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Wire-Cell Noise Model

Overview

WCT provides a “sampled spectrum noise model” which is parametererized by two types of information:

  • A set of “mean noise spectra”.
  • An associative mapping between channel and spectrum.

The WCT implementation is structured into two layers.

  1. A “model” component provides a mean noise spectrum given a key (channel or group).
  2. An “adder” component generates a noise waveform and associates it to one or more channels.

Noise generation

The kernel of the noise generation returns a fluctuated noise waveform given a real-valued mean amplitude spectrum. The sampling of the distribution characterized by the mean spectrum has two steps.

  1. Each frequency bin of the mean spectrum is fluctuated to produce a complex amplitude sample.
  2. An inverse discrete Fourier transform is applied to the spectrum of samples to produce the noise waveform.

The fluctuation of each frequency bin is produced by calculating from the mean $μ$ the mode $σ = \sqrt{\frac{2}{π}}μ$ and then drawing a complex number with both real and imaginary parts each and independently distributed according to the Gaussian distribution $\mathcal{N}(0, σ)$. Equivalently, the complex amplitude can be drawn from the Rayleigh distribution $\mathcal{R}(σ)$ and the complex phase from the uniform distribution $\mathcal{U}(0,2π)$. The resulting complex spectrum must be made Hermitian-symmetric with the zero-frequency bin and the Nyquist bin (if number of samples is even) real valued.

Noise adders

There are two types of noise adders:

IncoherentAddNoise
adds an independent noise waveform to each channel. The AddNoise type is an alias for this type.
CoherentAddNoise
adds an independent noise waveform to each in a group of channels.

Noise spectrum models

There are two types of noise spectrum models:

EmpiricalNoiseModel

The EmpiricalNoiseModel spectrum model is intended to be used with the IncoherentAddNoise noise adder. This model associates an given mean spectrum, with an optional post-hoc transformation to all channels in a given set. It accepts the following configuration parameters. Note, all values are in WCT system of units.

spectra_file
directly an array of spectral objects or the name of a file (JSON or Jsonnet format) that evaluates to such an array. See below for description of a spectral object.
nsamples
the number of waveform samples to generate
period
the sampling time period of the generated waveform (eg 0.5*wc.us).
wire_length_scale
the scale at which wire lengths are binned in (eg 1*wc.cm)

Each spectral object in the array provided directly by or pointed to by the spectra_file option has the following attributes. They provide meta data about a sub-sampling of a sampled mean spectrum.

plane
the WirePlaneId::index() of the anode plane for which the spectrum models.
gain
the preamplifier gain in units of voltage per charge (eg 14*wc.mV/wc.fC).
shaping
the preamplifier shaping time (eg 2*wc.us)
wirelen
the length of the wire for which the spectrum models
const
the constant, white-noise component of the overall noise spectrum that is independent of frequency, units of voltage.
nsamples
the number of samples of the spectrum
period
the sampling period of the spectrum
freqs
the frequencies at which the sampled spectrum was sub-sampled in units of 1/time (eg, {i/(N*0.5*wc.us)})
amps
the sub-sampled mean spectrum at these frequencies in units of voltage.

Notes

  • The order of this array must have increasing wirelen for each given plane. But, see issue 228.
  • The sizes of freqs and amps are equal and generally much less than nsamples. This sub-sampling allows for the fact that the spectral amplitude is itself very smooth and/or may have been “hand digitized” from plots.

The following post-hoc transformations are applied to each spectral object:

  • Two nearest neighbor sub-sampled spectra (both const and non-constant amps vs freqs parts) are interpolated to match a given channel’s wire length.
  • This sub-sampled spectra is inflated through a time-interpolation to match the sampling of nsamples and period configuration parameters given to the model (not to be confused with the parameters of the same name in each spectra object).
  • If a ChanStat object is provided and a given channel has wrong gain and shaping time then this response is corrected based on the gain and shaping parameters of the spectral object. Note, this currently hard-wires the cold electronics response. See issue 229.

GroupNoiseModel

This spectral model associates a channel to a group and a group to a spectrum for the purposes of applying a generated noise waveform coherently across all channels in the group. A time interpolation from given sub-sampled spectra to full sampled is performed and a multiplicative scaling may be applied to the amplitude. The configuration parameters that the model accepts are:

scale
a scale factor multipliplied to the spectral amplitude.
nsamples
the number of waveform samples to generate.
tick
the sampling time period of the generated waveform (eg 0.5*wc.us).
groups
directly an array of channel group objects or the name of a file (JSON or Jsonnet format) that evaluates to such an array. See below for group object structure.
spectra
directly an array of spectral objects or the name of a file (JSON or Jsonnet format) that evaluates to such an array. See below for spectral object structure.

The group object has the following attributes:

group
an integer identifying the group (groupID is also an accepted attribute name)
channels
an array of channel ident numbers that are in the group

The spectral object has the following attributes:

group
an integer identifying a channel group (groupID is also an accepted attributes name)
nsamples
the number of samples of the spectrum
period
the sampling period of the spectrum
freqs
the frequencies at which the sampled spectrum was sub-sampled in units of 1/time (eg, {i/(N*0.5*wc.us)})
amps
the sub-sampled mean spectrum at these frequencies in units of voltage.

Note, the last four have the same meaning as for the EmpiricalNoiseModel.

Comment on mode vs mean

Here we focus on the use of mode $σ$ (instead of mean $μ$) as the parameter of the Rayleigh, or equivalently the double-Gaussian, distribution and how confusing the two can potentially pose practical pitfalls. Care is needed in that we naturally form a measure of a noise spectrum with the following procedure:

  1. collect $N$ noise waveforms of common noise characteristics (eg, same channels, with selection to suppress signals)
  2. apply the Fourier transform to each waveform to get a complex spectrum
  3. calculate the amplitude (absolute value) of this spectrum
  4. accumulate these to form a spectral sum and divide by $N$ to form a spectral mean

This spectral mean must be reduced to the spectral mode by multiplying by $\sqrt{\frac{2}{π}}$. The fluctuated waveforms can then also be sent through the collection process and the spectral mean can be reconstructed (within statistics). One may then compare the mean Parseval energy of the waveforms, the mean Parseval energy of their spectra and the Parseval energy of their mean spectrum. While the first two will be (must be) identical, the third differs. It is approximately $\sqrt{\frac{2}{π}}$ smaller. The mean spectral energy is not the energy of the mean spectrum.