demonstrated in [@Shvydko-SB; @SLK06], the angular dispersion in the hard x-ray regime can be achieved by Bragg diffraction from asymmetrically cut crystals or from special arrangements of asymmetrically cut crystals [@SSM13; @Shvydko15], which are a hard x-ray analog of the optical diffraction gratings and optical prisms. In the space-domain echo-spectrometer proposed here, an image of a point-like x-ray source is defocused by a dispersing system comprised of asymmetrically cut Bragg diffracting crystals. The defocused image is refocused into a point (echo) in a time-reversal dispersing system. We show, if the defocused beam is inelastically scattered from a sample, the echo signal acquires a spatial distribution, which is a map of the energy transfer spectrum in the scattering process. The spectral resolution of the echo spectroscopy does not rely on the monochromaticity of the x-rays, thus ensuring strong signals along with a very high spectral resolution.

In the present paper, we use an analytical ray-transfer matrix approach and the dynamical theory of x-ray diffraction in crystals to calculate and analyze the performance of a generic echo spectrometer comprised of defocusing and refocusing dispersing elements, to derive conditions for refocusing and expressions for the spectral resolution of the echo spectrometer. Specific designs 