""" **Description**
An Infinite Impulse Response ( IIR ) filter realizes a discrete
difference equation as a function of a recursive coefficient array,
a forward coefficient array, and a state array of a specified order,
consuming an incident signal and producing a reference signal.
.. math::
y_{n} = \sum_{i = 1}^{N} a_{i,n} y_{n-i} + \sum_{i = 0}^{N} b_{i,n} x_{n-i} = \sum_{i = 1}^{N} (\ a_{i,n} b_{0,n} + b_{i,n}\ ) s_{i,n} + b_{0,n} x_{n}\qquad a_{0,n} = 0
.. math::
s_{1,n+1} = \sum_{i = 1}^{N} a_{i,n} s_{i,n} + x_{n}\qquad\qquad s_{i,n+1} = s_{i-1,n}
A reset may minimize edge effects at a discontinuity by assuming
persistent operation at a specified incident signal condition.
.. math::
s_{i,n} = \\frac{1.0 - b_{0,n}}{\sum_{i=1}^{N} a_{i,n} b_{0,n} + b_{i,n}}\ x_{n}
A frequency response is expressed as a function of a recursive
coefficient array and a forward coefficient array.
.. math::
H_{z,n} = \\frac{\sum_{i = 0}^{N} b_{i,n} z^{-i}}{{1 - \sum_{i = 1}^{N} a_{i,n} z^{-i}}}
A factory is defined to facilitate construction of an instance,
defining a recursive coefficient array, a forward coefficient array,
and a state array of a specified order, to realize specified
constraints. An instance, classification, frequency, order, count,
complement, and gain are specified.
Frequency corresponds to a -3 dB frequency response normalized relative
to Nyquist.
Classification is in ( 'Bessel', 'Butterworth', 'Chebyshev' ).
* | 'Bessel' filters demonstrate maximally linear phase response or
| constant group delay.
* | 'Butterworth' filters demonstrate maximally flat magnitude response.
* | 'Chebyshev' filters demonstrate minimally low magnitude response error
| and improved rate of attenuation in a Type I form, with a maximum in
| band ripple of 0.125 dB.
Count is a quantity of filters of a specified order concatenated to
form an aggregate frequency response in cascade form.
Complement effectively constructs a mirror image of a specified
frequency response.
**Example**
::
from diamondback import IirFilter
import numpy
# Create an instance from a Factory with constraints.
obj = IirFilter.Factory.instance( typ = IirFilter, classification = 'Chebyshev', frequency = 0.1, order = 8, count = 1 )
# Create an instance with coefficients.
obj = IirFilter( a = obj.a, b = obj.b )
# Estimate frequency response, group delay, and roots.
y, f = obj.response( length = 8192, count = 1 )
y, f = obj.delay( length = 8192, count = 1 )
p, z = obj.roots( count = 1 )
# Filter an incident signal.
x = numpy.random.rand( 128 ) * 2.0 - 1.0
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, 2018-01-23.
**Definition**
"""
from diamondback.filters.FirFilter import FirFilter
from diamondback.interfaces.IA import IA
from diamondback.transforms.ZTransform import ZTransform
import math
import numpy
import scipy.signal
import typing
import warnings
[docs]class IirFilter( FirFilter, IA ) :
""" Infinite Impulse Response ( IIR ) filter.
"""
[docs] class Factory( object ) :
""" Factory.
"""
_classification = ( 'Bessel', 'Butterworth', 'Chebyshev' )
[docs] @staticmethod
def _evaluate( classification : str, frequency : float, order : int ) -> typing.Tuple[ numpy.ndarray, numpy.ndarray ] :
""" Evaluates coefficients.
Arguments :
classification : str - in ( 'Bessel', 'Butterworth', 'Chebyshev' ).
frequency : float - relative to Nyquist in ( 0.0, 1.0 ).
order : int.
Returns :
a : numpy.ndarray - recursive coefficient.
b : numpy.ndarray - forward coefficient.
