How FFT Works: Understanding the Fast Fourier Transform
The Fast Fourier Transform (FFT) is one of the most important algorithms in signal processing. Through the FFT, engineers and researchers can analyze signals in the frequency domain to identify the frequency components that make up a signal in the time domain. Many real-world signals, such as vibration measurements, audio signals, and sensor data, are easier to understand when examined in terms of their frequency components.
This article explains the following.
What is the Fast Fourier Transform?
Why is it used in signal processing?
Method to convert a time domain signal into a frequency domain representation
I will also show you a simple example using a sine wave.
Time Domain vs. Frequency Domain
The signal can be analyzed in two different ways.
| Domain | Explanation |
|---|
| Time domain | It shows how the signal changes over time. |
| Frequency domain | It shows what frequencies exist in the signal. |
Most real signals are observed in the original time domain.
For example, a simple sine wave measured over time is as follows.
This waveform shows how the signal amplitude changes over time. However, it does not directly show the frequency components included in the signal. It is precisely for this reason that the Fourier transform is useful.
What is Fourier Transform?
The Fourier transform converts a signal in the time domain into the frequency domain. Mathematically, the Discrete Fourier Transform (DFT) is defined as follows
x [n] is a time-domain signal.
X [k] represents the frequency domain spectrum.
N is the number of samples.
The conversion result tells you the amount of energy present at each frequency.
However, directly calculating the DFT is computationally expensive. This is actually why the FFT is used.
The process of converting a time-domain signal into a frequency component
To understand the working principle of the FFT, it is helpful to think of the signal as a combination of multiple sine waves. Any periodic signal can be represented as the sum of multiple sine waves with different frequencies and amplitudes. This idea is known as the Fourier Series.
For example, a complex signal may be composed of
a low-frequency component
several higher-frequency components
different amplitudes and phases
The role of the Fourier Transform is to decompose the signal into these frequency components.
In simple terms, the transform checks how strongly the signal matches sine waves of different frequencies.
It does this by comparing the signal with sinusoidal basis functions such as
If the signal contains a particular frequency, the transform produces a strong response at that frequency.
If the signal does not contain that frequency, the response will be close to zero.
By repeating this process across many frequencies, the Fourier Transform produces a spectrum showing how much energy exists at each frequency.
This is how a waveform in the time domain becomes a frequency spectrum.
Fourier-based Relationship
CTFS (Continous Time Fourier Series) → Fourier Series → FS
CTFT (Continuous Time Fourier Transform) → Fourier Transform → FT
DTFS (Discrete Time Fourier Series) → DFT(Discrete Fourier Transform) → FFT
DTFT (Discrete Time Fourier Transfom)
Periodic Assumption in DFT(FFT)
DFT(FFT) assumes that the input signal repeats periodically. If this assumption is violated (e.g., due to truncation), discontinuities occur at the boundaries, leading to spectral leakage.
Periodic assumption in FFT: Comparing single and multiple Periods → Same FFT results, just different resolutions
Periodic assumption in FFT: Comparing delayed step function and its repetition → Same FFT results, just different resolutions
Why the Fast Fourier Transform (FFT) Is Used
The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT.
Instead of calculating every operation individually, FFT reduces the computational complexity from
O (N²) → O (N log N)
This improvement allows FFT to process signals with thousands or even millions of samples quickly.
Because of this efficiency, FFT is widely used in
Example: FFT of a Sine Wave
To demonstrate how the Fast Fourier Transform works, consider a simple sine wave signal in the time domain.
The figure below shows a sine wave plotted against time.
This signal represents a pure tone with a single frequency component.
100Hz sine wave (refer to Samples/simple FFT.mmj)
This signal represents a pure sine wave with a single frequency.
Applying FFT to this signal generates a frequency spectrum representing the frequency components of the signal.
In the frequency domain graph, one major peak appears around 100Hz.
This peak indicates that the original signal contains a strong frequency component around 100Hz.
Since the input signal is a pure sine wave with a single frequency, the FFT generates a spectrum with a single distinct spike rather than multiple peaks.
By analyzing the peak positions in the FFT spectrum, you can identify the frequencies present in the signal.
Why is FFT important in signal analysis?
FFT is widely used because it allows engineers to quickly identify important frequency components.
| Application | How to use FFT |
|---|
| Vibration Analysis | Detects machine defects |
| Audio Signal Analysis | Identify timbres and harmonics |
| Communication Systems | Spectral analysis and multiplexing |
| Sensor Data | Extracts periodic behavior |
Without FFT, analyzing these signals would be much more difficult.
Practical signal analysis using MALMIJAL
You can easily perform FFT analysis using the latest signal processing tools.
For example, in MALMIJAL , users can perform the following tasks.
This workflow allows engineers and researchers to rapidly analyze signals without writing complex code.
Conclusion
The Fast Fourier Transform (FFT) is a fundamental tool in signal processing.
The FFT converts signals from the time domain to the frequency domain, enabling engineers to identify frequency components, detect patterns, and better understand complex signals.
When analyzing vibration signals, audio recordings, or experimental data, FFT remains one of the most widely used techniques in modern engineering analysis.
Using tools like MALMIJAL , you can perform FFT analysis and visualize the results in just a few steps.
