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Back to Phase Noise
Overview
Many descriptions of phase noise may be
somewhat confusing to those on first encounter of the concept. Terms such as
Power Spectral Density (PSD) are used despite the notion that “phase” itself is
a unit-less term. Phase is expressed in radians or degrees, which has no
inherent “power” concept associated with it, so why refer it phase noise as a
power?
Introduction
Phase
noise is a characteristic term that expresses phase disturbances to a desired
signal. Associated with phase noise is time jitter of a signal, or
simply jitter. Although variations in phase or time of a signal are equivalent,
depending on the system, one is usually a more appropriate parameter to
describe the resulting errors that are produced in the system. The consequences
of phase noise are that additional frequencies are present in the spectrum of the power signal.
These extraneous frequencies signals, for example, could cause the velocity of
a Doppler radar system to be calculated incorrectly, or result in a radio
transmission channel interfering with an adjacent channel. The consequences of
jitter are that signal digital data could be “clocked” at an incorrect time,
resulting in false data bits.
To
illustrate the basic concept of phase noise, consider a single frequency signal
that is expressed as:
V
x
(t)=
V
0
sin(
ω
0
t+
ϕ
pm
(t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiabgUcaRiabew9aMnaaBaaaleaacaWGWbGaamyBaaqabaGccaGGOaGaamiDaiaacMcacaGGPaaaaa@4BE7@
- 1
Where
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
is a term that expresses that fact that there
are disturbances to the phase and timing of a signal at a constant angular
frequency
ω
0
t
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaaaa@396C@
. The effect of
this phase term for a pure sinusoidal noise source is shown as follows.
Let:
ϕ
pm
(t)=ksin(
ω
pm
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadUgaciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaadchacaWGTbaabeaakiaadshacaGGPaaaaa@46F7@
- 2
And substitute 2 into 1
V
x
(t)=
V
0
sin(
ω
0
t+ksin(
ω
pm
t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiabgUcaRiaadUgaciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaadchacaWGTbaabeaakiaadshacaGGPaGaaiykaaaa@4FB4@
- 3
This
expression can be expanded by standard trigonometry relations and the result
for
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
<< 1 is:
V
x
(t)=
V
0
[sin(
ω
0
t))+kcos(
ω
0
t)sin(
ω
pm
t)]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGccaGGBbGaci4CaiaacMgacaGGUbGaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshacaGGPaGaaiykaiabgUcaRiaadUgaciGGJbGaai4BaiaacohacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcaciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaadchacaWGTbaabeaakiaadshacaGGPaGaaiyxaaaa@5A03@
- 4
Or,
expanding again:
V
x
(t)=
V
0
[sin(
ω
0
t))+
k
2
(sin(
ω
0
t+
ω
pm
t)−sin(
ω
0
t−
ω
pm
t))]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@6782@
- 5
This
expression shows that sinusoidal phase noise voltages produce additional frequencies
about the signal frequency. These additional frequencies are called sideband
frequencies.
Note that the multiplication term of the noise signal with the signal
is in quadrature of the signal frequency. That is, the noise
signal is multiplied by
cos(
ω
0
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbGaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshacaGGPaaaaa@3D98@
The
importance of this, is that if it is assumed that the frequency of the signal
is related to its zero crossings, it can be determined from equation 4, that the
zero crossings of
V
x
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaaaa@3A16@
will be changed by
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
.This means that
if
V
x
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaaaa@3A16@
is passed through a zero crossing detector
and/or limiter in order to determine the fundamental signal frequency of the
signal, this frequency will be effected by
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
. That it, it is
a frequency error that can not be easily eliminated.
From
expression 5, it can also be determined that the ratio of the amplitude of the
sideband frequencies to the signal frequency amplitude is
R=
k
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2da9maalaaabaGaam4Aaaqaaiaaikdaaaaaaa@394F@
- 6
And
that, due to the sinusoidal nature of the assumed noise, the peak phase error
is
ϕ
pmpeak
=
k
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbGaamiCaiaadwgacaWGHbGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaadUgaaeaacaaIYaaaaaaa@4012@
- 7
Or
R=
V
0
k/2
V
0
=
ϕ
pmpeak
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2da9maalaaabaGaamOvamaaBaaaleaacaaIWaaabeaakiaadUgacaGGVaGaaGOmaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaaaOGaeyypa0Jaeqy1dy2aaSbaaSqaaiaadchacaWGTbGaamiCaiaadwgacaWGHbGaam4Aaaqabaaaaa@462E@
- 8
Where
R
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@368D@
is a characterisation number expressing the
ratio of the signal voltage (power) to sideband voltage (power). Therefore, phase noise, in radians, can be equated to the ratio of
signal power to noise power. Often, the phase noise is referred to
as a Power Spectral Density (PSD), but strictly speaking, phase is not a power,
but its relative frequency spectrum is still given by measurements of sideband
power to signal power. That is, the effect of a phase noise disturbance is that
additional frequencies are generated.
