Impact of beam-coupling impedance on the Schottky spectrum of bunched beam

The Schottky monitors of the Large Hadron Collider (LHC) can be used for non-invasive beam diagnostics to estimate various bunch characteristics, such as tune, chromaticity, bunch profile or synchrotron frequency distribution. However, collective effects, in particular beam-coupling impedance, can significantly affect Schottky spectra when large bunch charges are involved. In such conditions, the available interpretation methods are difficult to apply directly to the measured spectra, thus preventing the extraction of beam and machine parameters, which is possible for lower bunch charges. To study the impact of impedance on such spectra, we introduce a method for building Schottky spectra from macro-particle simulations performed with the PyHEADTAIL code, applied to LHC beam conditions. In this case, the use of a standard Fast Fourier Transform (FFT) algorithm to recover the spectral content of the beam becomes computationally intractable memory-wise, because of the relatively short bunch length compared to the large revolution period. To circumvent this difficulty, a semi-analytical method was developed to efficiently compute the Fourier transform. The simulated Schottky spectrum is then compared against theoretical formulas and measurements of Schottky signals previously obtained with lead ion beams in the LHC where impedance effects are expected to be limited. Furthermore, this study provides preliminary interpretations of the impact of beam-coupling impedance on proton Schottky spectra by incorporating longitudinal and transverse resonator-like impedance models into the simulations. A theoretical framework is also introduced for the case of the longitudinal impedance, allowing the extension of the existing theoretical formalism.


Introduction
Schottky signals contain information on various beam and machine parameters, such as momentum spread, betatron tune, synchrotron frequency and chromaticity, among others.The basic theory of Schottky signals predicts how all of these parameters express themselves in the recorded signal, after it is transformed to the frequency domain.This topic is treated in detail in a classical work by Boussard [1].The developed theory is however limited to a simple beam dynamics model and does not include, e.g., collective effects and beam interaction with the vacuum chamber through impedance.Due to the complex theoretical description of such effects, it is most suitable to study their impact on Schottky spectra using multi-particle simulations, as done in ref. [2], where the effect of space charge was investigated.
The study presented herein is based on simulations performed with PyHEADTAIL [3,4], a macro-particle code that can be used to track turn-by-turn the six-dimensional phase space evolution of a bunch, possibly including the effects of direct space charge and beam-coupling impedances.The following section will present the method used to post-process the data from PyHEADTAIL and reconstruct the Schottky spectra.The simulated spectra will be benchmarked against theoretical formalisms and measurements from ion beams in the Large Hadron Collider (LHC) where the effects of beam-coupling impedance are expected to be limited.
In the third section, simulations are performed for the case of LHC proton bunches where impedance effects are present.We explore the impact of longitudinal and transverse impedance on the Schottky spectrum by incorporating longitudinal and transverse resonator-like impedance models in the simulations.Furthermore, the amplitude-dependent synchrotron tune shift arising from longitudinal impedance is discussed theoretically.

Macro-particle simulation method
Obtaining a Schottky spectrum from multi-particle simulations in the case of the LHC, is particularly challenging computationally due to the highly sparse characteristic of the current signal produced by a single bunch.While discretising the current signal in time, and applying the Fast Fourier Transform (FFT) algorithm to retrieve the spectral content of the beam is manageable for smaller accelerators, typically having smaller revolution periods and longer bunches, like the CERN Proton Synchrotron (PS), this is unfortunately intractable memory-wise for the LHC.As a conservative estimate, for the LHC revolution frequency  0 ≈ 11 245.5 Hz, if we sample a 1 ns long bunch with 100 points (which corresponds to a sampling frequency of 100 GHz) over 10'000 turns, we would have to store an array of 10 11+4 /  0 ∼ 10 11 signal samples1 or around 1 TB of data, hence the need of an alternative approach.In the presented method, the beam's spectral content is calculated concurrently with the macro-particle simulation and stored efficiently, irrespective of the particle count.This approach enables the processing of one million macro-particles in the LHC over 10,000 revolutions within a few hours, using a standard computer.

