Math 658 - Fall 2008
Homework
Assignment 5 - Due Friday, November 7, 2008
- Read sections 9.1 and 9.4.
- Problems
- 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.
-
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, falias. Show that on
[−π, π],
f^alias(ω) = f^(ω − 2π) + f^(ω)
+ f^(ω + 2π),
and that f^alias = 0 otherwise.
- 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 falias(t) in this case? Use your favorite
software to make a plot showing f(t), falias(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.
- Section 7.5: 2, 3, 6(a,b).
- Find the n-dimensional Fourier transform of the Gaussian, g(r)
=(2π)− n/2 exp(− r2/2), where r =
|x|.
- Prove the three formulas listed at the end of section 2 on p. 3
of the
Notes on the Discrete Fourier Transform.
- Section 9.4: 3.
Updated 10/29/08 (fjn)