# Why Fourier series if Fourier transform can be calculated for both periodic and aperiodic?

While learning about Fourier Transform after Fourier Series, That we can calculate Fourier transform of periodic signals too. If we can take the Fourier transform of periodic signal too then my question is why we Fourier series if Fourier transform can be calculated for both periodic and aperiodic?

You can't take the Fourier transform of a periodic signal, the integral diverges for all multiples of the period.

This can be handled by the theory of distributions, but the Fourier series is a better fit.

• Can you please explain bit more – Aadnan Farooq A Oct 18 '15 at 6:25
• @AadnanFarooqA: What don't you understand ? A periodic signal just has no Fourier transform. – Yves Daoust Oct 18 '15 at 8:47
• I am using Oppenheim Signals and Systems book.. and in that I found that Fourier transform is extracted from the Fourier series equation which was used for the periodic signal. – Aadnan Farooq A Oct 18 '15 at 9:39
• @AadnanFarooqA: that doesn't tell me what you don't understand. – Yves Daoust Oct 18 '15 at 9:58
• Sorry I was just confuse with many things.. and i am clear now.. – Aadnan Farooq A Oct 18 '15 at 10:08

The difference between a series and a function is, algebraically speaking, the field over which you define the mapping, and important concepts like convergence and differentiability. Your choice of whether you're after the Fourier Series representation or the Fourier Transform doesn't depend on what you want to analyze, but what you want to do -- these are two separate concepts, and you shouldn't confuse them.

For example, in Physics you'll need the Fourier transform to convert a crystal grid from spatial domain to impulse domain -- although crystals are assumed to be perfectly periodic, having but a series won't cut it as soon as you want to do functional analysis on it.

On the other hand, in digital signal processing, you typically neither have infinite nor perfectly periodic signals, so your Discrete Fourier Transform (DFT) gives you a finite series.

The point here is that you write yourself:

While learning about Fourier Transform after Fourier Series

Keep learning. Sooner than later, you'll understand the significance of both concepts.

It's all the same thing. They are just different flavors based on signal properties to make the math work properly.

If a signal is periodic than its transform is discrete and vice versa. If the signal is aperiodic the transform is continuous and vice versa. So there are a total of four flavors

   time                frequency           name
cont. aperiodic      cont. aperiodic     Fourier Transform
cont. periodic       discrete aperiodic  Fourier Series
discrete aperiodic   cont. periodic      Discrete Fourier Transform
discrete periodic    discrete periodic   Discrete Fourier Series or Discrete Time Fourier Transform


The difference in the formulas is simply due to the fact that you need summation for discrete signals and integration for continuous signals.

Fourier series and Fourier transform basically involve the decomposition of the signals in terms of sinusoidal (or complex exponential) components. With such decomposition, a signal is said to be represented in the frequency domain. Most signals of practical interest can be decomposed into a sum of sinusoidal components. For the class of periodic signals, such a decomposition is called the Fourier Series. For the class of finite energy signals the decomposition is called a Fourier Transform.