Spectral Line Configurations
In semester 2013B observers have access to very flexible subbands within two 1 GHz baseband pairs (using the 8bit samplers). These capabilities may be summarized as follows:
 Two 1 GHz baseband pairs (using the 8bit samplers), independently tunable within the limits outlined in the section on VLA Frequency Bands and Tunability. These baseband pairs are referred to as A0/C0 and B0/D0, and correspond to the IF pairs in the preexpansion VLA.
 Up to 16 subband pairs (spectral windows) in each baseband pair
 Tuning, bandwidth, number of polarization products, and number of channels can be selected independently for each subband
 All subbands must share the same integration time
 No part of a subband can cross a 128 MHz boundary
 Subband bandwidths can be 128, 64, 32, ..., 0.03125 MHz (128 / 2^{n}, n=0, 1, ..., 12)
 The sum over subbands of channels times polarization products is limited to 16,384.
 These may be spread flexibly over subbands and polarization products, in powers of two: 64, 128, 256, 512, ..., 16384 crosscorrelation products.
 Assigning many channels to a given subband may reduce the total number of subbands available.
The remainder of this section discusses the various limitations in more detail, including some examples to show how they come up in practice.
Subband tuning restrictions
Each subband may be placed anywhere within a baseband, with the caveat that no subband may cross a 128 MHz boundary. Mathematically:
ν_{BB0} + n*128 MHz <= ν_{sbLow} <= ν_{sbHigh} <= ν_{BB0} + (n+1)*128 MHz 
where:
ν_{BB0}  the lower frequency edge of the baseband;  
n= 0, 1, ..., 7  (i.e., any integer between 0 and 7);  
ν_{sbLow}  the lower edge of the subband (i.e., the subband center frequency minus half the subband bandwidth);  
ν_{sbHigh}  the upper edge of the subband (i.e., the subband center frequency plus half the subband bandwidth). 
So for example, if the baseband were tuned to cover 1000011024 MHz, one could place a 64 MHz subband to cover 1057010634 MHz, but not to cover 1060010664 MHz (because that would cross the 128 MHz boundary at 10640 MHz).
The figure below illustrates these restrictions:
The black curve shows the analog filter response for an 8bit baseband covering 1024 MHz, starting at ν_{BB0}. The dashed blue vertical lines show the 128 MHz boundaries; no subband can cross those boundaries. 128 MHz subbands are thus constrained to cover a region between two of those boundaries, and no finer tuning is possible. Narrower subbands, like the 64 MHz subband shown here in red, can be shifted around arbitrarily within one of the 128 MHz "slots", but cannot cross any of these boundaries. (The dotted vertical red lines show the boundaries of the 64 MHz subband, while the solid curve shows an illustrative line within the subband.)
The analog filter shape defining the baseband rolls off severely at one edge of the baseband, so the 128 MHz "slot" at that edge has reduced sensitivity. The baseband edge is at the lowest sky frequency in the baseband when using upper sideband, and at the highest sky frequency in the baseband when using lower sideband.
Subband bandwidths & the digital filter response
The bandwidth for each subband may be selected independently, and can be any of 128/2^{n} MHz, for n= 0, 1, ..., 12: 128, 64, 32, 16, 8, 4, 2, or 1 MHz, or 500, 250, 125, 62.5, or 31.25 kHz.
The usable portion of the subband is set by three effects. First, as discussed above, the analog filters which define the baseband are not perfect, leading to lower sensitivity in the 128 MHz near the baseband edge for the 8bit samplers
Second, because the digital filters are not infinitely sharp, the rejected sideband leaks in at both edges of the subband. This leads to additional (aliased) noise, with a factor ~2 increase in the noise at the subband edges, dropping to a few per cent within a few per cent of the subband edge. The precise filter shape and noise increase is a complex but predictable function of the subband bandwidth and the subband tuning.
