UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 2

Transformer för beräkningar 2001 03 30 01trp2

2.1. Suppose we know that a ∈ l^{2}(Z_{N}) satisfies â(0) = 0, â(k) ≠ 0, k = 1, ..., N − 1. Can you determine the dimension of

{z ∈ l^{2}(Z_{N}); a ∗ z = 0}?

Can you determine the dimension of

{a ∗ z; z ∈ l^{2}(Z_{N})}?

Generalize to an arbitrary convolution equation a ∗ z = b, where a, b are given vectors, z an unknown vector. What is the rule connecting the two dimensions?

2.2. Calculate ẑ when z = (1, 2, 3, 4)^{T} ∈ l^{2}(Z_{4}). Calculate ŵ when

w = (1, 2, 3, 4, 5, 6)^{T} ∈ l^{2}(Z_{6}).

2.3. Let u = (1, 3)^{T}, v = (0, 4)^{T}, z = (1, 0, 3, 4)^{T}. Compute û, v̂ and then ẑ using the fast Fourier transform. Compute the Fourier transform of w = (0, 1, 4, 3)^{T}.

2.4. Compute the Fourier transform of w = (1, 0, 1, 0, 1, 0, 1, 0)^{T}—maybe using z = (1, 1, 1, 1)^{T}.

2.5. Let U, V and W be subspaces of an inner product space X, and suppose that U ⊥ V, U ⊥ W, and that U + V = U + W. Prove that V = W.

2.6. Prove that for any z ∈ l^{2}(Z_{N}), we have { \atop U(z)}(n) = ẑ(n) for all n, and that, if N is even, { \atop D(z)}(n) = \frac{1}{2}(ẑ(n) + ẑ(n + \frac{1}{2}N)) for all n. Here U and D denote upsampling and downsampling, respectively.

2.7. Let u ∈ l^{2}(Z_{N}) and define v ∈ l^{2}(Z_{N/2}) by v(n) = u(n) + u(n + \frac{1}{2}N), assuming N even. Prove that v̂(n) = û(2n) for all n.

2.8. Define z = (1, 2, 3, 4, 5, 6, 7, 8) ∈ l^{2}(Z_{8}). Calculate P_{−j}(z) and Q_{−j}(z), j = 1, 2, 3, in the Haar wavelet system. Notation as in Frazier.