Longitudinal spectrum
The longitudinal Schottky spectrum is given by the power spectral density (PSD) of the beam current.The current of a single particle  as seen from a pickup, can be expressed as where  is the Dirac delta distribution,   the number of turns,  the charge of the particle,  0 = 1/ 0 the LHC revolution frequency, and  , the arrival time difference between the particle  and the synchronous particle at turn .The current produced by one bunch is simply obtained by summing the contributions of the   particles constituting the bunch: We proceed by computing analytically the Fourier transform To evaluate numerically this last expression we would have to store in memory the   ×   numbers  , and to compute O (      ) exponentials, where   is the number of frequencies to be included in the spectrum.Both these requirements are highly limiting the number of particles we can simulate.We expect that the number of macro-particles required to make the PyHEADTAIL simulation realistic would be at least   ∼ 10 6 .Typical values for the number of turns and frequencies are   ∼   ∼ 10 4 and come respectively from the acquisition time and spectral resolution of the LHC Schottky Monitor.
The spectrum in eq.(2.1) can be computed more efficiently by replacing the right-hand exponential with its Taylor expansion.Since we are interested in the frequency domain of the LHC Schottky monitor working at the ℎ = 427725 harmonic of the revolution frequency, the spectrum will only be evaluated for a limited range around the central frequency   = 2ℎ  0 .By respectively defining  =   + , replacing the right-most exponential with its Taylor expansion, and inverting the summation order, eq.(2.1) becomes where R comes from the truncated part of the Taylor expansion.By introducing the variables L , and   () defined as and neglecting the rest R, the previous expression can be written in the compact form where we used the property ! (+  )! ≤ 1   ! for the last inequality.For typical LHC conditions,  max ∼ 1 ns and | max  max | ∼ 10 −5 (considering a longitudinal Schottky spectrum made of ∼ 25 -3 -satellites spaced by the nominal synchrotron frequency   ∼ 65 Hz).Table 1 presents the relative magnitude of the upper bound of the rest R max with respect to the minimal value of interest of the longitudinal spectrum | ()| for different value of   .For the LHC simulation parameters stated above,   = 3 or 4 is typically sufficient.2As an additional check, the full computation of the Schottky spectrum was also performed with lower values of   to verify that convergence was achieved.

Transverse spectrum
The horizontal and vertical transverse Schottky spectra are given by the PSD of the corresponding dipole moment of the beam.The developments discussed below are carried out for the horizontal plane  but are analogous in every aspect for the vertical plane.With the dipole moment of a single particle  defined as   () =   ()   (), for the full bunch we get: where  , =   ( =  0 +  , ) ≈   ( =  0 ) is the horizontal transverse position of the particle  when passing in front of the detector at turn .Following a similar derivation as for the longitudinal spectrum, we compute the Fourier transform of the signal, replace the right-most exponential by its Taylor expansion around   , and invert the summation order, yielding Using the definition ,      , ( , )  , we get a similar expression as eq.(2.2) where the longitudinal coefficients L are replaced by the transverse coefficients T (2.4) Similarly, an upper bound for the rest of the transverse spectrum R ′ is given by with for the LHC case  max ∼ 3 mm and | max  max | ∼ 3 × 10 −5 (considering a transverse Schottky sideband made of ∼ 25 satellites spaced by the nominal synchrotron frequency   ∼ 65 Hz and for a fractional part of the tune   ∼   ∼ 0.3).For these typical LHC simulation parameters, table 1 indicates that   = 3 or 4 is typically sufficient for the transverse case as well.
Besides the considerable gain in computation time, another advantage of the formalism of eqs.(2.2) and (2.4) is that the L and T coefficients can be calculated simultaneously with the tracking done in the PyHEADTAIL simulation.Hence there is no need to keep in memory the extremely large arrays of  , ,  , and  , for each turn and each particle -all the necessary information is contained in the   ×   coefficients L and T .Since the number of these coefficients does not depend on the number of macro-particles, we can increase the latter without any concern regarding memory.

