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Quiz: Symmetries of the Fourier transform

Page history last edited by RH 11 years, 11 months ago

This quiz is designed to test your knowledge of how the Fourier transform behaves under various transformations.

 

The quiz is phrased so that it does not matter what your normalization conventions are for the Fourier transform. Note: some of these questions lie outside the scope of Math 133. All abelian groups are assumed to be amenable (if you don't know what this means, ignore it). 

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. If Formula is a function on an abelian group Formula, and Formula is translated by a shift Formula, then the Fourier transform of Formula is
    1. Shifted in the direction Formula.
    2. Shifted in the direction Formula.
    3. Modulated by a character with frequency Formula or Formula (depending on conventions)
      • CORRECT.
    4. Dilated by Formula.
      • INCORRECT. This couldn't be right (consider the Formula case).
    5. Dilated by Formula.
      • INCORRECT. This couldn't be right (consider the Formula case).
    6. Convolved by Formula.
      • INCORRECT. This doesn't even make sense; Formula is a group element, not a function.
    7. Multiplied by Formula.
      • INCORRECT. This couldn't be right (consider the Formula case).
  2. If Formula is a function on an abelian group Formula, and Formula is modulated by a character with frequency Formula, then the Fourier transform of Formula is
    1. Modulated by the character with frequency Formula.
    2. Modulated by the character conjugate to Formula.
    3. Shifted in the direction Formula or Formula (depending on conventions)
      • CORRECT.
    4. Dilated by Formula.
    5. Dilated by Formula.
    6. Convolved by Formula.
    7. Multiplied by Formula.
  3. If Formula is a function on Formula, and Formula is dilated by a factor Formula (i.e. Formula is replaced by Formula, then the Fourier transform of Formula is
    1. Dilated by a factor Formula.
    2. Dilated by a factor Formula.
    3. Dilated by a factor Formula, and multiplied by Formula.
      • CORRECT.
    4. Dilated by a factor Formula, and multiplied by Formula.
    5. Dilated by a factor Formula, and multiplied by Formula.
    6. Dilated by a factor Formula, and multiplied by Formula.
    7. Multiplied by a factor Formula.
  4. If Formula is a function on Formula, and f is rotated by an orthogonal matrix Formula, then the Fourier transform of Formula is
    1. Rotated by Formula.
      • CORRECT.
    2. Rotated by Formula.
    3. Rotated by the transpose Formula.
    4. Rotated by Formula, and then conjugated.
    5. Rotated by the inverse of the transpose of Formula.
      • CORRECT. For orthogonal matrices, the inverse of the transpose Formula is the matrix Formula itself.
    6. Rotated by Formula, and then reflected around the origin.
    7. Rotated by the inverse of the transpose of Formula, and multiplied by Formula.
      • CORRECT. For orthogonal matrices, inverse transpose of Formula is Formula itself, and Formula.
  5. If Formula is a function on Formula, and Formula is composed with an invertible linear transformation Formula (thus Formula is replaced by Formula), then the Fourier transform of Formula is
    1. Composed with Formula
    2. Composed with the inverse of Formula, and multiplied by Formula
    3. Composed with the transpose of Formula
    4. Composed with the inverse transpose of Formula, and multiplied by Formula
      • CORRECT.
    5. Composed with the inverse transpose of Formula, and multiplied by Formula
    6. Composed with the transpose of Formula, and multiplied by Formula
    7. Composed with Formula, and multiplied by Formula
  6. If Formula is a function on an abelian group Formula, and Formula is restricted to a subgroup Formula of Formula, then the Fourier transform of Formula is (after identifying the dual group of Formula in a canonical manner)
    1. Projected from Formula to Formula.
      • CORRECT.
    2. Restricted from Formula to Formula.
    3. Convolved with the indicator function of Formula.
    4. Convolved with the normalized indicator function on Formula.
    5. Averaged over cosets of Formula.
    6. Divided by the index of Formula in Formula.
    7. Divided by the cardinality of Formula.
  7. If Formula is a function on an abelian group Formula, and Formula is projected onto a quotient Formula of Formula, then the Fourier transform of Formula is (after identifying the dual group of Formula in a canonical manner)
