Phase noise and jitter modeling for fractional-N PLLs...thesizers is of special interest in mm-wave...
Transcript of Phase noise and jitter modeling for fractional-N PLLs...thesizers is of special interest in mm-wave...
Adv. Radio Sci., 5, 313–320, 2007www.adv-radio-sci.net/5/313/2007/© Author(s) 2007. This work is licensedunder a Creative Commons License.
Advances inRadio 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 forfractional-N phase-locked loops (PLL) with emphasis on in-tegrated RF synthesizers in the GHz range. The noise of thecrystal reference, the voltage-controlled oscillator (VCO),the loop filter, the charge pump, and the sigma-delta mod-ulator (SDM) is filtered by the PLL operation. We expressthe rms phase error (jitter) in terms of phase noise of the ref-erence, the VCO phase noise and the third-order loop filterparameters. In addition, we consider OFDM systems, wherethe PLL phase noise is reduced by digital signal processingafter down-conversion of the RF signal to baseband. The rmsphase error is discussed as a function of the loop parameters.Our model drastically simplifies the noise optimization of thePLL loop dynamics.
1 Introduction
Frequency synthesizers are essential components in wire-less RF transceivers. A synthesizer provides a stable signalfor frequency conversion, modulation and demodulation. Afractional-N PLL gives a greater flexibility than an integer-N PLL due to the higher frequency resolution (Perrot etal., 2002). The phase noise is the most important perfor-mance criterion of the synthesizer for its stringent require-ment in many applications, e.g., in OFDM based systems(Stott, 1998; Herzel et al., 2005). The problem with phasenoise is exacerbated in high-frequency applications, since theoscillator phase noise at a given frequency offset typicallyincreases in proportion to the carrier frequency. For this rea-son, the minimization of the phase noise in frequency syn-thesizers is of special interest in mm-wave application, e.g.,in the 60 GHz band (Winkler et al., 2005). Many publica-tions on phase noise modeling of oscillators and PLLs have
Correspondence to:S. A. Osmany([email protected])
appeared during the last few years, e.g.Lee and Hajimiri(2000)–Rohde(2005). This paper is focussed on problemswhich are specific to integrated PLLs. They arise mainlyfrom the low quality factor of the available passives as wellas from the limited size of filter capacitors. Especially, thelow quality factor of integrated inductances causes a highnoise level of the voltage-controlled oscillator (VCO). Thisproblem is exacerbated by the fact that variations of the de-vice parameters with process and temperature require a rela-tively large VCO tuning range affecting oscillator noise andloop filter noise. In integrated PLLs, the values of the in-tegrated capacitances are limited by chip area and, possibly,yield. This can make the filter noise comparable to the VCOnoise, if the VCO gain is large. Another problem arises inRF synthesizers for mm-wave applications. The high oscil-lation frequency results in a large division factor enhancingthe reference noise to a level comparable to the phase noiseof integrated VCOs. The different dependencies of the noisecontribution on the PLL dynamics results in a subtle trade-offfor the loop bandwidth.
This paper describes the phase noise spectrum of a PLL,where emphasis is placed on the high-pass filtering in OFDMsystems 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 two-sided power spectral density (PSD) of the phase and thesingle-sideband phase noise. For moderate and large fre-quency offsets, these quantities are almost equal (Herzel,2004). For wideband systems, close-in phase noise is lessrelevant due to a large channel width and/or proper basebandfiltering explained below. Therefore, we will not distinguishbetween the single-sideband phase noise and the two-sidedpower spectral density of the phase in the remainder of thispaper. The phase noiseS can be determined from the single-
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
314 S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs
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Fig. 1. Schematic view of a fractional-N synthesizer architecture.
sideband phase noise£ specified in units of dBc/Hz accord-ing to
S = 10£/10 . (1)
In a charge-pump PLL as used in modern communicationsystems, the output phase of the VCO is divided by N andcompared with a reference phase in a phase-frequency detec-tor (PFD). A charge pump current proportional to the phaseerror is produced, low-pass filtered, and applied to the con-trol input of the VCO as shown in Fig.1. In fractional-NPLLs the divider ratio is controlled by a sigma-delta modu-lator (SDM).
