Imaging spectroscopy on the basis of measurements of both intensity and spatial relationships among several spectral-line features (e.g., emission-line ratios or emission-region morphology) is a powerful tool to extract physical information from starlight. Astronomical sources are typically faint, implying that instrument sensitivity is limited by electronics noise at modest spectral resolution (*R *≡ λ/Δλ∼10^{3}, where λ is the wavelength of the incident light. Spectral resolution refers to how finely the wavelength axis is sampled.) As a consequence, a key performance goal for astronomical-spectrometer development is to sample—in a detector-noise-limited context—an information volume (consisting of two spatial coordinates on the sky and a wavelength scale) with high efficiency in terms of both time and detector use. The resulting data products are often referred to as hyper- or multispectral datacubes. Several instrument designs and observing techniques produce such datacubes. However, these methods typically offer a multiplex advantage along only two axes of the information volume, and hence their efficiency suffers.

**Figure 1. **Datacube of the Trapezium stellar cluster, perpendicular to the instrument's slit. East is left and wavelength increases to the top. The bright Pβ spectral line appears as a horizontal dark line. The four prominent stars of the Trapezium system are depicted as four vertical strips. (

See video.)

Hadamard transformation^{1} enables significant reductions in the errors on a large number of measurements if they can be grouped effectively. Like the discrete Fourier transform, the Hadamard transform (*H*) can be expressed as a matrix and is symmetric. Unlike for the former, the *H* elements can only be +1 or −1. *H* is almost its own inverse, i.e., *H*·*H*=*nI*. Thus, if data (*D*) is taken with *D*=*H*·*S*, where *S* is some measure of the sky, then the sky's intrinsic properties can be recovered through *S*=*H*·*D/n*, which requires *n* observations. Therefore, if the observational noise is dominated by photon noise, Hadamard transformation is just as fast as when observing each point in sequence. However, if the measurements are readout-noise dominated, the variance of *H*·*D* and the corresponding signal are both greater by *n*. Hence, the signal-to-noise ratio is improved by .

Hadamard transforms can be applied to any set of 4*n* elements. However, its use is particularly simple for data sets containing 2^{k}elements (where *k* is an integer value). Hadamard matrices of order 2^{k} can be obtained recursively through

By initializing *H*_{1}=[1], an entire series of Hadamard matrices for order 2^{k} can be calculated.

Although *H* can only contain +1 and −1, shutters or mirrors can only be on or off (i.e., 1 or 0). Of course, instruments could be built to take advantage of this by recording both outputs at once. Nevertheless, even without a symmetric readout, the data can still be collected by obtaining two observations for each row of *H*, one for the +1 and one for the −1 elements. This approach reduces the measurement efficiency by a factor of two, however. The −1 observations can then be subtracted from the +1 data to yield the input for the Hadamard transform.

For the first row, all elements are +1. All other rows contain equal numbers of elements of +1 and −1. The observation strategy can still use + and − observations, but for the first observation the − observation is a dark exposure, made with all shutters closed. With a matrix of the proper order, one can open a row of the micromirror array for each +1 and close one for each −1. For the next observation, one can apply the reverse approach. Similar matrices exist that employ only the numbers 1 and 0, but these have approximately the same overall efficiency as the Hadamard-matrix-based observing strategy. The *H* elements are obtained by subtracting pairs of observations, which has the added advantage of effectively canceling any common error such as a mirror (or shutter) stuck in either the open or closed position, light leakage, and detector offsets. Finally, the pairs of observations can be added rather than subtracted. This provides an independent data set that should be identical for all pairs of observations, allowing for uncertainty estimation as well as providing a powerful data set to hunt for and possibly correct systematic errors and other instrumental effects.

IRMOS

The Infrared Multi-Object Spectrometer (IRMOS) was developed as a pathfinder for the James Webb Space Telescope's (JWST) Near-Infrared Echelle Spectrograph (NIRSpec). Although it uses reflective micromirrors instead of the microshutters adopted for JWST, IRMOS serves as a testbed for observational strategies using a highly selectable slitmask implemented with microelectromechanical-systems (MEMS) technology. It was designed for mounting on either the 2 or 4m telescopes at Kitt Peak National Observatory (KPNO).^{2,3} The heart of IRMOS is a Texas Instruments^{™} digital-micromirror-device array consisting of 848×640 16μm mirrors, spaced by 17μm each. Four passbands are available (*Z*, *J*, *H*, and *K*, centered at ~0.9, 1.2, 1.6, and 2.2μm, respectively) with three resolutions (*R*=300, 1000, and 3000) but only R1000 and R300 are used. IRMOS is now available to the astronomical community at the KPNO 4m telescope.

A set of Hadamard 128 10s observations was collected, each using a different combination of open mirrors. We subsequently differenced each pair, thus creating a datacube of 64×1024×1024elements. The observation number is given by the first index, the rows correspond to the east-west direction on the sky, and the columns to a combination of wavelength and north-south direction. The datacube is, in essence, a matrix of 64×1048576 (64×1024^{2})elements. We multiplied the datacube by the Hadamard matrix *H*_{64}, resulting in a data cube of 64×1024×1024 cells, but now the first index corresponds to the north-south direction. Rows still correspond to the east-west direction, and the third index corresponds to the frequency of the spectra. However, because the spectra are not aligned, the prominent H i (neutral hydrogen) Pβ(3–5) spectral line was used to generate a wavelength template. The peak positions are found for all spectra, which are then fit biquadratically to within one pixel. The Pβ(3–5) line was used for convenience. (Any night-sky or calibration-lamp lines observed in all spectra could be used.)

The point-spread function is several pixels across while the spectra were smeared by ~7 pixels since seven rows of mirrors were used for each element of the Hadamard transform. To reduce the final datacube, the data is combined in groups of 2×2 pixels, thus setting both the frequency axis and the scale in one of the spatial directions. The data are interpolated in the third direction to recover the same scale in the other sky direction. This leaves a datacube containing 253×512×512 elements. Figure 1 shows the datacube from one of the axes [video].

Thus, using the Hadamard observation mode meets the promise of 3D multiplex advantage, allowing up to 50% duty observation by all pixels in the telescope's view. In contrast to an image-slicing integral-field unit, Hadamard spectroscopy provides integral-field spectroscopy over very wide fields using the detector the same way as in a broadband imager. It implicitly makes the base-offset correction and corrects for the usual flat-field distortion, bad pixels, cosmic-ray hits, and detector errors. With proper observational selection, the resulting data is robust, even allowing for the catastrophic loss of an entire observation set (with modest measurement degradation). The Hadamard transform has software support at the 4m KPNO telescope equipped with IRMOS. We also hope to use this approach for NIRSpec observations when JWST becomes operational.

Dale Fixsen

Department of Astronomy

University of Maryland

College Park, MD

and

Goddard Space Flight Center

Greenbelt, MD

Matthew Greenhouse, John Mather

Goddard Space Flight Center

Greenbelt, MD

John MacKenty

Space Telescope Science Institute

Baltimore, MD