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A Simplified Method for
Phase Noise Calculation
Massoud Tohidian,
Ali Fotowat Ahmady*
and Mahmoud Kamarei
University of Tehran, *Sharif University of Technology, Tehran, Iran
Poster: T-18
Outline
• Introduction
• Preliminary Assumptions
– Noise Approximation
– Noise Frequency Translation
– PM and AM Noise Separation
– Output Phase Noise Spectra
• Phase Noise Calculation Method
– The Proposed Method
– Comparison with Hajimiri’s ISF Method
• Experimental Results
• Conclusion
2
Introduction
• Leeson’s Equation [1]
– A heuristic and historical model for phase noise of LC oscillators
• Hajimiri’s ISF Method [2]
– General
– Accurate
– Impulse Sensitivity Function (ISF)
• Transient-time simulation
m
mQV
kTRFL
2
4 0
2
3
Noise Approximation
• A noise signal is approximated by a series of several
sinusoidal signals [3].
4
No
ise
So
urc
e
Sp
ectr
a
ω
. . .
ω
No
ise
Ap
pro
xim
atio
n
. . .
Δω
Random Phase
Equal Power
Noise Frequency Translation
tωωAAα
tωωAAα
tωAαY
nn
nn
nnn
)2cos(3
)cos(2
cos
0
2
03
002
1
3
3
2
21 XαXαXαY
)cos(
)cos( 00
tA
tAX
nn
• Consider a general nonlinear system
– Input: carrier + noise
– Output: translated noise (neglecting the carrier and small terms)
5
“Prominent” Frequencies
• Only “prominent” frequencies translate to around the carrier.
ωn0 ωn1+ωn1- ωn2+ωn2- ωni+ωni-
Carrier
@ ω0
Ou
tpu
t
Sp
ectr
a
No
ise
So
urc
e
Sp
ectr
a
ω
ω
. . .
ωn0 + ω0
2ω0 - ωn1-
ωn1+
3ω0 - ωn2-
ωn2+ - ω0
6
PM and AM Noise Separation
• A single tone noise at an prominent frequency generates
two noise tones around the carrier.
• These two carriers generally modulate both Phase (PM)
and Amplitude (AM) of the carrier.
+ωs
-ωs
V0
AM
PM
ωω0 ω0+ωs
nnA
nnA
V0
ω0-ωs
Output Spectra Phasor Representation
7
PM and AM Noise Separation
• The output noise can be separated into PM and AM noise.
nnnnnnAM
nnnnnnPM
AAAAA
AAAAA
cos22
1
cos22
1
22
22
-ωs+ωs
V0PM
-ωs
+ωs
V0
AM
Pure PM Noise
Pure AM Noise
8
Output Phase Noise Spectra
• Leeson’s equation:
– Phase noise skirt around the carrier signal falls with 20dB/dec for
a white noise source [1].
• General phase noise equation:
22
22
2log10log10sm
nn
sig
noisem
V
iZ
P
PL
carrier
amplitude
total noise
translation gain noise current
frequency at
which Zn is
simulated
phase noise
offset frequency
9
Noise Translation Gain
2
3
2
2
2
1
2
0
2
3
2
3
2
2
2
2
2
1
2
1
2
0
2
222 nnnn
nnnnnnnn
zzzz
zzzzzzzZ
• The translation gains are calculable using simulation
sniniiniPMni
sniniiniPMni
iiVz
iiVz
0,,
0,,
,/
,/
10
ωn0 ωn1+ωn1- ωn2+ωn2- ωni+ωni-
Carrier
@ ω0
Noise @
ω0+ωs
Noise @
ω0-ωs
Ou
tpu
t
Sp
ectr
a
No
ise
So
urc
e
Sp
ectr
a
ω
ω
zn0zn1- zn1+ zn2-
zn2+zni- zni+
. . .
Noise Sources
• Stationary Noise Sources
– Resistors, constant current source, etc.
– Noise source is modeled with a single
tone (ST) source.
• Cyclostationary Noise Sources
– Modulated sources, switching
transistors, etc.
– Noise source is modeled with a single
tone source with amplitude modulation
in @
ωni
iout=f(id,in)
in @
ωni
id
11
Proposed ST Simulation Method
• Calculate the noise translation gain
– Applying the ST noise source at prominent frequencies, simulating
the circuit and measuring the output spectrum around the carrier.
