some interesting observations are possible from these plots. First, clearly the log likelihood varies substantially across frames, even at the highest bit rate of 40 Kbps. Second, even though designated as \"toll quality\" by ITU-T, G.726 at 32 Kbps has a few frames where the log likelihood ratio is larger than even the perceptually significant threshold. Third, as would be expected, as the bit rate is lowered, the number of frames that exceed each of the thresholds increases, and, in fact, at 16 Kbps, only the  
perceptually significant threshold is drawn on the figure, since the statistically significant threshold is so low in comparison. Fourth, the primary distortion heard in ADPCM is granular quantization noise, or a missing sound, and in lowering the rate from 32 to 24 to 16 Kbps, an increase in the hissing sound is audible, and a larger fraction of the log likelihood values exceeds the perceptually significant threshold. It is useful to associate a typical spectral match to a value of \(D\), so in Figures 2-4, we show the linear prediction spectra of three frames of speech coded at 24 Kbps: one frame with a \(D\) value less than the statistically significant threshold, one with a \(D\) value between the statistically significant and the perceptually significant threshold, and one with a \(D\) value greater than both thresholds. In Figure 2, \(D=11\), considerably below both thresholds, and visually, the spectral match appears to be good. In Figure 3, we have \(D=36.5>24\), and the spectral match is not good at all at high frequencies, with two peaks, called fornants, at higher frequencies, poorly reproduced. The linear prediction (LP) spectrum corresponding to a log likelihood value of \(D=104.5>72\) is shown in Figure 4, where the two highest frequency peaks are not reproduced at all in the coded speech, and the spectral shape is not well approximated either. Figures 2 and 4 would seem to validate, for these particular cases, the interpretation of \(D=24\) and \(D=72\) as statistically and perceptually significant thresholds, respectively, for the log likelihood ratio. The quality of the spectral match in Figure 3 would appear to be unsatisfactory and so the statistically significant threshold is also somewhat validated; however, it would be useful if something more interpretive or carrying more of a physical implication could be concluded for this \(D\) value. Figure 1: \(D\) values for G.726 at various bit rates for “We were away a year ago”. : Statistically significant threshold. : Perceptually significant threshold. Motivated by the variation in the values of the log likelihood ratio across frames, we calculate the percentage of frames that fall below the statistically significant threshold, in between the statistically significant and perceptually significant thresholds, and above the perceptually significant threshold for  
Figure 4: Comparison of LP spectra of original and 24 Kbps G.726 “We were away a year ago” where \(D=104.5\). Figure 3: Comparison of LP spectra of original and 24 Kbps G.726 \"We were away a year ago” where \(D=36.5\). Figure 2: Comparison of linear prediction (LP) spectra of original and 24 Kbps G.726 \"We were away a year ago” where \(D=11\).