Tutorial on Wave Digital Filters

Intro WDF Summary Appendix David Yeh Center for Computer Research in Music and Acoustics (CCRMA) Stanford University CCRMA DSP Seminar January 25, 2008

Outline Intro WDF Summary Appendix 1 Introduction Motivation Classical Network Theory 2 Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity 3 Summary Examples Conclusions 4 Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Outline Intro WDF Summary Appendix Motiv Netwrk Thry 1 Introduction Motivation Classical Network Theory 2 Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity 3 Summary Examples Conclusions 4 Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Intro WDF Summary Appendix Motiv Netwrk Thry Wave digital filters model circuits used for filtering. Overview Fettweis (1986), Wave Digital Filters: Theory and Practice. Wave Digital Filters (WDF) mimic structure of classical filter networks. Low sensitivity to component variation. Use wave variable representation to break delay free loop. WDF adaptors have low sensitivity to coefficient quantization. Direct form with second order section biquads are also robust Transfer function abstracts relationship between component and filter state WDF provides direct one-to-one mapping from physical component to filter state variable

Intro WDF Summary Appendix Motiv Netwrk Thry Wave digital filters model circuits used for filtering Applications Modeling physical systems with equivalent circuits. Piano hammer mass spring interaction Generally an ODE solver Element-wise discretization and connection strategy Real time model of loudspeaker driver with nonlinearity Multidimensional WDF solves PDEs Ideal for interfacing with digital waveguides (DWG).

Intro WDF Summary Appendix Motiv Netwrk Thry Classical Network Theory N-port linear system N port + V 1 + V 2... + V n I 1 I2 I n Describe a circuit in terms of voltages (across) and current (thru) variables General N-port network described by V and I of each port Impedance or admittance matrix relates V and I V 1 Z 11 Z 12... Z 1N I V 2. =. Z.. 1 21 Z2N I 2... V N Z N1... Z NN I N }{{} Z

Intro WDF Summary Appendix Motiv Netwrk Thry Classical Network Theory Element-wise discretization for digital computation For example, use Bilinear transform s = 2 T 1 z 1 1+z 1 Capacitor: Z (s) = 1 sc Z (z 1 ) = T 1 + z 1 2C 1 z 1 = V (z 1 ) I(z 1 ) v[n] = T (i[n] + i[n 1]) + v[n 1] 2C v[n] depends instantaneously on i[n] with R 0 = T 2C This causes problems when trying to make a signal processing algorithm Can also solve for solution using a matrix inverse (what SPICE does).

Intro WDF Summary Appendix Motiv Netwrk Thry Classical Network Theory Wave variable substitution and scattering A =V + RI B =V RI V = A+B 2 I = A B 2R Variable substitution from V and I to incident and reflected waves, A and B An N-port gives an N N scattering matrix Allows use of scattering concept of waves

Intro WDF Summary Appendix Motiv Netwrk Thry Classical Network Theory Two port is commonly used in microwave electronics to characterize amplifiers a 1 a 2 R 1 R 2 Input port (1) and output port (2) ( ) ( ) ( ) V1 Z11 Z = 12 I1 V 2 Z 21 Z 22 b 1 b 2 I 2 Represent as scattering matrix and wave variables ( ) ( ) ( ) b1 S11 S = 12 a1 b 2 S 21 S 22 Scattering matrix S determines reflected wave b n as a linear combination of N incident waves a 1,..., a n Guts of the circuit abstracted away into S or Z matrix a 2

Outline Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity 1 Introduction Motivation Classical Network Theory 2 Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity 3 Summary Examples Conclusions 4 Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements Basic one port elements Work with voltage wave variables b and a. Substitute into Kirchhoff circuit equations and solve for b as a function of as. Wave reflectance between two impedances is well known ρ = b a = R 2 R 1 R 2 + R 1 Define a port impedance R p Input wave comes from port and reflects off the element s impedances. Resistor Z R = R, ρ R (s) = 1 Rp/R 1+R p/r Capacitor Z C = 1 sc, ρ C(s) = 1 RpCs 1+R pcs Inductor Z L = sl, ρ L (s) = s Rp/L s+r pl

