In this paper, we study mixing sequences of modulated wideband converters (MWC). The MWC is a sub-Nyquist sampling system which mixes an input analog signal by multiple numbers of fast mixing sequences in parallel. When the mixing sequences are random cyclic shifts of a base sequence for a memory efficiency, the system turns into random partial Fourier structured MWC (RPFMWC). In RPFMWC, the spectrum distribution of a base sequence is important in reconstruction of the input signal. We show that a reconstruction of the input is guaranteed if and only if all the discrete Fourier transform (DFT) elements of a base sequence are nonzero. The sufficient and necessary conditions allow more flexible lengths for bipolar base sequences compared to existing results. In addition, we propose a measure for evaluating a base sequence in order to predict noisy performance of the RPFMWC.