sdsidebandsplit¶

sdsidebandsplit(outfile='', overwrite=False, signalshift='', imageshift='', getbothside=False, refchan=0.0, refval='', otherside=False, threshold=0.2)[source]

[EXPERIMENTAL] invoke sideband separation using FFT

[Description] [Examples] [Development] [Details]

Parameters
• imagename (pathVec=’’) - a list of names of input images

• outfile (string=’’) - Prefix of output image name

• overwrite (bool=False) - overwrite option

• signalshift (doubleVec=’’) - a list of channel number shifts in signal side band

• imageshift (doubleVec=’’) - a list of channel number shifts in image side band

• getbothside (bool=False) - sideband separation (True) or supression (False)

getbothside = True
• refchan (double=0.0) - reference channel of spectral axis in image sideband

• refval (string=’’) - frequency at the reference channel of spectral axis in image sideband (e.g., “100GHz”)

• otherside (bool=False) - solve the solution of the other side band side and subtract the solution

• threshold (double=0.2) - Rejection limit of solution

Description

Warning

WARNING: This task is EXPERIMENTAL. Interface and capabilities may change frequently.

The task sdsidebandsplit performs a sideband separation operation on data collected by double sideband (DSB) receivers. The task splits the emission from the signal and image sidebands by utilizing the feature that spectral lines in the two sidebands shift in different amounts between observations with different LO offsets. The algorithm used in the task is analogous to that of Emerson, Klein, & Haslam (1979) 1 with shifts in the frequency domain instead of spatial one as described in the paper. The details of algorithm is also discussed in the section "Brief description of the mathematics behind the task", below.

The task takes two or more images as inputs and is able to identify and split the contribution from the signal and image sidebands. The resulting output are separate image(s). When the parameter getbothside=False is set, only the signal sideband is solved for and stored as an image. When getbothside=True, both the signal and image sidebands are obtained and stored separately as two images. The name of output image(s) is defined by outfile and suffixed by ‘.signalband’ and ‘.imageband’ for the signal and image sidebands, respectively.

How to prepare input images

This task can only be used with spectral line data and not continuum. Therefore input images must be appropriately calibrated, for example, by using sdcal (and applycal), and any residual bandpass structure and continuum must be subtracted from the spectral line emmission using sdbaseline. Then an image must be created for each LO offset (e.g., sdimaging). The spatial and stokes coordinates must coincide with each other in the input images. It is recommended to use the frequency setting native to the observation when creating images to avoid adding complexity in the definition of the parameters, signalshift and imageshift. The default frequency parameters in sdimaging (nchan=-1, start=0, and width=1) help to avoid in adding this complexity.

Definition of signalshift and imageshift

Since the input images do not have information on how much the frequency is offset in the spectral window in each observation, sdsidebandsplit relies on user to provide it. Currently, the offset in each image should be defined in the unit of channel numbers of the image. In the future, the task may support other units such as frequency (Hz, MHz, GHz) or velocity (km/s). The parameter, signalshift, must be a list of offset channels of the signal sideband in corresponding elements of imagename, hence the number of elements in signalshift must be equal to that of imagename. The parameter imageshift is the same as signalshift but for the image sideband.

Note

signalshift and imageshift must be defined in the unit of channel numbers in the image. The sdsidebandsplit task relies on these values to shift back the spectra and construct a group of spectra whose signal (or image) sideband contribution are aligned. The solution significantly degrades if the values are inaccurate. It is the user’s responsibility to calculate and provide appropriate numbers of shifts especially in case the frequency coordinate of input images is different from the native observation, for example by regridding and/or by converting frequency frame.

Solution flag: otherside

There are two ways to obtain a spectrum of a sideband of interest in sdsidebandsplit. The parameter otherside allows a user to switch between the image or signal sideband. When solving for the signal (image) sideband with otherside=False, spectra are shifted back to construct a group of spectra in which the signal (image) sideband spectra are static in terms of channel and the spectrum of the signal (image) sideband is solved. When otherside=True, the signal (image) sideband spectrum is obtained by solving that of the other, image (signal), sideband and by subtracting it from the observed spectrum which contains contribution from both sidebands.

