Signal Processing Concepts and Engineering Insights.
Explore signal processing concepts, algorithm comparisons, and practical engineering insights. Topics include FFT vs STFT, FRF analysis, filtering techniques, and other signal processing methods used in real engineering workflows.
STFT vs FFT: Why FFT Cannot Track Time-Varying Signals
The Fast Fourier Transform (FFT) is one of the most widely used tools in signal processing for identifying frequency components within a signal.
However, FFT has a fundamental limitation. It assumes that the signal is stationary during the entire observation window. But in many real-world signals, this assumption does not hold.
Examples include
Machine vibration during speed changes
Speech signals
Musical audio signals
Transient mechanical events
In these cases, the frequency content changes over time, and a single FFT cannot reveal when those frequency changes occur. To analyze such signals, engineers often use the Short-Time Fourier Transform (STFT).
This article explains
Why FFT cannot track time-varying signals
How STFT solves this limitation
When to use FFT vs STFT in practical signal analysis
Why FFT Cannot Track Time-Varying Signals
The FFT converts a signal from the time domain to the frequency domain.
Mathematically
This transformation summarizes the entire signal window into a single spectrum. As a result, the FFT answers only one question.
Which frequencies exist in the signal?
But it cannot answer.
When did those frequencies occur?
Frequency Changing Over Time
Consider a signal where the frequency changes during the measurement.
0–1 sec → 100 Hz
1–2 sec → 200 Hz
2–3 sec → 400 Hz
If we compute a single FFT over the entire signal, the result will show
100 Hz
200 Hz
400 Hz
But the FFT cannot tell when each frequency occurred.
Below is an example of FFT analysis performed in MALMIJAL.
Sampling rate: 2000Hz
Duration: 6sec
4 segments
0 ~ 1.5sec: steady 80Hz sine tone
1.5 ~ 3sec: chirp signal from 120Hz to 400Hz
3 ~ 4.5sec: two-tone signal: 200Hz + 500Hz
4.5 ~ 6sec: 300Hz background tone, short 700Hz bursts around 4.7sec, 5.1sec, 5.5sec
Time domain signal
Time-varying signal in time domain, try playing back the sound (click ▶)
FFT result
FFT of time-varying signal, no time information exists
You just can know uniform energy distribution from 120 to 400Hz, peak at 80, 200, 500, and 700Hz
Even though the signal changes frequency over time, the FFT simply reports all frequencies present in the entire signal window. The temporal information is completely lost.
How STFT Solves This Problem
The Short-Time Fourier Transform (STFT) addresses this limitation by analyzing the signal in small time segments.
Instead of computing one FFT over the entire signal, STFT performs the following steps.
Divide the signal into short overlapping windows
Apply a window function
Compute FFT for each segment
Combine the results into a time-frequency representation
Mathematically
w(t) is the window function
τ represents the time position of the window
This produces a spectrogram, which shows how frequencies evolve over time.
Visualizing STFT Results
A typical STFT result is displayed as a spectrogram.
4 segments
0 ~ 1.5sec: horizontal line near 80Hz
1.5 ~ 3sec: upward slanted ridge from 120Hz to 400Hz
3 ~ 4.5sec: two horizontal bands, 200Hz + 500Hz
4.5 ~ 6sec: horizontal line at 300 Hz plus brief high-frequency events at 4.7, 5.1, 5.55sec
In this representation
Horizontal axis → Time
Vertical axis → Frequency
Color intensity → Signal magnitude (PSD in this case)
Unlike FFT, STFT clearly shows when specific frequencies appear and disappear.
Trade-Off: Time Resolution vs Frequency Resolution
STFT introduces an important trade-off.
Because the signal is divided into short windows:
Short windows → good time resolution
Long windows → good frequency resolution
This is known as the time-frequency resolution trade-off.
Window Length
Time Resolution
Frequency Resolution
Short window length
High
Low
Long window length
Low
High
Choosing the correct window size depends on the nature of the signal being analyzed.
FFT vs STFT: When to Use Each
Both FFT and STFT are essential tools in signal processing.
Method
Best Use Case
FFT
Stationary signals
STFT
Time-varying signals (non-stationary signals)
Use FFT when
The signal is stationary over time
You only need the overall frequency components
High frequency resolution is required
Examples
Rotating machinery vibration with constant rotating speed
Harmonic analysis
Frequency response measurements
Use STFT when
The signal changes over time
Transient events occur
Time-frequency behavior is important
Examples
Speech signals
Musical audio analysis
Machine startup vibration
Fault detection in rotating equipment
Practical Signal Analysis Example
In practical vibration analysis, a machine may operate under changing conditions.
Startup
Speed ramp
Load changes
During these events, frequency components evolve continuously. Using only FFT would show all frequencies together, but would not reveal when each component appears. STFT makes it possible to track these changes in real time.