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements Discretize the capacitor by bilinear transform Plug in bilinear transform b n = 1 R pc 2 T a n 1 + R p C 2 T 1 z 1 1+z 1 1 z 1 1+z 1 (1 + R p C 2 T )b[n] + (1 R pc 2 T )b[n 1] = (1 R p C 2 T )a[n] + (1 + R pc 2 T )a[n 1] Choose R p to eliminate depedence of b[n] on a[n], e.g., R p = T 2C, resulting in: b[n] = a[n 1] Note that chosen R p exactly the instantaneous resistance of the capacitor when discretized by the bilinear transform

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements T-ports and I-ports T-port. Port resistance can be chosen to perfectly match element resistance to eliminate instantaneous reflection and avoid delay-free loop. I-port. If port is not matched, b[n] depends on a[n] instantaneously. R p can be chosen as any positive value. Short circuit b[n] = a[n] Open circuit b[n] = a[n] Voltage source of voltage V b[n] = a[n] + 2V Current source of current I b[n] = a[n] 2R p I

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements One port summary for voltage waves Element Port Resistance Reflected wave Resistor R p = R b[n] = 0 Capacitor R p = T 2C b[n] = a[n 1] Inductor R p = 2L T b[n] = a[n 1] Short circuit R p b[n] = a[n] Open circuit R p b[n] = a[n] Voltage source V s R p b[n] = a[n] + 2V s Current source I s R p b[n] = a[n] + 2R p I s Terminated V s R p = R s b[n] = V s Terminated I s R p = R s b[n] = R p I s I s R s V + s R s

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements One port summary for current waves. Signs are flipped for some reflectances. Element Port Resistance Reflected wave Resistor R p = R b[n] = 0 Capacitor R p = T 2C b[n] = a[n 1] Inductor R p = 2L T b[n] = a[n 1] Short circuit R p b[n] = a[n] Open circuit R p b[n] = a[n] Voltage source V s R p b[n] = a[n] + 2V s Current source I s R p b[n] = a[n] + 2R p I s Terminated V s R p = R s b[n] = V s Terminated I s R p = R s b[n] = R p I s I s R s V + s R s

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Wave Digital Elements Two ports Series Parallel Transformer Unit element

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Adaptors Adaptors perform the signal processing calculations Treat connection of N circuit elements as an N-port Derive scattering junction from Kirchhoff s circuit laws and port impedances determined by the attached element Scattering matrix is N N Parallel and series connections can be simplified to linear complexity Signal flow diagrams to reduce number of multiply or add units Dependent port - one coefficient can be implied, Reflection free port - match impedance to eliminate reflection

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Two-Port Parallel Adaptor b 1 = a 2 + γ(a 2 a 1 ) b 2 = a 1 + γ(a 2 a 1 ) γ = (R 1 R 2 )/(R 1 + R 2 )

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity N-Port Parallel Adaptor G ν are the port conductances G n = G 1 + G 2 + + G n 1, G ν = 1/R ν Find scattering parameters Note γ sum to 2 γ ν = G ν G n, ν = 1 to n 1 γ 1 + γ 2 + + γ n = 2 Use intermediate variable to find reflected waves a o = γ 1 a 1 + γ 2 a 2 + + γ n a n b ν = a o a ν

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Parallel Adaptor with port n reflection free (RFP) G ν are the port conductances R n is set equal to equivalent resistance looking at all the other ports (their Rs in parallel) to make port n RFP G n = G 1 + G 2 + + G n 1, G ν = 1/R ν γ n = 1 γ 1 + γ 2 + + γ n 1 = 1 γ ν = G ν G n, ν = 1 to n 1 b n = γ 1 a 1 + γ 2 a 2 + + γ n 1 a n 1, RFP b ν = b n + a n a ν

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Series Adaptor with port n dependent R ν are the port resitances. Find scattering parameters γ ν = 2R ν R 1 + R 2 + + R n Note scattering parameters sum to 2 γ 1 + γ 2 + + γ n = 2 Use intermediate variable to find reflected waves a o = a 1 + a 2 + + a n b ν = a ν γ ν a o

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Series Adaptor with port n reflection free (RFP) R ν are the port resitances R n is set equal to equivalent resistance looking at all the other ports (their Rs in series) to make port n RFP R n = R 1 + R 2 + + R n 1 γ n = 1 γ 1 + γ 2 + + γ n 1 = 1 γ ν = R ν R n, ν = 1 to n 1 b n = (a 1 + a 2 + + a n 1 ), b ν = a ν γ ν (a n b n ) RFP

Adaptors Dependent ports Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Adaptors have property of low coefficient sensitivity, e.g., coefficients can be rounded or quantized. Dependent ports take advantage of property that γs sum to two. Use this fact along with quantization to ensure that adaptor is (pseudo-)passive.

