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Fourier Synthesis Character Generator

glen_kleinschmidt_fourier_characters

Quicklink. p5.js implementations can be found:

This directory contains an adaptation of Glen Kleinschmidt's "Fourier Synthesis Character Generator", which generates single-stroke vector glyphs using a custom analog circuit. In Kleinschmidt's circuit, a number of sinusoidal oscillators have frequencies set by resistor values, as depicted in his circuit diagram romschematic.pdf. Kleinschmidt's circuit only represents the sixteen characters, 0-9 and A-F. His circuit synthesizes each character as a 2-dimensional parametric curve, such as one would see on an oscilloscope, summing the contributions of five such oscillators in each of the X and Y dimension.

kleinschmidt_figure1.jpg

The table below is based on the ROM resistor matrix in romschematic.pdf, with the adjusted values used by the p5 sketch. The component reference is included in parentheses. Smaller resistor values produce larger terms because the summing weight is proportional to conductance, 1/R. Each X or Y deflection waveform is a passive sum of selected 1-volt peak source terms:

  • sinN means sin(Nwt).
  • -sinN means -sin(Nwt).
  • cosN means cos(Nwt).
  • -cosN means -cos(Nwt).
  • +dc and -dc are the +12 V and -12 V DC offset buses.

The visible character trace is six eighths of the waveform period; the remaining two eighths are the blanked return segment as described in his page. A local copy of his project page is here.

Extracted ROM Table

Char X terms Y terms
0 sin1 75k (R95); -sin2 130k (R96); -sin3 270k (R97); -sin4 560k (R98); -sin5 1M5 (R99) -cos1 56k (R100); cos2 180k (R101); cos3 470k (R102); cos4 820k (R103); cos5 2M2 (R104); -dc 4M7 (R105)
1 none cos5 62k (R106)
2 -sin1 680k (R107); -cos1 910k (R108); sin2 56k (R109); cos2 160k (R110); sin3 360k (R111); +dc 20M (R112) -sin1 56k (R113); -cos1 360k (R114); -cos2 430k (R115); sin3 510k (R116); cos3 2M2 (R117); -sin4 2M4 (R118); cos4 1M3 (R119); sin5 1M (R120); cos5 1M5 (R121); -dc 5M1 (R122)
3 -cos1 56k (R123); -cos2 180k (R124); cos3 150k (R125); -cos4 1M5 (R126); cos5 510k (R127); -dc 20M (R128) -sin1 62k (R129); -cos1 1M (R130); -sin2 300k (R131); sin3 390k (R132)
4 sin1 150k (R133); cos1 100k (R134); -sin2 110k (R135); cos2 430k (R136); -cos3 430k (R137); -sin4 2M7 (R138); sin5 2M (R139); +dc 3M6 (R140) -sin1 560k (R141); -cos1 75k (R142); -sin2 110k (R143); -cos2 820k (R144); -sin3 1M2 (R145); -cos3 2M7 (R146); -sin4 820k (R147); -sin5 2M7 (R148); +dc 12M (R149)
