Let the new proposed companded signal also follow Rayleigh distribution in the interval [0, [gamma]' [sigma]] and uniform distribution in the interval [[gamma]' [sigma], G'], where the parameter [gamma]' is defined as the ratio between the companding threshold and the root mean square level of the MCM signal,
Correspondingly, the PDF of the amplitude of the companded signal is given by
The CDF of the amplitude of the companded signal can be then represented as
On the other hand, the power of companded signal can be expressed as
Accordingly, the companded signals can be recovered correctly through the corresponding inversion function.
where [PAPR.sub.C] is the PAPR of the companded signal.
Using the Bussgang theorem [19][20], the companded OFDM signal can be represented as the sum of attenuated replica of the original signal and uncorrelated distortion, i.e.,
It is also shown in [19] that [alpha] is time invariant and equal to the correlation coefficient of original and companded signals.
3(b) and 3(c) show the power spectral density (PSD) plots of the companding noise, D = T (x) - x, and those of the companded signal, respectively.
This means that the waveform of the companded signal will closely follow that of the original signal.
* The compressed amplitudes in the companded signal are larger than [a.sub.m] in the original signal.
7 shows empirical curves of CCDF of PAPR for original and companded signals.