"""
bilinear = True
if ( classification == 'Bessel' ) :
bilinear = False
u, a = numpy.ones( 1 ), numpy.ones( 2 )
for ii in range( 2, order + 1 ) :
x = numpy.concatenate( ( u, numpy.zeros( 2 ) ) ) + numpy.concatenate( ( [ 0.0 ], ( ( 2.0 * ii ) - 1.0 ) * a ) )
u, a = a, x
elif ( classification == 'Butterworth' ) :
a = numpy.ones( 1 )
for ii in range( 1, ( order // 2 ) + 1 ) :
a = numpy.convolve( a, numpy.array( [ 1.0, -2.0 * math.cos( ( ( ( 2.0 * ii ) + order - 1.0 ) / ( 2.0 * order ) ) * math.pi ), 1.0 ] ) )
if ( order & 1 ) :
a = numpy.convolve( a, numpy.ones( 2 ) )
elif ( classification == 'Chebyshev' ) :
ripple = 0.125
u = numpy.array( [ numpy.exp( 1j * math.pi * x / ( 2.0 * order ) ) for x in range( 1, 2 * order, 2 ) ] )
v = math.asinh( 1.0 / ( ( 10.0 ** ( 0.1 * ripple ) - 1.0 ) ** 0.5 ) ) / order
a = ( numpy.poly( ( -math.sinh( v ) * u.imag + 1j * math.cosh( v ) * u.real ) * 2.0 * math.pi ) ).real
a /= a[ -1 ]
a, b = ZTransform.transform( a, [ 1.0 ], frequency, bilinear )
b = numpy.poly( -numpy.ones( order ) )
b *= ( 1.0 - sum( a ) ) / sum( b )
return a, b
[docs] @classmethod
def instance( cls, typ : type, classification : str, frequency : float, order : int, count : int = 1, complement : bool = False, gain : float = 1.0 ) -> typing.Any :
""" Constructs an instance.
Arguments :
typ : type - derived from IirFilter.
classification : str - in ( 'Bessel', 'Butterworth', 'Chebyshev' ).
frequency : float - relative to Nyquist in ( 0.0, 1.0 ).
order : int.
count : int.
complement : bool.
gain : float.
Returns :
instance : typ( ).
"""
if ( ( not typ ) or ( not issubclass( typ, IirFilter ) ) ) :
raise ValueError( f'Type = {typ}' )
if ( ( not classification ) or ( classification not in IirFilter.Factory._classification ) ) :
raise ValueError( f'Classification = {classification}' )
if ( ( frequency <= 0.0 ) or ( frequency >= 1.0 ) ) :
raise ValueError( f'Frequency = {frequency}' )
if ( order <= 0 ) :
raise ValueError( f'Order = {order}' )
if ( count <= 0 ) :
raise ValueError( f'Count = {count}' )
if ( complement ) :
frequency = 1.0 - frequency
beta, eps, error = 10.0, numpy.finfo( float ).eps, float( 'inf' )
index, mu, zeta = 500 * ( 1 + ( count > 2 ) ), 2.5e-2, 1.0
a, b = [ ], [ ]
for ii in range( 0, index ) :
u, v = IirFilter.Factory._evaluate( classification, zeta * frequency, order )
x = numpy.exp( 1j * math.pi * frequency )
e = ( 2.0 ** ( -0.5 ) ) - ( ( abs( numpy.polyval( v, x ) / numpy.polyval( numpy.concatenate( ( [ 1.0 ], -u[ 1 : ] ) ), x ) ) ) ** count )
if ( abs( e ) < error ) :
a, b, error = u, v, abs( e )
if ( error < ( 10.0 * eps ) ) :
break
zeta = max( zeta + mu * math.tanh( beta * e ), eps )
if ( complement ) :
a *= numpy.array( [ ( ( -1.0 ) ** x ) for x in range( 0, len( a ) ) ] )
b *= numpy.array( [ ( ( -1.0 ) ** x ) for x in range( 0, len( b ) ) ] )
b /= sum( b * numpy.array( [ ( ( -1.0 ) ** x ) for x in range( 0, len( b ) ) ] ) ) / sum( numpy.concatenate( ( [ 1.0 ], -a[ 1 : ] ) ) * numpy.array( [ ( ( -1.0 ) ** x ) for x in range( 0, len( a ) ) ] ) )
return typ( a, b * gain )
def __init__( self, a : typing.Union[ typing.List, numpy.ndarray ] = numpy.zeros( 1 ), b : typing.Union[ typing.List, numpy.ndarray ] = numpy.ones( 1 ), s : typing.Union[ typing.List, numpy.ndarray ] = numpy.zeros( 1 ) ) -> None :
""" Initialize.
Arguments :
a : typing.Union[ typing.List, numpy.ndarray ] - recursive coefficient, s-domain.
b : typing.Union[ typing.List, numpy.ndarray ] - forward coefficient.
s : typing.Union[ typing.List, numpy.ndarray ] - state.
"""
if ( ( not numpy.isscalar( a ) ) and ( not isinstance( a, numpy.ndarray ) ) ) :
a = numpy.array( list( a ) )
if ( ( len( a.shape ) != 1 ) or ( ( len( a ) > 0 ) and ( a[ 0 ] ) ) ) :
raise ValueError( f'A = {a}' )
if ( len( a ) < len( b ) ) :
a = numpy.concatenate( ( a, numpy.zeros( len( b ) - len( a ) ) ) )
if ( len( b ) < len( a ) ) :
b = numpy.concatenate( ( b, numpy.zeros( len( a ) - len( b ) ) ) )
if ( ( isinstance( a[ 0 ], complex ) ) or ( isinstance( b[ 0 ], complex ) ) ) :
a, b = numpy.array( a, complex ), numpy.array( b, complex )
if ( a[ 0 ] != 0.0 ) :
raise ValueError( f'A = {a}' )
super( ).__init__( b, s )
self.a = numpy.array( a )
[docs] def delay( self, length : int = 8192, count : int = 1 ) -> typing.Tuple[ numpy.ndarray, numpy.ndarray ] :
""" Estimates group delay and produces a reference signal.
