""" **Description**
A derivative filter realizes a discrete difference equation which
approximates a discrete derivative as a function of a forward coefficient
array and a state array of a specified order, consuming an incident signal
and producing a reference signal. A derivative is approximated relative
to a sample. A derivative is electively approximated relative to a second
by multiplying a reference signal by an absolute sampling frequency raised
to a derivative power.
.. math::
y_{n} = \sum_{i = 0}^{N} b_{i} x_{n-i} = \sum_{i = 1}^{N} b_{i} s_{i,n} + b_{0} x_{n}
s_{1,n+1} = x_{n}\qquad\quad s_{i,n+1} = s_{i-1,n}
A frequency response is expressed as a function of a forward coefficient
array.
.. math::
H_{z} = \sum_{i = 0}^{N} b_{i} z^{-i}
A factory is defined to facilitate construction of an instance, defining
a forward coefficient array and a state array of a specified order, to
satisfy specified constraints. An instance, derivative, and order are
specified.
.. math::
y_{n} = f^{D}\ \\frac{x_{n}}{D}\quad\quad\quad\quad\scriptsize{ f = 1.0 }
.. math::
b_{1,1} = \scriptsize{ [ \matrix{ 1 & -1 } ] }
.. math::
b_{1,2} = \scriptsize{ [ \matrix{ 1 & 0 & -1 } ]\ \\frac{1}{2} }
.. math::
b_{1,4} = \scriptsize{ [ \matrix{ -1 & 8 & 0 & -8 & 1 } ]\ \\frac{1}{12} }
.. math::
b_{2,2} = \scriptsize{ [ \matrix{ 1 & -2 & 1 } ] }
.. math::
b_{2,4} = \scriptsize{ [ \matrix{ 1 & 0 & -2 & 0 & 1 } ]\ \\frac{1}{4} }
.. math::
b_{2,6} = \scriptsize{ [ \matrix{ -1 & 8 & 1 & -16 & 1 & 8 & -1 } ]\ \\frac{1}{24} }
.. math::
b_{2,8} = \scriptsize{ [ \matrix{ 1 & -16 & 64 & 16 & -130 & 16 & 64 & -16 & 1 } ]\ \\frac{1}{144} }
.. math::
b_{3,4} = \scriptsize{ [ \matrix{ 1 & -2 & 0 & 2 & -1 } ]\ \\frac{1}{2} }
.. math::
b_{3,6} = \scriptsize{ [ \matrix{ 1 & 0 & -3 & 0 & 3 & 0 & 1 } ]\ \\frac{1}{8} }
.. math::
b_{3,8} = \scriptsize{ [ \matrix{ -1 & 8 & 2 & -24 & 0 & 24 & -2 & -8 & 1 } ]\ \\frac{1}{48} }
**Example**
::
from diamondback import ComplexExponentialFilter, DerivativeFilter
import numpy
# Create an instance from a Factory with constraints.
obj = DerivativeFilter.Factory.instance( typ = DerivativeFilter, derivative = 1, order = 2 )
# Filter an incident signal.
x = ComplexExponentialFilter( 0.0 ).filter( numpy.ones( 128 ) * 0.1 ).real
obj.reset( x[ 0 ] )
y = obj.filter( x )
**License**
`BSD-3C. <https://github.com/larryturner/diamondback/blob/master/license>`_
© 2018 - 2021 Larry Turner, Schneider Electric Industries SAS. All rights reserved.
**Author**
Larry Turner, Schneider Electric, Analytics & AI, 2018-02-06.
**Definition**
"""
from diamondback.filters.FirFilter import FirFilter
from typing import Any, List, Union
import numpy
[docs]class DerivativeFilter( FirFilter ) :
""" Derivative filter.
"""
[docs] class Factory( object ) :
""" Factory.
"""
_b = { 1 : { 1 : numpy.array( [ 1.0, -1.0 ] ),
2 : numpy.array( [ 1.0, 0.0, -1.0 ] ) * ( 1.0 / 2.0 ),
4 : numpy.array( [ -1.0, 8.0, 0.0, -8.0, 1.0 ] ) * ( 1.0 / 12.0 ) },
2 : { 2 : numpy.array( [ 1.0, -2.0, 1.0 ] ),
4 : numpy.array( [ 1.0, 0.0, -2.0, 0.0, 1.0 ] ) * ( 1.0 / 4.0 ),
6 : numpy.array( [ -1.0, 8.0, 1.0, -16.0, 1.0, 8.0, -1.0 ] ) * ( 1.0 / 24.0 ),
8 : numpy.array( [ 1.0, -16.0, 64.0, 16.0, -130.0, 16.0, 64.0, -16.0, 1.0 ] ) * ( 1.0 / 144.0 ) },
3 : { 4 : numpy.array( [ 1.0, -2.0, 0.0, 2.0, -1.0 ] ) * ( 1.0 / 2.0 ),
6 : numpy.array( [ 1.0, 0.0, -3.0, 0.0, 3.0, 0.0, -1.0 ] ) * ( 1.0 / 8.0 ),
8 : numpy.array( [ -1.0, 8.0, 2.0, -24.0, 0.0, 24.0, -2.0, -8.0, 1.0 ] ) * ( 1.0 / 48.0 ) } }
[docs] @classmethod
def instance( cls, typ : type, derivative : int, order : int ) -> Any :
""" Constructs an instance.
Arguments :
typ : type - derived from DerivativeFilter.
derivative : int - in [ 1, 3 ].
order : int.
Returns :
instance : typ( ).
"""
if ( ( not typ ) or ( not issubclass( typ, DerivativeFilter ) ) ) :
raise ValueError( f'Type = {typ}' )
if ( derivative not in DerivativeFilter.Factory._b ) :
raise ValueError( f'Derivative = {derivative}' )
b = DerivativeFilter.Factory._b[ derivative ]
if ( order not in b ) :
raise ValueError( f'Order = {order}' )
return typ( b[ order ] )
def __init__( self, b : Union[ List, numpy.ndarray ] = numpy.ones( 1 ) ) -> None :
""" Initialize.
Arguments :
b : Union[ List, numpy.ndarray ] - forward coefficient.
"""
super( ).__init__( b )
[docs] def filter( self, x : Union[ List, numpy.ndarray ] ) -> numpy.ndarray :
""" Filters an incident signal and produces a reference signal.
Arguments :
x : Union[ List, numpy.ndarray ] - incident signal.
Returns :
y : numpy.ndarray - reference signal.
"""
return super( ).filter( x )