typically less dominant at the higher frequencies and are thus easier (though still challening) to handle for CO or \[CII\] intensity mapping experiments. However, such experiments must also contend with the problem of interloper lines, where two spectral lines of different rest wavelengths may redshift into the same observation band, leading to confusion as to which spectral line has been observed.

In addition to astrophysical foregrounds, instrumental systematics must be well-controlled for a successful measurement of the cosmological signal. Among others, these systematics include beam-forming errors [@neben_et_al2016b], radio frequency interference [@offringa_et_al2013; @offringa_et_al2015; @huang_et_al2016], polarization leakage [@geil_et_al2011; @moore_et_al2013; @shaw_et_al2014b; @sutinjo_et_al2015; @asad_et_al2015; @moore_et_al2015; @kohn_et_al2016], calibration errors [@newburgh_et_al2014; @trott_and_wayth2016; @barry_et_al2016; @patil_et_al2016], and instrumental reflections [@neben_et_al2016a; @ewall-wice_et_al2016a; @thyagarajan_et_al2016].

In this paper, we focus specifically on measurements of the power spectrum $P(k)$ of spatial fluctuations in brightness temperature, where roughly speaking, the temperature field is Fourier transformed and then squared. In diagnosing the aforementioned systematics as they pertain to spatial fluctuation experiments, it is helpful to decompose the fluctuations into modes that separate purely angular fluctuations from purely radial fluctuations from those that are a mixture of both. In recent years, for example, simulations and measured upper limits of the $21\,\textrm{cm}$ power spectrum have often 