The output voltage of a PLL can be described as
Vout(t) = V0 cos[ω0t + φout(t)] , (2)
whereV0 is the amplitude,f0 = ω0/(2π) is the oscillationfrequency, andφout(t) is the excess phase. The dominatingnoise sources in our type of PLL are the VCO and the ref-erence. But also the low-pass filter (LPF) and charge pump(CP), if not properly designed, may significantly contributeto the overall phase noise. The problem is critical for in-tegrated PLLs due to the relatively small capacitances andthe large VCO gain. In fractional-N PLLs, another importantnoise source may significantly contribute to the overall phasenoise, namely, quantization noise of the6−1 modulator. Inthis paper, we will include the following five noise sources:
– reference noise,
– VCO noise,
– loop filter noise,
– charge pump noise,
– 6 − 1 quantization noise.
These are typically the dominant noise contributions in in-tegrated fractional-N RF synthesizers for the GHz range(Uang, 2004; Valero-Lopez, 2004). Digital noise is disre-garded in this paper.
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Fig. 3. 2nd order loop filter.
3 Linearized PLL model
Fig. 2 shows the linearized model of a PLL, where thePFD/CP gain is modeled byICP/2π [A/rad], the loop fil-ter transimpedance is denoted asF(s), and the VCO gain isKVCO/s. N is the divider ratio of the PLL. Note that the VCOgainKVCO=dω/dVctrl refers to the angular frequency and isgiven in units of rad/s/V. The closed-loop transfer functionreads
H =H0
1 + H0/N, (3)
where the forward path transfer functionH0 is
H0 =ICPF(s)KVCO
2πs. (4)
For the filter transimpedance calculation we assume thatno current flows into VCO control input, since the filter ca-pacitances are notably large. For the second-order filter ac-cording to Fig.3, we obtain the transimpedance
F(s) =
(R1 +
1
sC1
)||
1
sC2. (5)
In order to suppress reference spurs, an additional low-passfilter is often included as shown in Fig.4, resulting in a third-order loop filter. The current throughZ2 causes a voltage
Adv. Radio Sci., 5, 313–320, 2007 www.adv-radio-sci.net/5/313/2007/
S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs 315
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Fig. 4. 3rd order loop filter.
drop overC3, which can be calculated by using the currentdivider rule resulting in
F(s) =1
sC3
1/Z2
1/Z1 + 1/Z2, (6)
where
Z1 =
(R1 +
1
sC1
)||
1
sC2(7)
and
Z2 = R3 +1
sC3. (8)
The crossover frequency of the open loop transfer functionis the PLL bandwidthfT , defined by
mag
(H0(fT )
N
)= 1 . (9)
The PLL phase margin is given by
phase margin = phase
(H0(fT )
N
)+ 180◦. (10)
In the following five sections, the noise sources mentionedin Sect.2 will be analytically described in the framework ofthe linear PLL model. The transfer functions of the noisesources to the PLL output will be calculated.
4 Reference noise
A low-noise reference is crucial for a good PLL performance.The phase noise contribution of the reference oscillator as afunction of the frequency offsetf is modeled by
SREF(f ) = SREF(1f )(1f )2
f 2+ SREF,floor , (11)
whereSREF(1f ) is the phase noise at a specific offset1f inthe –20 dB/decade region of the spectrum andSREF,floor is thenoise floor of the reference. According to Fig.5, the transfer
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� Ë>Æ2Ì dZ�2g��� Ë Æ2Ì dZ¼F�2g dZ¼F�2g ð� ð �Ä� Ë Æ2Ì 3 5�"��æR� dA)�Q1g
1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
−150
−120
−90
−60
−30
0
Pha
se N
oise
[dB
c/H
z]
X HLPF
SPLL
SREFSREF
out
out
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Fig. 5. Reference noise shaping model.
