# Combining Multiple BPM Measurements for Precession AC Dipole Bump Closure Page: 4 of 5

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methods are not strictly DFT/FFT methods. As a matter

of fact the BPM magnitude and phase used by software

calculations used a sine/cosine fit, which can be thought

of as the evaluation of the discrete time Fourier transform

(DTFT) or z-transform at a single frequency.

o-

--

-9-

Figure 4: BPM spectrums of weakest (blue), strongest

(red), average (green), weighted average (black) and

vector average (purple) for AC dipole #4 at 90 Apk

(76 G-m) for PP at injection (23.8 GeV).

.2-

-15- -

3a- - - -

-35-

Figure 5: BPM spectrums of weakest (blue), strongest

(red), average (green), weighted average (black) and

vector average (purple) for AC dipole #4 at 117.9 Apk

(100 G-m) for PP at store (255 GeV).

Figures 4 and 5 show the weakest and strongest BPM

spectrums and the spectrums for different methods of

combining BPMs for polarized proton beam at injection

and store respectively when excited by a single AC dipole

(#4). In all cases the response is normalized to place the

AC dipole peak at the driven frequency (0.49) at 0 dB.

This is done as convenient way to visualize the signal to

noise ratio. BPM data was taken with 1024 turn data

records. For the 2013 run, 4096 turn records were

available and these longer records would improve the

signal to noise ratio by 6dB. Also note that the AC dipole

excitation is 2 dB stronger for the data taken at store. The

weakest BPM responses (blue traces) do change

proportionally with the change in energy or magnetic

rigidity (-20 dB). The strongest response (red traces)

changes less (~12dB), which is just a direct result of the

optics also being different. The weakest and strongest

responses were also produced by different BPMs.The average response (green trace) is just a simple

magnitude average:N

k ZIYnI

YL=i(1)

Yn is the frequency response for BPM n and ks is a scale

factor which normalizes the average to 1 (0 dB) when

excited by a single AC. Since this is a magnitude

response of all BPMs, all need to be zero for this to be

zero and this would only happen for a closed bump. The

noise floor for 1024 turn data at store was -45 dB.

The weighted average (black trace) is a magnitude

average weighted by each BPM's normalized signal to

noise ratio squared:N

"=Y(2)

Software calculations for an use the mean magnitude of

the DFT/FFT bins from 0.025 to 0.48 as the estimated

noise value. The RMS error from a fit could also be used.

The noise floor for 1024 turn data at store was -50 dB.

The vector average (purple trace) scales frequency

response of each BPM by the normalized signal to noise

ratio squared, just as the weighted average, but also

counter rotates the phase by measured phase:aN 1 n

in=yPhase #n is the measured phase of BPM n when it was

excited by a single AC dipole. The noise floor for 1024

turn data at store was -70 dB. While this might look like

a good candidate for determining bump closure, because

this is a linear combination of BPMs, it is possible for it

to be zero for combinations of only two AC dipoles and

therefore not useful for definitely trimming bump closure

(see eqn. 6 of next section).

Frequency Response of multiple AC dipoles

From previous treatments of transverse beam motion

due to a single AC dipole [5] the basic equation of motion

can be thought of as a discrete sampled time system, but

the actual sampling is done by a particle or bunch.

Extending the model for an arbitrary number of AC

dipoles in the time domain is:y0[n] J

y'[n] y'[n-1] e'[n]

(4)

(5)e[n]N

- acd -(3)

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Oddo, P.; Kewisch, J.; Bai, M.; Dawson, C.; Makdisi, Y.; Pai, C. et al. Combining Multiple BPM Measurements for Precession AC Dipole Bump Closure, article, September 29, 2013; United States. (digital.library.unt.edu/ark:/67531/metadc870973/m1/4/: accessed October 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.