Determining Fixed-Point Formats using the Worst-Case Peak Gain measure Anastasia Volkova , Thibault Hilaire, Christoph Lauter Sorbonne Universit´ es, University Pierre and Marie Curie, LIP6, Paris, France ASILOMAR 49 November 10, 2015 1/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation State-space, DFI, DFII, . . . 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation State-space, DFI, DFII, . . . Software or Hardware implementation 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation ! Coefficient quantization State-space, DFI, DFII, . . . Software or Hardware implementation 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation ! Coefficient quantization State-space, DFI, DFII, . . . ! Large variety of structures with no common quality criteria Software or Hardware implementation 2/18
Motivation 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 Filter implementation flow: Transfer function generation ! Coefficient quantization State-space, DFI, DFII, . . . ! Large variety of structures with no common quality criteria Software or Hardware implementation ! Constraints: power consumption, area, error, speed, etc. 2/18
Motivation: Automatized filter implementation flow 3/18
Motivation: Automatized filter implementation flow Focus on Fixed-Point realization. 3/18
Motivation: Automatized filter implementation flow Focus on Fixed-Point realization. Optimization of wordlengths: Take more - pay more Take less - overflow risk What we want in the end: Rigorous algorithm for Fixed-Point Formats (FxPF) determination Integration into automatized code generator for filters Multiple wordlength paradigm 3/18
LTI filters H := ( A , B , C , D ) is a Bounded Input Bounded Output LTI filter in state-space representation: � x ( k + 1) = Ax ( k ) + Bu ( k ) H y ( k ) = Cx ( k ) + Du ( k ) with q inputs, n states and p outputs and state matrices 4/18
Basic brick: interval through filter The Worst-Case Peak Gain theorem Input u ( k ) 8 k, | u ( k ) | ≤ ¯ u Amplitude Time 5/18
Basic brick: interval through filter The Worst-Case Peak Gain theorem Input u ( k ) 8 k, | u ( k ) | ≤ ¯ u Amplitude u ( k ) y ( k ) H Time amplification/attenuation 5/18
Basic brick: interval through filter The Worst-Case Peak Gain theorem Input u ( k ) Output y ( k ) 8 k, | u ( k ) | ≤ ¯ u Amplitude Amplitude u ( k ) y ( k ) H Time amplification/attenuation Time 5/18
Basic brick: interval through filter The Worst-Case Peak Gain theorem Input u ( k ) Output y ( k ) 8 k, | u ( k ) | ≤ ¯ u 8 k, | y ( k ) | ≤ hhHii ¯ u Amplitude Amplitude u ( k ) y ( k ) H Time amplification/attenuation Time Worst-Case Peak Gain hhHii = | D | + P ∞ k =0 | CA k B | 5/18
Two’s complement Fixed-Point arithmetic 2 0 2 − 1 − 2 m 2 m − 1 2 ℓ s m + 1 − ℓ w m − 1 � t = − 2 m t m + 2 i t i i = ℓ Wordlength: w M ost S ignificant B it position: m L east S ignificant B it position: ℓ := m − w + 1 6/18
Two’s complement Fixed-Point arithmetic 2 0 2 − 1 − 2 m 2 m − 1 2 ℓ s m + 1 − ℓ w m − 1 � t = − 2 m t m + 2 i t i i = ℓ quantization step: 2 ℓ t represented by integer T = t · 2 ℓ T ∈ [ − 2 m ; 2 m − 2 ℓ ] ∩ Z 6/18
Two’s complement Fixed-Point arithmetic 2 0 2 − 1 − 2 m 2 m − 1 2 ℓ s m + 1 − ℓ w m − 1 � t = − 2 m t m + 2 i t i i = ℓ y ( k ) ∈ R wordlength w bits minimal Fixed-Point Format (FPF) is the least m : y ( k ) ∈ [ − 2 m ; 2 m − 2 m − w +1 ] ∀ k, 6/18
Problem statement Let H := ( A , B , C , D ) be a LTI filter: � x ( k + 1) = Ax ( k ) + Bu ( k ) H y ( k ) = Cx ( k ) + Du ( k ) Given wordlength constraints w x and w y for each state and output variable input domain ¯ u we need to determine the minimal FPF for all variables of filter H , i.e. find the least m x and m y such that x i ( k ) ∈ [ − 2 m xi ; 2 m xi − 2 m xi − w xi +1 ] ∀ k, y i ( k ) ∈ [ − 2 m yi ; 2 m yi − 2 m yi − w yi +1 ] . ∀ k, 7/18
Modification of H � x ( k ) � Let ζ ( k ) := be a vertical concatenation of state and y ( k ) output vectors. 8/18