The third effect stems from the offset frequencies used for sideband rejection in the WIDAR correlator. The local oscillators at the individual antennas are tuned to slightly different frequencies, with those offsets taken out in the correlator. This means that each antenna observes a slightly different sky frequency, and thus some baselines will not give an interesting correlation near one edge of the subband. The maximum frequency shift is currently set to 32*f0, with the fundamental f0 being set to f0= max(25.6 kHz*sbBW/128 MHz, 100 Hz). Here sbBW is the smallest subband bandwidth within the baseband. For the wider subband bandwidths the maximum frequency shift corresponds to <1% of that bandwidth, but for narrower subbands the effect can be severe. For instance, a 31.25 kHz subband has f0= 100 Hz, and a maximum frequency shift of 3.2 kHz  10% of the subband may be lost on some baselines.
Spectral channels and polarization products
Each subband can have a different number of channels and polarization products, subject to two limitations:
 For the i^{th} subband, the number of spectral channels can be:
 64 n_{BlBP,i} with full polarization products (RR,RL,LR,LL)
 128 n_{BlBP,i} with dual polarization products (RR,LL)
 256 n_{BlBP,i} with a single polarization product (RR or LL)
 The sum over all subbands of n_{BlBP,i} must be less than or equal to 64. Equivalently, the sum over all subbands of spectral channels times polarization products is limited to 64x256= 16,384.
Baseline Boards are the boards in the WIDAR correlator where the actual crossmultiplications are done. There are 128 Baseline Boards arranged as 64 Baseline Board pairs (BlBPs). The limitations given here correspond to the capabilities of the individual boards, and the finite number of boards the correlator has.
Limitation #1 corresponds to the following table of the options for subband bandwidth and spectral resolution when using n_{BlBP} Baseline Board pairs for a subband:
Subband Bandwidth and Spectral Resolution Options  

Subband bandwidth & total velocity coverage 
Full polarization products (RR, RL, LR, LL) 64n_{BlBP} spectral channels Channel spacing: 
Dual polarization products (RR, LL) 128n_{BlBP} spectral channels Channel spacing: 
Single polarization product (RR or LL) 256n_{BlBP} spectral channels Channel spacing: 

128 MHz  38400/ν_{GHz} km/s  2000/n_{BlBP} kHz  600/n_{BlBP}/ν_{GHz} km/s  1000/n_{BlBP} kHz  300/n_{BlBP}/ν_{GHz} km/s  500/n_{BlBP} kHz  150/n_{BlBP}/ν_{GHz} km/s 
64  19200  1000 / n_{BlBP}  300 / n_{BlBP}  500 / n_{BlBP}  150 / n_{BlBP}  250 / n_{BlBP}  75 / n_{BlBP} 
32  9600  500 / n_{BlBP}  150 / n_{BlBP}  250 / n_{BlBP}  75 / n_{BlBP}  125 / n_{BlBP}  37.5 / n_{BlBP} 
16  4800  250 / n_{BlBP}  75 / n_{BlBP}  125 / n_{BlBP}  37.5 / n_{BlBP}  62.5 / n_{BlBP}  18.75 / n_{BlBP} 
8  2400  125 / n_{BlBP}  37.5 / n_{BlBP}  62.5 / n_{BlBP}  18.75 / n_{BlBP}  31.25 / n_{BlBP}  9.375 / n_{BlBP} 
4  1200  62.5 / n_{BlBP}  18.75 / n_{BlBP}  31.25 / n_{BlBP}  9.375 / n_{BlBP}  15.625/n_{BlBP}  4.687 /n _{BlBP} 
2  600  31.25 / n_{BlBP}  9.375 / n_{BlBP}  15.625/n_{BlBP}  4.687 / n_{BlBP}  7.8125 / n_{BlBP}  2.344 / n_{BlBP} 
1  300  15.625/n_{BlBP}  4.687 / n_{BlBP}  7.8125 / n_{BlBP}  2.344 / n_{BlBP}  3.906 / n_{BlBP}  1.172 / n_{BlBP} 
0.5  150  7.8125 / n_{BlBP}  2.344 / n_{BlBP}  3.906 / n_{BlBP}  1.172 / n_{BlBP}  1.953 / n_{BlBP}  0.586 / n_{BlBP} 
0.25  75  3.906 / n_{BlBP}  1.172 / n_{BlBP}  1.953 / n_{BlBP}  0.586 / n_{BlBP}  0.977 / n_{BlBP}  0.293 / n_{BlBP} 
0.125  37.5  1.953 / n_{BlBP}  0.586 / n_{BlBP}  0.977 / n_{BlBP}  0.293 / n_{BlBP}  0.488 / n_{BlBP}  0.146 / n_{BlBP} 
0.0625 
18.75  0.977 / n_{BlBP}  0.293 / n_{BlBP}  0.488 / n_{BlBP}  0.146 / n_{BlBP}  0.244 / n_{BlBP}  0.073 / n_{BlBP} 
0.03.25  9.375  0.488 / n_{BlBP}  0.146 / n_{BlBP}  0.244 / n_{BlBP}  0.073 / n_{BlBP}  0.122 / n_{BlBP}  0.037 / n_{BlBP} 
Subband bandwidth and spectral resolution options. Note that the table entries refer to the spacing between spectral channels  that spacing is before any frequency smoothing, so these channels are not independent.

Here are four examples of allowed general observing setups which use all 64 BlBPs to produce the maximum number of channels times polarization products:
Baseband  Subband 
Pol'n Products 
Spectral channels 
n_{BlBP}  
Example 1  A0/C0  sb0  RR  16384  64 
Example 2  A0/C0  sb0  RR  8192  32 
A0/C0  sb1  RR, LL  1024  8  
A0/C0  sb2  RR, LL  512  4  
B0/D0  sb0  RR, LL  2048  16  
B0/D0  sb1  RR,RL,LR,LL  256  4  
Example 3  A0/C0  sb0  RR  8192  32 
A0/C0  sb1  LL  1024  4  
A0/C0  sb2  RR, LL  1024  8  
A0/C0  sb3  RR,RL,LR,LL  1024  16  
A0/C0  sb4  RR,RL,LR,LL  256  4  
Example 4  A0/C0  sb07  RR,RL,LR,LL  64  8 x 1 
A0/C0  sb8  RR, LL  4096  1 x 32  
B0/D0  sb07  RR,RL,LR,LL  64  8 x 1  
B0/D0  sb8  RR, LL  2048  1 x 16 
Number of subbands and Baseline Board stacking
The number of subbands available in a baseband depends on the correlator resources available. This means that using Baseline Board stacking to increase the number of channels in one subband, may limit the number of subbands available in one or more of the basebands. Understanding how this works requires understanding some of the details of the correlator hardware. That understanding is built into the GOST, and observers may simply use that tool to find out whether their particular setup will, in fact, work. But the results can be confusing without some understanding of the hardware constraints from which they arise. These hardware constraints are complex, and most observers will not need to understand these details. This section is for those few who are attempting complex line experiments, and who find the GOST or the RCT restricting the number of subbands and/or channels they can use in unexpected ways.
First let us consider how the correlator hardware is organized. The crossmultiplications in the WIDAR correlator are spread across 64 Baseline Board (BlB) pairs, arranged into 4 quadrants of 16 BlB pairs each. Each baseband is connected directly to one of those quadrants. In the simplest mode, each of the 16 BlB pairs of a quadrant handles the correlations for one of the 16 subbands of the corresponding baseband. Four basebands and four quadrants are required, in order to handle the full 8 GHz of bandwidth per polarization provided by the 3bit (wideband) samplers: that 8 GHz is split into four basebands of 2 GHz each, with each baseband fed into a different BlB quadrant. Each BlB pair in that quadrant handles a subband of maximum bandwidth 128 MHz, so 16 BlB pairs handles 16 subbands for a total of 16x128 MHz= 2048 MHz.
