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Spectral Audio Signal Processing
    Time-Frequency Displays
       The Short-Time Fourier Transform
          Summary of STFT Computation Using the FFT

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Summary of STFT Computation Using the FFT

  1. Read $ M$ samples of the input signal $ x$ into a local buffer of length $ N\geq M$ which is initially zeroed

    $\displaystyle \tilde{x}_m(n) \isdef x(n+mR), \qquad n=-M_h, \ldots\,,-1,0,1,\ldots\,, M_h $

    We call $ x_m$ the $ m$th frame of the input signal, and $ \tilde{x}_m=x(n+mR)$ the $ m$th time normalized input frame (time-normalized by translating it to time zero). The frame length is $ M\isdef 2M_h+1$, which we assume to be odd for reasons to be discussed later. The time advance $ R$ (in samples) from one frame to the next is called the hop size or step size.

  2. Multiply the data frame pointwise by a length $ M$ spectrum analysis window $ w(n), n=-M_h,\ldots\,,M_h$ to obtain the $ m$th windowed data frame (time normalized):

    $\displaystyle \tilde{x}_m^w(n) \isdef \tilde{x}_m(n) w(n), \qquad n=-M_h,\ldots\,,M_h $

  3. Extend $ \tilde{x}_m^w$ with zeros on both sides to obtain a zero-padded frame:

    $\displaystyle \tilde{x}_m^{w,z}(n) \isdef \left\{\begin{array}{ll} \tilde{x}_m^... ...frac{N}{2}}-1 \\ [5pt] 0, & -{\frac{N}{2}}\leq n < -M_h \\ \end{array}\right. $

    where $ N$ is chosen to be a power of two larger than $ M$. The number $ N/M$ is the zero-padding factor. As discussed in §2.5.3, the zero-padding factor is the interpolation factor for the spectrum, i.e., each FFT bin is replaced by $ N/M$ bins, interpolating the spectrum using ideal bandlimited interpolation [248], where the ``band'' in this case is the $ M$-sample nonzero duration of $ \tilde{x}_m^w$ in the time domain.

  4. Take a length $ N$ FFT of $ \tilde{x}_m^{w,z}$ to obtain the time-normalized, frequency-sampled STFT at time $ m$:

    $\displaystyle \tilde{X}_m^{w,z}(\omega_k )=\sum _{n=-N/2}^{N/2-1} \tilde{x}_m^{w,z}(n) e^{-j\omega_k n T} $

    where $ \omega_k = 2\pi k f_s / N $, and $ f_s=1/T$ is the sampling rate in Hz. As in any FFT, we call $ k$ the bin number.

  5. If needed, time normalization may be removed using a linear phase term to yield the sampled STFT:

    $\displaystyle X^{w,z}_m(\omega_k) = e^{-j\omega_k mR}\tilde{X}_m^{w,z}(\omega_k ) $

    The (continuous-frequency) STFT may be approached arbitrarily closely by using more zero padding and/or other interpolation methods.

    Note that there is no irreversible time-aliasing when the STFT frequency axis $ \omega$ is sampled to the points $ \omega_k$, provided the FFT size $ N$ is greater than or equal to the window length $ M$.

Previous: Practical Computation of the STFT
Next: Two Dual Interpretations of the STFT

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

 

hsmdsp wrote:

3/2/2009
 
For vibratory string analysis, since the signal is almost periodic over short intervals (10 - 20 ms), windowing can be avoided. Meaning, a rectangular window suffices for most practical purposes. DO you have a different opinion ?
 

JOS wrote:

3/2/2009
 
In my experience, it works ok for the fundamental and first several partial overtones. It gets pretty noisy for the high partials, however. If the string stiffness is negligible, then you can resample the signal so its period is exactly an integer, and that gives best results. A "movie" made from a one-period plot (across time or partial number) is interesting to watch in this case.

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