Learn Prove finite transform identities, diagonalize circular filters and derive a bounded radix-two implementation with explicit normalization and arithmetic costs.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
Fourier coefficients combine rotations and magnitudes; the sign of the angle determines the convention.
It rotates by θ without changing magnitude. Conjugation reverses the angle; at θ=−π/2 the multiplier is −i.
Fourier inversion depends on orthogonality and the norm of each basis vector, not merely on having distinct vectors.
Taking the inner product with one basis vector kills the other terms. Divide by its squared norm to recover that coefficient; normalized vectors have squared norm one.
Loading saved mathematics work…
Transform conventions, singular filters and padding
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Derive the finite inverse transform, energy identity and real-input symmetry from exact geometric sums.
6 practice tasks.
Track cyclic indices through the Fourier transform, identify circulant eigenmodes and determine when padding turns a circular computation into an ordinary convolution.
6 practice tasks.
Split a negative-exponent Fourier transform into even and odd subsequences, prove the recursive index contract, and audit exact work and numerical checks.
6 practice tasks.