In the previous question,
Smoothing in frequency domain and impulse reponse of the filter in time-domain
I guess I have made a mistake in-terms for applying windowing to FFT.
After reading "Understanding Digital Signal Processing - Richard G. Lyons" few things are rather very clear now. Nevertheless to be certain I would like to ask whether my understanding is correct or not.
Frequency Domain Windowing is simply given by,
$$X_{\mathrm{three-term}}(m)=\alpha X(m) - \frac{\beta}{2}X(m-1)-\frac{\beta}{2}X(m+1)$$ where, $\alpha$ and $\beta$ are the co-efficients of Hamming window or Hann window.
Further $$X_{\mathrm{three-term}}(0)=\alpha X(0)-\frac{\beta}{2}X(N-1)-\frac{\beta}{2}X(1)$$ and $$X_{\mathrm{three-term}}(N-1)=\alpha X(N-1)-\frac{\beta}{2}X(N-2)-\frac{\beta}{2}X(0)$$
So after taking fft(X(m))
, the windowed N-point FFT is simply the above three equations?
All these formulas are from Understanding Digital Signal Processing - Richard G. Lyons and all the credits goes to the author of the book.
Some questions
Is Windowed FFT called smoothing?
Do I need to do anything else apart from
ifft(X_3term(m))
to perform inverse of $X_{\mathrm{three-term}}(m)$?