Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       FBS Window Constraints for R=1

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

FBS Window Constraints for R=1

Recall that in overlap-add (Chapter 8), perfect reconstruction required only that the analysis window $ w$ meet a constant overlap-add (COLA) constraint:

$\displaystyle \sum_{m=-\infty}^\infty w(n-mR) = c $

where $ c\neq 0$ is any constant (always true for $ R=1$).10.3

The Filter Bank Summation (FBS) is interpreted as a demodulation (frequency shift by $ -\omega_k$) and subsequent lowpass filtering by $ w$. Therefore, to resynthesize our original signal, we need to remodulate each baseband signal and sum up the channels. For $ R=1$ (no downsampling), this sum is given by [10]

$\displaystyle \hat{x}(m)$ $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} X_m(\omega_k)e^{j\omega_km}$  
  $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} \left[ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right] e^{j\omega_km}$  
  $\displaystyle =$ $\displaystyle \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} e^{j\omega_k(m-n)}$  
  $\displaystyle =$ $\displaystyle \sum_{n=-\infty}^\infty x(n)w(\underbrace{n-m}_{-rN}) N\sum_{r=-\infty}^\infty \delta(m-n-rN)$  
  $\displaystyle =$ $\displaystyle N\sum_{r=-\infty}^\infty {\tilde w}(rN)x(m-rN)$  
  $\displaystyle =$ $\displaystyle Nw(0) x(m) \qquad \hbox{if $w(rN)=0,\,\forall r\neq 0$} \protect$ (10.5)

We have thus derived the following sufficient condition for perfect reconstruction [196]:
$\textstyle \parbox{0.8\textwidth}{Perfect reconstruction is assured in the slid... ...bank sum provided the analysis window is zero at all nonzero multiples of $N$.}$
Since normally our windows are shorter than $ N$, this condition holds trivially. In the overlap-add context, we also had guaranteed perfect reconstruction in this case ($ R=1$), because every window $ w$ overlap-adds to a constant at displacement $ R=1$.

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Inverse DFT and the DFT Filter Bank Sum
Next: Nyquist(N) Windows
written by 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

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )