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The phaseless techniques have been discussed in the antenna measurements community and the theories behind these techniques are well explained in literature. The issue of the noise and the presence of measurement errors are not investigated in details to provide strong impetus to the importance of phaseless measurements. In this paper the near-fields of a number of different types of antennas with high, medium and low side lobes are simulated to create some realistic cases. The probe positioning error effects are investigated by implementing random errors in the position of the simulated probe samples along different axes. A novel method is also adopted to incorporate the probe-height positioning error in an actual near-field measurement of an array antenna. It is also illustrated how the positioning errors can distort the phase distributions. Through detailed characterizations of the constructed far-field patterns, robustness of the Iterative Fourier technique even at the presence of very high probe positioning errors is demonstrated. It is shown how the utilization of a phaseless technique can significantly reduce the effects of probe positioning errors. The results are compared with the results extracted from the commonly used amplitude and phase near-field measurement techniques and the clear improvements are illustrated.