5 sin1 430k (R150); cos1 560k (R151); -sin2 62k (R152); -sin3 560k (R153); cos3 430k (R154); -sin4 910k (R155); -cos4 1M5 (R156); sin5 1M5 (R157); cos5 430k (R158); -dc 20M (R159) -sin1 56k (R160); cos1 1M6 (R161); -sin2 430k (R162); cos2 430k (R163); sin3 430k (R164); -sin4 910k (R165); sin5 820k (R166); -cos5 2M7 (R167); +dc 6M8 (R168)
6 -sin1 470k (R169); cos1 220k (R170); sin2 56k (R171); sin3 430k (R172); cos3 240k (R173); -dc 30M (R174) sin1 100k (R175); cos1 200k (R176); -cos2 91k (R177); sin3 510k (R178); -dc 30M (R179)
7 -cos1 39k (R180); -cos2 36k (R181); -cos3 110k (R182); -cos4 750k (R183); -cos5 1M (R184); -dc 1M8 (R185) cos1 51k (R186); -cos2 180k (R187); cos3 270k (R188); cos4 1M (R189); cos5 680k (R190); +dc 1M5 (R191)
8 sin1 430k (R192); sin2 130k (R193); -sin3 68k (R194); -sin4 330k (R195); -sin5 910k (R196) -cos1 68k (R197); cos2 180k (R198); cos3 680k (R199); cos4 270k (R200); cos5 2M7 (R201); -dc 4M7 (R202)
9 sin1 470k (R203); -cos1 220k (R204); -sin2 56k (R205); -sin3 430k (R206); -cos3 240k (R207); +dc 30M (R208) -sin1 100k (R209); -cos1 200k (R210); cos2 91k (R211); -sin3 510k (R212); +dc 30M (R213)
A cos1 56k (R214); -cos2 150k (R215); cos3 510k (R216); cos4 750k (R217); cos5 1M8 (R218); +dc 2M (R219) sin1 560k (R220); -sin2 100k (R221); -sin3 510k (R222); -sin4 1M3 (R223); -dc 1M6 (R224)
B sin1 100k (R225); -cos1 82k (R226); -sin2 130k (R227); -cos2 150k (R228); -cos4 620k (R229); -cos5 1M5 (R230); -dc 1M6 (R231) -sin1 150k (R232); cos1 130k (R233); -sin2 91k (R234); cos2 150k (R235); sin3 1M5 (R236); -cos3 470k (R237); -sin4 1M8 (R238); cos4 1M8 (R239); -sin5 1M5 (R240); -cos5 2M2 (R241); -dc 4M7 (R242)
C cos1 43k (R243); +dc 3M3 (R244) sin1 100k (R245); -dc 1M5 (R246)
D -sin1 100k (R247); cos1 82k (R248); sin2 130k (R249); cos2 150k (R250); cos4 620k (R251); cos5 1M5 (R252); +dc 1M6 (R253) -sin1 150k (R254); cos1 130k (R255); -sin2 91k (R256); cos2 150k (R257); sin3 1M5 (R258); -cos3 470k (R259); -sin4 1M8 (R260); cos4 1M8 (R261); -sin5 1M5 (R262); -cos5 2M2 (R263); -dc 4M7 (R264)
E -sin1 470k (R265); cos1 220k (R266); sin2 56k (R267); sin3 430k (R268); cos3 240k (R269); -dc 30M (R270) -sin1 100k (R271); -cos1 200k (R272); cos2 91k (R273); -sin3 510k (R274); -dc 820k (R275)
F -sin1 180k (R276); cos1 68k (R277); -sin2 220k (R278); cos2 100k (R279); cos3 330k (R280); -sin5 1M (R281); cos5 3M3 (R282); +dc 30M (R283) -sin1 110k (R284); cos1 110k (R285); -sin2 910k (R286); -cos2 150k (R287); -sin3 750k (R288)

