Tutorial on Discrete Time Phase Noise Modeling
for Phase Locked Loops
Steffen Bittner, Student Member, IEEE, Stefan Krone, Student Member, IEEE,
and Gerhard Fettweis, Senior Member, IEEE
Email: {bittner,krone,fettweis}@ifn.et.tu-dresden.de
Abstract—Phase noise is one of the main impairments in
wireless communications systems, due to its strong distortion of
the transmitted signal. In this work, a discrete time phase noise
model for simulation environments is proposed. This model can
be used to obtain a better understanding of the effects of phase
noise on the behavior of communications systems. It has already
been shown that phase noise can be analytically described by
stochastic differential equations. In this contribution numerical
solutions for these stochastic differential equations are given for
different types of phase locked loops, allowing for the generation
of phase noise samples.
I. I NTRODUCTION
Phase Noise (PN) describes a phase fluctuation of an
oscillator with a noiseless state response. The impact of PN
is especially relevant in wireless communications systems
because PN increases with higher carrier frequency and causes
non-negligible distortion. Driven by these facts there exists
an increasing demand on simulating PN in different kinds of
simulation chains in order to investigate the performance of a
system or to evaluate different compensation approaches [1]–
[3]. PN simulation is usually performed using a model based
on a free running oscillator. However, in practice, phase locked
loops (PLLs) are more often used as frequency synthesizer
in RF communications systems. The analytic description of
PN in PLLs is well discussed [4] [5] using the assumption
that the overall circuit noise can be summarized using just
one white noise source. For the computation of the PLL
output spectrum, a nonlinear analysis of the voltage controlled
oscillator (VCO) in a feedback loop was studied. The obtained
spectra match very well with measured PLL output spectra.
In this contribution a discrete time model is introduced that
provides numerical solutions to the stochastic differential
equations of the PN process. Thus, time discrete PN samples
can be generated for different types of PLLs.
(a)
u(t)+∆u(t)
(b)
VCO
xs (t)
cvco
Rt
....dt
α(t)
o
ξ(t)
Fig. 1.
VCO: (a) schematic, (b) noise flow
fs = 1/∆t. Using the discrete time shift α[n] = α(n · ∆t) the
n-th PN sample is given as ϕ[n] = 2πfo α[n].
A. Free running VCO
The easiest and most common PN model is based on a free
running VCO as given in Fig. 1(a). The voltage fluctuation
∆u(t) at the input of the VCO results in a phase fluctuation at
the output. In this setup PN is considered as a Wiener Process,
given by the stochastic integral:
Z t
√
ϕ(t)
α(t) =
ξ(t0 )dt0 ,
(1)
= cvco
2πfo
0
where cvco is a constant, which determines the quality of
a given oscillator, defined via an offset frequency ∆f in
the Lorentzian power density spectrum L(f ) given in data
sheets. Combining all noise sources into one white noise
source, cvco measured in seconds can be approximated as:
2
cvco ≈ 10L(∆f )/(10dBc/Hz) (∆f /fo ) . Furthermore, ξ(t0 ) is
0
at each time instance t a realization of a Gaussian random variable (RV) with zero mean and variance one, i.e.
ξ(t0 ) ∼ N (0, 1). Solving (1) for α(t) results also in a standard
Gaussian RV with variance σα2 (t) = cvco t. In order to generate
time samples α[n], (1) is written as a sum of integrals:
Xn−1 Z ∆t
√
α[n] = α(n · ∆t) = cvco
ξ(t0 + i · ∆t)dt0 ,
i=0 0
{z
}
|
W (∆t)
(2)
II. P HASE N OISE M ODELS
In [4] PN is modeled by means of a random continuous time
shift α(t), resulting in an oscillator output x(t) = x0 (t+α(t)).
Here x0 (t) is the noiseless periodic steady state response of the
oscillator. The phase shift itself is given by: ϕ(t) = 2πfo α(t),
with fo representing the oscillator frequency. In the following
the stochastic differential equations (DE) discussed in [5] are
solved for a discrete time model, which allows for sample
generation in Monte-Carlo simulations. For denoting the continuous time “t” is used, whereas “n” and “i” are discrete
time indices. Furthermore, the sampling frequency is given as
√
where W (∆t) ∼ N (0, ∆t). Hence, samples of W (∆t) can
be generated by multiplying
samples coming from a standard
√
Gaussian RV with ∆t. The variance of α(n∆t) follows the
properties of the Wiener process and increases linearly in time,
i.e. σα2 (n∆t) = cvco n∆t.
