deconvolve.org

Deconvolution in Wire Cell

2D

The 2D deconvolution is in the prototype in the class =uBooNEData2DDeconvolutionFDS=.

Notation

The functionality is described below with this notation:

$A_xy$

A matrix (complex valued in general) of coordinates $x$ vs $y$. The first $x$ is in wire/channel or its Fourier periodicity space, the second $y$ is in time or frequency space.

$A_xz = FFT_y→{z}(A_xy)$

A Fourier transform that takes coordinate $y$ to its dual $z$. The transfer operates in a row-wise or column-wise manner. Note, changing index labels implies a new matrix value! See definition of indices below.

$F_x$

a filter function operating independently (row-wise or column-wise) on a single coordinate $x$.

Each plane is considered independently and so the notation is free of any plane indices. Wrapped wires are not currently considered.

Values are measured in discrete space (eg. wire pitch distance) and time (eg ticks). Fourier transforms on both of these coordinates and in both directions are taken. These labels are used:

$d$

channel, wire number or wire region number giving a proxy for the discrete distance along the wire pitch direction. For channels it runs $0 ≤ d < N_wires$ where for typical planes, $N_wires$ is two or three thousand. For wire regions it runs $0 ≤ d ≤ 2 N_regions$ to give $± N_regions$ on either side of the wire-of-interest plus one around the wire-of-interest.

$p$

the “periodicity number” or spatial frequency number corresponding to the Fourier transform over the distance measures $d$.

$t$

the discrete sample time or tick. The $t$ spans a readout which is typically a few ms sampled at 0.5 μs giving about ten thousand indices.

$f$

the temporal frequency number corresponding to the Fourier transform over the time measure $t$.

Field Response as Distance vs. Time

An element of the “field response matrix” $R_dt$ describes the instantaneous induced current on a given wire-of-interest at the time $t$ due to a drifting charge distributed uniformly across the region near a wire which is $d$ wires from the wire-of-interest. These are calculated on a finer time scale and are later down-sampled and extended to match the digitization sample time and extent.

Field Response as Distance vs Frequency

The Fourier transform is applied along the time coordinate independently for each wire region response of in $R_dt$ to form $R_df = FFT_t→{f}(R_dt)$.

Field Response as Periodicity vs Frequency

A second Fourier transform is applied across the spatial (wire region) coordinate of the field response matrix giving $R_pf = FFT_d→{p}(R_df)$. Here, $d$ or $p$ run across the $2 N_regions+1$ indices

Measured Current as Distance vs. Time

The full detector “frame” readout is represented by a matrix $M_dt$ (for a given plane) which represents the measured waveform in channel $d$ over digitization time ticks $t$.

Measured Current as Distance vs. Frequency

The Fourier transform is applied to each measured waveform to produce $M_df = FFT_t→{f}(M_dt)$.

Measured Current as Periodicity vs. Frequency

A second Fourier transform is applied across the distance coordinate to produce the “periodicity” (spatial frequencies) of the (temporal) frequency spectra. This is $M_pf = FFT_d→{p}(M_df)$.

Spatial and Temporal filter functions

Two functions operate as filters, one in the spatial frequency (periodicity) and one in the temporal frequency domains. Both are of the form of an falling exponential of some power of the coordinate. The spatial “wire filter” is $F_p$ and the temporal frequency filter is $F_f$.

Signal as Periodicity vs. Frequency with Filter

The deconvolution is performed in spatial and temporal frequency space as the ratio of the measured current and the response and a product of a filter which is composed of function of the wire dimension and one of the time dimension. $S_pf = \frac{M_pf}{R_pf}\frac{F_p}{N_wiresN_ticks}$.

Signal as Distance vs. Frequency

The signal is transformed from spatial frequency (periodicity) to distance $S_df = FFT_p→{d}(S_pf)$.

Signal as Distance vs. Time with Filter

The signal is finally transformed back to time after a frequency filter is applied. $S_dt = FFT_f→{t}(S_df F_f)$.

Baseline restoration

Finally the baseline is restored.

Implementation

The full 2D deconvolution in the above notation can be condensed to:

signal

$S_dt = FFT_f→{t}(FFT_p→{d}(FFT_d→{p}(FFT_t→{f}(M_dt)) * F_pf))$

filter

$F_pf = f_p * f_f / (N_wire N_tick * R_pf)$

With all variables being $N_wire × N_tick$ matrices:

$M_dt$

measured ADC and

$F_pf$

a filter in wire periodicity vs frequency space composed of:

$f_x$

a frequency space filter in either the time or wire dimension.

$R_pf$

total response, a product of:

$R_field,pf$

field response and

$R_elec,pf$

electronics response.