# Fourier Analysis - sawtooth

• freezer

## Homework Statement

Sawtooth signal with To = 1, at T=0, x = 0, at T=1, x =1

verify:
$a_{k} = \left\{\begin{matrix} \frac{1}{2}, for k=0; & \\\frac{j}{2\pi k}, for k \neq 0; & \end{matrix}\right.$

## Homework Equations

$\frac{1}{T_{0}} \int_{0}^{T_{0}} te^{-j(2\pi/T_{0}))kt}dt$

## The Attempt at a Solution

for k = 0

$a_{0} = \int_{0}^{1} t dt$

$a_{0} = \frac{1}{2} t^{2}$ from 0 to 1 = 1/2

for k != 0

$\int_{0}^{1} te^{-j(2\pi) kt}dt$

u = t
du = dt
dv = $e^(-j2\pi kt)$

$v = \frac{-1}{j2\pi k}e^{-j2\pi kt}$

$t * \frac{-1}{j2\pi k}e^{-j2\pi kt} - \int \frac{-1}{j2\pi k}e^{-j2\pi kt} dt$

$t * \frac{-1}{j2\pi k}e^{-j2\pi kt} - \frac{e^{-j2\pi kt}}{4\pi^2k^2}$

-1/j = j

$t * \frac{j}{2\pi k}e^{-j2\pi kt} - \frac{e^{-j2\pi kt}}{4\pi^2k^2}$

$e^{-j2\pi kt} (t \frac{j}{2\pi k} - \frac{1}{4\pi^2 k^2})$

getting close but not seeing where to go from here.

Last edited:
Check the integration by parts rules: You seem to have forgotten to evaluate the first part at the boundaries (in particular, if you integrate over t from 0 to 1, there is no way t should remain in the final expression)
$\int_a^b u(x)v'(x)\,dx = \left[u(x)v(x)\right]_a^b - \int_a^b u'(x)v(x)\,dx$,
first term on the right hand side.

Check the integration by parts rules: You seem to have forgotten to evaluate the first part at the boundaries (in particular, if you integrate over t from 0 to 1, there is no way t should remain in the final expression)
$\int_a^b u(x)v'(x)\,dx = \left[u(x)v(x)\right]_a^b - \int_a^b u'(x)v(x)\,dx$,
first term on the right hand side.

$\frac{je^{-j2\pi k} }{2\pi k} - \frac{e^{-j2\pi k} }{4\pi^2 k^2} - \frac{1}{4\pi^2 k^2}$

You seem to have a sign error. Also, remember that k is an integer (a periodic function is mapped into a series in Fourier space), and you should be able to arrive at the result.

Okay, see the sign error but still not seeing how that is going to get
the other terms to fall out leaving just j/(2pik).

k is an integer. What is $\exp(-j2\pi k)$ for k integer?

Thank you Paallikko, I did not have that one in my notes.