# What is the Difference Between FFT and STFT? An In-Depth Comparison

You might have heard the technical terms “FFT” and “STFT” if you’re familiar with audio processing. But, do you know what they stand for or what makes them different? FFT (Fast Fourier Transform) and STFT (Short Time Fourier Transform) are two techniques used for processing signals in audio and digital signal processing.

FFT is a technique that is commonly used for analysis in audio processing and is based on a mathematical algorithm that converts a time-domain signal into its frequency-domain representation. In contrast, STFT is a technique that focuses on spectral analysis of short, overlapping time intervals, also known as windows, of a signal. This makes it more suitable for analysis of signals that are time-varying, such as speech, music, or sound effects.

Both FFT and STFT are useful in different aspects of audio processing, and it’s important to understand their differences to better utilize them. FFT is ideal for analyzing steady-state signals, while STFT is best suited for signals that are time-varying. By understanding the difference between the two, you can choose the right method for your audio processing needs and achieve the best results.

## Fourier Transform

The Fourier Transform is a mathematical technique used to convert a time-domain signal into its frequency-domain representation. It is named after the French mathematician Joseph Fourier, who discovered that any periodic function can be represented by a sum of sine and cosine functions.

The Fourier Transform enables us to analyze the frequency content of a signal, and it has numerous applications in fields such as signal processing, communications, and audio engineering.

## What is the Difference between FFT and STFT?

• The Fast Fourier Transform (FFT) is a computationally efficient algorithm that computes the Fourier Transform of a signal using a divide-and-conquer approach. It is widely used in digital signal processing applications and is especially useful for real-time processing.
• The Short-Time Fourier Transform (STFT) is a variation of the Fourier Transform that allows us to analyze the frequency content of a signal over time. It breaks up the signal into small time segments and computes the Fourier Transform of each segment. This allows us to see how the frequency content of the signal changes over time.

## How does the Fourier Transform Work?

The Fourier Transform works by decomposing a signal into its component frequencies. It does this by expressing the signal as a sum of sine and cosine waves with different frequencies and amplitudes. The equation for the Fourier Transform of a signal f(t) is:

$\Large&space;\hat{f}(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega{t}}dt$

where ω is the frequency and &hat;f(ω) is the Fourier Transform of f(t). The inverse Fourier Transform is used to convert the Fourier Transform back into the time domain:

$\Large&space;f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega{t}}d\omega$

## Fourier Transform Example

Let’s take a simple example of a signal that changes over time:

Time (t) Signal Amplitude
0 1
1 2
2 3
3 2
4 1

To compute the Fourier Transform of this signal, we apply the equation:

$\Large&space;\hat{f}(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega{t}}dt$

for each frequency ω. The resulting Fourier Transform will give us the frequency content of the signal.

## Time-Frequency Analysis

Time-Frequency analysis is a signal processing technique that aims to analyze non-stationary signals that contain time-varying frequency content. The evolution of frequency components over time is what makes non-stationary signals different from stationary signals, where the frequency content does not change with time.

Time-Frequency analysis provides a time-varying representation of the signal’s frequency content and helps to identify the time at which the specific frequencies appear and disappear. This type of analysis is essential in many applications such as speech processing, radar, sonar, and biomedical signal analysis.

## What is the Difference between FFT and STFT?

• FFT: FFT (Fast Fourier Transform) is a technique used to analyze the frequency content of a stationary signal. It provides a static spectral analysis of a signal by decomposing it into its individual frequency components. The FFT is efficient, fast, and widely used because its computational complexity is O(N log N), where N is the number of samples. However, the FFT assumes stationary signals, which means that it cannot handle non-stationary signals like music, speech, and biomedical signals that change their frequency content over time.
• STFT: STFT (Short-time Fourier Transform) is a technique used to analyze the frequency content of non-stationary signals. It represents the local frequency content of a signal over time by applying FFT to short overlapping segments of the signal. STFT enables the analysis of the frequency content of a signal at different time intervals, providing a time-frequency representation of the signal. STFT has a higher computational cost than FFT because it requires multiple FFT processes, but it is the go-to technique for analyzing non-stationary signals.

