A PPM/PM Hybrid Modulation System

Abstract

The objective of a pulse-modulation (PM) system is to yield as high as possible an output signal-to-noise ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0}</tex> for a given channel signal-to-noise ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0}</tex> and bandwidth expansion factor β. In particular, one would like to design a modulation system in which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0}</tex> approaches that of the Shannon bound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0} \leq [1 + (S/N)_{c}\beta^{-1}]^{\beta}</tex> . For a fixed system such as the pulse-position modulation (PPM), the relationship between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{c}</tex> , must be linear above threshold. Thus to approach the β-power behavior of the Shannon bound, we must consider a family of PM systems. A class of systems is analyzed in which phase modulation of varying modulation indices is amplitude modulated by a PPM signal of fixed bandwidth. I t i s shown that when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{c}</tex> , is large, the performance is proportional to the square of the modulation index which can be increased without changing the signal bandwidth. However, as the modulation index is increased, more sidelobes appear in the output of likelihood receiver which lead to large so-called anomolous errors. The effect of the sidelobes can be decreased significantly by redesigning the PPM signal since its autocorrelation function amplitude modulates the sidelobes. A new upper bound on the mean-squared error is derived which takes into account the effects of the noise at these sidelobe peaks. The bound depends functionally on the waveform which is used in the PPM modulator, and this raises the question as to which signal is the best one to use. An optimization problem is formulated using state-space techniques and solved numerically using a known algorithm. It is shown that it is possible to reduce the threshold significantly using optimally designed waveforms. The Barankin bounds are applied to the PPM/PM waveform to give lower bounds on the mean-squared error. It is therefore possible to bracket the actual performance region of the PPM/PM modulation system. The lower bounds lead to upper bounds on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(S/N)_{0}</tex> and a sequence of bounds, each representing the best performance possible for a given modulation index, is obtained. The envelope of these bounds is then compared with the Shannon bound, and it is shown that although the PPM/PM modulation does not have performance closely approaching the Shannon bound, it does represent a substantial improvement over straight PPM.