Calculating
phase noise near the signal frequency
Consider
equation 1
V
x
(t)=
V
0
sin(
ω
0
t+
ϕ
pm
(t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiabgUcaRiabew9aMnaaBaaaleaacaWGWbGaamyBaaqabaGccaGGOaGaamiDaiaacMcacaGGPaaaaa@4BE7@
- 1
Now
suppose that a 90 degrees phase shifted signal, with no noise is available:
V
r0
(t)=
V
r0
cos(
ω
0
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGYbGaaGimaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcaWGwbWaaSbaaSqaaiaadkhacaaIWaaabeaakiGacogacaGGVbGaai4CaiaacIcacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaaiykaaaa@4674@
- 9
Then
a new signal can be constructed by multiplication:
V
xr
(t)=α
V
0
sin(
ω
0
t+
ϕ
pm
(t))
V
r
cos(
ω
0
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4bGaamOCaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcqaHXoqycaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaci4CaiaacMgacaGGUbGaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshacqGHRaWkcqaHvpGzdaWgaaWcbaGaamiCaiaad2gaaeqaaOGaaiikaiaadshacaGGPaGaaiykaiaadAfadaWgaaWcbaGaamOCaaqabaGcciGGJbGaai4BaiaacohacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcaaaa@5867@
- 10
Which
expands to:
V
xr
(t)=
α
V
0
V
r
2
[sin(2
ω
0
t)cos(
ϕ
pm
(t))+(1+cos(2
ω
0
t))sin(
ϕ
pm
(t))]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@6DF5@
- 11
If
this signal is passed through a low pass filter such that the frequencies of
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
are < 2
ω
0
t
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaaaa@396C@
, and if
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
is << 1, equation 11 is reduced to:
V
xr
(t)=
α
V
0
V
r
2
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4bGaamOCaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpdaWcaaqaaiabeg7aHjaadAfadaWgaaWcbaGaaGimaaqabaGccaWGwbWaaSbaaSqaaiaadkhaaeqaaaGcbaGaaGOmaaaacqaHvpGzdaWgaaWcbaGaamiCaiaad2gaaeqaaOGaaiikaiaadshacaGGPaaaaa@4888@
- 12
Or
Δ
ω
an
=
ω
0
−
ω
an
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeqyYdC3aaSbaaSqaaiaadggacaWGUbaabeaakiabg2da9iabeM8a3naaBaaaleaacaaIWaaabeaakiabgkHiTiabeM8a3naaBaaaleaacaWGHbGaamOBaaqabaaaaa@437A@
- 13
The
summary of this result, is that if a signal is
multiplied by a noise free signal that is 90 degrees shifted, and low passes
filtered what remains, bar a gain constant, is the phase spectrum of the phase noise.
General
Phase Noise Analysis
For
reference, the following expression is usually used as a starting point in analysing
the details of phase noise:
V
x
(t)=
V
0
sin(
w
0
t+
ϕ
pm
(t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaam4DamaaBaaaleaacaaIWaaabeaakiaadshacqGHRaWkcqaHvpGzdaWgaaWcbaGaamiCaiaad2gaaeqaaOGaaiikaiaadshacaGGPaGaaiykaaaa@4B16@
- 14
Where
V
am
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGHbGaamyBaaqabaGccaGGOaGaamiDaiaacMcaaaa@3AF1@
is an amplitude modulation (AM) noise term and
where
ϕ
pm
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaSbaaSqaaiaadchacaWGTbaabeaakiaacIcacaWG0bGaaiykaaaa@3BED@
is a phase modulation (PM) noise term.
Amplitude
Modulation (AM) Noise
From
equation 10, amplitude noise may be evaluated. This is useful as it more easily
clarifies the difference between AM noise and PM noise. Ignoring the PM terms
of the equation results in:
V
x
(t)=
V
0
(1+
V
am
(t))sin(
ω
0
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGccaGGOaGaaGymaiabgUcaRiaadAfadaWgaaWcbaGaamyyaiaad2gaaeqaaOGaaiikaiaadshacaGGPaGaaiykaiGacohacaGGPbGaaiOBaiaacIcacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaaiykaaaa@4CFF@
- 15
Suppose
that:
V
am
(t)=k
V
amp
sin(
ω
am
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGHbGaamyBaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcaWGRbGaamOvamaaBaaaleaacaWGHbGaamyBaiaadchaaeqaaOGaci4CaiaacMgacaGGUbGaaiikaiabeM8a3naaBaaaleaacaWGHbGaamyBaaqabaGccaWG0bGaaiykaaaa@49CA@
- 16
Then:
V
x
(t)=
V
0
(1+k
V
amp
sin(
ω
am
t))sin(
ω
0
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGccaGGOaGaaGymaiabgUcaRiaadUgacaWGwbWaaSbaaSqaaiaadggacaWGTbGaamiCaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaadggacaWGTbaabeaakiaadshacaGGPaGaaiykaiGacohacaGGPbGaaiOBaiaacIcacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaaiykaaaa@5597@
- 17
Expanding
this equation gives:
V
x
(t)=
V
0
(sin(
ω
0
t)+k
V
amp
sin(
ω
am
t)sin(
ω
0
t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGccaGGOaGaci4CaiaacMgacaGGUbGaaiikaiabeM8a3naaBaaaleaacaaIWaaabeaakiaadshacaGGPaGaey4kaSIaam4AaiaadAfadaWgaaWcbaGaamyyaiaad2gacaWGWbaabeaakiGacohacaGGPbGaaiOBaiaacIcacqaHjpWDdaWgaaWcbaGaamyyaiaad2gaaeqaaOGaamiDaiaacMcaciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcacaGGPaaaaa@5CC3@
- 18
V
x
(t)=
V
0
(sin(
ω
0
t)+k
V
amp
cos(
ω
0
t−
ω
am
t)−cos(
ω
0
t+
ω
am
t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@67FF@
- 19
Again,
also showing the presence of sideband frequencies. However, in contrast to the
phase modulation case, examination of equation 13 shows that
V
am
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGHbGaamyBaaqabaGccaGGOaGaamiDaiaacMcaaaa@3AF1@
does not change the zero crossings of
V
x
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaaaa@3A16@
generated by the signal. This is because for
any product function,
y(x)=u(x)v(x)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacIcacaWG4bGaaiykaiabg2da9iaadwhacaGGOaGaamiEaiaacMcacaWG2bGaaiikaiaadIhacaGGPaaaaa@40B1@
-20
If
u(x)=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacIcacaWG4bGaaiykaiabg2da9iaaicdaaaa@3AC6@
, then
y(x)=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacIcacaWG4bGaaiykaiabg2da9iaaicdaaaa@3ACA@
irrespective of
v(x)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacIcacaWG4bGaaiykaaaa@3907@
- 21
This
means that, for example, if
V
x
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaaaa@3A16@
is passed through a zero crossing detector
and/or limiter in order to determine the fundamental signal frequency, this
frequency will be unaffected by
V
am
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWGHbGaamyBaaqabaGccaGGOaGaamiDaiaacMcaaaa@3AF1@
.
