# Multimode Tree-Coding of Speech with Pre-/Post-Weighting +
## 14 Comparison with Rate Distortion Bounds  
Rate distortion functions provide the optimal performance theoretically attainable (OPTA) for compressing a given source with a selected distortion measure [41]. To derive such bounds, we need a good source model of a physical source and a meaningful distortion measure. Both have been stumbling blocks to obtaining useful rate distortion bounds.  
Models averaged over many sentences do not allow such bounds to be developed because an average source cannot produce a lower bound since one of more of the sources in the average will almost certainly have performance that is better than the average bound developed. Thus, a new model needs to be obtained for each sentence or utterance considered. Additionally, the distortion measure challenge is that it must be analytically tractable and physically meaningful. Most rate distortion functions have been derived for mean squared error or Hamming distortion measures.  
[PERSON] and [PERSON] used existing rate distortion functions for autoregressive sources for mean squared error distortion but extended those results using composite source models [29] and conditional rate distortion theory [43] to generate good source models for speech [5,44]. For the distortion measure, they experimentally obtained a mapping from mean squared error into PESQ-MOS. Thus, the need for an accurate source model and a meaningful distortion measure were both addressed.  
Three models for each sentence are obtained. One model is averaged over the entire utterance, one model that has two modes, voiced and silence, and a third model that has 5 modes, voiced, unvoiced, silence, onset, and hangover. The conditional rate distortion bound is derived for each subsource, along with the probability of that subsource, and these conditional rate distortion bounds are combined as indicated by conditional rate distortion theory [5,43].  
The mapping from MSE to PESQ-MOS is generated for each utterance using waveform codecs which try to track the time domain waveform but also allow the PESQ-MOS to be found. The waveform codecs need to have a range of coding rates so a mapping can be obtained. G.727 is one such codec and is one of the codecs used by [PERSON] and [PERSON] [5].  
The details of producing the composite source models and the MSE to PESQ-MOS mappings are left to the monograph by [PERSON] and [PERSON] [5]. We use their approach here to develop rate distortion bounds for two of our sentences, \"A lathe is a big tool\" and \"We were away a year ago\".  
Figures 15 and 16 show the PESQ and rate performance of G.726, G.727, G.728, and MMT with weighting and pitch predictor, and the rate distortion bounds for three different  
Figure 14: The operational rate distortion performance of narrowband speech am1s03 (Male Speaker).  
source models. In Figure 15, compared with the G.726 PESQ-rate performance curve, MMT with weighting and pitch predictor is about 0.8 bit/sample lower than G.726 when the PESQ are comparable. Compared with the rate distortion bounds using the 5 modes source model, MMT-WP is still 1.5-2 bits/sample above the theoretical rate distortion bound.  
Results for G.728 and G.728 with VAD/CNG are also shown in Figures 15 and 16. We see that G.728 at 2 bits/sample achieves a PESQ-MOS of near 4.0 for both sentences. When VAD/CNG is used with G.728, 0.5 bit/sample is saved for \"lathe\" as seen in Figure 15, but from Figure 16, there is no reduction in rate when using VAD/CNG with G.728 because \"we were away\" is an all-voiced sentence.