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The problem of synthesizing a time-domain signal with a given energy spectral density (ESD) often arises in the field of signal processing. Many solutions have been proposed and successfully used over the years. However the problem of synthesizing a time-domain signal with constraints for a given ESD has not been investigated sufficiently. We propose a solution to one such constraint where the amplitude of the complex-valued time-domain signal is required to be unity. This is equivalent to phase modulating a unit amplitude signal, such that its ESD matches a desired energy spectral density. We provide an algorithm for this solution and apply it to a real problem encountered in radars.