Homework Set 8
Math/ECE 430
due Friday, March 29, 2013
The page numbers below refer to the book by Gasquet and Witomski.
1. Compute the Fourier transform of the Gaussian f (t) = exp(−at2 ).
2. p. 156, # 18.5 (Shannon’s formula)
Suppose f is bandlimited, and in particular the Fourier transform Ff is supported
in the interval [−λc , λc ].
(a) We extend Ff to obtain a periodic function g as follows. For a such that
0 < a ≤ 1/(2λc ), let g = Ff on the interval (−1/(2a), 1/(2a)) and extend
g to be periodic with period 1/a. Show that the Fouirer coefficients of g are
cn (g) = af (−na) for n ∈ Z.
(b) For t real and fixed, we define another function h with period 1/a: h(λ) =
exp(2πiλt) for λ ∈ (−1/(2a), 1/(2a)). Show that the Fourier coefficients of h
are cn (h) = sinc[(t − na)π/a].
(c) Use the expression for the Fouirer coefficients of a product (exercise 5.13, from
Homework Set 2) to prove Shannon’s formula:
f (t) =
∞
X
f (na)sinc[(t − na)π/a]
n=−∞
1