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
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X HLPF
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SREFSREF
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Fig. 6. Reference noise at the PLL input and output. In addition,the solid line shows the total output noise spectrum.
function from the reference phase to the synthesizer outputphase is given by
φoutREF
φREF=
H0
1 + H0/N. (12)
From this, we obtain the reference noise spectrum at the out-put from the reference noise spectrum at the input accordingto
SoutREF = SREFN2
|HLPF|2 (13)
with the low-pass filter function
HLPF =H0/N
1 + H0/N. (14)
Note that the reference noise is low-pass filtered and ampli-fied by the divider ratio. Although the phase noise of thereference is typically lower than that of the VCO by manyorders of magnitude, it may become comparable to the VCOnoise, if a large division factor is employed. For instance, thevalue of N=1024 inWinkler et al.(2005) corresponds to anincrease of the phase noise by about 60 dB as it appears atthe PLL output. Figure6 shows the reference noise for typi-cal crystal oscillators (long dashed). Using the noise transferfunction (dashed) we obtain the corresponding phase noisecontribution (dot-dashed). Comparison with the total phasenoise (solid) shows that at offsets below 1 kHz the referencenoise dominates.
www.adv-radio-sci.net/5/313/2007/ Adv. Radio Sci., 5, 313–320, 2007
316 S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
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f�üvò&õ`ï�ðJñ/ï�òJö�û�ô���ö�÷/ïc���� üvñ&ö`þ/ñ/öFò/üvø�õ�ï�ö`üÄ�/ñ/ï4e�f�3�þ/÷júUõ`ïò/ü�ø�õ�ï|ø�õ���ø��vï�òq�Gô � ���"�Ë>Æ2Ì ���2Ë Æ2Ì ï )hºPè-O1Ç 2 ï ð dA)'�1gÿ�ø�ö`÷�ö`÷&ïÁû�üBÿh�OþjúUõ`õhSjû�ö�ï�î � ñ&ò/ó�ö`ø�ü�òpdA)"N�g"��}üvö�ïÁö`÷júUö�ö`÷/ï�e�f�3ò/ü�ø�õ�ï¥ø�õF÷/ø��v÷���þjúBõ�õFSjû�ö�ï�î`ï"��ø�ò�ö`÷&ïU����%�Ýí�ø��R���#õ�÷/üBÿ�õ�ö`÷/ïe�f�3Zþ/÷júBõ�ï³ò/ü�ø�õ`ïHdW��úBõ`÷&ï"�Rg�úBò���ø�ö`õ0ó�ü�òJö`î`ø��/ñ&ö`ø�üvò�ö�üÁö`÷/ï��%���üvñ&ö`þ/ñ/ö�þ/÷júUõ`ïQò/ü�ø�õ`ï�d û�ü�ò�����júUõ`÷/ï"�2g"�>-�÷/ïh����âò/üvø�õ`ï8õ`þ�ï�ó�ö`î�ñ/ýúBö�÷&ø���÷/ï�î�ü DWõ�ï�ö�ø�õ��/üvýFø�ò�úBö`ï@�F�GôFö`÷/ï�e�f�3 þ/÷júUõ`ï|ò/ü�ø�õ�ï1�*âãläCtiè�èxy{zQìWÃ�æ>y&çizÔèSìZx"y|#ï·ý'ü��/ï�û0ö`÷&ï�ò/üvø�õ`ôÜû�üJü�þ4Sjû�ö�ï�îH�Jô úFò/ü�ø�õ`ï·ó�ñ/î`î�ï�òJö¿ø�ò#þjúUî��úBû�û�ï�û�ÿ�ø�ö`÷6ú·ò/üvø�õ`ï+�Oû�ï�õ�õÔú �/ý'ø�ö�öLúBò&ó�ï�úUõ³õ`÷&üBÿ�ò�ø�òNí�ø��R�����h-�÷/ïö�ÿ�ü��õ`ø��/ï"��þ�üBÿ�ï<î�õ�þjï�ó<ö`îLúUû>�/ï�ò/õ�ø�ö�ô�ü � ò/ü�ø�õ�ï�ó�ñ/î�î`ï�òJö $~} ólúBòq�jïï"!Jþ/î�ï�õ`õ�ï"� úUõ ��2)� O d_É�g%�l;Z�)���û�gï1d���2)� O d_É�gAg>� dA)'�1g
Fig. 7. VCO noise shaping model.
