Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Fixed-Point implementation of Lattice Wave Digital Filter: comparison and error analysis Anastasia Volkova, Thibault Hilaire Sorbonne Universit´ es, University Pierre and Marie Curie, LIP6, Paris, France EUSIPCO 15 September 2, 2015 1/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation ? ? implementation Mathematical Target function Need to deal with Discretize functions and coefficients parametric errors computational errors Implementation under constraints software implementation hardware implementation 2/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Different filter structures: Direct Form I, Direct Form II State-space Wave, Lattice Wave, ... ρ -operator: ρ DFIIt, ρ Modal, ρ State-space... LGS, LCW, etc. Problem: They are equivalent in infinite precision but no more in finite precision. The finite precision degradation depends on the realization. 3/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Given transfer function and a target, we want: Represent various realizations Evaluate finite precision degradation Find an optimal realization Tradeoff: Error Quality Power consumption Speed etc. 4/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Given transfer function and a target, we want: Represent various realizations (in an easy way) Evaluate finite precision degradation Find an optimal realization Tradeoff: Error Quality Power consumption Speed etc. 4/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Given transfer function and a target, we want: Represent various realizations (in an easy way) Evaluate finite precision degradation (a priori/a posteriori) Find an optimal realization Tradeoff: Error Quality Power consumption Speed etc. 4/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Given transfer function and a target, we want: Represent various realizations (in an easy way) Evaluate finite precision degradation (a priori/a posteriori) Find an optimal realization (need to compare realizations) Tradeoff: Error Quality Power consumption Speed etc. 4/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Motivation Given transfer function and a target, we want: Represent various realizations (in an easy way) Evaluate finite precision degradation (a priori/a posteriori) Find an optimal realization (need to compare realizations) Tradeoff: Error Quality Power consumption Speed etc. Specialized Implicit Framework (SIF) 4/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Outline 1 Motivation 2 Specialized Implicit Framework 3 Lattice Wave Digital Filters 4 LWDF-to-SIF convertion 5 Example and comparison 6 Summary 5/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF SIF is: Macroscopic description Based on state-space Explicit all the computations and their order Any DFG can be transformed to this form Analytical derivation of measures Jt ( k + 1) = Mx ( k ) + N u ( k ) H x ( k + 1) = Kt ( k + 1) + P x ( k ) + Q u ( k ) y ( k ) = Lt ( k + 1) + Rx ( k ) + S u ( k ) Denote Z the matrix containing − J M N all the coefficients Z � K P Q L R S 6/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: measures Measures a priori measures � based on ∂H � transfer function sensitivity ∂ Z → stochastic measure, takes into account coefficient − wordlengths � � e.g based on ∂ | λ i | poles or zeros sensitivity for a pole λ i ∂ Z → stochastic measure, takes into account coefficient − wordlengths RNG, ... a posteriori measures Signal to Quantization Noise Ratio output error 7/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: measures Measures a priori measures � based on ∂H � transfer function sensitivity ∂ Z → stochastic measure, takes into account coefficient − wordlengths � � e.g based on ∂ | λ i | poles or zeros sensitivity for a pole λ i ∂ Z → stochastic measure, takes into account coefficient − wordlengths RNG, ... a posteriori measures Signal to Quantization Noise Ratio output error 7/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound WCPG theorem Let H = { A , B , C , D } be a BIBO stable MIMO state-space. If ∀ k u ( k ) � ¯ u component-wisely, then component-wisely y ( k ) � � ∀ k � H � � ¯ u , where � � H � � is the Worst-Case Peak Gain matrix of the system and can be computed as ∞ � � � � CA k B � � H � � := | D | + � . � � k =0 Note: we can compute � � H � � in arbitrary precision. 