# Signal Processing for Longitudinal parameters of the Tevatron Beam \*

S. Pordes^\#^, J. Crisp, B. Fellenz, R. Flora, A. Para, A.V. Tollestrup
, FNAL, Illinois 60510, U.S.A.

Abstract

We describe the system known as the Tevatron SBD \[1\] which is used to
provide information on the longitudinal parameters of coalesced beam
bunches in the Tevatron. The system has been upgraded over the past year
with a new digitizer and improved software. The quantities provided for
each proton and antiproton bunch include the intensity, the longitudinal
bunch profile, the timing of the bunch with respect to the low-level RF,
the momentum spread and the longitudinal emittance. The system is
capable of 2 Hz operation and is run at 1 Hz.

## General Description

A schematic is shown in Figure 1. The pick-up is a wide-band resistive
wall-current monitor (RWCM) \[2\] positioned where the proton and
antiproton bunches are maximally separated (\~200 ns). The signal from
the RWCM is brought from the tunnel and digitized in an oscilloscope
\[3\] located in a service building. The scope trigger is provided by
the low level RF system. The data are read from the oscilloscope over
ethernet and processing is performed in LabView \[4\] running on a
Macintosh G5 computer. Accelerator parameters such as the beam energy
and the RF voltage are read from the accelerator control system and the
longitudinal quantities are returned via ACNet.

## ![](media/image1.png){width="3.248611111111111in" height="1.5833333333333333in"}Signal Processing

### Signal Acquisition

The RWCM has broad (\>2 GHz) bandwidth with a 1.34 ohm resistance formed
by 88 120 ohm resistors across the ceramic gap. We use a Lecroy 6200
oscilloscope as digitizer. The RWCM output is brought to the service
building via 280 ft of 7/8" heliax cable and then split to provide two
copies of the signal, just upstream of the oscilloscope. The signals are
fed to two input channels with a gain ratio of \~8, the present ratio of
proton to antiproton intensities. The split is positioned so that any
reflections from one channel input will arrive at the other channel with
50 ns (2.5 buckets) delay.

The oscilloscope provides 8 bits of resolution. To use this range
efficiently, a DC offset in the scope sets the baseline to \~30 counts
(out of 255). The high gain channel accomodates the antiproton signal
and the proton signal is contained in the low gain channel. In practice,
we synthesize the proton signal from both channels thus improving the
resolution on the proton signal by \~ 8.

The signal is sampled at 5GS/s. To reduce the effect of digitizing
noise, a set of 32 sweeps is taken and averaged (within the 6200). Each
sweep covers 21 usecs, a full Tevatron period, and successive sweeps are
taken every \~ 42 usec, triggered by the Low Level RF proton marker.
(The 1.2 millisecond of acquisition is small compared to the synchrotron
period of the Tevatron.) A second set of sweeps triggered by the
antiproton marker is taken to obtain the antiproton RF timing. The data
from an acquisition (200 kbytes) are transferred via ethernet to the
Macintosh G5 for processing.

### Signal Processing

The (gain dependent) time offset between the channels is determined by a
simple convolution and linear interpolation is used to align the high
gain channel. We have not found any need to correct the gain ratio from
its nominal value. A 200 tap (40 ns) FIR filter is applied to remove the
effects of dispersion in the long cable \[5\]. Figure 2 shows a proton
bunch with two leading and one trailing\`satellites'.The effect of the
FIR filter, in particular the removal of the dispersive tail, is
evident.

![](media/image2.png){width="3.2534722222222223in"
height="2.3340277777777776in"}

Figure 2: a proton bunch signal (raw) and after application of the FIR
filter. The feature at the far right is a 3/4 % reflection. from one
channel through the splitter to the other. Full height of the main bunch
is \~5 amps.

The baseline is found by a histogramming technique in 18 separate
sections as shown in Figure 3.

![](media/image3.png){width="3.25in" height="2.192361111111111in"}

Figure 3: Baseline (solid line) around the ring (note the full range of
the plot is 0.1 Amps).

Aggregate quantities of the beam are derived from an aggregate proton
and an aggregate antiproton pulse constructed by summing the 36 sets of
signals in the 5 buckets centered on the bucket containing the main
bunch. (Linear interpolation is used to compensate for the fact that the
sampling clock has a slightly different relationship to the RF buckets
for the different bunches). The aggregate proton and antiproton bunches
are reproduced below. Note that the peak current varies from 220 amps in
the proton bunch to 0.5 amps in proton satellite -2 and from 27 amps in
the antiproton bunch to the 0.27 amps peak in the single antiproton
satellite.

![](media/image4.png){width="3.252083333333333in"
height="1.7715277777777778in"}

![](media/image5.png){width="2.361111111111111in"
height="1.6840277777777777in"}

Figure 4: The aggregate bunches for protons and antiprotons showing peak
currents and the calculated baseline (horizontal white line).

