Adv. Radio Sci., 5, 313–320, 2007
www.adv-radio-sci.net/5/313/2007/
© Author(s) 2007. This work is licensed
under a Creative Commons License.
Advances in
Radio Science
Phase noise and jitter modeling for fractional-N PLLs
S. A. Osmany, F. Herzel, K. Schmalz, and W. Winkler
IHP Im Technologiepark 25, 15236 Frankfurt (Oder), Germany
Abstract. We present an analytical phase noise model for
fractional-N phase-locked loops (PLL) with emphasis on integrated RF synthesizers in the GHz range. The noise of the
crystal reference, the voltage-controlled oscillator (VCO),
the loop filter, the charge pump, and the sigma-delta modulator (SDM) is filtered by the PLL operation. We express
the rms phase error (jitter) in terms of phase noise of the reference, the VCO phase noise and the third-order loop filter
parameters. In addition, we consider OFDM systems, where
the PLL phase noise is reduced by digital signal processing
after down-conversion of the RF signal to baseband. The rms
phase error is discussed as a function of the loop parameters.
Our model drastically simplifies the noise optimization of the
PLL loop dynamics.
1
Introduction
Frequency synthesizers are essential components in wireless RF transceivers. A synthesizer provides a stable signal
for frequency conversion, modulation and demodulation. A
fractional-N PLL gives a greater flexibility than an integerN PLL due to the higher frequency resolution (Perrot et
al., 2002). The phase noise is the most important performance criterion of the synthesizer for its stringent requirement in many applications, e.g., in OFDM based systems
(Stott, 1998; Herzel et al., 2005). The problem with phase
noise is exacerbated in high-frequency applications, since the
oscillator phase noise at a given frequency offset typically
increases in proportion to the carrier frequency. For this reason, the minimization of the phase noise in frequency synthesizers is of special interest in mm-wave application, e.g.,
in the 60 GHz band (Winkler et al., 2005). Many publications on phase noise modeling of oscillators and PLLs have
Correspondence to: S. A. Osmany
(osmany@ihp-microelectronics.com)
appeared during the last few years, e.g. Lee and Hajimiri
(2000)–Rohde (2005). This paper is focussed on problems
which are specific to integrated PLLs. They arise mainly
from the low quality factor of the available passives as well
as from the limited size of filter capacitors. Especially, the
low quality factor of integrated inductances causes a high
noise level of the voltage-controlled oscillator (VCO). This
problem is exacerbated by the fact that variations of the device parameters with process and temperature require a relatively large VCO tuning range affecting oscillator noise and
loop filter noise. In integrated PLLs, the values of the integrated capacitances are limited by chip area and, possibly,
yield. This can make the filter noise comparable to the VCO
noise, if the VCO gain is large. Another problem arises in
RF synthesizers for mm-wave applications. The high oscillation frequency results in a large division factor enhancing
the reference noise to a level comparable to the phase noise
of integrated VCOs. The different dependencies of the noise
contribution on the PLL dynamics results in a subtle trade-off
for the loop bandwidth.
This paper describes the phase noise spectrum of a PLL,
where emphasis is placed on the high-pass filtering in OFDM
systems due to the correction of the common phase error.
2
PLL phase noise sources
In the literature, two types of phase noise are used, the twosided power spectral density (PSD) of the phase and the
single-sideband phase noise. For moderate and large frequency offsets, these quantities are almost equal (Herzel,
2004). For wideband systems, close-in phase noise is less
relevant due to a large channel width and/or proper baseband
filtering explained below. Therefore, we will not distinguish
between the single-sideband phase noise and the two-sided
power spectral density of the phase in the remainder of this
paper. The phase noise S can be determined from the single-
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
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–
loop filter noise,
4J8 K
èéˆ
– charge pump noise,
,
è
)•iè
‹
‹"œ0ê
˜
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è
– 6 − 1 quantization noise.
è
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These are typically the dominant
contributions in integratedCfractional-N
RF
synthesizers
for
the GHz range
S
"
(Uang, 2004;
Valero-Lopez,
2004).
