What a topic for my first post ! I risk scaring away 90% of my sane audience. Yet I take the risk because I love this concept and its everywhere! Even people allergic to engineering might have come across it somewhere or the other. So do we really need another article about Fourier Transform on the internet? Probably not. But let me try to explain it to you as simple as it can get. This could very well be the first write-up of FT that has almost no mathematical equations!
So what exactly is it?
Well, the general idea is pretty simple. And fascinating too! Imagine having a secret potion from which you can make any food in the world ! All the food - from veggies to bacons - can be made by mixing this single potion in different ways ! The only ingredient in all cookbook recipes would be just one- the potion. That would be pretty amazing right? Well it turns out the world of wave-forms have such a magic potion. In other words, there is a magic 'waveform' from which all wave-forms in the world can be made . When i say waveform it includes all shapes of wave-forms you have ever seen - from giggly oscillating waves like the ripples on water to sharp steps like that of a staircase. And that magic waveform isss *drumroll* - the SINE wave ! Believe it or not, those wiggly oscillating sine waves can be added up to create any waveform we want ! And that is pretty much what Fourier Transform is all about. Given a waveform it tells you the recipe! It tells you exactly what kind of sine waves are to be added together to get the particular waveform.
No way! What about sharp edges?
If that question came to your mind, you are pretty smart. And more importantly, you understood what i was trying to say! :-) . Totally agree with you - how can we create sharp wave-forms (steps for example) by adding up smooth wiggly sine waves ? Well, you cannot. But wait, you can get pretty close! If you are careful enough, you can keep adding almost infinite (ouch!) sine waves until your result looks almost like a step for example. In fact, the following figure is very common in textbooks to illustrate this.
You can see that the black waveform is a perfect sine wave. The blue in fact is a sum of many sine waves. And the red is sum of even more sine waves. Notice how it gets closer and closer to a square wave ? Like i promised, i won't go into the details of how its done. Plenty of people have done that already. Google !
Interesting! But what's the big deal?
Good to know. But why would anyone want to waste time on something pointless as creating wave-forms from sine waves right? How does it affect me or things around me? Biased with my background, let me try to explain this from an engineering point of view. It so happens that there are many systems in engineering where you know how the system would respond if you try to feed it a sine wave. In fact, a large portion of systems would always output a sine wave if you give it a sine wave input - only the amplitude,frequency etc. change (no! not going there). Here lies the beauty! So, if you know how the system responds to sine waves, and any wave can be decomposed into sine waves, you can pretty much predict how the system would respond to any wave !! I rest my case.
So what exactly is it?
Well, the general idea is pretty simple. And fascinating too! Imagine having a secret potion from which you can make any food in the world ! All the food - from veggies to bacons - can be made by mixing this single potion in different ways ! The only ingredient in all cookbook recipes would be just one- the potion. That would be pretty amazing right? Well it turns out the world of wave-forms have such a magic potion. In other words, there is a magic 'waveform' from which all wave-forms in the world can be made . When i say waveform it includes all shapes of wave-forms you have ever seen - from giggly oscillating waves like the ripples on water to sharp steps like that of a staircase. And that magic waveform isss *drumroll* - the SINE wave ! Believe it or not, those wiggly oscillating sine waves can be added up to create any waveform we want ! And that is pretty much what Fourier Transform is all about. Given a waveform it tells you the recipe! It tells you exactly what kind of sine waves are to be added together to get the particular waveform.
No way! What about sharp edges?
If that question came to your mind, you are pretty smart. And more importantly, you understood what i was trying to say! :-) . Totally agree with you - how can we create sharp wave-forms (steps for example) by adding up smooth wiggly sine waves ? Well, you cannot. But wait, you can get pretty close! If you are careful enough, you can keep adding almost infinite (ouch!) sine waves until your result looks almost like a step for example. In fact, the following figure is very common in textbooks to illustrate this.
You can see that the black waveform is a perfect sine wave. The blue in fact is a sum of many sine waves. And the red is sum of even more sine waves. Notice how it gets closer and closer to a square wave ? Like i promised, i won't go into the details of how its done. Plenty of people have done that already. Google !
Interesting! But what's the big deal?
Good to know. But why would anyone want to waste time on something pointless as creating wave-forms from sine waves right? How does it affect me or things around me? Biased with my background, let me try to explain this from an engineering point of view. It so happens that there are many systems in engineering where you know how the system would respond if you try to feed it a sine wave. In fact, a large portion of systems would always output a sine wave if you give it a sine wave input - only the amplitude,frequency etc. change (no! not going there). Here lies the beauty! So, if you know how the system responds to sine waves, and any wave can be decomposed into sine waves, you can pretty much predict how the system would respond to any wave !! I rest my case.
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