Resolution Bandwidth (Resolving Sinusoids)

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Frequency Resolution

Minimizing Side-Lobe Level

Choosing Window Length to Resolve Sinusoids

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

A side lobe in the Fourier transform of a window function, such as the Chebyshev window, is any local maximum in the transform magnitude outside of the main lobe (response about dc). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

The main lobe in the Fourier transform of a window function, such as the Chebyshev window, is the central local maximum in the transform magnitude (response about dc). Outside of the main lobe are usually side lobes (ripples in the magnitude spectrum). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Search JOS Website, CCRMA Website, or Web

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Music 421 Overview

Frequency Resolution

Minimizing Side-Lobe Level

Choosing Window Length to Resolve Sinusoids

Resolution Bandwidth (Resolving Sinusoids)

Our ability to resolve two closely spaced sinusoids is determined by the main-lobe-width and sidelobe-level of our window's Fourier transform.

Let denote the main lobe width in Hz, with the main lobe width defined as the width between zero crossings:

$\epsfig{file=eps/WR2.eps,width=5in,height=3.5in}$

For the Rectangular Window (length ), we have

$\displaystyle W_R(\omega) = \hbox{asinc}_M(\omega) \mathrel{\stackrel{\Delta}{=}}\frac{ \sin \left( M\omega T/2 \right)}{\sin(\omega T/2)} = \frac{ \sin \left( M\pi f T \right)}{\sin(\pi f T)}$

Main lobe width is ``two sidelobes wide''

$\displaystyle \Rightarrow \quad \zbox{B_w = 2\frac{\Omega_M}{2\pi} = 2\frac{f_s}{M}}\quad \hbox{(\mbox{Hz})}$

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``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
[Automatic-links disclaimer]

Resolution Bandwidth (Resolving Sinusoids)

``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421). Copyright © 2020-06-27 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University [Automatic-links disclaimer]

``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
[Automatic-links disclaimer]