Modification of H � x ( k ) � Let ζ ( k ) := be a vertical concatenation of state and y ( k ) output vectors. Then the state-space takes the following form: x ( k + 1) = Ax ( k ) + Bu ( k ) � I � � 0 � H ζ ζ ( k ) = x ( k ) + u ( k ) C D 8/18
Modification of H � x ( k ) � Let ζ ( k ) := be a vertical concatenation of state and y ( k ) output vectors. Then the state-space takes the following form: x ( k + 1) = Ax ( k ) + Bu ( k ) � I � � 0 � H ζ ζ ( k ) = x ( k ) + u ( k ) C D We seek to determine the least m ζ such that | ζ i ( k ) | � 2 m ζi − 2 m ζi − w i +1 . ∀ k, 8/18
Computing the MSB using the WCPG theorem Applying the WCPG theorem on filter H ζ gives ∀ k, | ζ i ( k ) | � ( � � H ζ � � ¯ u ) i 9/18
Computing the MSB using the WCPG theorem Applying the WCPG theorem on filter H ζ gives ∀ k, | ζ i ( k ) | � ( � � H ζ � � ¯ u ) i Therefore, the smallest m ζ i satisfying u ) i � 2 m ζi − 2 m ζi − w i +1 , ( � � H ζ � � ¯ will satisfy the wordlength constraints. 9/18
Computing the MSB using the WCPG theorem Applying the WCPG theorem on filter H ζ gives ∀ k, | ζ i ( k ) | � ( � � H ζ � � ¯ u ) i Therefore, the smallest m ζ i satisfying u ) i � 2 m ζi − 2 m ζi − w i +1 , ( � � H ζ � � ¯ will satisfy the wordlength constraints. We can compute m ζ i with 1 − 2 1 − w ζi �� � � m ζ i = log 2 ( � � H ζ � � ¯ u ) i − log 2 . 9/18
Taking the quantization errors into account The exact filter H ζ is: x ( k + 1) = Ax ( k ) + Bu ( k ) � I � � 0 � H ζ ζ ( k ) = x + u ( k ) C D 10/18
Taking the quantization errors into account The actually implemented filter H ♦ ζ is: x ♦ ( k + 1) � Ax ♦ ( k ) + Bu ( k ) � = ♦ m x H ζ ♦ � � I � � 0 � � ζ ♦ ( k ) x ♦ ( k ) + = ♦ m ζ u ( k ) C D where ♦ m is some operator ensuring faithful rounding: | ♦ m ( x ) − x | � 2 m − w +1 . 10/18
Taking the quantization errors into account The actually implemented filter H ♦ ζ is: x ♦ ( k + 1) � Ax ♦ ( k ) + Bu ( k ) � = ♦ m x H ζ ♦ � � I � � 0 � � ζ ♦ ( k ) x ♦ ( k ) + = ♦ m ζ u ( k ) C D where ♦ m is some operator ensuring faithful rounding: | ♦ m ( x ) − x | � 2 m − w +1 . It holds x ♦ ( k + 1) Ax ♦ ( k ) + Bu ( k ) + ε x ( k ) = H ♦ � I � � 0 � � ε x ( k ) � ζ ♦ ( k ) x ♦ ( k ) + ζ = u ( k ) + C D ε y ( k ) with | ε x ( k ) | � 2 m x − w x +1 | ε y ( k ) | � 2 m y − w y +1 . and 10/18
Implemented filter decomposition u ( k ) ζ ♦ ( k ) H ♦ ζ 11/18
Implemented filter decomposition u ( k ) ζ ♦ ( k ) H ♦ ζ 11/18
Implemented filter decomposition u ( k ) ζ ♦ ( k ) H ♦ ζ A q − 1 X ( k + 1) X ( k ) 11/18
Implemented filter decomposition u ( k ) H ζ ζ ( k ) u ( k ) ζ ♦ ( k ) H ♦ ζ ζ ♦ ( k ) A q − 1 X ( k + 1) X ( k ) � � ε x ( k ) H ∆ ∆ ζ ( k ) ε y ( k ) 11/18
Implemented filter decomposition u ( k ) H ζ ζ ( k ) u ( k ) ζ ♦ ( k ) H ♦ ζ ζ ♦ ( k ) m ζ A q − 1 X ( k + 1) X ( k ) � � ε x ( k ) H ∆ ∆ ζ ( k ) ε y ( k ) ! Filter H ∆ depends on the MSBs of filter H ζ 11/18
Implemented filter decomposition u ( k ) H ζ ζ ( k ) u ( k ) ζ ♦ ( k ) H ♦ ζ ζ ♦ ( k ) m ζ A q − 1 X ( k + 1) X ( k ) � � ε x ( k ) H ∆ ∆ ζ ( k ) ε y ( k ) ! Filter H ∆ depends on the MSBs of filter H ζ 11/18
Implemented filter decomposition u ( k ) H ζ ζ ( k ) u ( k ) ζ ♦ ( k ) H ♦ ζ ζ ♦ ( k ) m ζ A q − 1 X ( k + 1) X ( k ) � � ε x ( k ) H ∆ ∆ ζ ( k ) ε y ( k ) ! Filter H ∆ depends on the MSBs of filter H ζ 11/18
Two step approach u ( k ) Step 1 Determine the MSBs m ζ H ζ ζ ( k ) for the exact filter H ζ , ζ ♦ ( k ) m ζ applying the WCPG � ε x ( k ) � H ∆ ∆ ζ ( k ) theorem; ε y ( k ) Step 2 Compute the error-filter, u ( k ) H ζ ζ ( k ) induced by the format m ζ ζ ♦ ( k ) m ζ and deduce the FPF of the implemented filter H ♦ � ε x ( k ) � H ∆ ∆ ζ ( k ) ε y ( k ) ζ 12/18
MSB computation error analysis 1 − 2 1 − w i �� � � m ζ i = log 2 ( h h H ζ i i ¯ u ) i + log 2 13/18
MSB computation error analysis 1 − 2 1 − w i �� � � m ζ i = log 2 ( h h H ζ i i ¯ u ) i + log 2 | {z } \ hhH ζ ii + ε W CP G 13/18
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