Windowed Fourier Transform

Fourier Analysis · Axiom Academy

LESSON Windowed Fourier Transform Understanding the Short-Time Fourier Transform (STFT) for time-frequency analysis Instead of analyzing the entire signal at once, we multiply it by a window function that's localized in time. We then slide this window along the signal and take the Fourier transform at each position. Think of it as looking at your signal through a moving magnifying glass - you can see what frequencies are present at each moment in time. The Short-Time Fourier Transform is defined mathematically as: Here, f(t) is your signal, w(t-τ) is the window centered at time τ, and we integrate over all time to get the frequency content at position τ. The result S(τ,ω) is a complex-valued function of two variables: time (τ) and frequency (ω). To visualize the STFT, we typically plot the spectrogram , which is the squared magnitude of the STFT: The spectrogram is a 2D representation where: Color/brightness: Energy at that time-frequency point 4. Fixed Time-Frequency Resolution A fundamental limitation of the STFT is that it has fixed resolution in both time and frequency, determined by the window width. The time-frequency plane is divided into tiles of equal size everywhere: Narrow window: Good time resolution, poor frequency resolution Wide window: Good frequency resolution, poor time resolution

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