– PM component:
– Partial translation gain:
– Total noise translation gain of the noise source:
• The total output phase noise contribution of the noise
source
2
3
2
2
2
1
2
0
2
3
2
3
2
2
2
2
2
1
2
1
2
0
2
222 nnnn
nnnnnnnn
zzzz
zzzzzzzZ
nnnnnnPM AAAAA cos22
1 22
22
22
2log10sm
nnm
V
iZL
niiniPMni iVz ,, /
12
Comparison with ISF Method
• Linear Time-Variant Model [2]
• It can be shown that the noise translation gains are
convertible to ISF and vice versa.
max
max
00
2
2
q
cVz
q
cVz
ini
n
ISF Fourier series
coefficients
22
max
22
0
1
00
max
0
log10
cos2
,
m
inm
i
i
i
q
ciL
icc
tuq
th
ISF
max. charge of
the node
13
)(ti
0 t
),( th)(t)(ti
)(t
0 t
ST vs. ISF
• ST Method
– Directly calculates noise contribution in frequency domain.
– The dominant noise sources and noise frequencies are found.
– Separately calculates the Phase and Amplitude Noise.
• Simulation Speed
– Accurate ISF requires several transient time simulations (time
consuming).
– ST method has improved the simulation speed using Harmonic-
Balanced simulation (HB) for just some few prominent frequencies.
• Flicker noise upconversion can be predicted by just one simulation
(considering zn0).
14
Experimental Results
• Case Study:
– QVCO designed for GPS application [4].
VDD VDD
Msw1 Msw2 Msw3 Msw4Mcp1 Mcp2 Mcp3 Mcp4
MtailMbias
Ibias
C1 C2
L1 L2
R1 R2 C3 C4
L3 L4
R3 R4
Ip In Qp Qn
Qp Qn In Ip
Cbp
15
Experimental Results
Die micrograph fabricated in
TSMC 1P6M+ 0.18-µm CMOSMeasured output phase noise spectra
16
• Simulated phase noise frequency contribution @ 1MHz
– The two methods almost predict the same results
• Time-domain ISF for Mtail
– DC + Fourth Harmonic
Phase Noise Frequency Contribution
0 1 2 3 4-160
-150
-140
-130
-120
-110
Noise Source Frequency (carrier hamonic index)
Outp
ut
Phase N
ois
e (
dB
c/H
z)
Msw1
Mcp1
Mtail
Mbias
R1
0 1 2 3 4-160
-150
-140
-130
-120
-110
Noise Source Frequency (carrier hamonic index)
Outp
ut
Phase N
ois
e (
dB
c/H
z)
Msw1
Mcp1
Mtail
Mbias
R1
ST
meth
od
ISF
meth
od0 pi/2 pi 1.5*pi 2*pi
0
0.2
0.4
0.6
0.8
1
Output Phase (rad)
Norm
aliz
ed I
SF
17
• Noise contributions are in good agreement for all
simulations methods and measurement.
Comparison
Noise Source Noise TypeMethod
Single Tone ISF SpectreRF
Msw1-4
Thermal -135.0 -135.0 -135.6
Flicker -141.8 -143.8 -148.2
Mcp1-4
Thermal -144.9 -145.8 -144.4
Flicker -131.8 -131.8 -134.9
Mtail
Thermal -138.8 -137.9 -139.3
Flicker -138.7 -137.6 -138.1
Mbias
Thermal -115.1 -115.3 -116.0
Flicker -114.2 -113.9 -114.7
R1-4 -136.8 -136.4 -137.2
Other Sources neglected neglected -128.2
Total -111.2 -111.1 -111.8
Measurement Result -109.5
Phase Noise @ 1MHz offset
18
Conclusion
• A faster method using nonlinear frequency domain
simulation
• ST directly shows the noise frequency translation
– Facilitates more intuitive design of high purity oscillators by
considering substantial noise frequencies
• Separate prediction of Phase and Amplitude Noise
• Simulation results show good agreement with other
methods and measurements
19
References
[1] D. B. Leeson, “A simple model of feedback oscillator
noise spectrum,” Proc. IEEE, pp. 329–330, Feb. 1966.
[2] A. Hajimiri and T. H. Lee, “A general theory of phase
noise in electrical oscillators,” IEEE J. Solid-State
Circuits, vol. 33, pp. 179-94, 1998.
[3] W.P. Robins, Phase Noise in Signal Sources, Peter
Peregrinus Ltd., London, 1982, pp. 6-8.
[4] F. Behbahani, et al., "A fully integrated low-IF CMOS
GPS radio with on-chip analog image rejection," IEEE J.
Solid-State Circuits, vol.37, no.12, pp. 1721-1727, Dec
2002.
20