Source code for diamondback.filters.IirFilter

""" **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
from typing import Any, List, Tuple, Union
import math
import numpy
import scipy.signal
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 ) -> 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 ) -> 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 : Union[ List, numpy.ndarray ] = numpy.zeros( 1 ), b : Union[ List, numpy.ndarray ] = numpy.ones( 1 ), s : Union[ List, numpy.ndarray ] = numpy.zeros( 1 ) ) -> None : """ Initialize. Arguments : a : Union[ List, numpy.ndarray ] - recursive coefficient, s-domain. b : Union[ List, numpy.ndarray ] - forward coefficient. s : Union[ 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 ) -> 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 : 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. """ 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 : Union[ complex, float ] ) -> None : """ Modifies a state to minimize edge effects by assuming persistent operation at a specified incident signal condition. Arguments : x : 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 ) -> 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 ) -> 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 ) ) ]