Summary of STFT Computation Using FFTs

1. Read $M$ samples of the input signal $x$ into a local buffer of length $N\geq M$ which is initially zeroed
   $$\tilde{x}_m(n) \isdef x(n+mR), \qquad n=-M_h, \ldots\,,-1,0,1,\ldots\,, M_h$$
   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):
   $$\tilde{x}_m^w(n) \isdef \tilde{x}_m(n) w(n), \qquad n=-M_h,\ldots\,,M_h$$

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

   $$\tilde{x}_m^{w,z}(n) \isdef \left\{\begin{array}{ll} \tilde{x}_m^w(n), & \left\vert n\right\vert\leq M_h\isdef {\frac{M-1}{2}} \\ [5pt] 0, & M_h< n \leq {\frac{N}{2}}-1 \\ [5pt] 0, & -{\frac{N}{2}}\leq n < -M_h \\ \end{array} \right.$$ (8.5)

   
   
   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 [264], 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 $m$ :

   $$\tilde{X}_m^{w,z}(\omega_k )=\sum _{n=-N/2}^{N/2-1} \tilde{x}_m^{w,z}(n) e^{-j\omega_k n T}$$ (8.6)

   
   
   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:

   $$X^{w,z}_m(\omega_k) = e^{-j\omega_k mR}\tilde{X}_m^{w,z}(\omega_k )$$ (8.7)

   
   
   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 $M$ .