Window Functions in Fourier Transform: Understanding the Role of Observation Time in Pitch Analysis

In the world of music analysis and signal processing, the Fourier transform is a powerful tool for understanding the frequency content of a sound. One of the key components of the Fourier transform is the window function, which controls how long the computer "looks" at the signal in order to determine the frequencies present.

In a perfect world, a longer observation time would provide better accuracy in identifying the frequencies present in a sound. However, in the real world, musical notes from instruments are not infinitely long and their sustain is not always perfectly flat and clear. As a result, the pitch of a note can fluctuate, which can lead to a noisy spectrum when analyzing the sound.

To address this problem, different window functions have been developed to control the observation time and minimize the impact of pitch fluctuations on the analysis. In music analysis, this principle can be applied to the frequency and time domain of a sound. A commonly used window function is the Hann window, which is a smooth, bell-shaped curve that tapers the edges of the observation time. This helps to reduce the impact of any sudden changes in the sound and minimize the creation of "spectral leakage," which can lead to inaccurate frequency analysis.

Other window functions such as Blackman, Hamming, or Kaiser also have specific properties that can be useful for different types of signals and purposes. 

It's important to note that, the window function does not only affect the precision and accuracy of the frequency analysis, but it also affects the time resolution of the transform. A wider window will provide a better frequency resolution but a worse time resolution, while a narrower window will provide a better time resolution but a worse frequency resolution, echoing the Heisenberg's Uncertainty Principle.

Dunn, F., Rossing, T., & Fletcher, N. H. (Eds.). (2015). Springer handbook of acoustics. Springer New York.

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