Math 658 - Fall 2008

Assignment 5 - Due Friday, November 7, 2008

• Read sections 9.1 and 9.4.
• Problems
  1. Let f(t) be band-limited, with its Fourier transform f^(ω) (f "hat") having support in [−2π, 2π]. The Nyquist rate for f is 2π/π = 2, and the Nyquist frequency is ½·2 = 1.
     1. Show that if f is sampled at the Nyquist frequency, instead of the Nyquist rate, then the sampling theorem doesn't recover f. Instead, it gives another band-limited function, f_alias. Show that on [−π, π],
        f^_alias(ω) = f^(ω − 2π) + f^(ω) + f^(ω + 2π),
        and that f^_alias = 0 otherwise.
     2. Referring to the previous part, consider the (single-frequency) band-limited function
        f(t) = sin(5πt/3+π/6),
        whose angular frequency 5π/3 is in the band [−2π, 2π]. What is f_alias(t) in this case? Use your favorite software to make a plot showing f(t), f_alias(t) on the interval − 6 ≤ t ≤ 6, and also use circles or x's to indicate samples of these two functions at integer values of t.
  2. Section 7.5: 2, 3, 6(a,b).
  3. Find the n-dimensional Fourier transform of the Gaussian, g(r) =(2π)^{− n/2} exp(− r²/2), where r = |x|.
  4. Prove the three formulas listed at the end of section 2 on p. 3 of the Notes on the Discrete Fourier Transform.
  5. Section 9.4: 3.

Updated 10/29/08 (fjn)