Digital Processing of Pick-up Signals for Position and Tune Determination

With the advent of fast high resolution Analog to Digital Converters (ADCs) and Field Programmable Gate Arrays (FPGAs), "all digital systems" for pick-up data processing to determine position and tune have become commonplace. This contribution compares the frequently used position estimators used in the digital systems in terms of measurement variance, bias and robustness to external interference. An analytical beam model, along with simulated pick-up signal and actual pick-up signal from the SIS-18 synchrotron are used for the comparison. The effect of precise position estimation on the tune spectra is discussed. INTRODUCTION High precision position estimation requires the optimization of the beam position measurement system in all the stages of development, which starts from EM simulations to optimize the pick-up design against unwanted resonances, establishing linearity while minimizing the cost of manufacturing [1]. The mechanical construction and installation of pick-ups with respect to the magnetic center of magnets within the specified tolerances is also a challenging task. Finally, the electronics required for acquisition and processing of the pick-up signals demand low noise and high dynamic range as well as periodic and precise calibration. The typical methods for signal processing and calibration are described in [2–4]. In the recent years, the signal processing have completely shifted to digital domain due to availability of fast high resolution ADCs and FPGAs and this contribution will focus on this aspect of position measurement system for circular accelerators. The concepts of pick-up position sensitivity and offset are presented along with the typical signal spectra for bunched and coasting beams in the next couple of sections. Following that, the frequently used digital position estimation methods are discussed and a new approach to position estimation based on "linear regression model" is introduced. All the presented methods are compared with an analytical beam model, simulated beam data and the pick-up signal from the SIS-18 synchrotron in terms of estimated position bias and variance. The effect of position estimation methods on tune spectra calculated from the turn-by-turn position is discussed. POSITION SENSITIVITY AND OFFSET The pick-up position sensitivity and offset estimates are obtained from the EM simulations [5] or on-bench wire based measurements [6, 7]. The uncertainty in the position sensitivity measurement is given by the precision of the measurement equipment used for bench measurements and simulation time/resources which are often < 0.1% of absolute values as shown in [5, 8]. In careful pick-up designs, the pick-up sensitivity is found to be constant within 0.1% of the absolute sensitivity value in the frequency region of interest [5]. Once the pick-up sensitivity and offset are known, the beam center-of-mass can be determined from the difference of the signal induced on opposite pick-up plates. There are two important features of pick-up signal which are relevant for digital position estimation a) Most of the pick-up types are "capacitive" or AC coupled, which leads to rejection of the DC component of the beam signal and b) The signal is sampled with fast ADCs such that many samples are acquired in each time interval for the position measurement. Thus the problem of position estimation is that of an overdetermined system whose low frequency components are significantly suppressed. The lower cut-off is given by the termination impedance of the pick-up [2]. PICK-UP SIGNAL SPECTRUM The pick-up signal spectrum of a ring accelerator is unique due to the periodic crossing of beam particles through the pick-up. A beam of particles traversing the synchrotron or f0 2f0 3f0 nf0 (n + 1)f0 −200 −180 −160 −140 −120 −100 −80 −60 −40 −20 2qI0 Amplifier noise Schottky Bands ∝ I0 Frequency P ow er d B m /H z Figure 1: A bunched beam power spectrum of a U28+ bunched beam with 109 particles at injection energy in SIS18. 2qI0 is the shot noise level while blue dashed line represents the electronics noise. The Schottky bands are shown for reference. storage ring with a constant energy is referred to as coasting beam. The beam is said to have no coherent longitudinal structure due to absence of any longitudinal focusing. However, due to finite momentum spread, finite number of particles and periodic traversal of particles through the BPM, signals proportional to the square root of number of particles are induced at the revolution frequencies. The power in each revolution band is given by 2qI0, where q is the charge Proceedings of IBIC2015, Melbourne, Australia TUPB010 BPMs and Beam Stability ISBN 978-3-95450-176-2 321 C op yr ig ht © 20 15 C C -B Y3. 0 an d by th e re sp ec tiv e au th or s state and I0 is the beam current. They are called Schottky signals due to their origin in shot noise [9]. When an external field imposes a longitudinal structure on the beam, power is transferred from the DC component of the beam to higher harmonics at the revolution frequency. Any harmonic with sufficient power in the difference signal spectrum can be used to obtain the position information. This coherent power is proportional to the number of particles and is usually large enough for calculation of bunch-by-bunch beam position measurement. Figure 1 shows the estimated power spectra of a U28+ bunched beam with 109 particles at injection energy. In comparison, the power in the Schottky bands of the unbunched beam with the same current is also shown. Though the beam position measurement usually implies bunched beam position measurement; with high beam intensities and long measurement times, Schottky signals can be utilized for beam position measurements of a coasting beam. Detailed introduction to pick-up signal spectra can be found in [10].