1. Find the Fourier series for the following 1-periodic function
.
2. Find the sum
(Hint: Consider the Fourier series for the function f(t) = t2 on and f(t + k) = f(t) for all integer k.)
3. Find a function g(t) such that: for any f(t), the convolution f ∗g is the ideal low pass filter that retains only the frequencies in the interval (−1,1).
4. Find the Fourier transform of the function
,
5. Compute the Discrete Fourier Transform of [1 1 2 2]T.
6. Prove the discrete convolution theorem:
AN(f ∗ g) = (ANf) · (ANg),
where f,g ∈ CN are vectors, AN ∈ CN×N is the discrete Fourier transform matrix, · is the entrywise multiplication, and ∗ is the discrete convolution defined by ( for m = 0,1,...,N − 1.
7. Let f be a vector and let τ(f) be the cyclic shift by 1 position to the right. What is F(τ(f)) in relation to F(f)? Here F(f) is the discrete Fourier transform of f.
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