SMA Technical Memo Number 112

Basic Correlator Configurations

Eric Keto

July 29, 1997

This memo outlines the basic capabilities of the SMA correlator from an observer's viewpoint. To exploit the full capability of the correlator it is necessary to understand a little about the hardware so the correlator will be explained in the context of the hardware. There are plans for extended capabilities for very high spectral resolution. These modes have not been finalized and their description is deferred to a future memo.

Receivers

The SMA will initially have three receivers: two low frequency receivers covering 177 - 256 GHz and 250 - 350 GHz, and a high frequency receiver covering 600 - 720 GHz. There are future plans to add three additional receivers, possibly a duplicate 250 - 350 GHz receiver for cross-polarization, and additional high frequency receivers covering 400 - 520 and 780 - 920 GHz. With the initial 3 receivers, it will be possible to use the high frequency receiver together with either of the two low frequency receivers, but the two low frequency receivers will not be used together. Each receiver can observe a maximum bandwidth of about 4 GHz in double sideband mode, which is to say 1.968 GHz in each of the upper and lower sidebands for a combined total bandwidth of about 8 GHz. The 2 GHz limit is set by the bandwidth of the correlator.

Frequency Coverage

The first IF has a bandwidth from 4 to 6 GHz, set by an analog filter. So the sideband separation is fixed at 10 GHz. Each of the 2 GHz bands is divided into 6 sub-bands of 328 MHz each. These 6 sub-bands may be moved in frequency, for example to cover different spectral lines, and each sub-band may have a different spectral resolution. These movable sub-bands, generally called "blocks" in the SMA technical memos, are defined in the hardware by a combination of an LO to move the sub-band and a square flat-topped filter to define the 328 MHz passband. The LO's driving these sub-bands are what astronomers call the second LO's in the IF system, the first LO's being associated with the mixers in the receivers. (In the SMA engineering documentation, these second LO's are sometimes referred to as the first correlator LO's.) Although there are different first LO's for the two receivers, the same six second LO's are used for both receivers. Thus both receivers will always have the same pattern of sub-bands across its own 2 GHz bandwidth, although the 2 GHz bands for each receiver will cover different frequencies. There is one further restriction in that three of the six sub-bands must be in the upper 1 GHz half of the 2 GHz band and the other three must be in the lower half. This restriction is set by the so called pre-selection filters which are used for image rejection.

Spectral Resolution

Each of the movable 328 MHz sub-bands is further divided into four sub-bands of 82 MHz each. These are not movable within the 328 MHz, but are significant in that the correlator lags may be distributed among the four 82 MHz bands to obtain different spectral resolutions. The 82 MHz bands, generally called "chunks" in the SMA technical memos, are defined by an LO, the third LO, which is not adjustable by the observer, and a flat-topped square filter. Because the filter is not exactly square, the bands are actually 104 MHz wide and stacked with some overlap across the 328 MHz sub-band. The 82 MHz width represents the usable flat-topped part of the filter passband. In calculations to determine the spectral resolution, one must use the full 104 MHz width since the correlation operations for each sub-band are done over the full 104 MHz. The 6 element SMA has a total of 1536 correlator chips. This will be increased for the planned 8 element array. Each chip has 128 lags available for correlation operations. Because the visibility is a complex number, 2 lags are required for each correlation. (The SMA correlator calculates complex numbers by computing the correlation with both positive and negative lags. The result of the Fourier transform which takes the data from lag space to frequency space is then in general a complex number.) The lags can be allocated to the individual 82 MHz sub-bands in many different configurations. For example, suppose we are observing with 6 antennas and two receivers, and we wish to cover the full 2 GHz of available bandwidth in each sideband (8 GHz total including both sidebands and both receivers). We require 24 sub-bands of 82 MHz placed end-to-end to cover the 2 GHz sideband of receiver 1 and another 24 for receiver 2. We then have 48 sub-bands of 82 MHz and 15 baselines to correlate. With the same spectral resolution in all sub-bands, we have 1536/(15*48) = 2 chips per sub-band or 128 complex lags. Remembering that each 82 MHz sub-band is really 104 MHz wide to allow for overlap, the spectral resolution will be 104 MHz/128 = 812.5 kHz. (See the note at the end of this memo for a clarification of spectral resolution.) In this case 64*48 = 3072 spectral channels will be correlated for each baseline, although only 2422 channels will be recorded because of the overlaps between the 104 MHz sub-bands.