Simulation procedure
Figure 1 sketches the full simulation procedure.Note that we use the smooth approximation for the machine optics, and (unless stated otherwise) a non-linear radio frequency (RF) system (hence leading to synchrotron frequency spread).
The spatial coordinates  , ,  , and  , of each marco-particle at each turn, obtained from PyHEADTAIL, are plugged into eqs.(2.2) and (2.4) to reconstruct the longitudinal and transverse Schottky spectra.Notably, the so-called instantaneous3 Schottky spectra generated from this simulation procedure will exhibit significant dissimilarities from one simulation to another.This is due to the fact that instantaneous Schottky signals are only a single realisation of a random process, implicitly depending on the discrete distribution of synchrotron and betatron amplitudes and phases among the particles.As described in refs.[6,7], both the standard deviation and mean value of the PSD are proportional to the number of simulated particles,4 and therefore increasing the latter will not make the spectrum converge to its true average.To estimate the expected value of the Schottky power spectrum, and reveal the inner structure of the Bessel satellites described by the theory, the averaging of instantaneous Schottky spectra is required, both in simulations and experimental measurements.
3The term instantaneous originates from the measurements where each spectrum is acquired over a given period (∼16,000 revolutions).Consecutive measured instantaneous spectra differ significantly due to the drift of the initial betatron and synchrotron phases.This behaviour is analogous in simulations where the initial bunch is a new random draw for each simulation.Therefore, the term instantaneous is used to characterise the spectrum produced by a single simulation.
4This is not true for the central satellite of the longitudinal band.

Experimental benchmarking
The simulation method presented in the previous section is benchmarked against experimental Schottky spectra obtained with lead ion beams during LHC Run 2. The Schottky data consists of horizontal and vertical measurements of beam 2 taken at injection energy during the fill 7443.Among the PyHEADTAIL input parameters described in figure 1, some are based on the fitting of the experimental spectrum as done in ref. [8,9] and allow to determine precise machine parameters of the specific fill we want to reproduce.The parameters based on this fitting procedure are horizontal and vertical tunes, horizontal and vertical chromaticities, synchrotron amplitude distribution and nominal synchrotron frequency.5 The other parameters needed by PyHEADTAIL such as emmitances or bunch intensity and energy, come from direct measurements or machine design.Table 2 summarises the values of these parameters.The fitting procedure of ref. [8] assumes that the synchrotron amplitudes  follows a Rice distribution and provides the two parameters  and  of this distribution.On the other hand, the macro-particle simulation requires to have an initial distribution in terms of position  and momentum deviation Δ.The amplitude of the momentum deviation Δ can be deduced from  by using eqs.( 4) and ( 11) from ref. [10], which yield with  0 the nominal momentum,  the slippage factor, Ω 0 the nominal angular synchrotron frequency, ℎ rf the radio frequency harmonic and  0 the angular revolution frequency of the LHC.The initial position with respect to the synchronous particle  and momentum deviation Δ of each particle can then be computed from their synchrotron amplitude and phase6 using eqs.( 10) and ( 12) of ref. [10]: 5Here we define the nominal synchrotron frequency as the limit of the synchrotron frequency for synchrotron amplitudes approaching zero.
6Synchrotron phases are assumed to be uniformly distributed.
-6 - The initial transverse positions and momenta are instead obtained assuming a 2D Gaussian distribution in phase space.Figure 2 presents an overall view of the simulated longitudinal and transverse Schottky spectra obtained using eqs.(2.2) and (2.4) respectively and plotted on the same frequency axis (frequencies in the plot have been shifted from the LHC Schottky harmonic 427725, to the first harmonic of the revolution frequency).Figure 3 is a detailed view of these spectra compared against the experimental measurements of fill 7443.As visible from the plots, the agreement between simulations and measurements is very good.One can also observe that the effect of chromaticity, which makes the upper betatron sideband higher and thinner than the lower sideband, is well reproduced by the simulation.