Digital
F
Mef3
noise is disre„S
garded in this paper.
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iAdv.
%Radio
Sci., 5, 313–320, 2007
Ä' 8} *,
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2π s
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For the filter transimpedance
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we assume that
ê
î
no current flows into VCO control input, since the filter caH
,
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pacitances are notably large. For the second-order filter according to Fig. 3, we obtain the transimpedance
è ‹ dZš g
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dA)I1g
É !#"%$& ˆ É ë ê
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1
1
F (s) = R1 +
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.
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j
sC
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"
sC2 0
ë
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d 1g
þrÿ
:|
HjAj
?
@
,m
In
spurs,
order tosuppress
an
additional low-pass
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2filter is often :
included
as
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Fig.
4, resulting
in a third%
2
"
order loop filter. The current through Z2 causes a voltage
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Fig. 6. Reference noise at the PLL input and output. In addition,
1
the solid line shows the total output noise spectrum.
.
Z2 =
R
(8)
3 +
è ‹œ0ê
Œ
sC3
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The crossover frequency of the open loop transfer function
¿
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I
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c
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A
0
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I
m
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J
q
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A
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spectrum
at
the
input
according
è
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d
š
g
m
e
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REF
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R
S
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–01#five
2 sections,
)•Uè ‹ the
œ0ê noise sources mentioned
In the following
with the low-pass filter function
:|
HjAj 2 will?be analytically described
@ framework of: ,m
in Sect.
in the
RG
j ghKi k<lnm
0
1
H0 /N
the linear
PLL
model.
The
transfer functions
of the noise
2
r
HLPF =
.ud
8
@Rg"
sources to the PLL output will be calculated.
o?p q (14)
1 + H0 /N
:%
2
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_
b
`
a
f
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Note that the reference noise is low-pass filtered and ampliƒ 01#2 ˆ›ƒ 01N2 ê
è-O1Ç 2
dA)=1g
divider
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"Rg@the
rf phase noise of the
fied by the
ï
ï
4 Reference
noise
'
c
<
d
e
h
S
reference is c
typically
lower
than
that
of
the VCO
d
Rg
D by many
F
o)F
orders
of
magnitude,
it
may
become
comparable
to
the
VCO
A low-noise reference is crucial
8
è ‹œ0ê for a Œ good PLL performance.
noise, if a large
the
è O?contribution
dA)"as
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Ç 2»ˆ
Ÿ <division
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The phase noise
of
the
reference
oscillator
)•Uè ‹ œ0ê
Winkler
et
al.
(2005)
corresponds
to
an
value
of
N=1024
in
function of the frequency offset f is modeled by
oz 60 dBZx"as
y it appears at
increase
of
the
phase
noise
by
about
h
uS
"
/
¿
2
the PLL output. Figure 6 shows the reference noise for typi(1f
)
S "P S
„ 2
3 F Using the noise
transfer +
SREF,floor ,
(11)
"‘
REF (f ) = SREF (1f )
cal crystal oscillators
dashed).
– Ë>(long
)
f2
Æ2Ì
Œ
„ef3 ˆ the
(dashed) we obtain
phase noise dA)G1g >
corresponding
function
6ef3
– Ë>Æ2Ì
)•Äè ‹ œ0ê
Awhere
) is
the
? phase noise
„ at a specific offset
1f in
SREF (1f
contribution
(dot-dashed).
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with
the
total phase
h
0
:ef3 the –201dB/decade
, region
H of the spectrum and S
" is the
noise
(solid)
shows
that
at
offsets
below
1
kHz
the
reference
REF,floor
f
c
Ä
4
e
f
3
noise dominates.
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r
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5, the transfer
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pers. The optimization
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ctothe
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hde
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noise
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S. A. Osmany et al.: Phase noise and jitter
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output reads
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Frequency Offset [Hz]
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Ç
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where γ =2/3 for long-channel devices. Adding the noise
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r
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is, the ratio
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the time difference between two comparison instants. Note R2
uct
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)=of
6- the charge pump current ICP and the filter
¿
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that the spectralÌ densities SiNMOS , SiPMOS
must
be
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R
.
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d_Ê4ÈHgAÅ
1
ƒ
N The noise
• currents produce
ð ˜
since gm varies
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a noise d_;1I1g the charge pump current noise must be carefully traded off
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voltage over the filter impedance, which is transferred to the
with the noise produced by the filter resistance R1 .
4ˆ&; œ = "
www.adv-radio-sci.net/5/313/2007/ 0
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Note that the SDM quantization noise spectrum is low-pass
FSR
"
R2)=6-
d_
Phase Noise [dBc/Hz]
j
âã‘ã‘ãlä /ì
‘æ Áê åæì Jæì3èSå ÔèSì
„/òøˆö`÷&ú ï‘î`îLï úUó‘ö`/øˆôJüvòjòúUúBý'û øˆólúBõ`ûˆôJûˆô–òJö`ö`÷/üï‘õ‡úBøˆóxù‘÷/ï‘îøˆï ’€ö‡ï ÷/ï'îxúBøˆòJó‘ö‡ö`øˆï üv€òjï‘úUî û /vø úBGûˆø ñ//ï<ï<õ î vúUû…ñ&ó<ûï–úUõø…õ
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ý:ò/ü€úUø…õ‡ò/ï ó‘Qï í&üvîQñ/ö‡ö|÷/ïö‡÷/ï¿õ‡Üø vý:ò/ü€ú ø…õ‡&ï³ï‘úBûˆòjöLúßúUûˆý'ôGõ‡ü øˆõ/€ñ/ÿû^úBï‚ö‡üvýFî‚ü ú &ï‘&û/õ™ö`÷&ðJï ñjúUòJö`øˆùlúBJö‡øˆôÞüvòú
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dithered dynamically to achieve fractional values. A classical
+ö »−80
stt›sžv>wx@yUz
_x"yÄ{/yy
x¬bw ø÷ >y
architecture uses an accumulator to peform the
* fractional-N
C{ ªv¬uwxÕ?Ö8yà wØ× >w
ZÁ
w
›z
Zx"y
out
dithering operation. But in this case, a periodic error signal
out
j
4
h
8S SREF
createsspurious
tones in the synthesizer
A better ap
—
" Coutput.
:
−100 nef3
"
™SPLL CS
"
proach
is to use a sigma-delta
modulatorrather
"M
"
than
u a single
S
F /
_S out "&A
accumulator
to perform the dithering operation. This results
SCP
−120
/
„
h
8S
"
in a muchbetter
spurious
But1the
X
out
:
performance.
‡
sigma-delta
SVCO
L
%
/
out
modulator adds quantization noise. For the PLL noise analy
−140
SFIL
sis,
we
model
the
SDM
by
a
divider
value
and
a
sigma-dela
0&
B—
Sout
SDM
noise spectrum as shown in Fig. 13. The quantization noise
–2"‘"d“’‚gˆiúù{û
’+þ O1Ç 2Od“’+þg–01#2Od“’OºM’+þWg>•
/
H
−160
spectrum for a m-th order MASH type sigma-delta modula‹ýü
#
CM
tor is given by Miller (1991),
−180
1~‡
B—
ù{
û W ’ þ 4ÿ Ç 2 1e+04
d’ W þ g–2Ë>ÆRÌhd’6
º4
1e+02
1e+06
W ’ þ g>• 1e+08
W
2(m−1)
2
‹
ü
1
(2π
)
E
r|587 πf
Frequency Offset [Hz]
SSDM =
(23)
2 sin
,
h
/12fREF
0
fREF
Ÿ/ <¡
¢1ù £®JB¢»
©@°
’+ã
û ä2¦
’+©"þ ´_§°
Ç ­"2 <´_d“’+§6þ“g´Zø?§,O?¥AÇ ªZ2 ¯_·
d¨c
’º4
þWg> <ªZ• ´>¥A­"°ª_¯_ ·
ªZ­"¯_´A¢
PLL phase noise spectrum and jitter
In a PLL the phase noise of the reference is low-pass filtered,
while the VCO noise is high-pass filtered and the filter noise j
is bandpass-filtered. The charge pump noise and sigma delta quantization noise is low-pass filtered in the PLL. This results in the total phase error
Z ∞
φout (t) = N
dt 0 hLPF (t 0 ) φREF (t − t 0 ) +
0
Adv. Radio Sci., 5, 313–320, 2007
d_;1G1g
dZ¼Fš2
d_; 1g
(26)
/
1
"
" , hBPF ,Fh̃
and ĥLPF8are
- the linear
where N hLPF , hHPF
LPF
phase
responses at the
0
" PLL
output referred to the phase
noise
„
source. Note that N hLPF (t 0 ) implies a multiplication by the
the reference noise.