## Applications of Time-Frequency Analysis

Time-frequency analysis has many real-world applications, from speech and audio processing to biomedical signal analysis. Some of its applications are:

• Speech Processing: The study of voice, speech, and language require time-frequency analysis to analyze the frequency content of speech signals and identify speech features such as formants or phonemes.
• Audio Processing: Time-frequency analysis is essential in music analysis, noise reduction, and speech recognition, among others.
• Biomedical Signal Analysis: Time-frequency analysis is widely used to analyze ECG, EEG, EMG signals, among others, to diagnose diseases or study human physiological systems.

## Conclusion

Time-frequency analysis is a powerful tool for analyzing time-varying signals with non-stationary frequency content. FFT and STFT are two of the most widely used techniques for frequency analysis, with different strengths and weaknesses. Understanding the difference between FFT and STFT is important, especially when analyzing non-stationary signals. The use of time-frequency analysis can benefit various fields such as speech processing, audio processing, and biomedical signal analysis, among others.

FFT STFT
Used to analyze frequency content of stationary signals. Used to analyze frequency content of non-stationary signals.
Provides a static spectral analysis of the signal. Provides a time-varying spectral analysis of the signal.
Computational complexity is O(N log N). Higher computational complexity than FFT because it requires multiple FFT processes.
Cannot handle non-stationary signals. Enables the analysis of frequency content of the signal in different time intervals.

The table above summarizes the differences between FFT and STFT.

## Window Function

A window function is used in both FFT and STFT to overcome a limitation of these algorithms, which is that they assume the data being analyzed is periodic. In reality, most signals are not perfectly periodic and will have discontinuities at the edges of the time window being analyzed. This can cause artifacts in the frequency domain that obscure the true signal.

A window function is a mathematical function that is multiplied by the signal being analyzed before FFT or STFT is performed. This has the effect of tapering the signal at the edges, reducing the impact of these discontinuities. The choice of window function can have a significant impact on the accuracy of the analysis.

There are many different window functions available, each with its own strengths and weaknesses. Some common types of window functions include:

• Rectangular window: a simple window that has a constant value of 1 over the entire time window
• Hanning window: a window that smoothly tapers the signal to zero at the edges
• Hamming window: similar to the Hanning window, but with a slightly different shape that can provide better frequency resolution

Other types of window functions include the Blackman window, the Kaiser window, and the Gaussian window. Choosing the right window function depends on the specific characteristics of the signal being analyzed, including its frequency content and the length of the time window being used.

Table: Comparison of Common Window Functions

Window Function Main Lobe Width Maximum Side Lobe
Rectangular 4N -13 dB
Hanning 4N -31 dB
Hamming 4N -43 dB
Blackman 6N -58 dB

Each window function has a trade-off between the width of the main lobe and the height of the side lobes. A wider main lobe provides better frequency resolution, but at the cost of greater spectral leakage from the side lobes. A narrower main lobe reduces spectral leakage, but at the cost of poorer frequency resolution.

## Spectrogram

In signal processing, a spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time. It is a two-dimensional representation of a signal’s spectrum. The horizontal axis represents time and the vertical axis represents frequency. It is a way to analyze and visualize the frequency content of a signal over time. Spectrograms are often used to analyze speech, music, and other sounds.

• The spectrogram can be generated using both FFT and STFT. It is essentially a visual representation of the spectral content of a signal over time.
• Spectrograms are commonly used in audio and speech processing applications, as they provide a quick and easy way to visualize the frequency content of a signal.
• Spectrograms can also be used in machine learning applications, such as automatic speech recognition or music classification, as they allow for the extraction of specific features from a signal that are not easily identified by other means.