Theoretical benchmarking
The simulations are compared against two theory-based methods suitable to reconstruct the Schottky spectrum.The first method is based on the Monte Carlo approach presented in ref. [11].It consists of drawing randomly, for each particle, a synchrotron oscillation time-amplitude   and an initial synchrotron phase    from respectively a Rice and a uniform distribution.Using the theoretical formula for the longitudinal Schottky signal of a given particle  [1] the Fourier transform of each individual particle can be directly deduced using the expression giving Ω   (   ) from ref. [10].The single particle spectrum exhibits a fine structure characterised by synchrotron satellites spaced by the synchrotron frequency Ω   .The magnitude of these lines depends on the argument  0   and order  of the Bessel function.In general, the synchrotron initial phase    is responsible for the complex nature of the spectrum.When considering more than one particle in the beam, the synchrotron frequency and phase evolve into random distributions, transforming individual lines into bands.As the Fourier transform is a linear operation, the bunch spectrum can be derived by summing the complex values of single-particle spectra using a sliding window, as done in ref. [11].Taking the squared modulus of this sum gives the Power Spectral Density (PSD) of the whole bunch (i.e. the Schottky spectrum).Finally, as done for PyHEADTAIL simulations, we need to average the PSD over several random draws for the initial bunch in order to obtain the mean spectrum.
The transverse spectrum is obtained in a similar fashion by drawing an additional initial betatron phase    from a uniform distribution and using the general expression for the dipole moment from ref. [9] and where  is the nominal transverse tune,  the chromaticity, and  the betatron oscillation amplitude.The amplitude of the momentum deviation Δ  is derived from   with eq.(2.5).
Figure 4 shows a comparison between the results of our proposed simulation-based method, the Monte Carlo method, the matrix formalism developed in refs.[8] and [9], and real measurements -7 -    -8 -for four different regions of the Schottky spectrum shown in figures 2 and 3.After normalisation, all the spectra are in good agreement with each other.

Discussion on the simulation methods
While the matrix formalism offers faster computations than macro-particle simulations, this theory is able to describe only relatively simple situations, excluding complex beam dynamics effects such as beam-coupling impedance.Conversely, macro-particle simulations can be more time consuming but provide means to explore the impact of impedance and many other effects.
Besides, the matrix formalism only yields the expected value of Schottky spectra, lacking information on instantaneous spectra or their statistical properties.In contrast, both macro-particle simulations and the Monte-Carlo approach allow the computation of instantaneous spectra, enabling cross-verification of theoretical predictions regarding statistical properties.However, the Monte-Carlo approach, similarly to the matrix formalism, requires an analytical description of the Schottky spectra and is therefore not suitable to study Schottky spectra in the presence of complex collective effects.

Beam-coupling impedance
In the preceding section, the simulation of an LHC ion bunch was conducted without including any impedance in the PyHEADTAIL simulations.Nevertheless, a good agreement with experimental measurements and existing theories (which also exclude impedance effects) was achieved, showing that the impact of impedance is negligible for LHC ion bunches at injection energy, which could have been anticipated as the bunch charge is relatively low.
-9 -However, for high-intensity LHC proton bunches, a quantitative and accurate interpretation of Schottky spectra can become more challenging due to the increased influence of beam-coupling impedance.Existing theoretical frameworks for Schottky spectrum reconstruction [1,8,9,11] do not account for impedance effects.This section investigates, both theoretically and through simulations, how the spectra are influenced by impedance, starting with the longitudinal case and subsequently exploring the transverse case.

Longitudinal impedance
Theoretical reconstructions of Schottky spectra, such as the matrix formalism or the Monte Carlo approach mentioned above, are based on the assumption that the synchrotron frequency distribution is known.Under certain conditions, one can derive an analytical relation between the amplitude of the synchrotron oscillation and its frequency (see below) allowing these methods to reconstruct the Schottky spectrum from the synchrotron amplitude distribution.However, this relation has to be modified when additional forces, such as the one coming from beam-coupling impedance, affect the longitudinal dynamics.This section will briefly present the available theory relating the amplitude of the synchrotron oscillation to its frequency, as well as a commonly adopted approximation.Subsequently, we will deal with the additional forces coming from impedance, extending the theory presented in ref. [12] to the case of a non-linear RF bucket.The developed theory will be applied to the particular case of a longitudinal broad-band resonator, and benchmarked against macro-particle simulations.