8/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound Exact filter: Jt ( k + 1)= Mx ( k )+ N u ( k ) H x ( k + 1)= Kt ( k + 1)+ P x ( k ) + Q u ( k ) y ( k )= Lt ( k + 1) + Rx ( k ) + S u ( k ) 9/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound Implemented filter: Jt ∗ ( k + 1)= Mx ∗ ( k )+ N u ( k )+ ε t ( k ) H ∗ x ∗ ( k + 1)= Kt ∗ ( k + 1)+ P x ∗ ( k ) + Q u ( k ) + ε x ( k ) y ∗ ( k )= Lt ∗ ( k + 1) + Rx ∗ ( k ) + S u ( k ) + ε y ( k ) where ε t ( k ), ε x ( k ) and ε y ( k ) are the computational errors. 9/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound Implemented filter: Jt ∗ ( k + 1)= Mx ∗ ( k )+ N u ( k )+ ε t ( k ) H ∗ x ∗ ( k + 1)= Kt ∗ ( k + 1)+ P x ∗ ( k ) + Q u ( k ) + ε x ( k ) y ∗ ( k )= Lt ∗ ( k + 1) + Rx ∗ ( k ) + S u ( k ) + ε y ( k ) where ε t ( k ), ε x ( k ) and ε y ( k ) are the computational errors. The output error ∆ y ( k ) � y ∗ ( k ) − y ( k ) can be seen as the output of a MIMO filter H ε . 9/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound Implemented filter: Jt ∗ ( k + 1)= Mx ∗ ( k )+ N u ( k )+ ε t ( k ) H ∗ x ∗ ( k + 1)= Kt ∗ ( k + 1)+ P x ∗ ( k ) + Q u ( k ) + ε x ( k ) y ∗ ( k )= Lt ∗ ( k + 1) + Rx ∗ ( k ) + S u ( k ) + ε y ( k ) where ε t ( k ), ε x ( k ) and ε y ( k ) are the computational errors. The output error ∆ y ( k ) � y ∗ ( k ) − y ( k ) can be seen as the output of a MIMO filter H ε . u ( k ) y ( k ) H y ∗ ( k ) ε ( k ) ∆ y ( k ) H ε + 9/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: the rigorous filter error bound Implemented filter: Jt ∗ ( k + 1)= Mx ∗ ( k )+ N u ( k )+ ε t ( k ) H ∗ x ∗ ( k + 1)= Kt ∗ ( k + 1)+ P x ∗ ( k ) + Q u ( k ) + ε x ( k ) y ∗ ( k )= Lt ∗ ( k + 1) + Rx ∗ ( k ) + S u ( k ) + ε y ( k ) where ε t ( k ), ε x ( k ) and ε y ( k ) are the computational errors. The output error ∆ y ( k ) � y ∗ ( k ) − y ( k ) can be seen as the output of a MIMO filter H ε . u ( k ) y ( k ) H y ∗ ( k ) ε ( k ) ∆ y ( k ) H ε + WCPG theorem on H ε gives the output error interval: ∆ y ( k ) � � � H ε � � ¯ ε 9/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: code generation WCPG theorem gives a rigorous way to compute Most Significant Bit: � � m y = log 2 ( � � H � � ¯ u ) + 1 WCPG-scaling, it guarantees that no Equivalent technique: overflows occur. 10/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: code generation WCPG theorem gives a rigorous way to compute Most Significant Bit: � � m y = log 2 ( � � H � � ¯ u ) + 1 WCPG-scaling, it guarantees that no Equivalent technique: overflows occur. Fixed Point Code Generator (FiPoGen) Given wordlength and evaluation scheme Generates bit-accurate fixed-point algorithms Given only evaluation scheme Optimizes the wordlength under certain criteria (e.g. area) Generates bit-accurate fixed-point algorithms 10/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: from transfer function to Fixed-Point code structures measures wordlengths target transfer code function Fixed-Point realization SIF FiPoGen algorithm choice 11/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: from transfer function to Fixed-Point code wordlengths target structures measures transfer code function Fixed-Point realization FiPoGen SIF algorithm choice 11/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary SIF: from transfer function to Fixed-Point code measures wordlengths target structures transfer code function realization Fixed-Point FiPoGen SIF choice algorithm 11/22
Motivation SIF LWDF LWDF-to-SIF Example and comparison Summary Lattice Wave Digital Filters Stage 2 Stage ( n − 1) z − 1 z − 1 γ 4 γ 2 · ( n − 1) Stage 0 z − 1 z − 1 z − 1 γ 0 γ 3 γ 2 · ( n − 1) − 1 High-pass output 1 / 2 + + Input − 1 γ 1 γ 5 γ 2 · n − 1 1 / 2 Low-pass output z − 1 z − 1 z − 1 γ 2 γ 6 γ 2 · n z − 1 z − 1 z − 1 Stage 1 Stage 3 Stage n 12/22
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