5 VCO noise
Oscillator phase noise has been the subject of numerous pa-pers. The optimization of the VCO phase noise is beyondthe scope of this paper. For low-noise microwave oscillatordesign we refer to the extensive literature cited, e.g., inRo-hde(2005). The phase noise of a free running oscillator ismodeled by
SVCO(f ) = SVCO(1f )(1f )2
f 2+ SVCO,floor , (15)
whereSVCO(1f ) is the phase noise at a specific offset1f
in the –20 dB/decade region of the spectrum, typically at1 MHz. SVCO,floor is the noise floor of the VCO. Note thatwe have disregarded flicker noise, since we focus on appli-cations, where the loop bandwidth is larger than the 1/f 3
noise corner frequency of the VCO, which is on the orderof 10 kHz for modern SiGe HBT technologies (Niu, 2005).Flicker noise in the VCO is then effectively suppressed dueto the high-pass filtering by the PLL. According to Fig.7,the transfer function from the VCO output phase to the PLLoutput reads
φoutVCO
φVCO=
1
1 + H0/N. (16)
Consequently, the PLL output noise to due VCO phase noiseis given by
SoutVCO = SVCO |1 − HLPF|
2 (17)
with the low-pass filter Function (14). Note that the VCOnoise is high-pass filtered in the PLL. Figure8 shows theVCO phase noise (dashed) and its contribution to the PLLoutput phase noise (long-dashed). The PLL noise spectrumat higher offset is dominated by the VCO phase noise.
6 Loop filter noise
We model the noisy loop filter by a noise current in parallelwith a noise-less admittance as shown in Fig.9. The two-sided power spectral density of noise currentin can be ex-pressed as
SFIL(s) = 2kBT Re(YFIL(s)) , (18)
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Fig. 8. VCO noise for open-loop (dashed) and closed-loop (long-dashed) case. In addition, the solid line shows the total output noisespectrum.
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Fig. 9. Filter noise model.
wherekB is Boltzmann’s constant,T the absolute tempera-ture, andYFIL is the complex admittance of the filter fromthe charge pump output to ground. Equation (18) is thewell-known Nyquist equation, generalized to a passive two-pole described by a complex admittance. For a second-orderloop filterYFIL(s) equals the inverse transimpedance 1/F (s),but for a third-order filter this is no longer the case. Wefind for this caseYFIL(s)=1/Z1+1/Z2, whereZ1 and Z2are given by (7) and (8). We assume|ZFIL | << ZCP
out and|ZFIL | <<ZVCO
in . This implies that the total filter noise cur-rent flows through the filter impedance. Using the filter tran-simpedance (6), we obtain a noise voltage PSD at the VCOinput. Exploiting the VCO phase noise transfer function, weobtain the noise spectrum at the output given by
SoutFIL(s) = SFIL(s) |F(s)|2
∣∣∣∣KV CO
s
∣∣∣∣2 |1 − HLPF|2 . (19)
Note that the noise transfer function for the loop filter has aband-pass characteristics. Figure10 shows the filter contri-bution for the integrated PLL. In the region around the PLLbandwidth, the filter noise may significantly contribute tothe overall phase noise due to the small filter capacitances.Its contribution can be much reduced by adding an external
Adv. Radio Sci., 5, 313–320, 2007 www.adv-radio-sci.net/5/313/2007/
S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs 317
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Fig. 10. Filter noise at the PLL output. In addition, the solid lineshows the total output noise spectrum.
capacitor which is, however, not always desirable for low-cost integrated solutions.
