3/13/2005 Phase Noise.
doc 1/6
Phase Noise
There are also short-term instabilities (e.g., msec to µ sec) in
oscillator frequency!
We can model these as:
vc (t ) = a cos ⎡⎣ω0t + φn (t )⎤⎦
where the relative phase φn (t ) is a random process called
phase noise.
Q: It looks a lot like phase modulation!
A: Essentially, it is.
The random process φn (t ) has a small magnitude, i.e.:
φn (t ) 1
Note since the phase changes as a function of time, the
frequency will as well! Specifically:
d (ω0t + φn (t ) )
ω (t ) =
dt
d φ (t )
= ω0 + n
dt
= ω0 + ωn (t )
Jim Stiles The Univ. of Kansas Dept. of EECS
3/13/2005 Phase Noise.doc 2/6
where:
d φn (t )
ωn (t ) =
dt
As a result, the frequency of the oscillator is also a random
process.
I.E., the oscillator frequency changes
randomly as a function of time!
This random fluctuation spreads the oscillator signal
spectrum.
In other words, instead of the spectrum of a perfect, “pure”
tone:
W
Hz
Pc
f
f0
Jim Stiles The Univ. of Kansas Dept. of EECS
3/13/2005 Phase Noise.doc 3/6
we get a wider, imperfect spectrum:
W
Hz
f0
In this case, we say our oscillator has
spectral impurities!
* Since the phenomenon of phase
noise is a random process, we must
describe the signal spectrum in terms
of its average spectral power density.
* Spectral Power Density is expressed in units of Watts/Hz.
* For white noise, the spectral power density is a constant
with respect to frequency:
Jim Stiles The Univ. of Kansas Dept. of EECS
3/13/2005 Phase Noise.doc 4/6
W
Hz
Average Spectral Power
Density of White Noise
* However, for phase noise, the resulting spectral power
density changes as a function of frequency!
Specifically, the average spectral power density of an
oscillator increases as frequency f nears the nominal signal
(i.e., carrier) frequency f0.
W
Hz
Average SPD
is relatively
large near f0
Average SPD
is small away
from f0
f0
Jim Stiles The Univ. of Kansas Dept. of EECS
3/13/2005 Phase Noise.doc 5/6
Now, although we typically express average spectral power
density in Watts/Hz or dBm/Hz, we generally express the
spectral power density of an oscillator output in dBc !
In other words, we are only concerned about the magnitude of
the phase noise spectral power density in comparison to the
oscillator signal power Pc !
* Note we have a mathematical problem here! Pc is in Watts,
and SPD is in Watts/Hz. Therefore, the ratio of the two is
not unitless!
* We get around this problem by specifying the noise as its
power in a 1 Hz bandwidth.
Æ Numerically, this is identical to the average spectral power
density of the noise!
For example, if the noise power has an average spectral power
density 2.0 µW Hz , then the noise power in a bandwidth of
1Hz is:
µW
2.0 (1 Hz ) = 2.0 µW
Hz
Thus, phase noise is expressed as a rather cumbersome:
dBc in a 1 Hz bandwidth
Jim Stiles The Univ. of Kansas Dept. of EECS
3/13/2005 Phase Noise.doc 6/6
Q: But phase noise is a function of frequency f . Do we have
to explicitly specify this function?
A: Generally speaking no. Phase noise is generally specified
by stating the value of the noise power at one or two specific
frequencies, with respect to the carrier frequency f0.
Typically, the frequencies where the phase noise is specified
ranges from 1 KHz to 100 KHz from the carrier.
For example, a typical oscillator spec might say:
-90 dBc in a 1 Hz bandwith at 1 KHz from the carrier, and
-120 dBc in a 1 Hz bandwith at 10 KHz from the carrier.
Pc
90 dB
120 dB
f0
f
f0 − 10 KHz f0 + 1 KHz
Make sure that you know how to proper specify the phase
noise of an oscillator. It is often incorrectly done, and the
source of many lost points on an exam or project!
Jim Stiles The Univ. of Kansas Dept. of EECS
