Number of Channels (Spectral Points)

The correlator design provides 2048 hardware lags for every baseline (1024 lead and 1024 lag multipliers). These lags can be apportioned amongst different basebands and polarization products (up to a maximum of 16 ways) in powers of 2. Each spectrum computed, corresponding to a single baseband/polarization product, would then have between 2048 and 128 lags, resulting in from 1024 to 64 channels (spectral points). This allows one to trade polarization products or basebands for channels (frequency resolution) in a fairly flexible way. One can also obtain an increase in the number of channels by reducing the input bandwidth into a sampler.

When used at the maximum bandwidth, full polarization, the correlator provides 64 spectral points (channels) across each of the 16 products (2 cross-hand and 2 parallel-hand polarization products for each of 4 BB pairs) for every baseline. There would thus be 256 channels across the entire 8 GHz band, corresponding to a resolution of 31.25 MHz.

Polarization products can be traded for channels (on a BB-by-BB basis). Thus one can double the number of channels (halve the frequency resolution) by producing only the parallel-hand products (XX, YY); or even gain a factor of 4, by producing only one of the parallel-hand products (e.g., XX). Similarly, using fewer BBs increases the number of channels proportionately. And finally, halving the bandwidth of each BB increases the number of channels proportionately, up to a maximum of a factor 32.

A few examples should clarify all this.

A.

16 GHz, single polarization: If the IF system could produce 16 GHz of one polarization, say X, in eight 2 GHz basebands, the 2048 correlator lags would be allocated evenly amongst those 8 BBs, giving a total of 1024 channels (128 per BB). The resolution would then be 16 GHz/1024 channels= 2 GHz/128 channels= 15.625 MHz/channel.

B.

8 GHz: Here one has 4 BB pairs, each covering 2 GHz in dual polarization.

• Full pol'n products: For full polarization information, the 2048 correlator lags are divided among 4 BB pairs times 4 polarization products= 16 different spectra, giving 128 lags (64 spectral points/channels) per 2 GHz spectrum. Each 8 GHz spectrum is covered by channels, yielding a resolution of 31.25 MHz/channel.
• XX only: Producing only 1 polarization product gains a factor 4 in the number of channels, because the 2048 lags are now divided only among four 2 GHz basebands, yielding 512 lags (256 channels) per 2 GHz spectrum. The full 8 GHz is covered by channels, each with a resolution of 7.8125 MHz.

C.

4 GHz, full pol'n products: In this case one could split the total bandwidth either between 2 BB pairs, each 2 GHz wide; or 4 BB pairs, each 1 GHz wide.

• BB pairs: With four polarization products the 2048 hardware lags are divided into 8 spectra, giving 256 lags (128 channels) per 2 GHz bandwidth, for a resolution of 15.625 MHz over the full 4 GHz bandwidth.
• BB pairs: With four polarization products the 2048 hardware lags are divided into 16 spectra, giving 128 lags (64 channels) per 1 GHz bandwidth; but halving the bandwidth of each baseband allows one to use the extra correlators (see above) to obtain twice as many channels. So one winds up with 256 lags (128 channels) per 1 GHz bandwidth, for a resolution of 7.8125 MHz over the full 4 GHz bandwidth.

In other words, because of the ability to recirculate signals when the samplers are run below their maximum rate of 4 Gsamples/sec, one always gains by splitting a given total bandwidth among the maximum number of basebands. Whether one would always want to do this, and to what extent this might argue for making the correlator smaller by employing more than eight basebands, is discussed in §3.4.

D.

The highest possible frequency resolution:

• Narrowing the bandwidth per BB: The number of channels goes up by a factor two for every halving of the bandwidth per baseband, up to a factor of 32. So, when using all 4 BB pairs and requesting full (four) correlator products, one can have 64 channels covering 2 GHz, or 128 covering 1 GHz, etc., up to 2048 channels covering 62.5 MHz. In this last case one would have full polarization products for each of channels (spectral points) covering a total of ( at 230 GHz), giving a resolution of 30.5 kHz ( at 230 GHz).
• Giving up BB pairs: The number of channels available for each baseband pair goes up by a factor of two for each halving of the number of baseband pairs. So, one could observe with only two baseband pairs, with a bandwidth of 62.5 MHz per BB pair, and obtain 4096 channels per baseband pair (with full polarization products). The total number of channels would still be 8192, but those would cover only , giving twice the frequency resolution of the last case discussed (i.e., 15.3 kHz).
• Maximizing the number of channels: What is the largest number of channels the correlator can produce, and what is the corresponding frequency resolution? Consider observing with a single 62.5 MHz baseband, producing only one correlator product (e.g., XX). This gains another factor 8 over the above case, yielding 32,768 channels covering 62.5 MHz, for a frequency resolution of . Of course one obtains only a single polarization product and a single baseband, so the total number of spectra (so to speak) has gone down by a corresponding factor. The velocity resolution for this case (at 230 GHz) is , over a total of .
• Even higher resolution: The highest resolution one can achieve is set by the minimum bandwidth presented to the samplers, which in turn in is given by the narrowest filters available in the IF system; in the current design, this is 31.25 MHz (Webber 1998, priv. comm.). Since the limit of recirculation has already been reached, cutting the bandwidth further does not produce more channels; all one gains is the corresponding increase in resolution. For this minimum bandwidth one still obtains 32,768 channels (assuming only a single polarization product is desired), and the resolution is 0.95 kHz per channel.
  Ignoring the IF system for a moment, the correlator itself could give virtually any desired spectral resolution, albeit over a limited bandwidth. So if one wants, e.g., 1 Hz resolution, one can only get a maximum of 32,768 channels (for one BB, one polarization product), so the total bandwidth covered would be 32,768 Hz. Similarly, if full polarization products are needed (which also requires using two BBs), one could cover only 8,192 Hz at 1 Hz resolution.

