We present a proof of principle method to reconstruct nuclear magnetic resonance spectra after perturbing the spin system with a series of Rabi modulated continuous wave excitations. This continuous wave method provides an exciting alternative to pulsed Fourier transform methods which dominate magnetic resonance techniques. Applications include the measurement of ultra-short relaxation samples, which would be beneficial for sodium imaging or the assessment of bone and connective tissues. It is known that under Rabi modulated continuous wave excitation the spin system will reach a substantial periodic orbit. In a first experiment, we confirm that these periodic orbits are affected by off-resonance effects, and therefore encode chemical shift information. Spectroscopy is posed as an inverse problem in a second experiment and we apply a nonlinear estimation model to obtain nuclear magnetic resonance spectra from experimental measurements.