The spectrogram is usually displayed using a colormap to indicate the spectral power of the signal at each point in time and frequency. The colormap is often displayed on a logarithmic scale to better show the dynamic range of the signal. Spectrograms can also be used to identify specific patterns or harmonics in a signal, making them useful in fields such as acoustics and vibration analysis.

Frequency (Hz) Time (s) Power (dB)
100 0.1 -20
200 0.2 -30
300 0.3 -15
400 0.4 -10
500 0.5 -25

In the above example, the table shows the power of a signal at each frequency and time point. The spectrogram for this signal would display the same information in a graphical format, with the frequency and time axes labeled and a colormap used to indicate the power at each point. This makes it easy to visualize the characteristics of the signal over time and frequency, and to identify any specific patterns or features that may be present.

## Alias Frequency

Aliasing is a common issue in signal processing that is caused by undersampling a signal. When aliasing occurs, the frequency components above the Nyquist frequency get folded back and appear as lower frequencies in the sampled signal. In the context of FFT and STFT, aliasing can lead to incorrect frequency analysis and affects the quality of the results.

• The Nyquist frequency is half the sampling frequency, which defines the highest frequency that can be accurately measured without aliasing.
• In FFT, the frequency resolution is defined by the sampling rate and the number of points in the signal window. However, increasing the number of points doesn’t necessarily improve the frequency resolution if there is aliasing present.
• STFT can avoid aliasing issues by using overlapping windows on the signal. By applying a window function that smoothly tapers towards zero, the frequency content that extends beyond the window boundaries gets reduced.

Table 1 shows an example of aliasing in a signal with a 200 Hz sine wave and a sampling rate of 500 Hz. The aliasing frequency components are highlighted in red and appear as a lower frequency in the sampled signal.

Time (s) Signal Value
0 0
0.002 0.98
0.004 0
0.006 -0.98
0.008 0
0.010 0.98
0.012 -0.98
0.014 0
0.016 0.98
0.018 0

To avoid aliasing issues, it’s important to carefully choose the sampling rate and the signal processing technique based on the frequency content of the signal. By understanding the Nyquist frequency and how aliasing affects the analysis, we can make informed decisions when analyzing signals using FFT and STFT.

## FFT vs DFT

Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT) are mathematical algorithms that convert signals from time domain to the frequency domain, allowing us to identify the components that constitute a signal. While they both are used to analyze signals and are related to each other, they possess fundamental differences that set them apart. Here, we will be discussing the differences between FFT and DFT.

• Speed: DFT is the standard method to calculate the Fourier transformation, however, it can be time-consuming and computationally expensive to compute. FFT, on the other hand, is a speedy algorithm that significantly reduces the computational complexity of DFT, allowing computations to be done in a much quicker time span.
• Windowing: In order to analyze signals with a fast Fourier transform, it is necessary to ‘window’ the signal. Windowing reduces the effects of spectral leakage (when a signal appears wider than it is) in the spectrum when selecting segments. While FFT automatically applies a windowing function, DFT requires manual selection of the window type which can be tedious and time-consuming.
• Input: FFT and DFT both generate frequency data from the input signal. However, DFT requires equally spaced samples in the time domain, whereas FFT can handle non-uniformly spaced samples.
• Accuracy: When computing the FFT, a small measurement error can occur due to the floating-point arithmetic used in the algorithm. DFT, on the other hand, is exact but can take a longer time to complete. For most applications, the FFT is precise enough and its speed makes it a better choice.
• Complexity: The FFT algorithm has an inherent complexity of O(n log n), which is the fastest possible computational time using current technology. DFT, however, possesses a complexity of O(n^2) which can be drastically slower.
• Applications: DFT has its advantages in smaller applications where computational speed is not a major requirement. This is because the algorithm is exact and can offer more precise frequency data compared to FFT which has a small error margin. FFT, on the other hand, is used in larger applications where speed is a primary consideration and the small error margin of magnitude isn’t of great consequence.