A single BlB pair produces 256 crosscorrelations per baseline for a single subband, which can be used for a single polarization product (e.g., RR with 256 spectral channels), or two (RR+LL with 128 spectral channels each), or four (RR,RL,LR,LL, with 64 spectral channels each).
Baseline Board (BlB) pair usage for a simple 4 x 2 GHz (3bit sampler) experiment. The different colors correspond to different baseband pairs. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup each subband is correlated using a single BlB pair, using all of the 64 available BlB pairs. 
When using the 8bit samplers, the total bandwidth is only 2 GHz per polarization, split into two basebands of 1 GHz each. The simplest continuum setup uses only two quadrants, since there are only two basebands; and only 8 subbands are required to span the 8x128 MHz= 1024 MHz of each baseband. Threequarters of the correlator BlB hardware remain unused.
Baseline Board (BlB) pair usage for a simple 2 x 1 GHz (8bit sampler) experiment. The cyan boxes correspond to the first baseband pair (IF A0/C0), fed directly into Q1; the yellow boxes to the second (IF B0/D0), fed directly into Q3. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup each subband is correlated using a single BlB pair, using only 16 of the 64 available BlB pairs. 
The spectral line mode allows access to these `extra' correlator resources through Baseline Board stacking: using multiple BlB pairs to process the same subband and produce more crosscorrelations for that subband. This is done using crossbar switches which make the data for a single subband available to several BlB pairs. Those BlB pairs can then be used to produce more spectral channels for that subband, with n BlB pairs producing 256*n crosscorrelations per baseline. The limit on the total number of crosscorrelations (16384) stems from the total number of BlB pairs (64): 64x256= 16384.
Unfortunately completely flexible crossbar switches are expensive, and could not be implemented in the VLA's new correlator. This means that one cannot route a given subband to a randomlychosen BlB pair. The routings which are possible, are as follows:
 A subband in a baseband can be routed to any BlB pair within the corresponding quadrant.
 Data coming into a given BlB pair in one quadrant, can be routed to the corresponding BlB pair in any other quadrant.
Routing option #1 means that one could use all the BlB pairs within a quadrant to correlate a single subband, yielding 16x256= 4096 crosscorrelations for that subband:
Baseline Board (BlB) pair usage when assigning all BlB pairs within a quadrant to a single subband within the corresponding baseband. The cyan boxes correspond to the first baseband pair (IF A0/C0), fed directly into Q1; the yellow boxes to the second (IF B0/D0), fed directly into Q3. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup each subband is correlated using 16 BlB pairs, yielding 16x256= 4096 crosscorrelations for those subbands, and using 32 of the 64 available BlB pairs. 
Routing option #2 means that one could use the BlB pairs in all 4 quadrants to correlate a single subband. One simple case would use 4 BlB pairs to correlate each of the 16 subbands in a single baseband, yielding 4x256= 1024 crosscorrelations for each of those subbands. Note that in this case, no BlB pairs are left to correlate any data from the second baseband.
Baseline Board (BlB) pair usage when assigning the BlB pairs in all 4 quadrants to correlate the corresponding subbands in a single baseband. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the baseband. In this setup each subband is correlated using 4 BlB pairs, yielding 4x256= 1024 crosscorrelations for those subbands, and using all of the 64 available BlB pairs. 