division ratio N, which enhances
) ?I
"‘
Œ
@ noise
"* ˆ sources are uncorrelated,
;%| Ç
the corre- d_; 1g
Since the
' five

‹
sponding noise spectra must be added
to obtain the overall
phasenoise
spectrum
"Ä @
out
out
out
SPLL
(f ) = SREF
(f ) + SVCO
(f ) +
out
out
S outF(f
M
FIL ) + SCP (f ) + SSDM (f ) ,
1~j
(27)
out (f ),
‚ the˜ carrier and SREF
where f is the frequency offset from
ˆ
out
out
out
out
(f
) and S (f ) are the phase noise d_;)1g
SVCO (f ), SLPF (f ), SCP
=1G?I SDM
(
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üûˆüJ€î‚üvþ'ö`÷/þïÁúBîLþ&úU÷jý'úBõ‡ï‘ïÁö‡ï‘ò/î`üvõ øˆõ‡üvï î³ú߂÷/û…üBøˆÿ õ³úBVxûˆøˆûˆö`üBö`ÿ‚ï<î õÝú úUõ`ö}üvþ/ö‡ø…ýFøˆùlúBö‡ø…ü€ò¿ü ö`÷/ï
:
-
‹
Fig. 15. Phase
noise spectrum and its contributors.
ˆ() œ š ‹
0
"
1h-
Ewww.adv-radio-sci.net/5/313/2007/
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MSR
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a
"‘ Osmany et al.:Phase
"‘
"‘ jitter modeling for fractional-N
S.A.
noise and
dZ¼Fš2gå æ PLLs
ˆ&) œ
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ƒ Ç
d_šRg
01N2
ˆ›ƒ
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ƒ
dZš2
•Äƒ five
Zd š2g6•Uƒ
dZš2g>˜
d_; 1g
ÆÇ previously
contributions
ofgthe
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noise sources
"‘
2Z O
)
Æ°í Å
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Phase Noise [dBc/Hz]
X
„
In the high-speed digital design community, the so-called absolute jitter is a common measure to quantify the timing error
)'?I (
Œ
ofa
PLL
by d_;)1g
' @ driven
"* ˆ by a cleanù reference.
û ;%| Ç#"O)‘ ItO is given
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σ
abs
T
="Ä ◦ σφ , 360
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) œ š Modeling
0 of RF
synthesizers for OFDM
E‹
10
"
/
"!
A low pll1h
jitter
is especially important for OFDM systems,
where phase
noise
may
easily
destroy
the
orthogonality
of
:
r †D
u
„3 O587
(30)
0 D
"
U
is called
the useful symbol time. In"state-of-the-art
baseband
W
æ
æ
æ
„
F
¼
0
š
å
6
processor implementations
part of the phase noise (com
) '<=B*#j „3 65r7
Tu = 1/(1fcarrier
@ 4
ž
™
mon phase error) can be eliminated after the signal has been
æ g process can be modelled
‚Cˆ to) baseband.
d_=1I1g
œ dZ¼Fš0å æWæ ¢ This
down-converted
by a high-pass filter according to Stott (1998)
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2í Å
#õ`øˆò/ó /‚
"
18j
319
−100
out
SPLL
out
SVCO
−120
−140
Sout
FIL
out
SREF
−160
out
SSDM
out
SCP
W
W
1e+04
1e+06
W
1e+08
Frequency Offset [Hz]
+Y
xw 4ë¡ì›ù1±<ª_§,¯_ <°
¡
¢
Fig. 16. Phase noise spectrum after OFDM filtering.