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Connection Strategy Parameter updates Parameter updates propagate from leaf through its parents to the root Each adaptor s RFP must be recalculated when a child s port resistance changes Parameter update is more complicated than solving the Kirchhoff s equations directly, where the parameters are just values in the resistance matrix.

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Connection Strategy Avoid delay-free loops with adaptors connected as a tree Sarti et. al., Binary Connection Tree - implement WDF with three-port adaptors Karjalainen, BlockCompiler - describe WDF in text, produces efficient C code Scheduling to compute scattering Directed tree with RFP of each node connected to the parent Label each node (a, b, c,...) Label downward going signals d by node and port number Label upward going signals u by node Start from leaves, calculate all u going up the tree Then start from root, calculate all d going down the tree

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Connection Strategy Automatic generation of WDF tree structure SPQR tree algorithms find biconnected and triconnected graphs (Fränken, Ochs, and Ochs, 2005. Generation of Wave Digital Structures for Networks Containing Multiport Elements.) Q nodes are one ports S and P nodes are Series and Parallel adaptors R nodes are triconnected elements Implemented with similarity transform of N N scattering matrix into two-port adaptors (Meerkötter and Fränken, Digital Realization of Connection Networks by Voltage-Wave Two-Port Adaptors Includes bridge connections and higher order connections Implemented in WDInt package for Matlab (http://www-nth.uni-paderborn.de/wdint/index.html)

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity WDF Interconnections More about the R nodes For example, bridge connection higher order triconnections also common N-port scattering junction can be reduced to implementation by two port adaptors using similarity transform - keeps robust properties for quantization reduces operations for filtering vs scatter matrix In general parameter update is complicated

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity R-Nodes Other strategies: Y Y transformations a Za Zb b a Zab b c Zc Zac c Zbc Circuit analysis technique to replace triconnected impedances with equivalents that can be connected in series or parallel. Must discretize general impedances, no longer correspondence between prototype circuit element and WDF element

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity R-Nodes Example: Guitar amplifier tone stack with input and output loading in C1 R4 100k 250p Rp1t (1-potc1)*250k out Rp1b Rs a C2 100n potc1*250k b RL in Rs R4 C1 Rs ZY2a a b ZY2b out C3 47n c Rp2 Rp3 potc2*10k potc3*250k ZY1a ZY2a ZY2b Rp1t out or ZY2c c ZY1c RL ZY2c Rp2 in C1 R4 Rp1t Rp1b out in a R4 C1 Rs a ZY1a ZY1c c ZY1b b RL Rs ZY1a c ZY1b ZY1c Rp1b b out Rp1t RL Rp2 Rp2

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Tone Stack Formulate blocks compatabile with scattering Observe that tone stack is a specific two-port Direct implementation of a 2-port scattering matrix Or convert into an equivalent circuit with impedances and use adaptors Tabulate the scattering parameters or impedances as they vary with parameter changes RT1 RT2 Rs RT RL

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Nonlinearity Literature Meerkötter and Scholz (1989), Digital Simulation of Nonlinear Circuits by Wave Digital Filter Principles. Sarti and De Poli (1999), Toward Nonlinear Wave Digital Filters. Karjalainen and Pakarinen (2006), Wave Digital Simulation of Vacuum-Tube Amplifier Petrausch and Rabenstein (2004), Wave Digital Filters with Multiple Nonlinearities Either conceive as nonlinear resistor or dependent source Introduces an I-port, may lead to delay-free loops DFL must be solved as a system of equations in wave variables

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Nonlinear Conductance Meerkötter and Scholz (1989). Current is a nonlinear function of voltage, i = i(v) In wave variables a = f (v) = v + R p i(v) b = g(v) = v R p i(v) Substituting wave variables into Kirchhoff variable definition of nonlinear resistance and solving for b(a) f 1 must exist b = b(a) = g(f 1 (a)) Port resistance R p can be chosen arbitrarily within constraints that f (v) be invertible Instantaneous dependence exists regardless of R p

