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Geometric Interpretation of Discrete Fourier Transform

Geometric Interpretation of Discrete Fourier Transform
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2020-05-30 13:24
VectorComplex NumberTransformation
This example demonstrates the principle of the Discrete Fourier Transform through geometric figures. It involves complex numbers, vectors, and transformations. Using GeoGebra commands such as Circle, Vector, If, Segment, and PolyLine, users can visually understand the process of the Discrete Fourier Transform. Suitable for university mathematics and signal processing courses.

GeoGebra Commands & Parameters

  1. Circle(A2, R3) → c3
  2. Vector(A1, P1) → v1
  3. Vector(A2, P1) → v2
  4. Vector(A3, P1) → v3
  5. If(n ≟ 1, v1, Vector[If[n ≟ 2, v2, v3]]) → v
  6. Circle((-4, -2.5), R1) → d1
  7. Circle((-4, -4.5), R2) → d2
  8. Circle((-4, -6.5), R3) → d3
  9. Segment(Center[d1], B1) → b1
  10. Segment(Center[d2], B2) → b2
  11. Segment(Center[d3], B3) → b3
  12. Vector(B1, (t, f1(t))) → u1
  13. Vector(B2, (t, f2(t))) → u2
  14. Vector(B3, (t, f3(t))) → u3
  15. Circle((x(A1), y(A1)), R2) → c2
  16. Circle((-4, -4.5), R1) → c1
  17. PolyLine((-4, -4.5), A1, A2, A3) → p
  18. Vector((-4, -4.5), ((-4, -4.5) * (1 - sep)) + (sep * A1)) → vs2
  19. Translate(d2, vs2) → c20
  20. Vector((-4, -6.5), ((-4, -6.5) * (1 - sep)) + (sep * A2)) → vs3
  21. Translate(d3, vs3) → c30
  22. Vector((-4, -2.5), (-4, (-4.5 * sep) + ((1 - sep) * (-2.5)))) → vs1
  23. Vector((-4, -2.5), (-4, (-4.5 * sep) + ((1 - sep) * (-2.5)))) → vs10
  24. Translate(f1, vs10) → ff1
  25. Vector((-4, -6.5), (-4, (-4.5 * sep) + ((1 - sep) * (-6.5)))) → vs30
  26. Translate(d1, vs1) → c10
  27. Translate(f3, vs30) → ff3