More Insight
However, FFT provides a meaningful frequency representation only if the signal is approximately stationary within the analysis window. For non-stationary signals whose frequency content changes over time, time–frequency methods such as the Short-Time Fourier Transform (STFT) are more appropriate.
Suggested Further Reading
##You may also find these topics helpful:
How FFT Works: Understanding the Fast Fourier Transform
The Fast Fourier Transform (FFT) is one of the most important algorithms in signal processing. Through the FFT, engineers and researchers can analyze signals in the frequency domain to identify the frequency components that make up a signal in the time domain. Many real-world signals, such as vibration measurements, audio signals, and sensor data, are easier to understand when examined in terms of their frequency components.
This article explains the following.
What is the Fast Fourier Transform?
Why is it used in signal processing?
Method to convert a time domain signal into a frequency domain representation
I will also show you a simple example using a sine wave.
Time Domain vs. Frequency Domain
The signal can be analyzed in two different ways.
Most real signals are observed in the original time domain.
For example, a simple sine wave measured over time is as follows.
This waveform shows how the signal amplitude changes over time. However, it does not directly show the frequency components included in the signal. It is precisely for this reason that the Fourier transform is useful.
What is Fourier Transform?
The Fourier transform converts a signal in the time domain into the frequency domain. Mathematically, the Discrete Fourier Transform (DFT) is defined as follows
x [n] is a time-domain signal.
X [k] represents the frequency domain spectrum.
N is the number of samples.
The conversion result tells you the amount of energy present at each frequency.
However, directly calculating the DFT is computationally expensive. This is actually why the FFT is used.
The process of converting a time-domain signal into a frequency component
To understand the working principle of the FFT, it is helpful to think of the signal as a combination of multiple sine waves. Any periodic signal can be represented as the sum of multiple sine waves with different frequencies and amplitudes. This idea is known as the Fourier Series.
For example, a complex signal may be composed of
a low-frequency component
several higher-frequency components
different amplitudes and phases
The role of the Fourier Transform is to decompose the signal into these frequency components.
In simple terms, the transform checks how strongly the signal matches sine waves of different frequencies.
It does this by comparing the signal with sinusoidal basis functions such as
If the signal contains a particular frequency, the transform produces a strong response at that frequency.
If the signal does not contain that frequency, the response will be close to zero.
By repeating this process across many frequencies, the Fourier Transform produces a spectrum showing how much energy exists at each frequency.
This is how a waveform in the time domain becomes a frequency spectrum.
Fourier-based Relationship
CTFS (Continous Time Fourier Series) → Fourier Series → FS
CTFT (Continuous Time Fourier Transform) → Fourier Transform → FT
DTFS (Discrete Time Fourier Series) → DFT(Discrete Fourier Transform) → FFT
DTFT (Discrete Time Fourier Transfom)
Periodic Assumption in DFT(FFT)
DFT(FFT) assumes that the input signal repeats periodically. If this assumption is violated (e.g., due to truncation), discontinuities occur at the boundaries, leading to spectral leakage.
Why the Fast Fourier Transform (FFT) Is Used
The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT.
Instead of calculating every operation individually, FFT reduces the computational complexity from
This improvement allows FFT to process signals with thousands or even millions of samples quickly.
Because of this efficiency, FFT is widely used in
vibration analysis
audio processing
communication systems
radar and sensor analysis
Example: FFT of a Sine Wave
To demonstrate how the Fast Fourier Transform works, consider a simple sine wave signal in the time domain.
The figure below shows a sine wave plotted against time.
This signal represents a pure tone with a single frequency component.
This signal represents a pure sine wave with a single frequency.
Applying FFT to this signal generates a frequency spectrum representing the frequency components of the signal.
In the frequency domain graph, one major peak appears around 100Hz.
This peak indicates that the original signal contains a strong frequency component around 100Hz.
Since the input signal is a pure sine wave with a single frequency, the FFT generates a spectrum with a single distinct spike rather than multiple peaks.
By analyzing the peak positions in the FFT spectrum, you can identify the frequencies present in the signal.
Why is FFT important in signal analysis?
FFT is widely used because it allows engineers to quickly identify important frequency components.
Without FFT, analyzing these signals would be much more difficult.
Practical signal analysis using MALMIJAL
You can easily perform FFT analysis using the latest signal processing tools.
For example, in MALMIJAL , users can perform the following tasks.
Load signal data
Visualize waveforms
Calculates the FFT in a single step.
Check the frequency spectrum immediately.
This workflow allows engineers and researchers to rapidly analyze signals without writing complex code.
Conclusion
The Fast Fourier Transform (FFT) is a fundamental tool in signal processing.
The FFT converts signals from the time domain to the frequency domain, enabling engineers to identify frequency components, detect patterns, and better understand complex signals.
When analyzing vibration signals, audio recordings, or experimental data, FFT remains one of the most widely used techniques in modern engineering analysis.
Using tools like MALMIJAL , you can perform FFT analysis and visualize the results in just a few steps.
More Insight
However, FFT provides a meaningful frequency representation only if the signal is approximately stationary within the analysis window. For non-stationary signals whose frequency content changes over time, time–frequency methods such as the Short-Time Fourier Transform (STFT) are more appropriate.
Suggested Further Reading
##You may also find these topics helpful: