explain to me step by step what is going on here and also explain to me the mathematical intricasies and methods used 
/ This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
// © DonovanWall

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//@version=4
study(title=“Gaussian Channel [DW]”, shorttitle=“GC [DW]”, overlay=true)

// This study is an experiment utilizing the Ehlers Gaussian Filter technique combined with lag reduction techniques and true range to analyze trend activity.
// Gaussian filters, as Ehlers explains it, are simply exponential moving averages applied multiple times.
// First, beta and alpha are calculated based on the sampling period and number of poles specified. The maximum number of poles available in this script is 9.
// Next, the data being analyzed is given a truncation option for reduced lag, which can be enabled with “Reduced Lag Mode”.
// Then the alpha and source values are used to calculate the filter and filtered true range of the dataset.
// Filtered true range with a specified multiplier is then added to and subtracted from the filter, generating a channel.
// Lastly, a one pole filter with a N pole alpha is averaged with the filter to generate a faster filter, which can be enabled with “Fast Response Mode”.

//Custom bar colors are included.

//Note: Both the sampling period and number of poles directly affect how much lag the indicator has, and how smooth the output is.
// Larger inputs will result in smoother outputs with increased lag, and smaller inputs will have noisier outputs with reduced lag.
// For the best results, I recommend not setting the sampling period any lower than the number of poles + 1. Going lower truncates the equation.

//-----------------------------------------------------------------------------------------------------------------------------------------------------------------
//Updates:
// Huge shoutout to @e2e4mfck for taking the time to improve the calculation method!
// -> migrated to v4
// -> pi is now calculated using trig identities rather than being explicitly defined.
// -> The filter calculations are now organized into functions rather than being individually defined.
// -> Revamped color scheme.

//-----------------------------------------------------------------------------------------------------------------------------------------------------------------
//Functions - courtesy of @e2e4mfck
//-----------------------------------------------------------------------------------------------------------------------------------------------------------------

//Filter function
f_filt9x (_a, _s, _i) =>
int _m2 = 0, int _m3 = 0, int _m4 = 0, int _m5 = 0, int _m6 = 0,
int _m7 = 0, int _m8 = 0, int _m9 = 0, float _f = .0, _x = (1 - _a)
// Weights.
// Initial weight _m1 is a pole number and equal to _i
_m2 := _i == 9 ? 36 : _i == 8 ? 28 : _i == 7 ? 21 : _i == 6 ? 15 : _i == 5 ? 10 : _i == 4 ? 6 : _i == 3 ? 3 : _i == 2 ? 1 : 0
_m3 := _i == 9 ? 84 : _i == 8 ? 56 : _i == 7 ? 35 : _i == 6 ? 20 : _i == 5 ? 10 : _i == 4 ? 4 : _i == 3 ? 1 : 0
_m4 := _i == 9 ? 126 : _i == 8 ? 70 : _i == 7 ? 35 : _i == 6 ? 15 : _i == 5 ? 5 : _i == 4 ? 1 : 0
_m5 := _i == 9 ? 126 : _i == 8 ? 56 : _i == 7 ? 21 : _i == 6 ? 6 : _i == 5 ? 1 : 0
_m6 := _i == 9 ? 84 : _i == 8 ? 28 : _i == 7 ? 7 : _i == 6 ? 1 : 0
_m7 := _i == 9 ? 36 : _i == 8 ? 8 : _i == 7 ? 1 : 0
_m8 := _i == 9 ? 9 : _i == 8 ? 1 : 0
_m9 := _i == 9 ? 1 : 0
// filter
_f := pow(_a, _i) * nz(_s) +
_i * _x * nz(_f[1]) - (_i >= 2 ?
_m2 * pow(_x, 2) * nz(_f[2]) : 0) + (_i >= 3 ?
_m3 * pow(_x, 3) * nz(_f[3]) : 0) - (_i >= 4 ?
_m4 * pow(_x, 4) * nz(_f[4]) : 0) + (_i >= 5 ?
_m5 * pow(_x, 5) * nz(_f[5]) : 0) - (_i >= 6 ?
_m6 * pow(_x, 6) * nz(_f[6]) : 0) + (_i >= 7 ?
_m7 * pow(_x, 7) * nz(_f[7]) : 0) - (_i >= 8 ?
_m8 * pow(_x, 8) * nz(_f[8]) : 0) + (_i == 9 ?
_m9 * pow(_x, 9) * nz(_f[9]) : 0)

