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### Sidelobe Roll-Off Rate

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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