13.1 Short-time Fourier transform and Gabor transform

Time-frequency analysis lets us see how a signal's frequency content changes over time. The short-time Fourier transform (STFT) and Gabor transform are key tools for this, breaking signals into short segments and analyzing their frequency components.

These transforms help us understand complex signals better by showing how their frequencies evolve. They're super useful in fields like speech processing, music analysis, and biomedical signal processing, where we need to track changing frequencies or spot brief events.

Short-time Fourier Transform

Definition and Computation

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• Short-time Fourier transform (STFT) is a time-frequency analysis technique that divides a signal into short segments and applies the Fourier transform to each segment
• Computed by multiplying the signal with a window function and then applying the Fourier transform to the resulting windowed signal
• Window function is a function that is non-zero for a short period of time and zero elsewhere (rectangular, Hamming, Hanning, or Gaussian windows)
• Choice of window function affects the time-frequency resolution and the presence of spectral leakage

Time-Frequency Representation

• STFT provides a time-frequency representation of the signal, allowing for the analysis of the signal's frequency content as it changes over time
• Sliding window approach is used, where the window function is shifted along the time axis, and the STFT is computed for each position of the window
• Resulting STFT coefficients represent the signal's frequency content at different time instances
• Time-frequency representation enables the identification of time-varying frequency components and transient events in the signal

Gabor Transform

Definition and Properties

• Gabor transform is a special case of the STFT that uses a Gaussian window function
• Named after Dennis Gabor, who introduced the concept of time-frequency analysis
• Gaussian window function has optimal time-frequency localization properties, minimizing the uncertainty principle
• Gabor transform provides a balanced trade-off between time and frequency resolution

Time-Frequency Resolution

• Time-frequency resolution refers to the ability to distinguish between different frequencies and time instances in the time-frequency representation
• Gabor transform achieves a good balance between time and frequency resolution due to the use of the Gaussian window function
• Increasing the window size improves frequency resolution but reduces time resolution, while decreasing the window size improves time resolution but reduces frequency resolution
• Choice of window size depends on the specific requirements of the application and the characteristics of the signal being analyzed

Applications and Techniques

Spectrogram

• Spectrogram is a visual representation of the STFT or Gabor transform, displaying the time-frequency content of a signal
• Plotted as a 2D image, with time on the x-axis, frequency on the y-axis, and the magnitude or power of the STFT coefficients represented by color or intensity
• Spectrograms are widely used in various fields, such as speech processing (speech recognition, speaker identification), music analysis (pitch tracking, instrument classification), and biomedical signal processing (EEG, ECG analysis)

Overlap-Add Method

• Overlap-add method is a technique used to efficiently compute the STFT or Gabor transform of long signals
• Signal is divided into overlapping segments, and the STFT is computed for each segment
• Resulting STFT coefficients are then added together with appropriate time shifts to obtain the final time-frequency representation
• Overlap-add method reduces the computational complexity and memory requirements compared to computing the STFT on the entire signal at once
• Amount of overlap between segments affects the smoothness of the time-frequency representation and the presence of artifacts (50% overlap is commonly used)