Modern signal processing tools such as MALMIJAL allow engineers to visualize both
FFT spectra
STFT spectrograms
making it easier to understand complex time-varying signals.
Conclusion
The FFT is an extremely powerful tool for frequency analysis, but it has an important limitation.
Because FFT analyzes the entire signal window at once, it cannot track how frequency components change over time.
The Short-Time Fourier Transform (STFT) solves this problem by computing FFT over short time segments, producing a time-frequency representation of the signal.
Understanding the difference between FFT and STFT allows engineers to choose the right tool for analyzing both stationary and time-varying signals.
STFT vs FFT: Why FFT Cannot Track Time-Varying Signals
The Fast Fourier Transform (FFT) is one of the most widely used tools in signal processing for identifying frequency components within a signal.
However, FFT has a fundamental limitation. It assumes that the signal is stationary during the entire observation window. But in many real-world signals, this assumption does not hold.
Examples include
Machine vibration during speed changes
Speech signals
Musical audio signals
Transient mechanical events
In these cases, the frequency content changes over time, and a single FFT cannot reveal when those frequency changes occur. To analyze such signals, engineers often use the Short-Time Fourier Transform (STFT).
This article explains
Why FFT cannot track time-varying signals
How STFT solves this limitation
When to use FFT vs STFT in practical signal analysis
Why FFT Cannot Track Time-Varying Signals
The FFT converts a signal from the time domain to the frequency domain.
Mathematically
This transformation summarizes the entire signal window into a single spectrum. As a result, the FFT answers only one question.
Which frequencies exist in the signal?
But it cannot answer.
When did those frequencies occur?
Frequency Changing Over Time
Consider a signal where the frequency changes during the measurement.
0–1 sec → 100 Hz
1–2 sec → 200 Hz
2–3 sec → 400 Hz
If we compute a single FFT over the entire signal, the result will show
100 Hz
200 Hz
400 Hz
But the FFT cannot tell when each frequency occurred.
This is because the FFT assumes the signal is periodic and stationary within the observation window
Example: FFT Analysis of a Time-Varying Signal
Below is an example of FFT analysis performed in MALMIJAL.
Sampling rate: 2000Hz
Duration: 6sec
4 segments
Time domain signal
Time-varying signal in time domain, try playing back the sound (click ▶)
FFT result
You just can know uniform energy distribution from 120 to 400Hz, peak at 80, 200, 500, and 700Hz
Even though the signal changes frequency over time, the FFT simply reports all frequencies present in the entire signal window. The temporal information is completely lost.
How STFT Solves This Problem
The Short-Time Fourier Transform (STFT) addresses this limitation by analyzing the signal in small time segments.
Instead of computing one FFT over the entire signal, STFT performs the following steps.
Divide the signal into short overlapping windows
Apply a window function
Compute FFT for each segment
Combine the results into a time-frequency representation
Mathematically
w(t) is the window function
τ represents the time position of the window
This produces a spectrogram, which shows how frequencies evolve over time.
Visualizing STFT Results
A typical STFT result is displayed as a spectrogram.
4 segments
In this representation
Horizontal axis → Time
Vertical axis → Frequency
Color intensity → Signal magnitude (PSD in this case)
Unlike FFT, STFT clearly shows when specific frequencies appear and disappear.
Trade-Off: Time Resolution vs Frequency Resolution
STFT introduces an important trade-off.
Because the signal is divided into short windows:
Short windows → good time resolution
Long windows → good frequency resolution
This is known as the time-frequency resolution trade-off.
Choosing the correct window size depends on the nature of the signal being analyzed.
FFT vs STFT: When to Use Each
Both FFT and STFT are essential tools in signal processing.
Use FFT when
The signal is stationary over time
You only need the overall frequency components
High frequency resolution is required
Examples
Rotating machinery vibration with constant rotating speed
Harmonic analysis
Frequency response measurements
Use STFT when
The signal changes over time
Transient events occur
Time-frequency behavior is important
Examples
Speech signals
Musical audio analysis
Machine startup vibration
Fault detection in rotating equipment
Practical Signal Analysis Example
In practical vibration analysis, a machine may operate under changing conditions.
Startup
Speed ramp
Load changes
During these events, frequency components evolve continuously. Using only FFT would show all frequencies together, but would not reveal when each component appears. STFT makes it possible to track these changes in real time.
Modern signal processing tools such as MALMIJAL allow engineers to visualize both
FFT spectra
STFT spectrograms
making it easier to understand complex time-varying signals.
Conclusion
The FFT is an extremely powerful tool for frequency analysis, but it has an important limitation.
Because FFT analyzes the entire signal window at once, it cannot track how frequency components change over time.
The Short-Time Fourier Transform (STFT) solves this problem by computing FFT over short time segments, producing a time-frequency representation of the signal.
Understanding the difference between FFT and STFT allows engineers to choose the right tool for analyzing both stationary and time-varying signals.
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