Rotationally-invariant concatenated (turbo) TCM codes

We provide analysis of the rotational properties of turbo coded trellis modulation (TTCM) systems in various structures and with different signal constellation rotational symmetries. Design rules for making both parallel and serial concatenated turbo coding systems rotationally invariant are given. Generally TTCM rotational invariance requires that one or two of its underlying constituent finite-state encoders be rotationally-invariant depending on the TTCM system structure. Based on state transition diagrams, we provide a method to construct rotationally invariant constituent trellis encoders for M-QAM signal constellations. We also prove that any rotationally invariant encoder has an underlying state transition graph that falls into the framework of our method.