In general, if the first
derivatives of a continuous function
exist (*i.e.*, they are finite and uniquely defined), then its
Fourier Transform magnitude is asymptotically proportional to

- Thus, we have the following rule-of-thumb:

- This is also
dB per
*decade*. - To apply this result, we normally only need to look at the window's
*endpoints*. The interior of the window is usually differentiable of all orders.

- Amplitude discontinuity dB/octave roll-off
- Slope discontinuity dB/octave roll-off
- Curvature discontinuity dB/octave roll-off

In summary, the DTFT of the
-sample **rectangular window** is
proportional to the `aliased sinc function':

Some important points **(rect window transform)**:

- Zero crossings at integer multiples of

( freq. sampling interval used by a length DFT) - Main lobe width is
- As
gets bigger, the main-lobe narrows

(better frequency resolution) -
has
*no effect on the height of the side lobes*

(Same as the ``Gibbs phenomenon'' for Fourier series) - First side lobe only 13 dB down from main-lobe peak
- Side lobes roll off at approximately 6dB per octave
- A
*linear phase term*arises when we shift the window to make it*causal*, while the window transform is real in the zero-centered case (i.e., when the window is an*even function*of )

Download Intro421.pdf

Download Intro421_2up.pdf

Download Intro421_4up.pdf

[Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]