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We saw in §5.4.1 that our ability to resolve two closely
spaced sinusoids is determined by the mainlobe width of the
window transform we are using. We will now study this relationship in
more detail.
For starters, let's define mainlobe bandwidth very simply (and
somewhat crudely) as the distance between the first
zerocrossings on either side of the main lobe, as shown in
Fig.5.10 for a rectangularwindow transform. Let
denote this width in Hz. In normalized radian frequency units, as
used in the frequency axis of Fig.5.10,
Hz translates to
radians per sample, where
denotes the sampling rate in Hz.
Figure 5.10:
Window transform with mainlobe width marked.

For the length
unitamplitude rectangular window defined in
§3.1, the DTFT is given analytically by

(6.23) 
where
is frequency in Hz, and
is the sampling interval in
seconds (
). The main lobe of the rectangularwindow
transform is thus ``two side lobes wide,'' or

(6.24) 
as can be seen in Fig.5.10.
Recall from §3.1.1 that the sidelobe width in a
rectangularwindow transform (
Hz) is given in radians
per sample by

(6.25) 
As Fig.5.10 illustrates, the rectangularwindow transform
mainlobe width is
radians per sample (two sidelobe
widths). Table 5.1 lists the mainlobe widths for a
variety of window types (which are defined and discussed further in
Chapter 3).
Table 5.1:
Mainlobe bandwidth for various windows.
Window Type 
MainLobe Width
(rad/sample) 
Rectangular 

Hamming 

Hann 

Generalized Hamming 

Blackman 

term BlackmanHarris 

Kaiser 
depends on

Chebyshev 
depends on ripple spec 

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