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Quiz: Symmetries of the Fourier transform

Page history last edited by RH 15 years, 4 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.)

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

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