Frequency Resolution
The
frequency resolution of a
spectrum-analysis window is
determined by its
main-lobe width (Chapter
3) in the
frequency domain,
where a typical main lobe is illustrated in Fig.
5.6
(top). For maximum frequency resolution, we desire the narrowest
possible main-lobe width, which calls for the
rectangular
window (§
3.1), the transform of which is shown in
Fig.
3.3. When we cannot be fooled by the large
side-lobes
of the rectangular window transform (
e.g., when the
sinusoids under
analysis are known to be well separated in frequency), the rectangular
window truly is the optimal window for the estimation of frequency,
amplitude, and phase of a
sinusoid in the presence of stationary
noise
[
230,
120,
121].

The rectangular window has only one parameter (aside from
amplitude)--its
length. The next section looks at the effect
of an increased window length on our ability to
resolve two
sinusoids.
Figure
5.7 shows a
spectrum analysis of two
cosines
 |
(6.17) |
where

and

, and
the frequency separation

is

radians per sample. The zero-padded
Fourier analysis uses
rectangular windows of lengths

,

,

, and

(

, where

).
The length
FFT output is divided by

so that the ideal
height of each spectral peak is

.
Figure:
DTFT of two closely
spaced in-phase sinusoids, various rectangular-window lengths
.
![\includegraphics[width=\twidth]{eps/resolvedSines}](http://www.dsprelated.com/josimages_new/sasp2/img951.png) |
The longest window (

) resolves the
sinusoids very well, while
the shortest case (

) does not resolve them at all (only one
``lump'' appears in the
spectrum analysis). In difference-frequency
cycles, the analysis windows are two cycles and half a cycle in these
cases, respectively. It can be debated whether or not the other two
cases are resolved, and we will return to them shortly.
Figure
5.8 shows a similar
spectrum analysis of two
sinusoids
 |
(6.18) |
using the same frequency separation and window lengths. However, now
the
sinusoids are 90 degrees out of phase (one sine and one cosine).
Curiously, the top-left case (

) now appears to be resolved! However, closer inspection (see
Fig.
5.9) reveals that the ``resolved'' spectral peaks
are significantly far away from the
sinusoidal frequencies. Another
curious observation is that the lower-left case (

) appears worse off than it did in
Fig.
5.7, and worse than the shorter-window analysis at
the top right of
Fig.
5.8. Only the well resolved case at the lower right
(spanning two full cycles of the difference frequency) appears
unaffected by the relative phase of the two sinusoids under analysis.
Figure
5.9 shows the same plots as in
Fig.
5.8, but overlaid. From this we can see that the peak
locations are
biased in under-resolved cases, both in amplitude
and frequency.
Figure:
Overlay of the plots in Fig.5.8.
![\includegraphics[width=\textwidth ]{eps/resolvedSinesC2C}](http://www.dsprelated.com/josimages_new/sasp2/img957.png) |
The preceding figures suggest that, for a rectangular window of length

, two sinusoids are well
resolved when they are separated in
frequency by
 |
(6.19) |
where the frequency-separation

is in radians per sample. In
cycles per sample, the inequality becomes
 |
(6.20) |
where the

denotes normalized frequency in
cycles per sample. In Hz, we have
 |
(6.21) |
or
 |
(6.22) |
Note that

is the number of samples in one
period of a
sinusoid at frequency

Hz, sampled at

Hz. Therefore, we have
derived a rule of thumb for frequency resolution that requires at
least
two full cycles of the difference-frequency under the
rectangular window.
A more detailed study [
1] reveals that

cycles
of the difference-frequency is sufficient to enable fully accurate
peak-frequency measurement under the rectangular window by means of
finding
FFT peaks. In §
5.5.2 below, additional minimum duration
specifications for resolving closely spaced sinusoids are given for
other window types as well.
In principle, we can resolve
arbitrarily small frequency
separations, provided
- there is no noise, and
- we are sure we are looking at the sum of two ideal sinusoids under the window.
One method for doing this is described in §
5.7.2.
However, in practice, there is almost always some
noise and/or
interference from other
signals, so we normally prefer to require
sinusoidal frequency separation by on the order of one
main-lobe
width or more.
The rectangular window provides an abrupt transition at its edge.
While it remains the optimal window for sinusoidal peak estimation, it
is by no means optimal in all
spectrum analysis and/or signal
processing applications involving spectral processing. As discussed in
Chapter
3, windows with a
more gradual transition to zero have lower
side-lobe levels, and this
is beneficial for spectral displays and various signal processing
applications based on FFT methods. We will encounter such applications in
later chapters.
Next Section: Other Definitions of Main Lobe WidthPrevious Section: Nonlinear Optimization in Matlab