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## Resolving Sinusoids

We saw in §5.4.1 that our ability to resolve two closely spaced sinusoids is determined by the main-lobe width of the window transform we are using. We will now study this relationship in more detail.

For starters, let's define main-lobe bandwidth very simply (and somewhat crudely) as the distance between the first zero-crossings on either side of the main lobe, as shown in Fig.5.10 for a rectangular-window 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.

For the length- unit-amplitude 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 rectangular-window 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 side-lobe width in a rectangular-window transform ( Hz) is given in radians per sample by

 (6.25)

As Fig.5.10 illustrates, the rectangular-window transform main-lobe width is radians per sample (two side-lobe widths). Table 5.1 lists the main-lobe widths for a variety of window types (which are defined and discussed further in Chapter 3).

Table 5.1: Main-lobe bandwidth for various windows.
 Window Type Main-Lobe Width (rad/sample) Rectangular Hamming Hann Generalized Hamming Blackman -term Blackman-Harris Kaiser depends on Chebyshev depends on ripple spec

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