//9 var declaration fun
f_pole (_a, _s, _i) =>
_f1 = f_filt9x(_a, _s, 1), _f2 = (_i >= 2 ? f_filt9x(_a, _s, 2) : 0), _f3 = (_i >= 3 ? f_filt9x(_a, _s, 3) : 0)
_f4 = (_i >= 4 ? f_filt9x(_a, _s, 4) : 0), _f5 = (_i >= 5 ? f_filt9x(_a, _s, 5) : 0), _f6 = (_i >= 6 ? f_filt9x(_a, _s, 6) : 0)
_f7 = (_i >= 2 ? f_filt9x(_a, _s, 7) : 0), _f8 = (_i >= 8 ? f_filt9x(_a, _s, 8) : 0), _f9 = (_i == 9 ? f_filt9x(_a, _s, 9) : 0)
_fn = _i == 1 ? _f1 : _i == 2 ? _f2 : _i == 3 ? _f3 :
_i == 4 ? _f4 : _i == 5 ? _f5 : _i == 6 ? _f6 :
_i == 7 ? _f7 : _i == 8 ? _f8 : _i == 9 ? _f9 : na
[_fn, _f1]

//-----------------------------------------------------------------------------------------------------------------------------------------------------------------
//Inputs
//-----------------------------------------------------------------------------------------------------------------------------------------------------------------

//Source
src = input(defval=hlc3, title=“Source”)

//Poles
int N = input(defval=4, title=“Poles”, minval=1, maxval=9)

//Period
int per = input(defval=144, title=“Sampling Period”, minval=2)

//True Range Multiplier
float mult = input(defval=1.414, title=“Filtered True Range Multiplier”, minval=0)

//Lag Reduction
bool modeLag = input(defval=false, title=“Reduced Lag Mode”)
bool modeFast = input(defval=false, title=“Fast Response Mode”)

//-----------------------------------------------------------------------------------------------------------------------------------------------------------------
//Definitions
//-----------------------------------------------------------------------------------------------------------------------------------------------------------------

//Beta and Alpha Components
beta = (1 - cos(4asin(1)/per)) / (pow(1.414, 2/N) - 1)
alpha = - beta + sqrt(pow(beta, 2) + 2beta)

//Lag
lag = (per - 1)/(2N)

//Data
srcdata = modeLag ? src + (src - src[lag]) : src
trdata = modeLag ? tr(true) + (tr(true) - tr(true)[lag]) : tr(true)

//Filtered Values
[filtn, filt1] = f_pole(alpha, srcdata, N)
[filtntr, filt1tr] = f_pole(alpha, trdata, N)

//Lag Reduction
filt = modeFast ? (filtn + filt1)/2 : filtn
filttr = modeFast ? (filtntr + filt1tr)/2 : filtntr

//Bands
hband = filt + filttrmult
lband = filt - filttr*mult

// Colors
color1 = #0aff68
color2 = #00752d
color3 = #ff0a5a
color4 = #990032
fcolor = filt > filt[1] ? #0aff68 : filt < filt[1] ? #ff0a5a : #cccccc
barcolor = (src > src[1]) and (src > filt) and (src < hband) ? #0aff68 : (src > src[1]) and (src >= hband) ? #0aff1b : (src <= src[1]) and (src > filt) ? #00752d :
(src < src[1]) and (src < filt) and (src > lband) ? #ff0a5a : (src < src[1]) and (src <= lband) ? #ff0a11 : (src >= src[1]) and (src < filt) ? #990032 : #cccccc

//-----------------------------------------------------------------------------------------------------------------------------------------------------------------
//Outputs
//-----------------------------------------------------------------------------------------------------------------------------------------------------------------

//Filter Plot
filtplot = plot(filt, title=“Filter”, color=fcolor, linewidth=3)

//Band Plots
hbandplot = plot(hband, title=“Filtered True Range High Band”, color=fcolor)
lbandplot = plot(lband, title=“Filtered True Range Low Band”, color=fcolor)

//Channel Fill
fill(hbandplot, lbandplot, title=“Channel Fill”, color=fcolor, transp=80)

//Bar Color
barcolor(barcolor)