In conclusion, while FFT and DFT share some similarities, they possess significant differences. FFT is faster, requires less manual efforts, can handle non-uniform spacing, is error-prone yet computationally precise, has an inherent limitation, and is more used in larger applications. While DFT is slower, precisely accurate, manual labour-intensive, computationally more challenging, and used in smaller applications. Understanding their differences will help you make a better choice for your specific data analysis application.

## Real-time vs Offline Analysis

One of the key differences between FFT (Fast Fourier Transform) and STFT (Short-Time Fourier Transform) is their suitability for real-time vs offline analysis. In real-time analysis, the data is processed as it is being recorded, allowing for immediate feedback and adjustments. In contrast, offline analysis involves processing data after it has been recorded, allowing for more time and resources to be dedicated to the analysis.

• Real-time analysis: FFT is often favored for real-time analysis because it can quickly and efficiently transform a signal into the frequency domain, allowing for immediate analysis of the harmonics and frequencies present. This makes it well suited for applications where real-time decisions must be made based on the data being collected.
• Offline analysis: STFT, on the other hand, is typically used for offline analysis because it allows for more detailed analysis over a longer period of time. STFT breaks the signal into smaller windows, allowing for analysis of the variations in frequency content over time. This makes it more suitable for situations where a deeper understanding of the signal and its variations is required, such as in studying the effects of a particular stimulus on brain activity.

Ultimately, the choice between FFT and STFT will depend on the specific application and the goals of the analysis. Real-time applications will generally favor FFT for its speed and efficiency, while more detailed analysis will require the use of STFT.

It is important to note that both FFT and STFT can be used in both real-time and offline analysis, but the choice of which to use will depend on the specific goals of the analysis. For example, in studying the effects of a particular stimulus on brain activity in real-time, FFT may be used to quickly identify changes in frequency content, while in offline analysis, STFT may be used to analyze the variations in frequency content over a longer period of time.

FFT STFT
Transforms a signal into the frequency domain quickly and efficiently Breaks the signal into smaller windows, allowing for analysis of the variations in frequency content over time
Well suited for real-time applications Generally used for offline analysis
Ideal for situations where real-time decisions must be made based on the data being collected More suitable for situations where a deeper understanding of the signal and its variations is required

In summary, the choice between FFT and STFT for real-time vs offline analysis will depend on the goals of the analysis and the specific application. FFT is generally favored for real-time analysis due to its speed and efficiency, while STFT is typically used for more detailed analysis over a longer period of time.

## What is the difference between FFT and STFT?

Q: What is FFT?
A: FFT stands for fast Fourier transform. It is a mathematical algorithm used to convert time domain data to frequency domain data. To put it simply, it allows us to analyze the frequency content of a signal.

Q: What is STFT?
A: STFT stands for short-time Fourier transform. It is a type of Fourier transform that allows us to analyze the frequency content of a signal over time. STFT breaks the signal up into smaller, overlapping segments and applies the Fourier Transform to each segment.

Q: What is the main difference between FFT and STFT?
A: The main difference between FFT and STFT is that FFT provides information about the frequency content of a signal at a single point in time, while STFT provides information about the frequency content of a signal over time.

Q: When should I use FFT and when should I use STFT?
A: FFT is useful when you want to analyze the frequency content of a signal at a specific point in time. STFT is useful when you want to analyze how the frequency content of a signal changes over time.

Q: Can I use FFT and STFT together?
A: Yes, you can use FFT and STFT together. In fact, this is a common technique used in signal processing. By applying STFT to a signal and then using FFT on each segment, you can analyze both the time-varying and frequency content of the signal.

## Closing thoughts

Thanks for reading! Understanding the differences between FFT and STFT is important in signal processing and can help you choose the best approach for your specific analysis needs. If you have any further questions or comments, please feel free to leave them below. Don’t forget to come back and check for more informative articles!