Several issues related to the practical synthesis of ternary sequences with specified spectra are addressed in this paper. Specifically, sequences with harmonic multiples of two and three suppressed are studied, given their relevance when testing and characterizing nonlinear systems. In particular, the effect of nonuniform digital-to-analog converter (DAC) levels on the spectral properties of the generated signal is analyzed. It is analytically shown that the DAC nonuniform levels result in degraded harmonic suppression performance. Moreover, a new approach is proposed for designing ternary sequences, which is flexible and can be adapted to suit different requirements. The resulting sequences, denoted as randomized constrained sequences, are characterized theoretically by deriving an analytical expression of the power spectral density. Furthermore, they are extensively compared with three synthesis approaches proposed in the literature. The approach is validated by numerical simulations and experimental results, showing the potential to achieve harmonic suppression performance of approximately 100 dB.