Rectangular Window Summary
The rectangular window was discussed in Chapter 5 (§3.1). Here we summarize the results of that discussion.Definition ( odd):
(4.12) |
Transform:
(4.13) |
The DTFT of a rectangular window is shown in Fig.3.7. Properties:
- Zero crossings at integer multiples of
(4.14)
- Main lobe width is .
- As increases, the main lobe narrows (better frequency resolution).
- has no effect on the height of the side lobes (same as the ``Gibbs phenomenon'' for truncated Fourier series expansions).
- First side lobe only 13 dB down from the main-lobe peak.
- Side lobes roll off at approximately 6dB per octave.
- A phase term arises when we shift the window to make it causal, while the window transform is real in the zero-phase case (i.e., centered about time 0).
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Rectangular Window Side-Lobes