Coefficient Equations

The following equations use the adjusted resistor values above. Coefficients are normalized conductance weights: sinusoidal terms use 100k / R, while DC terms use 12 * 100k / R. The variable t is the phase angle over one waveform period.

Char X equation Y equation
0 x = 1.33*sin(1*t) - 0.77*sin(2*t) - 0.37*sin(3*t) - 0.18*sin(4*t) - 0.07*sin(5*t) y = -1.79*cos(1*t) + 0.56*cos(2*t) + 0.21*cos(3*t) + 0.12*cos(4*t) + 0.05*cos(5*t) - 0.26
1 x = 0 y = 1.61*cos(5*t)
2 x = -0.15*sin(1*t) - 0.11*cos(1*t) + 1.79*sin(2*t) + 0.63*cos(2*t) + 0.28*sin(3*t) + 0.06 y = -1.79*sin(1*t) - 0.28*cos(1*t) - 0.23*cos(2*t) + 0.2*sin(3*t) + 0.05*cos(3*t) - 0.04*sin(4*t) + 0.08*cos(4*t) + 0.1*sin(5*t) + 0.07*cos(5*t) - 0.24
3 x = -1.79*cos(1*t) - 0.56*cos(2*t) + 0.67*cos(3*t) - 0.07*cos(4*t) + 0.2*cos(5*t) - 0.06 y = -1.61*sin(1*t) - 0.1*cos(1*t) - 0.33*sin(2*t) + 0.26*sin(3*t)
4 x = 0.67*sin(1*t) + 1*cos(1*t) - 0.91*sin(2*t) + 0.23*cos(2*t) - 0.23*cos(3*t) - 0.04*sin(4*t) + 0.05*sin(5*t) + 0.33 y = -0.18*sin(1*t) - 1.33*cos(1*t) - 0.91*sin(2*t) - 0.12*cos(2*t) - 0.08*sin(3*t) - 0.04*cos(3*t) - 0.12*sin(4*t) - 0.04*sin(5*t) + 0.1
5 x = 0.23*sin(1*t) + 0.18*cos(1*t) - 1.61*sin(2*t) - 0.18*sin(3*t) + 0.23*cos(3*t) - 0.11*sin(4*t) - 0.07*cos(4*t) + 0.07*sin(5*t) + 0.23*cos(5*t) - 0.06 y = -1.79*sin(1*t) + 0.06*cos(1*t) - 0.23*sin(2*t) + 0.23*cos(2*t) + 0.23*sin(3*t) - 0.11*sin(4*t) + 0.12*sin(5*t) - 0.04*cos(5*t) + 0.18
6 x = -0.21*sin(1*t) + 0.45*cos(1*t) + 1.79*sin(2*t) + 0.23*sin(3*t) + 0.42*cos(3*t) - 0.04 y = 1*sin(1*t) + 0.5*cos(1*t) - 1.1*cos(2*t) + 0.2*sin(3*t) - 0.04
7 x = -2.56*cos(1*t) - 2.78*cos(2*t) - 0.91*cos(3*t) - 0.13*cos(4*t) - 0.1*cos(5*t) - 0.67 y = 1.96*cos(1*t) - 0.56*cos(2*t) + 0.37*cos(3*t) + 0.1*cos(4*t) + 0.15*cos(5*t) + 0.8
8 x = 0.23*sin(1*t) + 0.77*sin(2*t) - 1.47*sin(3*t) - 0.3*sin(4*t) - 0.11*sin(5*t) y = -1.47*cos(1*t) + 0.56*cos(2*t) + 0.15*cos(3*t) + 0.37*cos(4*t) + 0.04*cos(5*t) - 0.26
9 x = 0.21*sin(1*t) - 0.45*cos(1*t) - 1.79*sin(2*t) - 0.23*sin(3*t) - 0.42*cos(3*t) + 0.04 y = -1*sin(1*t) - 0.5*cos(1*t) + 1.1*cos(2*t) - 0.2*sin(3*t) + 0.04
A x = 1.79*cos(1*t) - 0.67*cos(2*t) + 0.2*cos(3*t) + 0.13*cos(4*t) + 0.06*cos(5*t) + 0.6 y = 0.18*sin(1*t) - 1*sin(2*t) - 0.2*sin(3*t) - 0.08*sin(4*t) - 0.75
B x = 1*sin(1*t) - 1.22*cos(1*t) - 0.77*sin(2*t) - 0.67*cos(2*t) - 0.16*cos(4*t) - 0.07*cos(5*t) - 0.75 y = -0.67*sin(1*t) + 0.77*cos(1*t) - 1.1*sin(2*t) + 0.67*cos(2*t) + 0.07*sin(3*t) - 0.21*cos(3*t) - 0.06*sin(4*t) + 0.06*cos(4*t) - 0.07*sin(5*t) - 0.05*cos(5*t) - 0.26
C x = 2.33*cos(1*t) + 0.36 y = 1*sin(1*t) - 0.8
D x = -1*sin(1*t) + 1.22*cos(1*t) + 0.77*sin(2*t) + 0.67*cos(2*t) + 0.16*cos(4*t) + 0.07*cos(5*t) + 0.75 y = -0.67*sin(1*t) + 0.77*cos(1*t) - 1.1*sin(2*t) + 0.67*cos(2*t) + 0.07*sin(3*t) - 0.21*cos(3*t) - 0.06*sin(4*t) + 0.06*cos(4*t) - 0.07*sin(5*t) - 0.05*cos(5*t) - 0.26
E x = -0.21*sin(1*t) + 0.45*cos(1*t) + 1.79*sin(2*t) + 0.23*sin(3*t) + 0.42*cos(3*t) - 0.04 y = -1*sin(1*t) - 0.5*cos(1*t) + 1.1*cos(2*t) - 0.2*sin(3*t) - 1.46
F x = -0.56*sin(1*t) + 1.47*cos(1*t) - 0.45*sin(2*t) + 1*cos(2*t) + 0.3*cos(3*t) - 0.1*sin(5*t) + 0.03*cos(5*t) + 0.04 y = -0.91*sin(1*t) + 0.91*cos(1*t) - 0.11*sin(2*t) - 0.67*cos(2*t) - 0.13*sin(3*t)

p5.js Sketch

A p5.js implementation for this project can be found:

This sketch contains the same table as data and evaluates the glyphs as Fourier sums:

x(t) = sum(X character terms)
y(t) = sum(Y character terms)

No character is stored as a point list. The sketch samples the functions only while drawing. The bright trace covers the visible six-eighths of the waveform, and the dim trace shows the blanked return interval.