7 Charge pump noise
For a PLL to work at a given bandwidth, a large charge pumpcurrent is generally favorable in order to reduce the loop filterresistanceR. The large current, however, produces signifi-cant current noise during the time, where the charge pump isactive. The time between two phase comparison instants isreferred to astcomp=1/fcomp, wherefcomp is the comparisonfrequency at the PFD input. In order to prevent a dead zone,the charge pump currents are usually activated during a timetCP, which is in the order of 100 ps to 1 ns in the steady statefor an integer-N PLL. For a fractional-N PLL the activationtime is typically larger due to the momentary frequency errorinherent in this type of PLL (Perrot et al., 2002). In a typicalCMOS charge pump as shown in Fig.11, an N-MOSFET anda P-MOSFET are connected to the filter. Their thermal andflicker noise is transferred to the PLL output causing phasenoise. We write the drain current noise PSD of a MOSFETas described inAllen (1987):
SMOSi = γ 4kBT gm +
(KF)ID
f COXL2, (20)
whereγ=2/3 for long-channel devices. Adding the noisecontributions of the two transistors, we obtain
SCP =
[SNMOS
i + SPMOSi
]α , (21)
whereα=tCP/tcomp is the duty cycle of the charge pump, thatis, the ratio of the average charge pump activation time andthe time difference between two comparison instants. Notethat the spectral densitiesSNMOS
i , SPMOSi must be averaged,
sincegm varies with time. The noise currents produce a noisevoltage over the filter impedance, which is transferred to the
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Fig. 11. Charge pump noise model.
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�2���"�"d���g�i�úù{û�ýü �+þ � O1Ç 2Od��+þ�g���0�1#2Od��OºM�+þWg>�ù{û� ü � þ �4ÿ Ç 2 d�� þ g��2Ë>ÆRÌhd��6º4� þ g>�ù û� ü �+þ � � Ç 2 d��+þ�g $ O?Ç 2 d��º4�+þWg>�ù û� ü �+þ��� O1Ç 2�d��+þWg $ Æ/Ç d��º4�+þWg>�ù{û� ü � þ��� O1Ç 2�d�� þ g�� Æôí Å d��OºM� þ g>� d_;1G1gÿ�÷/ï�î�ï:� � O?Ç 26� � ÿ Ç 2�� � � Ç 2O���� O1Ç 2#úBò�� �� O1Ç 2¥úUî`ï·ö`÷/ï·û�ø�ò/ïlúBîþ/÷júUõ`ïFî`ï�õ�þjü�ò/õ`ï�õ'úUö|ö`÷/ï�����rü�ñ/ö`þ&ñ/öÁî`ï � ï�î`î�ï"�rö`ü�ö`÷&ï'þ/÷júUõ`ïò/ü�ø�õ�ï�õ`ü�ñ/î`ó�ï?��³ü�ö`ï}ö`÷júUö� � O1Ç 2 d�� þ g8ø�ý'þ/û�ø�ï�õiúÔý�ñ/û�ö`ø�þ/û�ø�ólúBö�ø�üBò�JôÁö`÷&ï��/ø��Gø�õ�ø�ü�òÁîLúUö`ø�ü��C�vÿ�÷&ø�óx÷·ï�ò/÷júUò/ó�ï�õ8ö`÷/ï�î�ï � ï�î�ï�ò/ó�ï|ò/ü�ø�õ�ï1�
Fig. 12.Charge pump noise at the PLL output. In addition, the solidline shows the total output noise spectrum.
PLL output phase according to the VCO noise transfer func-tion. As a result, we obtain the noise spectrum at the PLLoutput
SoutCP = SCP |F(s)|2
∣∣∣∣KVCO
s
∣∣∣∣2 |1 − HLPF|2 . (22)
Figure12 shows the contribution of the charge pump to thetotal PLL phase noise. Flicker noise contribution is disre-garded here. The PLL bandwidth is proportional to the prod-uct of the charge pump currentICP and the filter resistanceR1. Consequently, if a certain PLL bandwidth is required,the charge pump current noise must be carefully traded offwith the noise produced by the filter resistanceR1.
