Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

One Sine and One Cosine
(``Phase Quadrature'' Case)
All Four Resolutions Overlaid

\begin{psfrags}\psfrag{Frequency wT (rad/sample)}{
Frequency $\omega T$\ (rad/sample)}\psfrag{w}{\footnotesize $\omega$}
\epsfig{file=eps/resolvedSinesC2C.eps,width=6in} \\

The preceding figures suggest that, for a rectangular window of length $ M$ , two sinusoids can be most reliably resolved when they are separated in frequency by a full main-lobe width:

$\displaystyle \zbox{\Delta\omega\geq 2\Omega_M} \qquad \left(\Omega_M \isdef \frac{2\pi}{M}\right)

This implies there must be at least two full cycles of the difference-frequency under the window. (We'll see later that this is an overly conservative requirement--a more careful study reveals that $ 1.44$ cycles is sufficient for the rectangular window.)

Resolving two closely spaced sinusoids is equivalent to AM demodulation:

$\displaystyle \cos(\omega_c t + \omega_d t) + \cos(\omega_c t - \omega_d t) = 2\cos(\omega_c t) \cos( \omega_d t )

where $ 2\omega_d$ is the difference frequency in rad/s.

In principle, arbitrarily small frequency separations can be resolved if

However, in practice, there is almost always some noise and/or interference, so we prefer to require sinusoidal frequency separation by at least one main-lobe width (of the sinc-function in this case, or the window transform more generally) whenever possible.

The rectangular window provides an abrupt transition at its edge. We will soon look at some other windows which have a more gradual transition. This is usually done to reduce the height of the side lobes.

Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download Intro421.pdf
Download Intro421_2up.pdf
Download Intro421_4up.pdf
[Comment on this page via email]

``Music 421 Overview'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2018-04-12 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]