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, /. /ç /å-ë
ö -S
2Ã x
110 Conclusions
"
x
"!
‚m "
0
"
We have presented
analytical
1 expressions
Fef3:for
the phase
ûS noise /
spectrum
of
phase-locked
loops.
We
have
included
white
,
/
F¸ÄºÀ¼
1noise of the reference,
the
VCO,
the
loop
filter
and
the
charge
h
@
u
u
B
pump, and the 6 − 1 quantization noise. The filter noise
„
"
@!
j
was described on the basis of the Nyquist equation general
S
1
ized to complex resistances. It may contribute significantly to
h
"
"
"
the overall noise, especially for low capacitance values used
"
/
H
in fully integrated frequency synthesizers. The reduction of
phase noise and jitter due tocorrection
q R the "common
of
phase /
c3 error
65r7 by digital baseband processing
in
" OFDM systems
—was
uj also included
in the
, model. If the carrier
q spacing
/
islarger
#
than the
loop bandwidth, the correction of the common phase
S
in a significant reduction
@ error results
of the rms phase jitter.
y y >y
2y2x
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Perrot, M. H., Trott, M. D., and Sodini, C. G.: A Modeling Ap‚g
U
"
where
the
sinc
function
is
defined
by
¤0ª_proach
¯_©" <¡"¦0ªW®for
­"¯W¬26©"¯Z−ã81
³­"Fractional-N
<´Z§+'°
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´ 0213335Synthesizers
4768 9 ;:<6>=@Allow?$6: A>B;C?
D
:
i
E
"
4
@ Psinc(x)=
sin(π x)/(πx).
The resulting improved phase
3E:ing
D
Straightforward
69"A
1«"­"±¢"÷ Noise
ø
ø¢1£ Analysis,
á ²£ "÷ IEEE
F'·
¡Journal
¢á á0¢ of Solid-State
"noise
F spectrum
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37,
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Stott, J.: The effects of phase noise in COFDM, BBC Research and
out
è Ì (f ) =
d_šRSg%PLL
ˆ (f
)hº )|1 − sinc
d_š (f Tg>u˜ )| .
d_=)(32)
g
SOFDM,eff
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Development, EBU Technical Review, Summer 1998.
Herzel,
F.,
Piz,
M.,
and
Grass,
E.:
Frequency
Synthesis
for
60
GHz
Integrating this spectrum, we obtain the rms phase error
OFDM Systems, in:
Proceedings of the 10th International
given by Stott (1998), Herzel et al. (2005)
OFDM Workshop (InOWo’05), Hamburg, Germany, 303–307,
s
Z B/2
2005.
180◦
Winkler, W., Borngräber, J., Heinemann, B., and Herzel, F.: A
σφ [degree] =
2
df SOFDM,eff (f) .
(33)
π
Fully Integrated BiCMOS PLL for 60 GHz Wireless Applica0
tions, ISSCC Digest of Technical Papers, San Francisco, 406–
The upper integration limit is half the bandwidth of the
407, 2005.
whole OFDM band. This corresponds to the middle-carrier
Lee, T. H. and Hajimiri, A.: Oscillator Phase Noise: A Tutorial,
weighting function as representative for the whole OFDM
IEEE Journal of Solid-State Circuits, 35, 326–336, 2000.
signal. Equation (33) allows the effective phase error σφ
Mehrotra, A.: Noise Analysis of Phase-locked Loops, IEEE Transto be calculated from circuit parameters and carrier spacing
actions on Circuits and Systems-I: Fundamental Theory and Ap(1f )car =1/Tu . The result is shown in Fig. 16.
plications, 49, 1309–1316, 2002.
