The method of discrete Fourier transform (DFT) is applied to obtain the exact one- or bi-magnonic states in several finite one-dimensional spin systems. The advantage of DFT method, compared to Bethe ansatz, is that no assumption is made on the form of the wave function and that its limit of infinite system reduces to the standard approach to excitation spectra in condensed matter. So the method has, in principle, no limitation on lattice dimensionality and its physical interpretation is relatively transparent. It is demonstrated that the two approaches (DFT and BA) give identical results for the solution of the Schrodinger equation on the 1D lattice, although the structure of the methods is rather different. The excitation spectrum of the XXZ chain with arbitrary end fields is analyzed in detail and an analogy with atomic wires is briefly discussed.