Theoretical description
Synchrotron oscillation.For an impedance-free environment, the equation of motion for the RF phase 7 of a given particle is (eq.(9.51) of ref. [13]): assuming that the synchronous phase   is such that sin   is small enough to be neglected (i.e.no acceleration or energy loss compensation).The nominal synchrotron frequency reads where  is the elementary charge, and where the relevant machine parameters are: the revolution frequency  0 , the slippage factor , the amplitude of the RF voltage , the RF harmonic number ℎ rf , the relativistic factor , and the reference energy  0 .Note that by convention  is positive above transition, such that  and cos   always have opposite sign.Equation (3.1) is identical to the non-linear pendulum equation studied in [14], hence the synchrotron frequency Ω  of the particle can be written 7Above transition  has to be taken as the difference between the RF phase of the particle and .
-10 -  where  is the RF phase amplitude of the synchrotron oscillation and K is the complete elliptic integral of the first kind given by eq. ( 8.112.1) of ref. [15].Equation (3.1) can also be approximated by replacing the sine function with its Maclaurin series expansion up to the third order, which yields

JINST 19 P03017
This last equation has been studied in [14], and an approximation of the oscillation frequency is given by . (3.5) Figure 5 illustrates how the exact solution from eq. ( 3.3) compares with the approximation from eq. (3.5).The amplitude distribution corresponding to a Gaussian bunch profile of standard deviation  = 0.31 ns is also shown in order to compare the most populated region with the range where the approximation is valid.As can be seen, the approximation holds for an amplitude  ≲ 0.8, which is the region where the vast majority of the particles are.Noting that since  = ℎ rf  0 , with  the arrival time difference between a given particle and the synchronous particle, we can also express the amplitude of the synchrotron oscillation in terms of the time amplitude given by Impedance effect on the longitudinal dynamics.In this section, we investigate the influence of impedance on the longitudinal dynamics of a particle and, more specifically, how it can affect its oscillation frequency.A given force   acting longitudinally on the particle will impact the phase dynamics with -11 -where we used the fact that  0 is constant, the definition of the slippage factor, and eq.(3.6).The equation of motion with an additional external force is given by combining eqs.(3.1) and (3.7), which yields In this study, we focus on the force arising from the longitudinal beam-coupling impedance, which is given by eq. ( 4.18) of ref. [16] where  stands for the circumference of the accelerator,  for the number of particles in the bunch,  ∥ () for the longitudinal impedance, and Λ() for the beam spectrum normalised by the bunch intensity.The force has to be taken at the current position of the particle, i.e. at a distance  =  behind the synchronous particle.Note that the bunch spectrum () in eq. ( 4.18) of ref. [16], has been replaced by the beam spectrum Λ() in order to take into account the possible multi-turn wake (this approach is equivalent to adding the forces coming from different turns).The beam spectrum is the Fourier transform of the beam profile.For a single bunch in a circular accelerator, the latter can be expressed as the convolution of the bunch profile () with a Dirac comb of the revolution period  0 .Using the convolution theorem, the beam spectrum can be written = () 0 X  0 (), (3.10)where * stands for the convolution operation and where the Dirac comb of period  0 is defined as follows Substituting eq.(3.10) in eq.(3.9) , we have where we used the compact notation  ∥ ( ) ≡  ∥ (  0 ), ( ) ≡ (  0 ), and the average current eq. (3.12) can be written in the compact form with the coefficients   defined by and In the last equality, we used the fact that ( ),   ∥ ( ) , and   ∥ ( ) are respectively even, even, and odd functions.The expansion performed above is motivated by the assumption that for small oscillation amplitudes, only the first-order term plays a significant role.In this context, the shift of the nominal synchrotron frequency is determined by the term represented by  1 .On the other hand, the higher-order, non-linear terms contribute to an amplitude-dependent synchrotron frequency shift.
Note that in a fully self-consistent picture, the forces arising from impedance would actually modify the bunch profile, hence influencing the impedance force in return.The analysis in this section proceeds under the assumption that the modification of the bunch profile due to impedance is sufficiently negligible, allowing the impedance force to be defined from the original Gaussian longitudinal bunch profile.
Broad-band resonator.We will now apply eq.(3.13) to the particular case of a broad-band resonator described by the following function where  ∥ is the shunt impedance,   the angular cut-off frequency, and  the quality factor.The values we will use for these parameters are shown in table 3. The even terms in eq.(3.13) are responsible for the synchronous phase shift and, in the particular case of a broad-band resonator, their contribution can be neglected (see appendix A).Expanding eq.(3.13) up to the third order gives -13 -which is similar to eq. (3.4) where the factors 1 and − 1 6 have been generalised to the arbitrary coefficients  1 and  3 .
Substituting  1 () = √  2 () in eq.(3.4) yields and since the two functions  1 () and  2 () only differ by a constant factor √ , they have the same oscillation frequency and  1 = √   2 : By defining ) is identical to eq. (3.16) and the synchrotron frequency in presence of impedance is given by: This last equation can be used to extend the theoretical matrix formalism introduced in refs.[8,9] to include longitudinal impedance effects.The original relation between synchrotron amplitudes and frequencies given by eq.(3.5) is replaced by eq.(3.17) to take into account the effect of impedance.

