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Motivated by 5G application requirements that challenge the use of orthogonal frequency division multiplexing (OFDM), non-orthogonal multi-carriers are being investigated. Unlike OFDM that takes advantage of orthogonal pilot observation, in non-orthogonal waveforms, pilots are contaminated by interference from multiple dimensions, i.e., inter-subsymbol-, inter-carrier-, and inter-antenna-interference, when multipleinput-multiple-output (MIMO) is also part of the transmission. Employing cyclic-prefix (CP) in multi-carrier systems not only protects the signal from inter-symbol-interference but also allows circular interpretations of the channel, which simplifies the estimation and equalization techniques. Nevertheless, the CP information is usually discarded at the receiver side. In this paper, by considering the fact that non-orthogonal waveforms suffer from multiple dimensions of interference, we derive a MIMO linear-minimum-mean-squared-error (LMMSE)-based parallelinterference-cancellation (PIC) method for joint channel estimation and equalization of non-orthogonal waveforms. Unlike the common practice, by properly localizing the pilots in time domain, we also use the pilots' information from CP. We apply our proposed algorithm to a flexible non-orthogonal waveform known as generalized frequency division multiplexing (GFDM). Taking advantage of block-circularity of GFDM, we investigate the complexity aspects for such CP-aided LMMSE-PIC channel estimation. Through simulation results, we show that using CP information of pilots for GFDM gains up to 2.4-dB better frame error rate performance than an OFDM signal.