www.adv-radio-sci.net/5/313/2007/ Adv. Radio Sci., 5, 313–320, 2007
318 S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−160
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−100
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−40
−10
Pha
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[dB
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X HLPF
SSDM
SPLL
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�2���"�"d���g�i�úù{û�ýü �+þ � O1Ç 2Od��+þ�g���0�1#2Od��OºM�+þWg>�ù{û� ü � þ �4ÿ Ç 2 d�� þ g��2Ë>ÆRÌhd��6º4� þ g>�ù û� ü �+þ � � Ç 2 d��+þ�g $ O?Ç 2 d��º4�+þWg>�ù û� ü �+þ��� O1Ç 2�d��+þWg $ Æ/Ç d��º4�+þWg>�ù{û� ü � þ��� O1Ç 2�d�� þ g�� Æôí Å d��OºM� þ g>� d_;1G1gÿ�÷/ï�î�ï:� � O?Ç 26� � ÿ Ç 2�� � � Ç 2O���� O1Ç 2#úBò�� �� O1Ç 2¥úUî`ï·ö`÷/ï·û�ø�ò/ïlúBîþ/÷júUõ`ïFî`ï�õ�þjü�ò/õ`ï�õ'úUö|ö`÷/ï�����rü�ñ/ö`þ&ñ/öÁî`ï � ï�î`î�ï"�rö`ü�ö`÷&ï'þ/÷júUõ`ïò/ü�ø�õ�ï�õ`ü�ñ/î`ó�ï?��³ü�ö`ï}ö`÷júUö� � O1Ç 2 d�� þ g8ø�ý'þ/û�ø�ï�õiúÔý�ñ/û�ö`ø�þ/û�ø�ólúBö�ø�üBò�JôÁö`÷&ï��/ø��Gø�õ�ø�ü�òÁîLúUö`ø�ü��C�vÿ�÷&ø�óx÷·ï�ò/÷júUò/ó�ï�õ8ö`÷/ï�î�ï � ï�î�ï�ò/ó�ï|ò/ü�ø�õ�ï1�
Fig. 13. SDM noise shaping model.
8 Sigma-delta quantization noise
In a fractional-N synthesizer, the integer divider value isdithered dynamically to achieve fractional values. A classicalfractional-N architecture uses an accumulator to peform thedithering operation. But in this case, a periodic error signalcreates spurious tones in the synthesizer output. A better ap-proach is to use a sigma-delta modulator rather than a singleaccumulator to perform the dithering operation. This resultsin a much better spurious performance. But the sigma-deltamodulator adds quantization noise. For the PLL noise analy-sis, we model the SDM by a divider value and a sigma-delanoise spectrum as shown in Fig.13. The quantization noisespectrum for a m-th order MASH type sigma-delta modula-tor is given byMiller (1991),
SSDM =(2π)2
12fREF
[2 sin
(πf
fREF
)]2(m−1)
, (23)
wherem is the order of the modulator andfREF is the ref-erence frequency. The transfer function for the noisy phasefrom the SDM output to the PLL output reads
φoutSDM
φSDM=
H0/N
1 + H0/N. (24)
Consequently, we obtain the SDM noise spectrum at the PLLoutput
SoutSDM = SSDM |HLPF|
2 . (25)
Note that the SDM quantization noise spectrum is low-passfiltered in the PLL. Figure14shows the resulting phase noisecontribution.
9 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 noiseis bandpass-filtered. The charge pump noise and sigma deltaquantization noise is low-pass filtered in the PLL. This re-sults in the total phase error
φout(t) = N∫
∞
0dt ′hLPF(t
′) φREF(t − t ′) +
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−160
−130
−100
−70
−40
−10
Pha
se N
oise
[dB
c/H
z]
X HLPF
SSDM
SPLL
SSDM
out
out
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Fig. 14. SDM noise at divider input and at PLL output. In addition,the solid line shows the total output noise spectrum.
1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
−160
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Pha
se N
oise
[dB
c/H
z]X
SVCO
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SFIL
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out
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
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se N
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Fig. 15. Phase noise spectrum and its contributors.
∫∞
0dt ′hHPF(t
′) φVCO(t − t ′) +∫∞
0dt ′hBPF(t
′) iLPF(t − t ′) +∫∞
0dt ′hLPF(t
′) iCP(t − t ′) +∫∞
0dt ′hLPF(t
′) φSDM(t − t ′) , (26)
where NhLPF, hHPF, hBPF, hLPF and hLPF are the linearphase responses at the PLL output referred to the phase noisesource. Note that NhLPF(t
′) implies a multiplication by thedivision ratio N, which enhances the reference noise.