Figure 1. Correlator setup to cover the maximum bandwidth.

The previous example allowed for the maximum bandwidth. Let us now determine the basic configuration for high spectral resolution. In the basic correlator configurations, it is possible to allocate up to 16 chips to one 82 MHz sub-band. This is because there are 32 chips on each correlator board, and each board is responsible for two baselines, two receivers, and one movable 328 MHz sub-band (block) made up of four 82 MHz sub-bands (chunks). If we use only one receiver and one 82 MHz sub-band per board, we have 16 chips to allocate to each baseline and each 82 MHz sub-band. There are 1024 complex lags available from the 16 chips, so the spectral resolution will be 104 MHz/1024 = 102 kHz. In this case the bandwidth is 6*82 MHz = 492 MHz because we can stack six 82 MHz sub-bands end to end, one from each of the six 328 MHz movable sub-bands.

Figure 2. Basic correlator setup for high spectral resolution over reduced bandwidth.

In the basic correlator configurations one can allocate 1,2,4,8, or 16 chips to an 82 MHz sub-band until all of the 1536 chips are used, keeping in mind the constraint that each board of 32 chips serves two receivers, two baselines, and one 328 MHz sub-band of data. For example, in the high resolution case above, this constraint means that we can use only 1440 of the total of 1536 chips. It is possible to achieve higher resolution with the SMA correlator if the correlation calculations are extended across boards. Ultimately, it will be possible to obtain a maximum resolution of 25 kHz. However, these so called special correlator modes will not be available initially. They will be described in a future memo.

Hanning smoothing and cross-polarization measurements will each reduce the spectral resolution by a factor of two. Hanning smoothing averages together adjacent channels to reduce the ringing caused by the finite extent of the time lags. Cross-polarization requires twice as many correlations, XX, YY, XY, and YX, as single polarization measurements in order to determine the 4 Stokes parameters needed to fully describe the polarization.

Note on the Definition of Spectral Resolution

In the examples we have computed the spectral resolution as the bandwidth divided by the number of lags which is the same as the inverse of the time delay in the correlator. This relation is a simple, useful definition for the spectral resolution and exactly equivalent to the Rayleigh criterion for the spectral resolution of a grating spectrometer or a Fourier transform spectrometer (FTS). Briefly, the Rayleigh criterion says that two lines, as seen convolved with the point spread function of an instrument, are resolved if the maximum of one line falls on the first null of the second. This is an approximate definition valid for any device such as a grating or FTS whose response is described by a function of the form,

Since the first zero occurs when the argument of the sin term is equal to , we have the relation . The Rayleigh criterion yields the same resolution for a spectrometer which measures intensity as for a digital spectrometer which derives the power spectrum as the Fourier transform of the correlation function. In the first case the response is characterized by a , while in the latter case by the unsquared sinc. This property is not true of other descriptions of the resolution such as the full width at half maximum (FWHM). For example, if the spectrometer measures the intensity, the FWHM of the sinc function squared is . But for a digital spectrometer, the FWHM of the unsquared sinc is . This is numerically similar to the well known relationship for the Rayleigh criterion for a circular aperture, . However, this latter is the full width between first nulls of the Airy function .

The Rayleigh criterion is used here in the description of the resolution of the SMA correlator because of the simplicity of the result . Knowing that this is the adopted definition, it is a simple matter to apply a factor of 1.2 or 20% to move to the FWHM description. As Richard Feynmann remarks, ``In any case it seems a little pedantic to put such precision into the resolving power formula. This is because Rayleigh's criterion is a rough idea in the first place.'' Indeed it is. As a mathematical statement, if the question is simply whether a diffracted image is made by one source or two, the answer to that question can always be determined no matter how broad the convolving function. But in real instruments, noise and digitization will make any figure of merit approximate.