If you are an electronic engineer, DSP or acoustics graduate, you may have come across the term “Fourier Transform” in your studies.
Typically, the term is followed by pages of maths, which makes it seem overly complicated and many never stop to ponder the meaning of the Fourier Transform.
To be honest, the term washed over me when I was in university and I never stopped to grasp it. It was not until I started working in the audio industry, did this term start to make sense.
In the simplest terms, the Fourier Transform is a mathematical function that allows us to convert a signal from the time domain to the frequency domain. The time domain is the representation of a signal as a function of time, while the frequency domain is the representation of a signal as a function of frequency.
Let’s dive deeper into what this actually means and how we can apply it to real-world applications.
I will be avoiding all the maths in this article, as I want to discuss the fundamental understanding of the principle without the maths, but I will show you how you can dig deeper into the maths if you choose to.
What Is The Fourier Transform?
The Fourier Transform is a mathematical function that allows us to convert a signal from the time domain to the frequency domain.
The time domain is the representation of a signal as a function of time, while the frequency domain is the representation of a signal as a function of frequency.
In the simplest form, you could describe the Fourier Transform as the formula that lets you reverse engineer a signal.
Let’s take an example. Imagine your friend cooks a fabulous curry. (This represents our signal).
We want to cook this curry too, but we don’t have the recipe. The Fourier Transform is the tool that allows us to break down the curry and work out what the individual ingredients are.
The Fourier Transform will take the curry (the signal) and give us the recipe.
In simple terms, the Fourier Transform allows us to understand what components make up a signal.
In essence, it says that any complex wave-like signal you care to measure, that fluctuates over time or space, can be broken down into a sum of the familiar, regular, sine waves – the type that roll across the tops of oceans or vibrate along strings.
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How Does The Fourier Transform Work?
As mentioned, complex maths equations define the formula for the Fourier Transform and how it works.
But how does the Fourier transform work in basic terms? What exactly are those complex mathematical formulas doing?
The Fourier Transform is able to convert a signal from the time domain to the frequency domain by breaking it down into a series of sinusoidal waves. It does this by taking samples of the signal at fixed intervals and then calculating the spectral coefficients for each sample.
The spectral coefficients represent the amplitude and phase of each sinusoidal wave at a specific frequency.
If that does not make sense, here is another way to think about it.
The concept of Jean-Baptiste Joseph Fourier who invented the Fourier Transform believed that every signal can be broken down into circular paths.
Although this took years for mathematicians to fully accept, he was correct.
Even complex triangular or square waves can be drawn using circles. If you can’t picture this, below is a great video that shows you how to visualise the Fourier Series using circles.
Once you accept this concept that even the most irregular and complex signal can be recreated using circles, the Fourier Transform becomes much easier to understand.
To create a circular path, Euler’s formula is used. Here is what Euler’s formula looks like:
Euler’s formula will help us define our circular path and break down whatever pattern our signal has into circular paths. To learn more about Euler’s formula and the Fourier Transform, check out this site which describes it brilliantly.
When I learned about how to break down a signal into circular paths, it was a breakthrough in my understanding of the Fourier Transform.
Below is a visual of how a signal can be decomposed into circular paths. [source]
I highly recommend checking out BetterExplained.com, to see this signal animated with a full description of how to break down your signal.
It is this breaking down of any signal into its component parts that is at the core of how the Fourier Transform works.
Why Is The Fourier Transform Important?
The Fourier Transform is important as it is a powerful tool that can be used to analyse any signal.
For example, when dealing with audio signals, the Fourier Transform can be used to determine the frequency content of a signal.
This is because all sounds are made up of different frequencies that create the unique timbre of the sound.
All modern digital audio processing depends heavily on the Fourier Transform.
When we know the frequency of a signal, we can easily calculate so much such as:
- What notes are playing
- Pitch manipulation
- What instruments and sounds are playing
For anyone working in the world of digital signal processing, the Fourier Transform is an essential tool for analysing audio signals.
What Are Real-World Applications Of The Fourier Transform?
Some real-world applications of the Fourier Transform include:
- Audio signal analysis
- Noise cancellation
- Signal generation
- Signal processing
In the real world, we can see some practical applications of the Fourier Transform in the fields of:
- Signal filtering
- Signal sampling
- Modulation
In a nutshell, Fourier Transform can be used to study any complex signal that fluctuates, from sound to stock prices.
It can be used to remove noise from data collected and to contribute to the equations that are used to compress digital images, such as JPEG or sound files.
Final Thoughts
The Fourier Transform is a powerful tool that can be used to analyse any signal. It is essential for anyone working in the world of digital signal processing and has many real-world applications.
In a nutshell, the Fourier Transform can be used to study any complex signal that fluctuates, from sound to stock prices.
It can be used to remove noise from data collected or to contribute to the equations that are used to compress digital images, such as JPEG or sound files. Understanding how the Fourier Transform works is essential for anyone working with digital signals.