The leading satellites suggest an error in the baseline of about 0.05
amps compared to the 220 amp peak of the proton current and 25 amp peak
of the antiproton current. The effect of the FIR filter is most evident
in the trailing proton satellite; it seems that we have a slight
overcompensation.

The moments (intensity, centroid, rms width, skewness and kurtosis) for
each bunch are calculated within windows defined by the region where the
appropriate aggregate is positive. The intensity measurement has a
resolution of better than 0.5%; the centroid and the rms width
measurements have a resolution of about 20 ps.

The proton and antiproton bunch centroids under steady state conditions
are shown in Figure 5; the shape is believed to be due to beam loading.

![](media/image6.png){width="3.2527777777777778in"
height="1.6034722222222222in"}

Figure 5: Centroids of the proton and antiproton bunches.

The centroids have proved useful for example in determining instability
modes as shown in Figure 6.

![](media/image7.png){width="3.2534722222222223in"
height="1.9979166666666666in"}

Figure 6: Proton centroids during a mode 1 instability; the full scale
is +/- 0.2 ns.

### Emittance Calculation

A new calculation of the beam momentum spread and longitudinal emittance
has been implemented. Earlier calculations assuming that the beam is
Gaussian in time and momentum spread were in contradiction with the
observed time distribution and implied that there was a significant
change in the longitudinal emittance between 150 GeV and 980 GeV despite
no obvious mechanism for, or symptom of, such a change. A technique
\[6\] that treats any stable phase-space distribution has been developed
by A. Tollestrup and implemented in LabVIEW by A. Para.

The technique depends on the fact that, for stable beam, the particle
density is uniform along an ellipse of constant action in phase space.
The procedure starts by dividing the bucket area into a number of
constant action ellipses such that the difference in phase space between
successive ellipses is the same (Figure 7a). The projection of the phase
space between successive ellipses on the time (or z) axis is shown in
Figure 7b. The SBD time distribution is the sum of these difference
projections each multiplied by the beam density for its particular phase
space interval. The longitudinal emittance distribution is found by
fitting the time distribution to a sum of these projections

![](media/image8.png){width="3.247916666666667in"
height="1.6409722222222223in"}

Figure 7a: Curves of constant action; the area between successive curves
is 10% of the total area.

![](media/image9.png){width="3.2527777777777778in" height="1.5in"}

Figure 7b: projection of the region between successive curves of Fig.
7a.

In practice, 25 regions are used. The difference functions are evaluated
numerically and the values for beam at 150 GeV and 980 GeV are
pre-calculated and stored in the SBD program. Figure 8 shows how the
bunch width, momentum spread and emittance change between 150 GeV and
980 GeV.

![](media/image10.png){width="3.067361111111111in"
height="2.323611111111111in"}

Figure 8:Longitudinal quantities at 150 and 980 GeV.

Figure 9 shows a comparison of the momentum spread as determined by the
1.7 GHz TeVatron Schottky \[7\] with the momentum spread reported by the
SBD. There are no arbitrary scales involved and the agreement is quite
satisfactory except at the time of the beam blow-up.

![](media/image11.png){width="3.25in" height="2.0416666666666665in"}

Figure 9: Comparison of the momentum spread reported by the SBD system
and the TeV Schottky.

## Summary

The Tevatron SBD is now capable of providing reliable values for the
longitudinal parameters of the Tevatron beam. The capability of the
system has been significantly enhanced with its new digitizer, more
sophisticated signal processing and a new algorithm for deriving the
momentum spread and longitudinal emittance. There is a small
overcompensation in the FIR filter and a small bunch shape dependence in
the intensity values which is being worked on.

## Acknowledgements

A. Hahn developed the previous SBD on which this system is based. Little
of the work reported here is due to the first author of this paper. F.
DeJongh has contributed a useful data viewer. Particular acknowledgement
must be made of A.V. Tollestrup who has so stimulated us in this effort
and pushed us to our, but not his, limit.

## References

\[1\] SBD stands for Sampled Bunch Display.

\[2\] B. Fellenz and J. Crisp, **BIW 98** *An improved Resistive Wall
Monitor* & AIP Conf. Proceedings 451 pp. 446-453

\[3\] Lecroy Corporation Waverunner 6200A.

\[4\] LabVIEW - National Instruments Corporation.

\[5\] S. Pordes et al., **PAC 2003** *Measurement of Proton and
Anti-proton Intensities in the Tevatron Collider* IEEE Catalog Number
03CH37423C, pp 2491-2493

\[6\] A.V. Tollestrup, **Fermilab Beams Document Data Base**
(http://beamdocs.fnal.gov/) #541, 548 and 881

\[7\] R. Pasquinelli et al., **PAC2003,** *A 1.7 GHz Waveguide Schottky
Detector System* IEEE Catalog Number 03CH37423C, pp 3431-3433