Accordingly, the decision thresholds and the representation levels of the uniform quantizer, denoted by [Q.sub.u] in the block diagram of the compander shown in Fig.
where N denotes the number of representation levels of the optimal compander.
By using (17) and (18) the decision thresholds [t.sub.m] and the representation levels [y.sub.m], of our optimal compander are then determined as the solutions of the following equations:
Table III shows the results of applying the proposed modified BTC algorithm to an audio signal and its comparison to forward adaptive optimal compander (FAOC) [13], and logarithmic PCM system (LOGPCM) [12] for very close bit rate values.
This type of control over the profile of compander is not possible in the amplitude distribution modification method used in previous works [8] [9][13] [14].
In order to study the relationship between the compander's profile with the amount of OBI, we have used a simple experiment using a transform with one inflexion point:
This means that the compander can be conveniently configured to achieve any PAPR at the output.
The profiles of the proposed transforms for the compander and decompander are shown in Fig.
(7), will now be used to describe amplitude statistics in respective subsets instead of Rayleigh distribution and the compander parameters for each subset will be calculated using these distributions, so the compander parameters will be optimized according to the value of S for symbols in each subset.
Now the compander is designed for each of the J subsets separately using the conditional distributions given in Eq.
If D is a measure of companding distortion for a certain companding function and all of the J companders have similar mathematical form, design specifications and constraints, then, by virtue of the law of total expectation, the over-all mean companding distortion remains same as for the fixed compander.
The adaptive compander will operate by first calculating the statistic S for the input OFDM symbol, then selecting its parameter values from the compander parameter set and then the selected function will transform the signal amplitudes.