E.

1 GHz total bandwidth, full polarization products: To maximize the number of channels, one would use 4 BB pairs, each covering 250 MHz. The 2048 hardware lags would be split among 16 spectra, while recirculation would increase the number of channels by a factor 2 GHz/250 MHz= 8, yielding 512 channels (spectral points) per 250 MHz, or a total of 2048 channels over 1 GHz. If on the other hand one required the full 1 GHz within a single baseband pair, for instance to avoid calibration difficulties in splitting a single broad line up into several basebands, recirculation would only give a factor 2 (rather than 8), and one would obtain only 512 channels over 1 GHz.

The astute reader will have noticed that all the tradeoffs discussed so far have involved factors of two (halving the bandwidth or the number of basebands; asking for two rather than four correlator products). This is probably not absolutely necessary, but allowing for other than binary trade-offs would force one to support an even larger number of modes, making the correlator even more complex. So far there has been no compelling scientific argument that the additional flexibility would be worth it.

The correlator modes used to process each baseband or baseband pair can be selected independently. Such (sub-)modes should also be powers of 2 (instead of 7 basebands at one resolution and 1 at another, each sub-mode should use 1/8, 1/4, or 1/2 of the correlator), and to avoid complexity the current design envisions a maximum of four different sub-modes in use at the same time. Some examples of this are:

F.

Cover 4 GHz with full polarization (2 BB pairs), and another 4 GHz with parallel-hand polarization products only (2 BB pairs). Each BB pair has 2048/4=512 hardware lags available.

• The full polarization BB pairs split those into four polarization products, yielding 128 lags 64 channels (spectral points) over each 2 GHz bandwidth.
• The parallel-only BB pairs gain a factor 2, yielding 128 channels over each 2 GHz bandwidth.

So in this mode the correlator would produce full polarization products for channels, and parallel-hand products for another channels.

G.

Use 3 BB pairs to cover 6 GHz, producing full polarization products; use one of the remaining BBs to cover 500 MHz, producing a single polarization product (e.g., XX).

• Each of the three 2 GHz BB pairs has 512 lags available, split amongst four polarization products gives the usual 64 channels over each 2 GHz bandwidth.
• The 500 MHz BB has 512 lags times a recirculation factor of 2 GHz/500 MHz= 4, for a total of 2048 lags available. Producing only a single polarization product, this gives 1024 channels (spectral points) across 500 MHz.

In this way the correlator would produce XX, YY, XY, and YX for each of channels covering 6 GHz, plus e.g. XX alone for another channels covering 500 MHz.

H.

Suppose one wants to do a survey over 500 MHz producing YY only, while observing one transition in dual polarization over 250 MHz and another over 62.5 MHz; meanwhile one wishes also to zoom in on the central MHz of one of these transitions for an experiment requiring all polarization products. This might be organized as follows:

• 500 MHz, YY only: use 2 basebands of 250 MHz, each with lags (the factor 8 comes from recirculation), producing a YY spectrum with channels.
• 250 MHz, XX & YY: use 1 BB pair with lags split between two polarization products, to give two spectra (XX and YY) each having channels (spectral points).
• 62.5 MHz, XX & YY: use another BB pair with lags, again split between two polarization products. From this pair one obtains XX and YY spectra, each with channels.
• 1 MHz, all four pol'n products: The final BB pair also has lags, since a factor 32 is the limit of the gain for recirculation. Those lags are split between four polarization products, giving 2048 channels (spectral points) over each of the four (XX, YY, XY, YX) 1 MHz spectra, for a spectral resolution of 0.5 kHz.

Table 1 summarizes these examples.