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In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampledsignals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimatorsof Fourier transforms of the analysed signals. ; The estimators are unbiased inside arbitrarily wide frequency ranges,regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, wecalculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator isderived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrumestimation. A couple of numerical examples illustrate the main thesis of the paper.