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Extension to nonlinear reactances Nonlinear Capacitor. Sarti and De Poli (1999). Use a mutator to integrate the Kirchhoff variable so that the nonlinear reactance can be defined in terms of wave variables. b 1 Capacitor/Mutator a 1 a 2 R 1 R 2 b 2 Voltage current Define wave variable such that port resistance can be an impedance. Voltage charge A(s) = V (s) + R(s)I(s) B(s) = V (s) R(s)I(s) Recall that standard wave definitions for a one port such as a capacitor are A 1 = V 1 + R 1 I 1 B 1 = V 1 R 1 I 1

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Extension to nonlinear reactances Nonlinear Capacitor. Sarti and De Poli (1999). For the capacitor with the usual single resistive port, define a second port across the capacitor with its port impedance R(s) = 1 sc A 2 (s) = V 2 (s) + 1 sc I 2(s) B 2 (s) = V 2 (s) 1 sc I 2(s) This looks like the usual wave variable definitions if I(s) is replaced by its integral Q(s), charge, and port impedance R 2 = 1/C. A 2 (s) = V 2 (s) + 1 C Q(s) B 2 (s) = V 2 (s) 1 C Q(s) A 2 and B 2 can be substituted into the definition of a generic nonlinear capacitance Q = f (V ) = C(V ) V to find the nonlinear reflection as it is done for the nonlinear resistor. R 2 can be chosen rather arbitrarily as before.

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Extension to nonlinear reactances Voltage-current to voltage-charge wave conversion (Felderhoff 1996). Compute scattering relations between the resistive and the integrated port using two relations: consistency of voltage for the two ports V 1 = V 2, and I 1 + I 2 = 0 across junction. V = A 1+B 1 2 = A 2+B 2 2 (1) bilinear transform s z I 1 = A 1 B 1 2R 1 = A 2 B 2 2R(s) = A 2 B 2 2 1 sc (2) A 1 B 1 R 1 = A 2 B 2 1 C T 1+z 1 2 1 z 1 (1 + z 1 )(A 1 B 1 )= 2 T CR 1(1 z 1 )(A 2 B 2 ) (A 1 + z 1 A 1 B 1 z 1 B 1 )= 2 T CR 1(A 2 z 1 A 2 B 2 + z 1 B 2 ) a 1 [n] + a 1 [n 1] b 1 [n] b 1 [n 1]= 2 T CR 1(a 2 [n] a 2 [n 1] b 2 [n] + b 2 [n 1]) substitute b 2 = a 1 + b 1 a 2 using (1) a 1 [n] + a 1 [n 1] b 1 [n] b 1 [n 1] = 2 T CR 1(a 2 [n] a 2 [n 1] (a 1 [n] + b 1 [n] a 2 [n]) + a 1 [n 1] + b 1 [n 1] a 2 [n 1])

Intro WDF Summary Appendix WD 1-Ports Adaptors Nonlinearity Extension to nonlinear reactances Voltage-current to voltage-charge wave conversion. Capacitor/Mutator Set R 1 = T /(2C) to eliminate reflection at port 1. a 1 [n 1] b 1 [n] = a 2 [n] + a 2 [n 1] a 1 a 2 R 1 R 2 b 1 b 2 Voltage current Voltage charge Resulting scattering junction (or mutator according to Sarti and De Poli) converts between voltage-current and voltage-charge waves: b 2 = a 1 + (a 1 [n 1] a 2 [n 1]) b 1 = a 2 + (a 1 [n 1] a 2 [n 1]) Port resistance for new mutated waves corresponding to voltage and charge is R 2 = 1 C. Port resistance for usual waves corresponding to voltage and current is R 1 = T 2C = T 2 R 2.

Outline Intro WDF Summary Appendix Examples Conclusions 1 Introduction Motivation Classical Network Theory 2 Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity 3 Summary Examples Conclusions 4 Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Intro WDF Summary Appendix Examples Conclusions Parametrized Linear Circuit Example Volume pot with bright switch Rs + Vs C Rt Rv Vo Output is voltage over R v, V o = (d 1S +u Rt ) 2 R V = R pot (vol), R t = R pot (1 vol). Changes in vol require recomputation of γ s starting from bottom of tree. Use RFP to allow open circuit when C is disconnected. Rt u Rt u P d C P d 2P d1p u S Rv u Rv S d P u C d 2S d u 1S V Rs V C u Rv = u Rt = 0, u S = (u Rv + u V ) u P = γ 1P u Rt + γ 2P u S u C = d P [n 1] u V = V d P = u P + γ P (u C u P ) d C = u C + γ P (u C u P ), or d C = u P d 1P = u P + d C u Rt, d 2P = u P + d C u S d 1S = u Rv γ 1S (d 2P u S ), d 2S = u V γ 2S (d 2P u S ), don t care output value don t care

