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.