Short-time Fourier transform

From Academic Kids

The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time.

Simply described, in the continuous case, a window function, which is nonzero for only a short period of time, is multiplied by the function to be transformed. The Fourier transform (a one-dimensional function) of the resulting signal is taken as the window is slid along the time axis, resulting in a two-dimensional representation of the signal. Mathematically, this is written as:

<math>STFT(t,\omega)=\int_{-\infty}^{\infty} x(\tau)w^*(\tau-t)e^{-j \omega \tau}\, d\tau<math>

where w(t) is the window function, commonly a lone cosinusoid or gaussian "hill" centered around zero, and x(t) is the signal to be transformed. STFT(t,ω) is then a complex function representing the phase and magnitude of the signal over time and frequency.

In the discrete time case, the data to be transformed is broken up into chunks (which usually overlap each other). Each chunk is Fourier transformed, and the complex result is added to a matrix, which records magnitude and phase for each point in time and frequency. This can be written as:

<math>STFT[n,\omega]=\sum_{m=-\infty}^{\infty} x[n+m]w[m]e^{-j \omega m}<math>

likewise, with signal x[n] and window w[n]. In this case, n is discrete and omega is continuous, but in most typical applications the STFT is performed on a computer using the Fast Fourier Transform, so both variables are discrete and quantized.

One of the downfalls of the STFT is that it has a fixed resolution. Choosing a wide window gives better frequency resolution but poor time resolution. A narrower window gives good time resolution but poor frequency resolution. These are called narrowband and wideband transforms, respectively. If the window is narrow, it is said to be compactly supported. This is one of the reasons for the creation of the wavelet transform (or multiresolution analysis in general), which can give good time resolution for high frequency events, and good frequency resolution for low frequency events, which is the type of analysis best suited for many real signals.

This property is related to the Heisenberg uncertainty principle, but it is not a direct relationship. The product of the standard deviation in time and frequency is limited. The boundary of the uncertainty principle (best simultaneous resolution of both) is reached with a gaussian window function.

The magnitude of the STFT yields the spectrogram of the function:

<math>spectrogram(t,\omega)=\left|STFT(t,\omega)\right|^2<math>

The STFT is invertible, that is, the original signal can be recovered from the transform by the Inverse STFT.

See also

Other time-frequency transforms

External links

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