Arguments :
length : int.
count : int.
Returns :
y : numpy.ndarray - reference signal.
f : numpy.ndarray - relative to Nyquist in [ -1.0, 1.0 ).
"""
if ( length <= 0 ) :
raise ValueError( f'Length = {length}' )
if ( count <= 0 ) :
raise ValueError( f'Count = {count}' )
with warnings.catch_warnings( ) :
warnings.simplefilter( 'ignore' )
y, f = scipy.signal.group_delay( ( self.b, numpy.concatenate( ( [ 1.0 ], -self.a[ 1 : ] ) ) ), length, True )[ 1 ], numpy.linspace( -1.0, 1.0 - 2.0 / length, length )
y = numpy.concatenate( ( y[ len( y ) // 2 : ], y[ : len( y ) // 2 ] ) )
if ( length > 2 ) :
y[ 0 ] = y[ 1 ] * 2.0 - y[ 2 ]
return y, f
[docs] def filter( self, x : typing.Union[ typing.List, numpy.ndarray ] ) -> numpy.ndarray :
""" Filters an incident signal and produces a reference signal.
Arguments :
x : typing.Union[ typing.List, numpy.ndarray ] - incident signal.
Returns :
y : numpy.ndarray - reference signal.
"""
if ( ( not numpy.isscalar( x ) ) and ( not isinstance( x, numpy.ndarray ) ) ) :
x = numpy.array( list( x ) )
if ( ( len( x.shape ) != 1 ) or ( len( x ) == 0 ) ) :
raise ValueError( f'X = {x}' )
y = numpy.zeros( len( x ), type( self.b[ 0 ] ) )
for ii in range( 0, len( x ) ) :
self.s[ 0 ] = x[ ii ]
c = self.a * self.b[ 0 ] + self.b
y[ ii ] = c.dot( self.s )
if ( len( self.s ) > 1 ) :
z = self.a.dot( self.s )
self.s[ 1 : ] = self.s[ : -1 ]
self.s[ 1 ] += z
return y
[docs] def reset( self, x : typing.Union[ complex, float ] ) -> None :
""" Modifies a state to minimize edge effects by assuming persistent
operation at a specified incident signal condition.
Arguments :
x : typing.Union[ complex, float ] - incident signal.
"""
if ( not numpy.isscalar( x ) ) :
raise ValueError( f'X = {x}' )
if ( len( self.s ) > 1 ) :
self.s.fill( x * ( 1.0 - self.b[ 0 ] ) / ( self.a[ 1 : ] * self.b[ 0 ] + self.b[ 1 : ] ).sum( ) )
[docs] def response( self, length = 8192, count = 1 ) -> typing.Tuple[ numpy.ndarray, numpy.ndarray ] :
""" Estimates frequency response and produces a reference signal.
Arguments :
length : int.
count : int.
Returns :
y : numpy.ndarray - reference signal.
f : numpy.ndarray - relative to Nyquist in [ -1.0, 1.0 ).
"""
if ( length <= 0 ) :
raise ValueError( f'Length = {length}' )
if ( count <= 0 ) :
raise ValueError( f'Count = {count}' )
with warnings.catch_warnings( ) :
warnings.simplefilter( 'ignore' )
y, f = scipy.signal.freqz( self.b, numpy.concatenate( ( [ 1.0 ], -self.a[ 1 : ] ) ), length, True )[ 1 ], numpy.linspace( -1.0, 1.0 - 2.0 / length, length )
y = numpy.concatenate( ( y[ len( y ) // 2 : ], y[ : len( y ) // 2 ] ) ) ** count
return y, f
[docs] def roots( self, count = 1 ) -> typing.Tuple[ numpy.ndarray, numpy.ndarray ] :
""" Estimates roots of a frequency response in poles and zeros.
Arguments :
count : int.
Returns :
p : numpy.ndarray - poles.
z : numpy.ndarray - zeros.
"""
p, z = numpy.tile( numpy.roots( numpy.concatenate( ( [ 1.0 ], -self.a[ 1 : ] ) ) ), count ), numpy.tile( numpy.roots( self.b ), count )
return p[ numpy.argsort( abs( p ) ) ], z[ numpy.argsort( abs( z ) ) ]