Editing Short-time Fourier transform (section)

==== Constraint ====
a. <math>\Delta_t \Delta_f = \tfrac{1}{N}</math>, where <math>N</math> is an integer

b. <math>N \geq 2Q+1</math>

c. Nyquist criterion (avoiding the aliasing effect):

:<math>\Delta_t < \frac{1}{2\Omega}</math>, <math>\Omega</math> is the bandwidth of <math>x(\tau) w(t-\tau)</math>

d. Only for implementing the [[Rectangular mask short-time Fourier transform|rectangular-STFT]]

Rectangular window imposes the constraint
:<math>w((n - p)\Delta_t) = 1 </math>
Substitution gives:
:<math>
\begin{align}
X(n\Delta_t, m\Delta_f) &= \sum_{p=n-Q}^{n+Q} w((n - p)\Delta_t)&x(p\Delta_t) e^{-\frac{j2\pi pm}{N}}\Delta_t \\
                        &= \sum_{p=n-Q}^{n+Q}                   &x(p\Delta_t) e^{-\frac{j2\pi pm}{N}}\Delta_t \\
\end{align}
</math>

Change of variable {{math|''n''-1}} for {{math|''n''}}:
:<math>
X((n-1)\Delta_t, m\Delta_f) = \sum_{p=n-1-Q}^{n-1+Q} x(p\Delta_t) e^{-\frac{j2\pi pm}{N}}\Delta_t 
</math>

Calculate <math>X(\min{n}\Delta_t, m\Delta_f)</math> by the ''N''-point FFT:

:<math>X(n_0\Delta_t, m\Delta_f) = \Delta_t e^{\frac{j 2\pi(Q-n_0)m}{N}} \sum_{q=0}^{N-1} x_1(q) e^{-j\frac{2\pi qm}{N}}, \qquad n_0=\min{(n)}</math>

where

:<math> x_1(q) = \begin{cases} x((n - Q + q)\Delta_t) &  q \leq 2Q\\ 0 & q >2Q \end{cases}</math>

Applying the recursive formula to calculate <math>X(n\Delta_t, m\Delta_f)</math>

:<math>X(n\Delta_t, m\Delta_f) = X((n-1)\Delta_t, m\Delta_f) - x((n - Q -1)\Delta_t) e^{-\frac{j 2\pi(n-Q-1)m}{N}}\Delta_t + x((n+Q)\Delta_t)e^{-\frac{j 2\pi(n+Q)m}{N}}\Delta_t</math>