F
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the different subcarriers. A strong correlation between the S
yà phase error and
&{the
>
v>yRx ZÁ
y rate
x of an OFDM
CÂ system
rms
bit-error
has been demonstrated in Herzel et al. (2005). In OFDM
ž3 6587
systems, different signals are used at frequencies separated
„
1
in frequency by the sub-carrier spacing 1fcarrier . The inverse h
q
of that
CÂ
6-
W
(29)
FM
=1G?I
−180
1e+02
@
where T =1/f0 is the average oscillation period. The jitter
1~j
can be calculated numerically using our analytic expression
for the phase noise. This
allows a fast optimization of the
‚ jitter.
loop parameters
for
a
low
ˆ
d_;)1g
˜
H3 O587
−80
referred to the PLL output. The rms phase error, sometimes
š
D
/
called absolute phase jitter, is given by
0"1N‘ 2 dZš2g"2ƒ Ë ‚@Æ2‘ Ì dZš2g"2ƒ O1"Ç ‘ 2 d_šRsg@RZƒ Æ/"Ç ‘ dZš2g /ƒ Æ°í"‘ Å d_šRg
M
"
180◦ ∞FSR out "
2 SPLL df .8σφ [degree]
=
(28)
"
F
π
0
"
D
4d_=1=?g
"
0
HOFDM (f ) = 1 − sinc2 (f Tu ) ,
www.adv-radio-sci.net/5/313/2007/
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Adv. Radio Sci., 5, 313–320, 2007
320
S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs
Lee, D. C.: Analysis of jitter in phase-locked loops, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal
Processing, 49, 704–711, 2002.
Demir, A.: Phase Noise and Timing Jitter in Oscillators with
Colored-Noise Sources, IEEE Transactions on Circuits and
Systems-I: Fundamental Theory and Applications, 49, 1782–
1791, 2002.
Kundert, K.: Predicting the Phase Noise and Jitter of PLLBased Frequency Synthesizers, in: Phase-Locking in HighPerformance Systems, edited by: Razavi, B., John Wiley & Sons,
46–69, 2003.
Centurelli, F., Ercolani, A., Scotti, G., Tommasino, P., and Trifiletti,
A.: “Behavioral Model of a Noisy VCO for Efficient TimeDomain Simulation, Microwave and Optical Technology Letters,
40, 352–354, 2004.
Herzel, F., Winkler, W., and Borngräber, J.: Jitter and Phase Noise
in Oscillators and Phase-locked Loops, in: Proc. SPIE Symp.
Fluctuations and Noise, Maspalomas, Spain, 5473, 16–26, 2004.
Navid, R., Jungemann, C., Lee, T. H., and Dutton, R. W.: Closein Phase Noise in Electrical Oscillators, in: Proc. SPIE Symp.
Fluctuations and Noise, Maspalomas, Spain, 5473, 27–37, 2004.
Adv. Radio Sci., 5, 313–320, 2007
Rohde, U. L., Poddar, A. K., and Böck, G.: The Design of Modern
Microwave Oscillators for Wireless Applications, John Wiley &
Sons, 2005.
Uang, R., Bourde, C., and Bagwell, T.: A SiGe 23 GHz
Fractional-N Frequency Synthesizer, online available: http://
www.mpdigest.com/Articles/2004/May2004/centellax, 2004.
Estimating PLL Phase Noise, online available: http://www.
rfengineer.net/pdf/pll pn magisnetworks.pdf, 1998.
Valero-Lopez, A. Y.: Design of Frequency Synthesizers for Short
Range Wireless Transceivers, Ph. D. Dissertation, Texas A & M,
Sep. 2004.
Niu, G.: Noise in SiGe HBT RF Technology: Physics, Modelling,
and Circuit Implications, Proceedings of the IEEE, 93, 1583–
1597, 2005.
Allen, P. E. and Holberg, D. R.: CMOS Analog Circuit Design,
Oxford University Press, New York, p.118, 1987.
Miller, B.: A Multiple Modulator Fractional Divider, IEEE Transactions on Instrumentation and Measurements, 40, 578–583, 1991.
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