# What is difference between outputs of Fourier transform and Fourier series of a periodic square waveform?

We can use Fourier transform of an aperiodic signal and Fourier series of periodic signal. But we can use Fourier transform formula for periodic function also.

Now, let us consider a periodic square wave with fundamental period $T$ . Then I want to ask is

What is difference between the outputs of Fourier transform and Fourier series of a periodic square waveform?

• You've asked many many question on similar topics here and on math.SE but you hardly ever accept an answer. Are all answers you got so far unsatisfactory? Jun 19 '15 at 8:44
• @Matt L. I admit that I have asked many questions but what happen is, in my previous questions I expected the explanation with addition of an example or mathematical equation . I write this as note in my question but I didn't get reply . but when I remove this "note", I get many answers but in generalised format. so when I try to use it for particular example ,I get confusion. Then I post new question for particular example . This process goes on and on. Jun 19 '15 at 9:22
• one simple answer to your question "What is difference between Fourier transform and Fourier series of a periodic square waveform?" is nothing. but, probably, the best answer is that the output of the Fourier series is a set of coefficients. i.e. the mapping is from a $T$-periodic $x(t)$ (which needs only one period to fully define it) to a countably infinite set of coefficients $c_n$, whereas the Fourier transform will map the $T$-periodic $x(t)$ to a function of frequency $X(\omega)$ which is comprised of equally spaced Dirac impulses each scaled by $c_n$. Jun 19 '15 at 15:28
• @pandu: Why don't you just compute the Fourier coefficients of that square wave and use the formula I showed you in my answer??? Jun 19 '15 at 21:20
• @pan, please get a textbook on Applied Mathematics or Engineering Mathematics or similar with an introductory chapter on Fourier Series. my old textbook is an old edition of Kreyszig. it would be educational and good for you to learn how to determine Fourier series coefficients of a general $T$-periodic waveform, not just the square wave. Jun 20 '15 at 17:32

The Fourier transform $X(\omega)$ of a $T$-periodic function $x(t)$

$$x(t+T) = x(t) \quad \quad \forall t$$

having complex Fourier coefficients $c_n$

$$x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j 2 \pi n t/T}$$

$$c_n = \frac{1}{T} \int_{0}^T x(t) e^{-j 2 \pi n t/T} \ dt \tag{1}$$

can be expressed as a weighted sum of Dirac impulses, where the weights are given by the complex Fourier coefficients:

$$X(\omega) \triangleq \mathcal{F}\left\{ x(t) \right\} = 2 \pi \sum_{n=-\infty}^{\infty} c_n \delta\left(\omega - \frac{2\pi n}{T}\right) \tag{2}$$