Example Problem Set 7

Parts 2 and 3 are difficult to do if you just use the lectured material as your reference. Nevertheless, you may like to try them anyway, although you will probably need to look things up elsewhere.

1.

Show that for all families of functions which are ``self-Fourier" (i.e. equivalent in functional form to their own Fourier transforms), closure of a family under multiplication entails also their closure under convolution, and vice versa. (Hint: closure of a set of functions under an operation means that applying that operation to any member of the set creates a function which is also a member of the set.)

2.

A periodic square wave, which alternates between the constants $+\pi/4$and $-\pi/4$ with period $2\pi$ has the following Fourier series, using just all odd integers n:

$$f(x) = \sum_{{\rm odd \ }n=1}^{\infty} \frac{1}{n} \sin(nx)$$

Derive from this the Fourier series for a periodic triangular wave, which ramps up and down with slopes $+\pi/4$ and $-\pi/4$ and with period $2\pi$.

3.

Any real-valued function f(x) can be represented as the sum of one function f_e(x) that has even symmetry (it is unchanged after being flipped around the origin x=0) so that f_e(x) = f_e(-x), plus one function f_o(x) that has odd symmetry, so that f_o(x) = -f_o(-x). Such a decomposition of any function f(x) into f_e(x)+f_o(x) is illustrated by

$$f_{e}(x) = \frac{1}{2} f(x) + \frac{1}{2} f(-x)$$

$$f_{o}(x) = \frac{1}{2} f(x) - \frac{1}{2} f(-x)$$

Use this type of decomposition to explain why the Fourier transform of any real-valued function has Hermitian symmetry: its real-part has even symmetry, and its imaginary-part has odd symmetry.

Comment on how this redundancy can be exploited to simplify computation of Fourier transforms of real-valued, as opposed to complex-valued, data.