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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
Proof: Look up ``roll-off rate'' in text index.
- Thus, we have the following rule-of-thumb:
(since
).
- 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.
Examples:
- Amplitude discontinuity
dB/octave roll-off
- Slope discontinuity
dB/octave roll-off
- Curvature discontinuity
dB/octave roll-off
For discrete-time windows, the roll-off rate slows down at high
frequencies due to aliasing.
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
)
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Download Intro421.pdf
Download Intro421_2up.pdf
Download Intro421_4up.pdf
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