To reduce both the complexity and the delay of demodulation and decoding for Multi-h Continuous Phase Modulation(Multi-h CPM) signal，a Soft Output Viterbi Algorithm with Low Complexity(LC-SOVA) is proposed，which can be utilized to M-ary Multi-h CPM signal with partial phase response. Based on the tilt phase-frequency(FPT) and pulse truncation low complexity demodulation of Multi-h CPM signal, the symbol soft information in the Viterbi states is stored and updated by discarding the smaller path to reduce the computational complexity. In the proposed algorithm, the soft information update process can be processed in parallel with half computation compared with the existing SOVA algorithm, and the time delay is equivalent with that of the Viterbi algorithm. The simulation results show that the proposed LC-SOVA algorithm is generally applied to any Multi-h signal and its performance degradation can be neglected compared to the Max-log-MAP algorithm.