### FAST PIBETA TRIGGER - AN UPDATE

```From: Dinko Pocanic
Date: 7 Feb 1997

The problem:
-----------

As you all know, the LRS 2373 memory lookup unit (MLU) currently used in
our fast trigger logic, has shown unsatisfactory timing characteristics,
i.e., there is a significant jitter between input and output signal timing
in the ``overlap'' mode.  For this reason, we have decided to substitute
the MLU with conventional units.

Current use of the MLU:
----------------------

MLU's are used in two logic steps in the formation of our fast trigger:

(1) in making up the pi-beta coincidence and the e-nu OR, and

(2) in making up the final trigger from the available gate signals and
calorimeter logic signals (pi-beta-HI, pi-beta-LO, e-nu-HI, e-nu-LO, etc.).

The latter function (2) can be accomplished in a straightforward way by
using the LRS 2365 instead of the MLU.  Below I discuss the formation of
the calorimeter logic for a pi-beta or e-nu event (both HI and LO).

Labeling:
--------

We start by labeling all of the superclusters according to their central
pentagon with numbers 0 - 9.  The numbering is chosen such that the
supercluster directly opposite SC-0 is SC-5, that opposite SC-1 is SC-6,
and so on.

Pi-beta event logic:
-------------------

Our accepted pi-beta event logic has been to require a coincidence between
a specific supercluster and a logical OR of all non-neighboring
superclusters (the complementary hemisphere).  This relaxed requirement is
due to having a broad unbiased sample of the non-pi-beta background.  The
lax requirement could, conceivably, be tightened, if necessary, for higher
rate running.  However, if we want to measure radiative decays with a broad
angular correlation coverage, we need to preserve this option.  The
corresponding logic matrix is given below (all numbers correspond to
superclusters):

0*0_conj = 0*(9 + 5 + 6 + 2 + 3)  -
1*1_conj = 1*(5 + 6 + 7 + 3 + 4)   |
2*2_conj = 2*(6 + 7 + 8 + 4 + 0)   |  for SC's in the ``northern'' hemisphere
3*3_conj = 3*(7 + 8 + 9 + 0 + 1)   |
4*4_conj = 4*(8 + 9 + 5 + 1 + 2)  -

5*5_conj = 5*(4 + 0 + 1 + 7 + 8)  -
6*6_conj = 6*(0 + 1 + 2 + 8 + 9)   |
7*7_conj = 7*(1 + 2 + 3 + 9 + 5)   |  for SC's in the ``southern'' hemisphere
8*8_conj = 8*(2 + 3 + 4 + 5 + 6)   |
9*9_conj = 9*(3 + 4 + 0 + 6 + 7)  -

To form the pi-beta trigger, we OR logically all of the ANDs of i*i_conj.

This logic matrix as written has the property of symmetry, i.e., if we
require, we can scale and monitor individual coincidences of the type
i*i_conj.  However, if we don't have an MLU to make all of the logic ANDs
for us, we must reduce the redundances found above.  In the expressions
listed above you will find 50 pairwise ANDs, of which only 25 are unique.

There are several ways to remove the redundances.  I have found one to be
the most economical that I could come up with.  I encourage other
collaborators to come up with alternate solutions.  The immutable fact
remains that, one way or another, we have to make 25 pairwise ANDs.

D.P.'s minimal (asymmetric) solution equivalent to the solution above with
no redundances:

0*0_conj' = 0*(9 + 5 + 6 + 3)  -
1*1_conj' = 1*(5 + 6 + 7 + 4)   |
2*2_conj' = 2*(6 + 7 + 8 + 0)   |  for SC's in the ``northern'' hemisphere
3*3_conj' = 3*(7 + 8 + 9 + 1)   |
4*4_conj' = 4*(8 + 9 + 5 + 2)  -

5*5_conj' = 5*(7 + 8)  -
6*6_conj' = 6*(8 + 9)   |         for SC's in the ``southern'' hemisphere
7*7_conj' = 7*9        -

We note that the north-south symmetry is completely gone.  However, there
is still full symmetry (for rate monitoring) of coincidences referenced to
one hemisphere, given that the conjugate OR's are reduced from 5 to 4
elements each.

One significant feature of this solution is that the final OR involves only
8 signals (pairwise ANDs) that will turn out to save us 2 modules.

D.P.'s suggested implementation:
-------------------------------

We form the minimal (asymmetric) conjugate OR's, as well as a straight OR
of all SC's (the e-nu signal) in one unit of LRS 2365.  Thus, TWO such
units are required (one for the HI, the other for the LO signals).

The final pairwise AND's as well as the overall OR to form the pi-beta
logic condition are all done in ONE unit of LRS 4516 for both pi-beta-HI
(input/output channels 1-8) and pi-beta-LO (input/output channels 9-16).

This scheme requires an additional split of the individual SC signals.  For
that purpose we need TWO modules LRS 4518 (or 4518/100).  These are common,
and we already own a number.  We need to ascertain whether or not new ones
are needed.

One concern is that the two eightfold OR's of the LRS 4516 are indicated as
NIM outputs in the LeCroy catalog.  I have not had a chance to confirm
this.  If the outputs are indeed NIM, we need to do one of two things: (i)
run them to a NIM/ECL converter and then into the master trigger unit LRS
2365, or (ii) get another 4564 and OR the individual outputs in ECL.

Either way, my estimate of the total propagation time is that we will wind
up well within the total of ~75 ns required by the present chain of two
MLU's.  In our critical path are: one 4518 (or not, if we can use the
shaped outputs of the 4564's), one 2365 (12 ns prop. time), one 4516 (12
ns), one NIM/ECL conv. or another 4564 (12ns) and, finally another 2365 (12
ns).  Adding some cable delays, I come up with about 50 ns, vs. the current
~80 ns.

The wiring of the supercluster 4564's can be done to our advantage.  We can
get pairwise AND's of directly opposite clusters (restrictive pi-beta ANDs)
if we wire the units appropriately (e.g., A=0, B=5, C=1, D=6, and so on),
since the 4564's give as outputs: A*B, C*D and A*B+C*D.  This way we can
monitor the rates of restrictive pi-beta coincidences on line.  We may also
wish to form an on-line restrictive pi-beta event bit.

One issue is that without external adjustment the timing of the SC-i and
SC-i_conj at the inputs of LRS_4516_1 will be off by about 12ns + 1-2ns due
to the extra cable length.  This could be solved in several ways:

(1) The inelegant way by having ~15 ns long twisted pair cables going from
one set of 4518_1 outputs to the 4516_1.  This should work in principle -
we just have to measure the correct delay due to the 2365 before cutting
the cables.

(2) The expensive way, by using an intermediate LRS 4518 unit to give us
solution has the advantage of giving us two extra copies of each SC signal.

(3) The cheap, elegant way: use the shaped outputs of the 4564's to feed
into the 2365's.  Then, the proper delay is set in the 4518_1 once and for
all.  The catch: we have to ensure that the shaped outputs of the 4564
update correctly so that we do not loose the proper logic condition in case
of pulses closely spaced in time.  This requires a test.

A crude sketch (reflecting my poor mastery of xfig) of the corresponding
circuit diagram is provided below.  Comments and criticism are invited.
``` In case you have trouble with the above bitmap image, here's a Postscript version of the same figure.

D. Pocanic, 7 Feb. 1997.