Additive
Noise (AN) Noise
Consider the case of adding
noise to a single frequency signal, which can be referred to as additive noise,
or AN:
V
x
(t)=
V
0
sin(
ω
0
t)+
V
ap
(t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcacqGHRaWkcaWGwbWaaSbaaSqaaiaadggacaWGWbaabeaakiaacIcacaWG0bGaaiykaaaa@4AEE@
- 22
For a sinusoidal noise
source:
V
x
(t)=
V
0
sin(
ω
0
t)+
V
ap
sin(
ω
an
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcacqGHRaWkcaWGwbWaaSbaaSqaaiaadggacaWGWbaabeaakiGacohacaGGPbGaaiOBaiaacIcacqaHjpWDdaWgaaWcbaGaamyyaiaad6gaaeqaaOGaamiDaiaacMcaaaa@51A2@
- 23
This equation can be
restated, with reference to frequency offsets from the main signal, as:
V
x
(t)=
V
0
sin(
ω
0
t)+
V
ap
sin(
ω
0
t+Δ
ω
an
t)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBaaaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9iaadAfadaWgaaWcbaGaaGimaaqabaGcciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaamiDaiaacMcacqGHRaWkcaWGwbWaaSbaaSqaaiaadggacaWGWbaabeaakiGacohacaGGPbGaaiOBaiaacIcacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaWG0bGaey4kaSIaeuiLdqKaeqyYdC3aaSbaaSqaaiaadggacaWGUbaabeaakiaadshacaGGPaaaaa@57A0@
- 24
where
Δ
ω
an
=
ω
0
−
ω
an
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeqyYdC3aaSbaaSqaaiaadggacaWGUbaabeaakiabg2da9iabeM8a3naaBaaaleaacaaIWaaabeaakiabgkHiTiabeM8a3naaBaaaleaacaWGHbGaamOBaaqabaaaaa@437A@
Expanding
20 gives:
V
x
(t)=
V
0
sin(
ω
0
t)(1+
V
ap
2
V
0
cos(Δ
ω
an
t))+
V
0
(sin(
ω
0
t)+
V
ap
2
V
0
cos(
ω
0
t)sin(Δ
ω
an
t))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=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@7C4E@
- 25
Which
on comparison to the results for AM and PM, it can be seen that AN can be reformulated
as equal parts of equivalent AM and PM noise.
Clippers,
Clampers, Comparators and Limiters
A
distinction will now be made being clampers, clippers, simple limiters, and
comparators.
Typical
the term “limiter” may refer to simple clipping/clamping of a signal or to high
gain amplifiers/comparators which limit their output when input signals reach
certain levels. To avoid confusion, clippers and clampers will be regarded as
simple lopping of the tops and bottoms of the amplitudes of a signal, and
limiters and comparators will be taken to be relatively high gain amplifiers
that, essentially, form zero crossing detectors.
Clipper/Clampers
Consider
passing a signal through a memory less clipper, where the clipping levels are
sufficiently distant from the zero crossings of the signal. In this instance,
the zero crossings of the signal are unaffected and hence, the output phase
noise of the clipper system will not be changed. This is the often given
example of eliminating AM noise in a PN sensitive system by “limiting” the
amplitude variations.
Limiters/Comparators
Limiters/Comparators
are a bit more difficult to analyse. One standard approach is to model the gm
of the amplifier as a square wave and multiply the signal by the Fourier
expansion of the square wave. However, the simplest method is to simply use a
simulation program.
Cadence
Spectre RF - Cadence Spectre RF is a software program that solves the
complicated non-linear differential equations that occur when analysing phase
noise. It is known to reliably calculate phase noise over a wide range of
conditions.
Conventional
wisdom e.g. C.Samori et al “spectrum folding and phase noise in LC tuned
oscilators”, asserts that the signal to phase noise at the output to an ideal
comparator is the same as its input signal to noise ratio. However, this is only true when the input and output are band limited. In reality, it is often extremely difficult to
filter the signals such that that condition can be fulfilled, and for the
internal noise of a transistor or resistor, it is simply impossible to
filter.
Real High Gain Limiters/Comparators Are Intrinsically Noisy
Spectre
RF simulations show that if a source has wideband noise, for example a simple
sum of a signal and resister thermal noise, without input band limiting close to
the signal frequency, the output phase noise around the carrier, will intrinsically
be, some
12dB/
Hz
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaikdacaWGKbGaamOqaiaac+cadaGcaaqaaiaadIeacaWG6baaleqaaaaa@3B77@
worse. This is due to folding of harmonically
mixed noise. This will always be so for internal noise of the input active devices
of the comparator as they cannot be band limited. Internal device noise is thus
effectively amplified by 12db. The practical result is that a high gain limiter amplifier in a well designed
oscillator system used to “square up” a low noise oscillator is the single most dominate source of noise degradation to
the oscillator signal. That is, in a well designed oscillator
system, if the comparator is not the dominant noise source, there is probably
something wrong in the design.
© Kevin
Aylward 2013
All
rights reserved
Website
last modified 28th June 2013
www.kevinaylward.co.uk