短時傅利葉轉換
维基百科，自由的百科全书
短時傅利葉轉換或短時傅利葉變換(英文：short-time Fourier transform, STFT，又稱short-term Fourier transform)是和傅利葉轉換相關的一種數學轉換關係，用以決定時變訊號其局部段落之弦波成份的頻率與相位。
簡單來說，在連續時間的例子，一個函數可以先乘上僅在一段時間不為零的窗函數(window function)再進行一維的傅利葉轉換。再將這個窗函數沿著時間軸挪移，所得到一系列的傅利葉轉換結果排開則成為二維表象。數學上，這樣的操作可寫為：
其中w(t)是窗函數，通常是翰氏窗函數(Hann window)或高斯函數的「丘型」分布，中心點在零，而x(t)是待轉換的訊號。X(τ,ω)本質上是x(t)w(t − τ)的傅利葉轉換，乃一個複函數代表了訊號在時間與頻率上的強度與相位。Often phase unwrapping is employed along either or both the time axis, τ and frequency axis, ω, to suppress any jump discontinuity of the phase result of the STFT. The time index τ is normally considered to be "slow" time and usually not expressed in as high resolution as time t.
In the discrete time case, the data to be transformed could be broken up into chunks or frames (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:
![\mathbf{STFT} \left \{ x[ \ ] \right \} \equiv X(m,\omega) = \sum_{n=-\infty}^{\infty} x[n]w[n-m]e^{-j \omega n}](../../../math/5/5/4/5549ce7a9fd260343e80f98ed70c3a95.png)
likewise, with signal x[n] and window w[n]. In this case, m is discrete and ω 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. Again, the discrete-time index m is normally considered to be "slow" time and usually not expressed in as high resolution as time n.
The magnitude squared of the STFT yields the spectrogram of the function:
