We prove that a sufficient and necessary condition for HeiΘ(s) = -ieiΘ(s), where H is Hilbert transformation, Θ is a continuous and strictly increasing function with Θ(R) = 2π, is that d Θ(s) is a harmonic measure on the line. The counterpart result for the periodic case is also established. The study is motivated by, and has significant impact to time-frequency analysis, especially to aspects of analytic signals inducing instantaneous amplitude and frequency. As a by-product we introduce the theory of Hardy-space-preserving weighted trigonometric series and Fourier transformations induced by harmonic measures in the respective contexts.