Signal Phase Cannot Be Determined from a Fourier Transform of Sampled Data

Abstract

The Discrete Fourier Transform is a well-established tool for obtaining a frequency spectrum from a set of sampled data. The coefficients obtained from the DFT are complex-valued and so both phase and amplitude can be extracted. However, David Gibson asserts that if the sampled data originated in the real world (say, as a recording from a broadband radio antenna) the phase of the spectral components cannot be determined. This, he claims, is an 'obvious' observation, but one that is not emphasised in textbooks. The details of the problem are left for the reader to work out, as an interesting 'examination question' in DSP.