  Knowledge of longitudinal electron bunch profiles is vital to optimize the
performance of plasma wakefield accelerators and x-ray free electron laser
linacs. Because of their importance to these novel applications, noninvasive
frequency domain techniques are often employed to reconstruct longitudinal
bunch profiles from coherent synchrotron, transition, or undulator radiation
measurements. In this paper, we detail several common reconstruction techniques
involving the Kramers-Kronig phase relationship and Gerchberg-Saxton algorithm.
Through statistical analysis, we draw general conclusions about the accuracy of
these reconstruction techniques and the most suitable candidate for
longitudinal bunch reconstruction from spectroscopic data.
