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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 $ B_w$ denote this width in Hz. In normalized radian frequency units, as used in the frequency axis of Fig.5.10, $ B_w$ Hz translates to $ 2\pi B_w / f_s$ radians per sample, where $ f_s$ denotes the sampling rate in Hz.

Figure 5.10: Window transform with main-lobe width marked.

For the length-$ M$ unit-amplitude rectangular window defined in §3.1, the DTFT is given analytically by

$\displaystyle W_R(\omega) = M\cdot\hbox{asinc}_M(\omega) \isdef \frac{ \sin \left( M\omega T/2 \right)}{\sin(\omega T/2)} = \frac{ \sin \left( M\pi f T \right)}{\sin(\pi f T)}, \protect$ (6.23)

where $ f$ is frequency in Hz, and $ T$ is the sampling interval in seconds ($ T=1/f_s$ ). The main lobe of the rectangular-window transform is thus ``two side lobes wide,'' or

$\displaystyle \zbox {B_w = 2\frac{f_s}{M}}\quad \hbox{(\hbox{Hz})}$ (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 ($ f_s/M$ Hz) is given in radians per sample by

$\displaystyle \zbox {\Omega_M \isdef \frac{2\pi}{M}.}$ (6.25)

As Fig.5.10 illustrates, the rectangular-window transform main-lobe width is $ 2\Omega_M$ 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 $ K\Omega_M$ (rad/sample)
Rectangular $ 2\Omega_M$
Hamming $ 4\Omega_M$
Hann $ 4\Omega_M$
Generalized Hamming $ 4\Omega_M$
Blackman $ 6\Omega_M$
$ L$ -term Blackman-Harris $ 2L\Omega_M$
Kaiser depends on $ \beta $
Chebyshev depends on ripple spec

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University