    1. Projected from Formula to Formula.
    2. Restricted from Formula to Formula.
      • CORRECT.
    3. Convolved with the indicator function of Formula.
    4. Convolved with the normalized indicator function on Formula.
    5. Averaged over cosets of Formula.
    6. Divided by the index of Formula in Formula.
    7. Divided by the cardinality of Formula.
  8. If Formula is a function on Formula, and Formula is complex conjugated, then the Fourier transform of Formula is
    1. Complex conjugated.
    2. Complex conjugated, and reflected around the origin.
      • CORRECT.
    3. Reflected around the origin.
    4. Multiplied by Formula.
    5. Reflected around the Formula-axis.
    6. Reflected around both the Formula-axis and the origin.
    7. Complex conjugated, and multiplied by Formula.
  9. If Formula is a function on Formula, and Formula is reflected around the origin, then the Fourier transform of Formula is
    1. Complex conjugated.
    2. Complex conjugated, and reflected around the origin.
    3. Reflected around the origin.
      • CORRECT.
    4. Multiplied by Formula.
    5. Reflected around the Formula-axis.
    6. Reflected around both the Formula-axis and the origin.
    7. Complex conjugated, and multiplied by Formula.
  10. If Formula is a function on Formula, and Formula is both complex conjugated and reflected around the origin, then the Fourier transform of Formula is
    1. Complex conjugated.
      • CORRECT.
    2. Complex conjugated, and reflected around the origin.
    3. Reflected around the origin.
    4. Multiplied by Formula.
    5. Reflected around the Formula-axis.
    6. Reflected around both the Formula-axis and the origin.
    7. Complex conjugated, and multiplied by Formula.
  11. If Formula is a function on Formula, and Formula is differentiated, then the Fourier transform of Formula is
    1. Multiplied by some multiple of Formula.
      • CORRECT.
    2. Differentiated.
    3. Integrated.
    4. Reflected around the origin.
    5. Divided by some multiple of Formula.
    6. Multiplied by a constant.
  12. If Formula is a function on Formula, and Formula is multiplied by the identity function Formula, then the Fourier transform of Formula is
    1. Differentiated, and multiplied by a constant.
      • CORRECT.
    2. Unchanged.
    3. Inverted.
    4. Reflected around the line Formula.
    5. Integrated, and multiplied by a constant.
    6. Multiplied by Formula.
    7. Divided by Formula.
  13. If Formula and Formula are two functions on an abelian group Formula, then the Fourier transform of the convolution of Formula with Formula is (up to normalization constants)
    1. The convolution of the Fourier transforms of Formula and Formula.
    2. The pointwise product of the Fourier transforms of Formula and Formula.
      • CORRECT.
    3. The composition of the Fourier transforms of Formula and Formula.
    4. The inner product of the Fourier transforms of Formula and Formula.
    5. The sum of the Fourier transforms of Formula and Formula.
    6. The tensor product of the Fourier transforms of Formula and Formula.
  14. If Formula and Formula are two functions on an abelian group Formula, then the Fourier transform of the pointwise product of Formula with Formula is (up to normalization constants)
    1. The convolution of the Fourier transforms of Formula and Formula.
      • CORRECT.
    2. The pointwise product of the Fourier transforms of Formula and Formula.
    3. The composition of the Fourier transforms of Formula and Formula.
    4. The inner product of the Fourier transforms of Formula and Formula.
    5. The sum of the Fourier transforms of Formula and g$.
    6. The tensor product of the Fourier transforms of Formula and Formula.
  15. If Formula and Formula are two functions on two abelian groups Formula and Formula, then the Fourier transform of the tensor product of Formula with Formula is (up to normalization constants)
    1. The convolution of the Fourier transforms of Formula and Formula.
    2. The pointwise product of the Fourier transforms of Formula and Formula.
    3. The composition of the Fourier transforms of Formula and Formula.
    4. The inner product of the Fourier transforms of Formula and Formula.
    5. The sum of the Fourier transforms of Formula and Formula.
    6. The tensor product of the Fourier transforms of Formula and Formula.
      • CORRECT.

 

 

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