Macro-particle simulations
PyHEADTAIL simulations of an LHC proton bunch at injection energy have been performed following the simulation method presented above.The relevant simulation parameters are given in table 3 and different longitudinal broad-band resonators of varying shunt impedance  ∥ have been included in the simulations.6 presents an overall view of the simulated longitudinal and transverse Schottky spectra together with a detailed view of some regions.The spectra calculated with the matrix formalism -14 -including impedance effects are also shown for comparison.It can be observed that the broad-band resonator induces a shift of the nominal synchrotron frequency (all satellites moving toward the central one within each band).This shift is due to the term  1 and the new nominal synchrotron frequency is Ω 0 √  1 .The broad-band resonator will reduce the nominal synchrotron frequency for a machine operating above transition, as can be seen from eq. (3.14) (the opposite happens below transition).One can also observe in figure 6 an amplitude-dependent synchrotron frequency shift, as the alterations in the satellite shapes can not be explained by a simple linear transformation involving shifting and scaling of the original form.This effect is due to the higher order terms  2+1 ,  ≥ 1 in eq.(3.13).To highlight the amplitude-dependent effects of the synchrotron tune shift, figure 7 illustrates spectra calculated using the matrix formalism, retaining either the first-order or both the first and third-order terms in eq.(3.17).The dotted grey line corresponds to the matrix formalism where only the first order term of the impedance contribution has been kept (i.e. 1 ≠ 0 and  3 = 0, which corresponds to  1 = 1 −  1 and  3 = −1/6 in eq.(3.17)).As can be seen, the nominal synchrotron frequency shift is well reproduced by the theory, while the shift for non-zero amplitude particles, requires higher order terms.The dashed orange line also includes the third order term of the impedance contribution (i.e. 1 ≠ 0 and  3 ≠ 0, which corresponds to  1 = 1 −  1 and  3 = −1/6 −  3 ).With the third order impedance term, the theory is in good agreement with the simulation.
One can also probe the validity of eq.(3.17) by directly extracting the relation Ω  ( ) from the macro-particle simulation.This was done in figure 8 where each black dot corresponds to the synchrotron amplitude and frequency of a given macro-particle.As before, one can observe that the nominal synchrotron frequency shift is well reproduced by the first order term  1 , while the shift for larger amplitude particles is correct with the third order theory, up to a certain amplitude  ∼ 1.25 rad.In order to extend the region where eq.(3.17) is valid, one would need to take into account the fifth order term  5 , in the equation of motion.However, this is not crucial since the majority of the particles are well described, as can be seen from the amplitude distribution.3.17) including impedance terms   up to the first (red) and third (green) order, against macro-particle simulation (black dots).The simulation has been conducted with a longitudinal BB resonator of shunt impedance  ∥ = 30 kΩ.

Transverse impedance
In this last section, a preliminary study on the effects of transverse impedance is conducted through simulations for an LHC proton bunch.The simulation parameters are identical to the one presented in table 3, but the longitudinal broad-band resonator is replaced by a transverse one whose parameters are given in table 4. The transverse broad-band resonator is placed in the horizontal plane and is defined by the following impedance function: -16 - It is worth noting that only the quadrupolar part of the impedance affects the Schottky spectrum.Indeed, as the dipolar wake is proportional to the transverse offset of the leading particle, the total dipolar wake of the bunch cancels out for a stable beam where the bunch is symmetrically centered on the design orbit.The situation is different for the quadrupolar wake, which is proportional to the offset of the trailing particle and does not depend on the leading particle's position.
The effects on the Schottky spectrum can be observed in figure 9.The longitudinal band is not affected by the transverse impedance, while a betatron tune shift is visible on the transverse bands (all satellites in a given transverse sideband are displaced in the same direction).The direction of the satellite's shift -toward the right (resp.left) for the lower (resp.upper) sideband -indicates that the broad-band resonator decreases the betatron tune.Similarly to the case of the longitudinal impedance, the satellites are not simply shifted but their internal structure is also affected by impedance.-17 -