Intro WDF Summary Appendix Examples Conclusions Nonlinear Circuit Example Diode clipper Vi u_s u_v d_1s R_V 2.2k S Cc 0.47u u_p R_D d_1p u_ch d_2s D P d_d d_2p u_cl 0.01u Cl Vo Diode clipper circuit found in guitar distortion pedals Treat diodes together as single nonlinear one-port I(V ) = 2I s sinh (V /V d ) Solve for b(a), a b = 2I 2R s sinh a+b p 2V d Isolate nonlinearity at root of tree Incorporate resistor into voltage source V Ch

Intro WDF Summary Appendix Examples Conclusions Nonlinear Circuit Example Diode clipper: Computational algorithm Cc Vi Vo 2.2k 0.47u 0.01u u_p R_D D d_d u V = V, u Ch = d 2S [n 1] u S = (u V + u Ch ), u Cl = d 2P [n 1]) u P = γ 1P u S + γ 2P u Cl d D = f (u P ), nonlinear function u_s d_1p P d_2p u_cl d 1P = u P + d D u S d 2P = u P + d D u Cl u_v d_1s R_V S u_ch d_2s Cl d 1S = u V γ 1S (d 1P u S ), d 2S = u Ch γ 2S (d 1P u S ) don t care V Ch

Intro WDF Summary Appendix Examples Conclusions Nonlinear Musical Acoustics Example WDF hammer and nonlinear felt Draw mass/spring/waveguide system in terms of equivalent circuits Waveguides look like resistors to the lumped hammer. Waves enter lumped junction directly. WDF result in tree like structures with adaptors/scattering junctions at the nodes, and elements at the leaves. The root of the tree allowed to have instantaneous reflections Nonlinearity gives instantaneous reflection, WDF handles only 1 nonlinearity naturally. Compression (d = u nl d nl from next slide) 2R nl must 0, otherwise hammer is not in contact. u m m u nl u P d nl d k P d 2P d1p u S R1 NL v + 1 v 1 S v + 2 R2 v 2 To left and right waveguides

Intro WDF Summary Appendix Examples Conclusions Nonlinear Musical Acoustics Example Computations v + 1, v 2 from waveguides u S = (v + 1 + v 2 ) u m = d 1P [n 1] u P = γ 1P u m + γ 2P u S u nl = u P + (u P [n 1] d nl [n 1]) n γ o d nl : solve unl +d nl = k unl d nl 2 2R nl d k = d nl + (u P [n 1] d nl [n 1]) d 1P = u P + d k u m d 2P = u P + d k u S v 1 = v + 1 γ 1S (d 2P u S ) v + 2 = v 2 γ 2S(d 2P u S ) u m m u nl u P d nl d k P d 2P d1p u S R1 NL v + 1 v 1 S v + 2 R2 v 2 To left and right waveguides

Intro WDF Summary Appendix Examples Conclusions Observations Scattering formulation works well with DWG - DWG looks like resistor in WDF For a standalone simulation of nonlinear circuits, may not be the best choice More difficult for parameter update than direct solving Root node is special - easy to implement nonlinearity or parameter changes May be able to design circuits without bridge connections that have equivalent transfer function

Summary Intro WDF Summary Appendix Examples Conclusions Wave digital formulation uses matched (reflection-free) ports to eliminate reflections and avoid delay-free loops Elements are connected in tree structure with reflection-free ports connected to the parent node. Useful for building up a model component-wise.

Outline Intro WDF Summary Appendix Scattering Mechanical 1 Introduction Motivation Classical Network Theory 2 Wave Digital Formulation Wave Digital One-Ports Derivation Wave Digital Adaptors Nonlinearity 3 Summary Examples Conclusions 4 Appendix Scattering Junction Derivations Mechanical Impedance Analogues

Intro WDF Summary Appendix Scattering Mechanical Two-port adaptor reflectance and transmittance

