Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am having some trouble knowing how to correctly start a problem of finding the Fourier Coefficients using complex exponential form. The problem is given below:

$$g_1(t)=\begin{cases} 1,~~\qquad t<1\\2-t,\quad t\ge 1\end{cases}$$


$$g_2(t)=\begin{cases} 1+2t, \qquad -0.5<t\le 0.5\\\tfrac{(7-2t)}{3},~~~\,\qquad0.5<t<3.5\\0,~~~\qquad\qquad \mbox{elsewhere}\end{cases}$$

over the interval of $0\le t\le2$.

General complex Fourier series representation is: $g(t)=\displaystyle\sum_{n=-\infty}^{\infty} C_ne^{int}$

I tried using the formula to compute the DC term or zeroth term from the following:

$C_0=\displaystyle\frac{1}{T_0}\int_{<T_0>} x(t)e^{int}\,\mathrm{d}t$, where $n=0$. (zero coefficient, DC term)

The result I get for $g_1(t)$ from this is: $C_0=\displaystyle \frac{1}{2}\int_0^2 1\,\mathrm{d}t+\frac{1}{2}\int_0^2 (2-t)\,\mathrm{d}t=2$, where $T_0$ is the interval in which it runs from $[0,2]$, but not sure if its correct and how to go about setting up for the $C_n$ term.

I have seen different forms of how to compute the coefficients and the general series representation in books, but I do not know which ones to appropriately apply to this problem. The other problem is defining a period for the constant function such as $1$ in the case of $g_1(t)$. Being that this is needed to computer coefficients of the series.

share|cite|improve this question
up vote 1 down vote accepted

First, you need to fix your limits of integration. For $g_1$, you should have

$$C_0 = \frac{1}{2} \int_0^1 dt + \frac{1}{2} \int_1^2 (2-t) dt$$

Second, your definition of $C_n$ (which you cited for the $n=0$ case) isn't quite right. This doesn't effect the $n=0$ case, but for general $C_n$ you should use

$$C_n = \frac{1}{|T|} \int_T g(t) e^{-2\pi i n t/|T|} \, dt$$

where $|T|$ is the length of the interval $T$ over which your function is defined.

See this Wikipedia page

share|cite|improve this answer
Just to add to that, because it can be confusing that people sometimes define T (or L) differently. You can sometimes see it in the form $$C_n = \frac{1}{2T} \int_{-T}^T f(t) e^{-\frac{\pi int}{T}} dt$$ with T being half the period over the functions definition too. – Magpie Aug 8 '12 at 20:54

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.