Since the five noise sources are uncorrelated, the corre-sponding noise spectra must be added to obtain the overallphase noise spectrum
SoutPLL(f ) = Sout
REF(f ) + SoutVCO(f ) +
SoutFIL(f ) + Sout
CP(f ) + SoutSDM(f ) , (27)
wheref is the frequency offset from the carrier andSoutREF(f ),
SoutVCO(f ), Sout
LPF(f ), SoutCP(f ) andSout
SDM(f ) are the phase noise
Adv. Radio Sci., 5, 313–320, 2007 www.adv-radio-sci.net/5/313/2007/
S. A. Osmany et al.: Phase noise and jitter modeling for fractional-N PLLs 319
contributions of the five previously discussed noise sourcesreferred to the PLL output. The rms phase error, sometimescalled absolute phase jitter, is given by
σφ[degree] =180◦
π
√∫∞
02 Sout
PLL df . (28)
In the high-speed digital design community, the so-called ab-solute jitter is a common measure to quantify the timing errorof a PLL driven by a clean reference. It is given by
σabs=T
360◦σφ , (29)
whereT =1/f0 is the average oscillation period. The jittercan be calculated numerically using our analytic expressionfor the phase noise. This allows a fast optimization of theloop parameters for a low jitter.
10 Modeling of RF synthesizers for OFDM
A low pll jitter is especially important for OFDM systems,where phase noise may easily destroy the orthogonality ofthe different subcarriers. A strong correlation between therms phase error and the bit-error rate of an OFDM systemhas been demonstrated inHerzel et al.(2005). In OFDMsystems, different signals are used at frequencies separatedin frequency by the sub-carrier spacing1fcarrier. The inverseof that
Tu = 1/(1fcarrier) (30)
is called the useful symbol time. In state-of-the-art basebandprocessor implementations part of the phase noise (com-mon phase error) can be eliminated after the signal has beendown-converted to baseband. This process can be modelledby a high-pass filter according toStott(1998)
HOFDM(f ) = 1 − sinc2(f Tu) , (31)
where the sinc function is defined bysinc(x)= sin(πx)/(πx). The resulting improved phasenoise spectrum reads
SOFDM,eff(f ) = SoutPLL(f )|1 − sinc2(f Tu)| . (32)
Integrating this spectrum, we obtain the rms phase errorgiven byStott(1998), Herzel et al.(2005)
σφ[degree] =180◦
π
√2
∫ B/2
0df SOFDM,eff(f) . (33)
The upper integration limit is half the bandwidth of thewhole OFDM band. This corresponds to the middle-carrierweighting function as representative for the whole OFDMsignal. Equation (33) allows the effective phase errorσφ
to be calculated from circuit parameters and carrier spacing(1f )car=1/Tu. The result is shown in Fig.16.
1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
−160
−140
−120
−100
−80
Pha
se N
oise
[dB
c/H
z]
XSVCO
SCP
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SSDM
SPLL
SREF
out
out
out
out
out
out
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1e+02W 1e+04W 1e+06W 1e+08WFrequency Offset [Hz]
−180
−160
−140
−120
−100
−80
Pha
se N
oise
[dB
c/H
z]
X SPLL
SVCO
SSDM
SFIL
SCP
SREFout
outout
out
out
out
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Fig. 16. Phase noise spectrum after OFDM filtering.
11 Conclusions
We have presented analytical expressions for the phase noisespectrum of phase-locked loops. We have included whitenoise of the reference, the VCO, the loop filter and the chargepump, and the6 − 1 quantization noise. The filter noisewas described on the basis of the Nyquist equation general-ized to complex resistances. It may contribute significantly tothe overall noise, especially for low capacitance values usedin fully integrated frequency synthesizers. The reduction ofphase noise and jitter due to correction of the common phaseerror by digital baseband processing in OFDM systems wasalso included in the model. If the carrier spacing is largerthan the loop bandwidth, the correction of the common phaseerror results in a significant reduction of the rms phase jitter.
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