Transform conventions, singular filters and padding
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Use indices starting at zero, ω=exp(−2πi/n), forward Xk=Σj xjω^(jk) and inverse division by n. State any different convention before calculating.
These are original public practice problems. Verify changed calculations and prove the identities used; saved responses are not independently graded examinations.
Loading saved assessment responses…
0 of 3 tasks have recorded responses. No overall mastery score is assigned.
For x=(2,0,−1,3), compute the four-point DFT by direct summation and by an even/odd FFT split. Identify every root butterfly and explain the paired output indices. Prove the general root-of-unity identity that justifies inversion, verify Parseval for this vector and diagnose an inverse routine that uses the opposite sign but omits division by n. Give the exact FFT complex-operation counts under the lesson convention.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On four periodic samples let h=(1,0,−1,0), and define yj=Σl hl x(j−l mod4). Prove that the Fourier basis diagonalizes this operator and compute its eigenvalues, kernel and range. Solve for every x producing y=(2,−4,−2,4), then use orthogonality to select the unique minimum-Euclidean-norm solution. Give a changed incompatible output and a Fourier certificate of its impossibility. Explain why an FFT cannot recover information destroyed by a zero filter eigenvalue.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Multiply the finite sequences x=(1,2,1) and h=(1,−1,2) by ordinary linear convolution. Compare an unpadded four-point circular convolution with an eight-point zero-padded FFT product and inverse. Prove the padding threshold for general lengths m and l using polynomial coefficients, and explain why four points alias in this example. Derive the FFT recurrence and exact complex-operation counts for length eight under the full butterfly convention; compare with one direct eight-point DFT, but keep the cost of a complete convolution pipeline separate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For x=(2,0,−1,3), compute the four-point DFT by direct summation and by an even/odd FFT split. Identify every root butterfly and explain the paired output indices. Prove the general root-of-unity identity that justifies inversion, verify Parseval for this vector and diagnose an inverse routine that uses the opposite sign but omits division by n. Give the exact FFT complex-operation counts under the lesson convention.
On four periodic samples let h=(1,0,−1,0), and define yj=Σl hl x(j−l mod4). Prove that the Fourier basis diagonalizes this operator and compute its eigenvalues, kernel and range. Solve for every x producing y=(2,−4,−2,4), then use orthogonality to select the unique minimum-Euclidean-norm solution. Give a changed incompatible output and a Fourier certificate of its impossibility. Explain why an FFT cannot recover information destroyed by a zero filter eigenvalue.
Multiply the finite sequences x=(1,2,1) and h=(1,−1,2) by ordinary linear convolution. Compare an unpadded four-point circular convolution with an eight-point zero-padded FFT product and inverse. Prove the padding threshold for general lengths m and l using polynomial coefficients, and explain why four points alias in this example. Derive the FFT recurrence and exact complex-operation counts for length eight under the full butterfly convention; compare with one direct eight-point DFT, but keep the cost of a complete convolution pipeline separate.