Setting otherside=True may have an advantage of removing residual offsets in a spectrum. This is because the current algorithm does not take into account the sideband ratio and the offset component is assigned to the sideband which is originally solved. Therefore, solving with otherside=False doubles the offset components by assigning to both sidebands and breaks the conservation of flux between the original and derived spectra. This is indeed inappropriate but the capability is now exposed for testing purposes. In the future, this should be corrected, for example, by accepting the sideband ratio as an input. Note, setting otherside=True may cause over subtraction. If an emission line in a sideband is strong and wide, it causes significant ghost emission in the solution of the other sideband. When this ghost emission in addition to the offset component is subtracted from the original spectrum (otherside=True), it may cause a negative offset in the derived spectrum.

Frequency definition of image sideband

Since the input images do not have information of the frequency settings of the output image of the image sideband, sdsidebandsplit relies on user inputs when solving for the image sideband (getbothside=True). The frequency information of the image consists of the reference channel in the output image (refpix) and the frequency at the reference channel (refval). The frequency increment is defined as the same amount as that of signal sideband but with the opposite sign. If the frequency increment of the signal sideband is 4880kHz, that of image sideband is defined as -4880kHz. See the Examples tab for a sample use case showing how to specify refpix and refval.

Brief description of the mathematics behind the task

The algorithm to split signals from two sidebands is based on the following criteria:

• The sign of the frequency increment for the image sideband is opposite to that for the signal sideband (Note that “signal sideband” and “image sideband” are the nominal terms that physically correspond to either an upper sideband or a lower sideband so if the increment for one sideband is positive, the other sideband is negative.)

• By shifting the LO frequency, the corresponding sky frequency for each spectral channel is shifted accordingly. Because of the opposite sign of the frequency increment, the amount of shifts in terms of channel occur in opposite directions: if the corresponding channel shift in the signal sideband is positive, the shift for the image sideband is negative.

• In the Fourier (time) domain, the frequency shift is represented as a modulation, which is a multiplication of a sinusoidal wave whose frequency is equal to the amount of the frequency shift.

Suppose that $$h$$ is an output spectrum of DSB system and $$f$$, $$g$$ represent contributions from signal and image sidebands, respectively. Then,

$$h_{m k} = f_{m k} + g_{m k}$$, $$k=0,1,2,...,N-1$$,

where $$k$$ denotes channel index and $$N$$ is a number of spectral channels. If LO frequency shift by x causes $$f_{m k}$$ and $$g_{m k}$$ to shift by $$\Delta^{m x}_{m f}$$ and $$\Delta^{m x}_{m g}$$ with respect to its original spectra, respectively, output spectrum with shift is wrtten as,

$$h^{m x}_{m k} = f_{m k - \Delta^x_f} + g_{m k - \Delta^x_g}$$.

We can shift $$h^{m x}_{m k}$$ as if the contribution from image sideband, $$g$$, is being unshifted. By shifting $$h^{m x}_{m k}$$ by $$-\Delta^{m x}_{m g}$$, we can construct such spectrum,

$$h^{m x,imag}_{m k} = f_{m k - \Delta^x} + g_{m k}$$,

where $$\Delta^{m x} = \Delta^{m x}_{m f} - \Delta^{m x}_{m g}$$. Channel shift in the signal sideband is represented as a modulation in Fourier (time) domain. Thus, Fourier transform of the above is written as,

$$H^{m x,imag}_{m t} = F_{m t} \exp(-i \frac{2\pi t \Delta^{m x}}{N}) + G_{m t}$$,

where $$H^{m x,imag}_{m t}$$, $$F_{m t}$$, and $$G_{m t}$$ are Fourier transform of $$h^{m x,imag}_{m k}$$, $$f_{m k}$$, and $$g_{m k}$$, respectively. Applying similar procedure for the different LO frequency offset, y, we can obtain another result:

$$H^{m y,imag}_{m t} = F_{m t} \exp(-i \frac{2\pi t \Delta^{m y}}{N}) + G_{m t}$$.

we can obtain $$G_{m t}$$, Fourier transform of the contribution from image sideband, $$g_{m k}$$, from the above two results,

$$G_{m t} = \frac{1}{2} (H^{m x,imag}_{m t} + H^{m y,imag}_{m t}) + \frac{1}{2} \frac{\cos\theta}{i\sin\theta} (H^{m x,imag}_{m t} - H^{m y,imag}_{m t})$$,

where $$\theta = 2\pi t (\Delta^{m x} - \Delta^{m y}) / N$$.