Conclusion
We proposed a new method for computing Schottky spectra from macro-particle simulations, facilitating future investigations into the impact of collective effects, such as beam-coupling impedances, on measured spectra.After discussing the high computational resource requirements of traditional FFT algorithms when performing simulations for LHC beam and machine conditions, we showed how one can achieve a substantial reduction of its computational complexity.The obtained results were shown to be in good agreement with reference measurements as well as with other theory-based methods, and reproduce the overall shape of the spectrum together with the detailed internal structure of the synchrotron satellites, thus validating the method.Going one step further, we then explored the influence of longitudinal and transverse impedance on the Schottky spectrum, presenting a generalised theory applicable to any kind of longitudinal impedance, and demonstrating agreement with simulations.The findings reveal that longitudinal impedance can induce amplitude-dependent shifts in the synchrotron frequency while the presence of a transverse impedance leads to betatron tune shifts.Both longitudinal and transverse impedance also alter the internal structure of transverse satellites, as shown on various examples using broadband resonators, and also confirmed by the theory presented in the longitudinal case.
with   , the initial phase of the synchrotron oscillation.The average value of the time-dependent part of eq.(A.where  =  is the total charge of the bunch.The  1 factor in eq.(A.4) accounts for the first order potential well distortion due to impedance.This expression is consistent with other formulas available in the literature, such as eq.(4.35) in [16].
In the case of the broad-band resonator defined in table 3 with  ∥ = 60 kΩ, we have Δ  = −1.87mrad with eq.(A.3), in relative agreement with Δ  = −2.14mrad from the macro-particle simulation as shown in figure 10.Such a phase shift is small enough to be neglected, hence we can drop the even terms in eq.(3.13).-19 -

Figure 2 .
Figure 2. Simulated longitudinal (E, F, G, H), transverse horizontal (A, B, C, D), and transverse vertical (I, J, K, L) Schottky spectra for the machine and beam configuration of LHC fill 7443.The dotted lines indicate respectively (from left to right): (1 −  / )  0 ,  0 and (1 +  / )  0 .A detailed view of the shaded region with the corresponding letters is provided in figure 3.

Figure 3 .
Figure 3. Measured (blue lines) longitudinal and transverse Schottky spectra compared with the simulated spectra (orange lines) for LHC fill 7443.The A-K region labels correspond to the frequency ranges of the shaded regions on figure 2.

Figure 4 .
Figure 4. Comparison between measured, simulated, and theoretical, longitudinal (F, G), transverse horizontal (C), and transverse vertical (K) Schottky spectra for LHC fill 7443 (the letters correspond to those defined in Figs 2 and 3).

Figure 5 .
Figure 5.Comparison between the exact (eq.(3.3)) and approximate (eq.(3.5)) expressions of the synchrotron frequency as a function of the oscillation amplitude.The distribution of amplitudes for a Gaussian bunch of RMS  = 0.31 ns is also shown.

3 IIFigure 6 .
Figure 6.Simulated longitudinal (b) and transverse (a,c) Schottky spectra impacted by a longitudinal broad-band resonator for various shunt impedance.The dashed lines in the two lower plots are obtained from the matrix formalism.

Figure 7 .
Figure 7.Comparison of the longitudinal Schottky spectrum from the macro-particle simulation against the adapted matrix formalism, with eq.(3.17) including impedance terms   up to the first and third order.

Figure 8 .
Figure 8.Comparison of eq.(3.17) including impedance terms   up to the first (red) and third (green) order, against macro-particle simulation (black dots).The simulation has been conducted with a longitudinal BB resonator of shunt impedance  ∥ = 30 kΩ.

Figure 9 .
Figure 9. Simulated longitudinal (b), and horizontal (a -lower sideband -and c -upper sideband) Schottky spectra without (blue) and with (orange and green) a transverse BB resonator.Frequencies are shifted from the LHC Schottky harmonic 427725, to the 1 st harmonic.

Table 1 .
Upper bound relative precision of the simulation for various number of terms in the Taylor expansion.