In spectroscopic imaging, spin frequency in the rotating frame (ω) is modulated by a magnetic field gradient and chemical shift:

where γ is the gyromagnetic ratio, δ

_{i} is the chemical shift of species

*i*, B

_{0} is the main magnetic field strength, G is the gradient strength and x is the spin position In the multi-band frequency encoding technique, the readout gradient amplitude is set such that the minimum chemical shift separation among species present (Δδ

_{min}) exceeds the gradient field difference across the FOV (i.e. Δδ

_{min}*B*_{0} >

*G · FOV*), whereby unique modulation occurs for all species at all locations, allowing determination of all spectral-spatial components by FE. The FOV here is defined only as the true object FOV, not an extended version containing the entire frequency range. Following Fourier transformation, images corresponding to each spectral component appear side-by-side within different FE bands (). To measure all of these components, the readout filter is opened wider than the conventional imaging setting of γ

*G · FOV* ; instead, the minimum setting is

where δ

_{min} and δ

_{max} are the minimum and maximum chemical shifts among the chemical species present. Due to practical sequence considerations, the readout bandwidth is likely to be the smallest multiple of Δδ

_{min}γ

*B*_{0} exceeding this minimum. During the reconstruction procedure, metabolite images are shifted along x to their proper locations, based on known chemical shift differences. For example, if the maximum value of G is used, the reconstruction shift for metabolite

*i* is (δ

_{i} − δ

_{min})

*N* / Δδ

_{min} pixels (where N= number of pixels across the FOV).

While in conventional imaging the readout gradient strength is usually maximized in order to minimize distortion (and also, to maximize imaging speed), in this method it is typically much lower than maximum. This is acceptable as long as the degree of misregistration and blurring remain small, and the imaging time remains sufficiently short.

The maximum FE bandwidth per pixel depends on the resolution and the minimum chemical shift separation:

which is also equal to the inverse of the minimum readout time for full Fourier sampling, determining the minimum TE and TR (and thus total scan time). B

_{0} misregistration (i.e. in mm per ppm inhomogeneity) scales with

*FOV* /Δδ

_{min}. In this initial implementation for animal experiments as detailed below, the maximum misregistration due to a typical inhomogeneity of ±0.25 ppm is ±1 mm or one-third of a voxel.

If the T

_{2}* relaxation time is on the order of or less than the duration of the readout window, T2* blurring may occur along the readout direction, which could reduce the true spatial resolution in this dimension, and there could also be degradation of SNR. The corresponding k-space filter, which combines this asymmetric exponential decay filter with symmetric windowed sampling of the Fourier data (

13), is

where W is the width of the spatial frequency sampling window, and the function “rect” is defined as a boxcar function equal to 1 when the argument lies between −1/2 and 1/2, and 0 otherwise. After inverse Fourier transformation (see

Appendix), the point spread function is

where Δx is the nominal spatial resolution (equal to 1 / W), and τ

_{read} is the readout time (equal to W / γG). To assess the extent of this effect on the experiments described in this study, this function was computed (Methods) over a range of reasonable representative values of T

_{2}*, and resultant loss of spatial resolution and SNR were estimated.

This method has not been applied for ^{1}H spectroscopic imaging because it would result in excessive distortion due to low FE bandwidth in the presence of B_{0} inhomogeneity. Much larger minimum chemical shift separation in many ^{13}C applications should allow much higher spatial resolution, pixel bandwidth, and imaging speed with this method for ^{13}C. Comparing the minimum spectral separation for ^{1}H MRSI of the brain (choline-creatine, 0.2 ppm) to hyperpolarized ^{13}C studies of [1-^{13}C]pyruvate (pyruvate- alanine, 5.7 ppm), pixel misregistration due to a fixed ppm difference in B_{0} would be 28× lower for ^{13}C (e.g. for a 4 cm FOV, misregistration due to 0.1 ppm inhomogeneity is 20 mm for ^{1}H vs. just 0.7 mm for ^{13}C), and the scan time is also 7 times faster.