Windowed N-point FFT
&quot;Is Windowed FFT called smoothing?&quot; No.
&quot;Do I need to do anything else apart from ifft(X_3term(m)) to perform inverse of Xthree−term(m)&quot; No.
Here is what is going on:
Suppose
$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} nk } $$
This is the conventional definition of the DFT which is unnormalized.
Now do the following with it:
$$
\begin{aligned}
Y[k] &amp;=  \alpha X[k] - \frac{\beta}{2}X[k-1]-\frac{\beta}{2}X[k+1] \\
&amp;= \alpha \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} nk } 
- \frac{\beta}{2} \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} n(k-1) } 
- \frac{\beta}{2} \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} n(k+1) } \\
&amp;= \alpha \sum_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} nk } 
- \frac{\beta}{2}  \sum_{n=0}^{N-1} e^{ i \frac{2\pi}{N}n} x[n] e^{-i \frac{2\pi}{N} nk } 
- \frac{\beta}{2}  \sum_{n=0}^{N-1} e^{-i \frac{2\pi}{N}n} x[n] e^{-i \frac{2\pi}{N} nk } \\
&amp;=  \sum_{n=0}^{N-1} \left[ \alpha - \frac{\beta}{2} e^{ i \frac{2\pi}{N}n} - \frac{\beta}{2} e^{-i \frac{2\pi}{N}n} \right] x[n] e^{-i \frac{2\pi}{N} nk } \\
&amp;=  \sum_{n=0}^{N-1} w[n] x[n] e^{-i \frac{2\pi}{N} nk } \\
\end{aligned}
$$
Where
$$
\begin{aligned}
w[n]  &amp;=\alpha - \frac{\beta}{2} e^{ i \frac{2\pi}{N}n} - \frac{\beta}{2} e^{-i \frac{2\pi}{N}n}  \\
&amp;= \alpha - \beta \cos \left( \frac{2\pi}{N}n \right) \\
\end{aligned}
$$
So
$$ y[n] = \text{DFT}^{-1} ( Y[k] ) =  w[n] \cdot x[n] $$
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