Using routing option #2 does come with a subtle cost: assigning a BlB pair in quadrant X to correlate a subband corresponding to quadrant Y, removes that BlB pair from use in the baseband corresponding to quadrant X...and hence also removes the corresponding subband in that baseband. So getting more channels for a subband in one baseband, may prevent the use of a subband in a different baseband. To take a simple example, consider an experiment where one wishes to observe a single line in dual polarization with 512 channels (requiring 4 BlB pairs), plus as much continuum bandwidth as possible. Naively one would say there are 16 subbands in each baseband; one is used for the spectral line, so that leaves 16+15= 31 subbands, and with the widest subband bandwidth (128 MHz) the total available continuum should be 31x128 MHz= 3968 MHz per polarization. Actually however there are only 15+15 subbands available, or 30x128 MHz= 3840 MHz per polarization, because the spectral line subband has "eaten" one BlB pair corresponding to the other baseband:
Baseline Board (BlB) pair usage for a continuum plus single line experiment. The cyan boxes correspond to the first baseband pair (IF A0/C0), fed directly into Q1; the yellow boxes to the second (IF B0/D0), fed directly into Q3. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup subband #15 of A0/C0 is correlated using 4 BlB pairs, yielding 4x256= 1024 crosscorrelations for that subband, while the other 15 subbands in each baseband are correlated using a single BlB pair each. The use of 4 BlB pairs for a subband in baseband A0/C0 has eliminated the corresponding subband of the other baseband (B0/D0). Note that only 30+4= 34 of the 64 available BlB pairs have been used  there is plenty of correlator hardware available, but the signal for the spectral line subband cannot be routed to them. 
If the same spectral line required twice as many channels, this result in the loss of two subbands in both of the basebands:
Baseline Board (BlB) pair usage for a continuum plus single line experiment. The cyan boxes correspond to the first baseband pair (IF A0/C0), fed directly into Q1; the yellow boxes to the second (IF B0/D0), fed directly into Q3. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup subband #14 of A0/C0 is correlated using 8 BlB pairs, yielding 8x256= 2048 crosscorrelations for that subband, while the other 14 subbands in each baseband are correlated using a single BlB pair each. The use of 8 BlB pairs for a subband in baseband A0/C0 has eliminated one additional subband in baseband A0/C0, and two subbands in B0/D0. Note that only 28+8= 36 of the 64 available BlB pairs have been used  there is no way to route the signals from the "missing" subbands to that hardware. 
In some cases one may want to use a different routing, to use up subbands in one baseband in preference to another. For instance, the same spectral line setup (2048 crosscorrelations for a single spectral line subband, plus as much continuum as possible) could be set up to allow 13 "continuum" subbands in the A0/C0 baseband, and the full 16 "continuum" subbands in B0/D0:
Baseline Board (BlB) pair usage for a continuum plus single line experiment. The cyan boxes correspond to the first baseband pair (IF A0/C0), fed directly into Q1; the yellow boxes to the second (IF B0/D0), fed directly into Q3. The BlB quadrants are indicated along the lefthand side, while the numbers across the top represent the BlB pairs within those quadrants. Numerals within the colored boxes label the subbands within the basebands. In this setup subband #13 of A0/C0 is correlated using 8 BlB pairs, yielding 8x256= 2048 crosscorrelations for that subband, while the other 13 subbands in A0/C0 and 16 subbands in B0/D0 are correlated using a single BlB pair each. In this setup the use of 8 BlB pairs for a subband in baseband A0/C0 has eliminated two additional subbands in baseband A0/C0, while retaining all 16 subbands in B0/D0. Note that only 29+8= 37 of the 64 available BlB pairs have been used. [Shared risk observations can assign a number of BlB pairs which is not a power of two, and hence could assign one additional BlBP (Q4BlBP 15) to the spectral line subband.] 
Understanding these confusing constraints can help observers set up the VLA more effectively to achieve their scientific goals. For instance, for a mixed line+continuum experiment it works best to use the resource tools to set up the baseband tunings and subband channelization for the most important lines first, then add the desired continuum, and then see what correlator resources remain for any lines of secondary interest. The Shared Risk Observing (SRO) programs provide even more options (see the ones for SRO and RSRO) and hence benefit even more from a clear understanding of the correlator hardware.
Once again, the GOST implements all of these constraints, and is generally smart enough to figure out the routing scheme that works best for your particular request.