Intro WDF Summary Appendix Scattering Mechanical N-port parallel adaptor

Intro WDF Summary Appendix Scattering Mechanical N-port series adaptor

Capacitor Intro WDF Summary Appendix Scattering Mechanical

Inductor Intro WDF Summary Appendix Scattering Mechanical

Intro WDF Summary Appendix Scattering Mechanical Mechanical Impedance Analogues from Music 420 The following mechanical examples are taken from: Lumped Elements, One-Ports, and Passive Impedances, by Julius O. Smith III, (From Lecture Overheads, Music 420). http://ccrma.stanford.edu/ jos/oneports/ http://ccrma.stanford.edu/~jos/oneports/ Copyright 2007-02-22 by Julius O. Smith III

Intro WDF Summary Appendix Scattering Mechanical Dashpot Ideal dashpot characterized by a constant impedance µ f(t) v(t) µ Dynamic friction law f (t) µv(t) Impedance Ohm slaw (Force = Friction_coefficient Velocity) R µ (s) µ 0 Dashpot = gain for force input and velocity output Electrical analogue: Resistor R = µ More generally, losses due to friction are frequency dependent hysteretic

Mass Intro WDF Summary Appendix Scattering Mechanical v(t) f(t) m 0 x(t) Ideal mass of m kilograms sliding on a frictionless surface Newton s 2nd Law f (t) = ma(t) m v(t) mẍ(t)(force = Mass Acceleration) Differentiation Theorem F (s) = m[sv (s) v(0)] = msv (s) for Laplace Transform when v(0) = 0. Impedance R m (s) F(s) V (s) = ms

Mass Intro WDF Summary Appendix Scattering Mechanical F(s) Γ(s) V(s) Black Box Description Admittance Γ m (s) 1 R m (s) = Impulse Response (unit-momentum input) 1 ms γ m (t) L 1 {Γ m (s)} = 1 m u(t) Frequency Response Γ m (jω) = 1 mjω Mass admittance = Integrator (for force input, velocity output) Electrical analogue: Inductor L = m.

Intro WDF Summary Appendix Scattering Mechanical Spring (Hooke s Law) f(t) k 0 x(t) Ideal spring Hooke s law f (t) = kx(t) k t 0 v(τ)dτ(force = Stiffness Displacement) Impedance R k (s) F(s) V (s) = k s Frequency Response Γ k (jω) = jω k Spring = differentiator (force input, velocity output) Velocity v(t) = compression velocity Electrical analogue: Capacitor C = 1/k (1/stiffness = compliance )

Intro WDF Summary Appendix Scattering Mechanical Series Connection of One-Ports F(s) = F 1 (s) + F 2 (s) + F 1 (s) Γ 1 (s) - V(s) = V 1 (s) = V 2 (s) + F 2 (s) Γ 2 (s) - Series Impedances Sum: Admittance: Physical Reasoning: Γ(s) = R(s) = R 1 (s) + R 2 (s) 1 1 Γ 1 (s) + 1 Γ 2 (s) = Γ 1Γ 2 Γ 1 + Γ 2 Common Velocity Series connection Summing Forces Series connection

Intro WDF Summary Appendix Scattering Mechanical Parallel Combination of One-Ports V(s) = V 1 (s) + V 2 (s) V 1 (s) V 2 (s) Γ 1 (s) + F(s) - Γ 2 (s) F(s) = F 1 (s) = F 2 (s) Parallel Admittances Sum Γ(s) = Γ 1 (s) + Γ 2 (s) Impedance: R(s) = 1 1 R 1 (s) + 1 R 2 (s) = R 1R 2 R 1 + R 2 or, for EEs, R = R 1 R 2 Physical Reasoning: Common Force Parallel connection Summing Velocities Parallel connection

Intro WDF Summary Appendix Scattering Mechanical Mass-Spring-Wall (Series) Physical Diagram: Electrical Equivalent Circuit: f ( t) + f ( t) + f ( t) 0 e xt m k = v ( t) v ( t) m = f ext ( t) m f k ( t) f m ( t) 0 k x(t) k v m (t) = v k (t) + + f m (t) fext(t) - voltage source f k (t) - + - Mass m Inductance L = m (impedance R m (s) = ms) Spring k Capacitance C = 1 k (impedance R k (s) = k s )

Intro WDF Summary Appendix Scattering Mechanical Spring-Mass (Parallel) f ext ( t) (frictionless, massless, vertical guide rod) Physical Diagram: Electrical Equivalent Circuit: f ( t) = f ( t) = f k k m m v ( t) vm ( t) + v ( t) v m ( t) = k ext fext(t) + - vext(t) + f k (t) - v k (t) + f m (t) m 1 k v m (t) -