There are two ways to obtain the contribution from signal sideband. One is to solve signal sideband exactly same procedure with the above. By doing that, we obtain,

$$F_{m t} = \frac{1}{2} (H^{m x,sig}_{m t} + H^{m y,sig}_{m t}) - \frac{1}{2} \frac{\cos\theta}{i\sin\theta} (H^{m x,sig}_{m t} - H^{m y,sig}_{m t})$$,

where the quantity with superscript “sig” corresponds to the shifted spectrum so that contribution from the signal sideband remain fixed. This is what the sdsidebandsplit does when otherside=True. Another way is to subtract the contribution of image sideband from the output spectrum. If otherside=False, contribution from signal sideband is estimated in that way.

In principle, the task can split contributions from signal and image sidebands if only two images with different LO shifts are given. However, the task accepts more than two images to obtain better result. If $$m$$ images are given and all images are based on independent LO shifts, there are $$m(m-1)/2$$ combinations to obtain the solution of splitted spectra. In that case, the task takes average of those solutions to get a final solution.

Note that, when $$\Delta^{m x}$$ and $$\Delta^{m y}$$ are so close that $$\theta$$ becomes almost zero, the above solution could diverge. Such a solution must be avoided to obtain a finite result. The parameter threshold is introduced for this purpose. It should range from 0.0 to 1.0. The solution will be excluded from the process if $$|\sin(\theta)|$$ is less than threshold.

Bibliography

1

Emerson, Klein, & Haslam 1979, A&A, 76, 92 ADS

Examples

Obtain an image of signal sideband (side band supression):

sdsidebandsplit(imagename=['shift_0ch.image', 'shift_132ch.image',
'shift_neg81ch.image'], outfile='separated.image',
signalshift=[0.0, +132.0, -81.0],
imageshift=[0.0, -132.0, +81.0])


The output image is ‘separated.image.signalband’.

To solve both signal and image sidebands, set frequency of image sideband explicitly in addtion to getbothside=True.

sdsidebandsplit(imagename=['shift_0ch.image', 'shift_132ch.image',
'shift_neg81ch.image'], outfile='separated.image',
signalshift=[0.0, +132.0, -81.0],
imageshift=[0.0, -132.0, +81.0], getbothside=True,
refpix=0.0, refval='805.8869GHz')


The output images are ‘separated.image.signalband’ and ‘separated.image.imageband’ for signal and image sideband, respectively.

To obtain signal sideband image by solving image sideband, set otherside=True:

sdsidebandsplit(imagename=['shift_0ch.image', 'shift_132ch.image',
'shift_neg81ch.image'], outfile='separated.image',
signalshift=[0.0, +132.0, -81.0],
imageshift=[0.0, -132.0, +81.0], otherside=True)


Solution of image sideband is obtained and subtracted from the original (double sideband) spectra to derive spectra of signal sideband. The output image is ‘separated.image.signalband’.

Development

Parameter Details

Detailed descriptions of each function parameter

imagename (pathVec='') - a list of names of input images. At least two valid images are required for processing
outfile (string='') - Prefix of output image name.
A suffix, “.signalband” or “.imageband” is added to
output image name depending on the side band side being solved.
overwrite (bool=False) - overwrite option
signalshift (doubleVec='') - a list of channel number shifts in signal side band.
The number of elements must be equal to that of imagename
imageshift (doubleVec='') - a list of channel number shifts in image side band.
The number of elements must be either zero or equal to that of imagename.
In case of zero length array, the values are obtained from signalshift
assuming the shifts are the same magnitude in opposite direction.
getbothside (bool=False) - sideband separation (True) or supression (False)
refchan (double=0.0) - reference channel of spectral axis in image sideband
refval (string='') - frequency at the reference channel of spectral axis in image sideband (e.g., “100GHz”)
otherside (bool=False) - solve the solution of the other side band side and subtract the solution
threshold (double=0.2) - Rejection limit of solution. The value must be greater than 0.0 and less than 1.0.