Fourier Transform - Made Easy to Understand

 

 

Fourier Transform - "Made Easy to Understand" 

Download Power Point Presentation for the Educational Purposes using below mentioned link:

https://drive.google.com/file/d/1CN8PjBkYTbvGl33h0c7vpnXfTEav4Bbq/view?usp=sharing



Download PDF file format for the Educational Purposes using below mentioned link:

 https://drive.google.com/file/d/1cfdXWOWLYtodRAK_E6hXhffqxJT3jyQh/view?usp=sharing

Transcription for the above video

Today in this video I am presenting about fundamentals of Fourier transform and its applications through analogy, Piano example, radio signal, working principle of radio knobs like Bass and Treble.

How the Images are processed using Photoshop or any other software’s.

As well as few more applications of the Fourier Transform.

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:

Rather than jumping into the symbols, let's experience the key idea firsthand in this video. Here's a plain-English metaphor:

Imagine that one day afternoon you feel hungry and come home after outing, feeling like eating.

Luckily your mom had prepared your favorite carrot sambar.

Unfortunately on that day, your mom had added more asafoetida. Actually you don’t like asafoetida.

Suppose you don’t like vegetable or onion you can remove it easily. But it is tough to remove asafoetida from the cooked sambar, because it’s a solvent.

Let us imagine, suppose at your home, there is a machine having several filters to remove each of the ingredients like mustard, salt, sambar powder, onion, vegetable and asafoetida etc.

South Indian Sambar

• What does the Fourier Transform do? Given a sambar, it finds the recipe.
• How? Run the Sambar through filters to extract each ingredient.
• Why? Recipes are easier to analyze, compare, and modify than the smoothie itself.
• How do we get the sambar back? Blend the ingredients.

Well, imagine you had a few filters lying around:

• Pour through the "vegetable" filter.
• Pour through the "salt" filter.
• Pour through the "asafoetida"
• Pour through the "sambar powder etc."

We can reverse-engineer the recipe by filtering each ingredient.

·         Filters must be independent. The vegetable filter needs to capture only vegetables

·         Filters must be complete. Our collection of filters must catch every possible ingredient.

·         Ingredients must be combine-able. Sambar can be separated and re-combined without issue based on our requirement.

How good would it be?

Now it becomes easy for us to remove the constituent/asafoetida which we don’t like from the sambar

Similarly the same kind of machine or approach can be used in TV, computer and mobile or other kind of electronic devices in our daily life. It’s a mathematical concept named Fourier Transform.

Let us consider the piano having 88 keys, all the left end keys will generate low frequency signal, similarly right side keys will generate high frequency signals.

If you press the left side key, that could generate (30 Hz) low frequency signal having 30 cycles per second.

You will hear 30 times of the same kind of signal in a minute. So we could hear low pitch sound.

Similar, if we press the right side key, it could generate 5000Hz high frequency sound signal. You will hear 5000 times of the same kind of signal in a second. That’s why you could hear high pitch.

From the above example, you could understand that both of the signals are clear sine waves or a simple signal. Hence it is easy to process.

Suppose consider the complex signal, processing such a signal would be little complicated process than processing simple sine waves.

If we understand how the complex signal are formed or generated, it becomes easy for us to process.

Suppose if we press 3 keys at a time in the piano, that could generate three different sound signals, but we could hear the combined signals of all the three.

Now processing such a kind of complicated signal becomes equivalent to the separation of asafoetida from the sambar as I mentioned earlier.

But, if we could separate each signal, signal processing becomes easier.

Separating complicated signal into separate signal could be done using Fourier Transform.

The complicated signal is termed as Time domain Output Signal, after the separation of each signal is termed as Frequency Domain Output.

In a simple terms signal processing with time domain signal is complicated than frequency domain signal.

Fourier Transform helps us in converting time domain signal to frequency domain signal.

However the signals are complicated, we can use Fourier Transform to convert it into simple sine waves.

Let us consider the radio signal; at the receiver end original audio signal is combined/mixed with noise signal during the transmission.

But removal of the noise signal becomes difficult when we deal with the time domain signal i.e original signal.

Hence, we must use the Fourier Transform to convert the time domain to frequency domain signal.

Now it becomes easy to remove the nose signal. After this process we could hear the radio programs without noise signal.

Similarly, when you adjust the (bass and terrible knobs in your radio), it converts the input signal into frequency domain using Fourier transform, it multiplies each signal with different values as you adjust, and then it sends to the speaker.

Some of the applications of Fourier transforms are; in our computer we used to convert the raw image into JPEG for the size reduction or to compress the image. Some time we try to improve the picture quality.

The way we separate the complicated audio signal into multiple sine waves, we can separate complex image signal into multiple sine waves.

Suppose in this picture we would like to remove the colors; then we have to use Fourier Transform to convert into frequency domain. (Mathematically this process is otherwise called convolution)

Subsequently, we can remove the signals which are relevant to colors. Now the image is converted into grey scale image.  (Mathematically this process is called de-convolution to work on the separated signals, finally combining the signal is termed as inverse Fourier transform)

Without Fourier Transform we can’t use TV, Radio and Image processing with Photoshop’s. Fourier transform is used in aerospace and quantum physics too.

I hope you would have understood and enjoyed learning the applications and fundamentals of Fourier Transform.

I thank “Let us Make Engineering Simple” (LMES team and Mr. Premanad) for courtesy for reusing this video for the educational purposes.

1. An Interactive Guide To The Fourier Transform
https://betterexplained.com

2. Lets Make Engineering Simple (LMES)
 https://www.youtube.com/watch?v=DBsGC