Symbolic Computation in System Simulation and Design

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Brian L. Evans

Dept. of Electrical Engineering and Computer Sciences University of California, Berkeley, CA 94720-1770 [email protected]

Ptolemy Mini-Conference, March 10, 1995 University of California, Berkeley, CA

1

| Introduction |

Role of Symbolic Computation INPUT Numeric Signal

Symbolic Description

OUTPUT Simulation

-

Numeric Signal

- Symbolic Description

Transformation

2

| System Simulation |

Symbolic Parameter Calculation Truncated Fourier Series Computation

x^(t)  m

N X

Cm = T1

j  mT t C e ,N m 2

=

T

Z2

, T2

x(t) ej  mT t dt 2

Fixed Fourier Coecient Formula Approximate a Square Wave by a Finite Number of Sinusoids parameter_map: gain = 4/(PI*(2*instance_number-1))

singen

Gain collect

bus

distribute

collect

bus Add

XMgraph

etc.

etc. MapGr SrcGr

parameter_map: frequency = 2*PI*(2*instance_number-1)/period

Compute Fourier Coecient Formula From x(t) 3

| System Simulation |

Numeric Parameter Optimization Optimization of an Existing Filter Design  Deviation from an ideal magnitude response  Linear phase response in the passband  Quality factors of second-order sections  Peak overshoot in the step response

The Optimization Problem  Sequential Quadratic Programming  Dierentiable objective functions  Filter speci cations to dierentiable constraints

4

| System Simulation |

Numeric Parameter Optimization Code generation  De ne the objective function and constraints  Compute gradients of both symbolically  Generate source code (C, Fortran, or Matlab)  Generate main program (Matlab)

5

| System Simulation |

Numeric Parameter Optimization Example: Fourth-Order All-Pole Filter  Speci cations:

at wp = 20 rad/sec, p = 0:21 at ws = 30 rad/sec, s = 0:31

 Initial lter is Butterworth  Pole locations initial: ,8:415  20:315 ,20:315  8:415 nal: ,7:792  22:898 ,19:562  0:626  Objective function

reduced from 1:17 to 4:7  10,5

 Final gradients of objective function 3:1  10,5, 4:2  10,5, ,2:3  10,5, and ,5:5  10,6

6

| System Simulation |

Numeric Parameter Optimization Example: Fourth-Order All-Pole Filter Phase Response 5

10

15

20

ω

-0.5 -1 -1.5 -2 -2.5

- - - dashed lines represent the initial Butterworth lter | solid lines represent the lter optimized for linear phase

response in the passband and for overshoot of the step response 7

| System Simulation |

Numeric Parameter Optimization Example: Fourth-Order All-Pole Filter Step Response 1 0.8 0.6 0.4 0.2

0.2

0.4

0.6

0.8

- - - dashed lines represent the initial Butterworth lter | solid lines represent the lter optimized for linear phase

1

t

response in the passband and for overshoot of the step response 8

| System Simulation |

Numeric Parameter Optimization Example: Fourth-Order All-Pole Filter Magnitude Response (dB) 20

40

60

80

ω

-10 -20 -30 -40 -50

- - - dashed lines represent the initial Butterworth lter | solid lines represent the lter optimized for linear phase

response in the passband and for overshoot of the step response 9

| System Design |

Non-Uniform Filter Bank x[n]

- "n 2 - h0[n] - #n 3

- "n 3 - g0[n] - #n 2

?x -n   6 +

- h1[n] - #n 3

- "n 3 - g1[n]

Flow graph of a two-channel non-uniform lter bank upperchannel = Downsample[2,n][ Convolve[n][ g0[n], Upsample[3,n][ Downsample[3,n][ Convolve[n][h0[n], Upsample[2,n][x[n]]]]]]] lowerchannel = Convolve[n][ g1[n], Upsample[3,n][ Downsample[3,n][ Convolve[n][h1[n], x[n]]]]]

Algebraic description of the lter bank 10

^[ ]

| System Design |

Non-Uniform Filter Bank X^(z ) = 16 (G0(,pz)H0(,pz) + G0(pz )H0(pz) + 2G1(z )H1(z ))X (z ) + pz )H (e  pz ) + G (pz)H (e  pz ) + 2G (z )H (e  z ) X (e  z ) + 1 G ( , 0 0 0 0 1 1 6  p p p p 1    z ) + G0(, z)H0(e z ) + 2G1(z )H1(e z ) X (e  z ) 6 G0 ( z )H0 (e 4i 3

i

3

2i 3

5i 3

2i 3

4i 3

2i 3

4i 3

Symbolic analysis of input-output relationship h0[n] = FIR[n, Hold[ReadList["ptolemy/h0", Number]]]; h1[n] = FIR[n, Hold[ReadList["ptolemy/h1", Number]]]; g0[n] = FIR[n, Hold[ReadList["ptolemy/g0", Number]]]; g1[n] = FIR[n, Hold[ReadList["ptolemy/g1", Number]]]; x[n] = Cos[2 Pi n / 3] Sinc[Pi n / 6] / 3; PtolemySimulation[ upperchannel + lowerchannel, {n, 1, 100} ] >> "!ptcl"

Transformation of algebraic description to Ptcl

11

| System Design |

Evaluating Alternate Implementations Rearrangement Rules  Rules based on interaction between operators  Based on properties of signals and systems

Cost Functions  Based on implementation costs  Require feedback from synthesis tools

Heuristic Searches  Search through space of alternate implementations

12

| System Design |

Multidimensional Signal Processing Multidimensional Signals De ned On Grid of Points Multidimensional Periodic Signals n] = x[n + N r]

x[

General Multidimensional DFT X[

k] = Xn x[n]e,j 2  kT N, n 1

Smith Form Decompositions N = U  V =) N,1 = V,1 ,1 U,1 T , , , X X [k] = x[n]e,j 2  (k V )  (U n) n X ^] = x[U n^]e,j 2  ^kT , n^ X [k n^ 1

1

1

13

1

| System Design |

Multidimensional Rearrangement Rules - # UM - # M - # VM

- " VL - " L - " UL

-

Cascade in Smith Form - # UM - # M

- " L - " UL

-

Simpli ed cascade if VM = VL - # UM - " L

- # M - " UL

-

Reversing order of operations if M and L are coprime - " LUM,1

- # M UL,1

-

Combining operations

Four Equivalent Forms of a Downsampler and Upsampler in Cascade 14

| System Design |

Multidimensional Rearrangement Rules

s

"S

s

z n0

s

#S

s

0

()

Up/downsampling by S when the shift vector n0 62 sublattice(S); i.e., S ,1n0 is not an integer vector

s

"S

s

z n0

s

#S

s

()

s

S,1 n0

z

s

Up/downsampling by S when the shift vector n0 2 sublattice(S); i.e., S ,1n0 is an integer vector

s

"L

s

z n0

s

#M

s

()

s

z nL

s

"L

s

#M

s

z nM

s

For any L and M , n0 can be rewritten as n0 = LnL + M nM

Interaction between Upsamplers, Shifters, and Downsamplers in Cascade 15

Conclusion System Simulation  Symbolic parameter calculation  Numeric parameter optimization

System Design  Symbolic analysis and transformation  Evaluating alternative implementations

Future Work  Allow parameters to calculated symbolically  Explore optimization of other behavioral models  Encode Synchronous Data ow (SDF) system rewriting in the

Design Methodology Management (DMM) Domain  Implement Multidimensional SDF system rewriting in DMM Domain, esp. non-separable resampling operations

16