Main-Lobe Bandwidth

We saw in the previous section 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 width very simply (and somewhat crudely) as the distance between the first zero-crossings on either side of the main lobe, as shown in Fig.1.16 for a rectangular-window transform. Let denote this width in Hz. In normalized radian frequency units, as used in the frequency axis of Fig.1.16, Hz translates to radians per sample, where denotes the sampling rate in Hz.

For the length- unit-amplitude rectangular window defined in §1.5, the DTFT is given analytically by

where is frequency in Hz, and is the sampling interval in seconds (). The main lobe of the rectangular-window transform is thus ``two sidelobes wide,'' or

as can be seen in Fig.1.16.

Recall from §1.5.1 that the side-lobe width in a rectangular-window transform ( Hz) is given in radians per sample by

As Fig.1.16 illustrates, the rectangular-window transform main-lobe width is radians per sample (two side-lobe widths). Table 1.1 lists the main-lobe widths for a variety of window types (which are defined and discussed further in Chapter 3).

---

Subsections

• Other Definitions of Main Lobe Width

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: One Sine and One Cosine ``Phase Quadrature'' Case
Next: Other Definitions of Main Lobe Width

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.