B. Phase Locked Loops
PLLs are used if low PN and large frequency range are
required. As shown in Fig. 2(a), a general PLL is realized by
a combination of a VCO and a high quality reference crystal
(a)
xxtl (t)
∆ũ(t)
xpll (t)
Phase ∆u(t)
Low Pass Filter
VCO
Detector
Crystal
1/N
(b)
Phase Detector
Voltage Controlled Oscillator
β(t)
αxtl (t)
hLP
−kpd
γ(t)
√
Rt
αpll (t)
....dt
ξvco (t)
cvco
Rt
....dt
assuming that the sample rate is high enough. Hence, the
discrete time model follows as:
√
β[n] = β[(n − 1)] + ccontr ∆tγ[(n − 1)]
√
√
+ cvco · Wvco (∆t) − cxtl · Wxtl (∆t).
(8)
αvco (t)
0
General PLL: (a) concatenation with crystal, (b) noise flow
operating at a low frequency. The depicted low pass filter is
optional.
Figure 2(b) shows the noise flow in the feedback loop of
a locked PLL system. The stochastic time shift at the output
of the PLL αpll (t) can be considered as a sum of a Wiener
process αxtl (t) and a one dimensional Ornstein-Uhlenbeck
process β(t):
αpll (t) = αxtl (t) + β(t).
(3)
The PN properties of the crystal are basically the same as for
the free running VCO and have been summarized for discrete
time in Sec. II-A. The stochastic DE of the process β(t) is
given as [5]:
β̇(t) = α̇pll (t) − α̇xtl (t)
√
√
√
= ccontr γ(t) + cvco ξvco (t) − cxtl ξxtl (t)
Z t
Z t
√
ccontr
γ(t0 )dt0 + cvco
ξvco (t)dt0
0
0
Z t
√
− cxtl
ξxtl (t0 )dt0 .
γ[n] = −kpd β[n].
√
(5)
The numerical solution for the 2nd and 3rd part of the right
hand side of (5) for an interval ∆t is:
Z ∆t
ξ(.) (t0 )dt0 = W(.) (n∆t) − W(.) ((n − 1)∆t) = W(.) (∆t).
(9)
2) Second Order PLL: One main drawback of the first
order PLL is that at high offset frequencies the PLL output
spectrum is higher than that of a free running oscillator. In
order to remove these higher frequencies, a low pass loop
filter is included leading to a second order PLL. The general
linear DE of a first order low pass filter is given by:
(4)
0
0
The generation of PN samples can now be based on (8).
1) First Order PLL: The first order PLL is the simplest
implementation of a PLL. In this case no loop filter is in the
feedback loop. Here the input noise of the VCO γ[n] is simply
given as:
y(t) +
and is known in physics as the Langevin equation. Here cxtl
and cvco are the crystal and the VCO oscillator constants. The
principle PLL structure can be seen as a filter. To be more
precise, for the reference signal αxtl (t) the PLL acts like a
low pass filter, whereas for the VCO signal the PLL acts like
a high pass filter. The 3dB corner frequency for both filters is
identical: ωGpll = 2πfGpll . It is desirable to choose fGpll as
high as possible in order to minimize the PN at the PLL output.
However, to guaranty a stable system fGpll should not be
higher than 1/10 of the crystal frequency. Together with a given
phase detector constant kpd , the constant at the control node of
√
the VCO ccontr can be computed as: ccontr = 2πfGpll /kpd .
For the derivation of the discrete time model, (4) is rewritten
as:
β(t) =
≈∆t·γ(i·∆t)=∆t·γ[i]
0
√
Fig. 2.
ccontr
Again, samples of W(.) (∆t) are generated by multiplying
√
samples coming from a standard Gaussian RV with ∆t. For
the 1st part of (5) we use the following approach:
Z t
Xn−1 Z ∆t
0
0
γ(t )dt =
γ(t0 + i · ∆t)dt0
(7)
i=0 0
0
|
{z
}
ẏ(t)
= x(t),
ωlp
(10)
where x(t) and y(t) are the input and output signals of the
filter. The 3dB corner frequency is defined as: flp = 2π/ωlp .
Applying this value to the system in Fig. 2(b) and transforming
(10) into discrete time leads to the following equation:
γ[n] = (1 − ωlp ∆t)γ[(n − 1)] − kpd ωlp ∆tβ[(n − 1)]. (11)
3) Charge Pump PLL: Both, the first order PLL and
the second order PLL have the disadvantage that the phase
difference between the crystal and the VCO output cannot
be canceled completely. The so called steady state error,
which describes the phase offset if t → ∞, is not zero [5].
However, zero steady state error can be achieved if the PLL
loop filter transfer function contains at least one pole, which
is the case for the Charge Pump (CP) PLL. Here the phase
detector acts as a current source charging a capacitor. The
capacitor voltage ucp (t) itself controls the VCO. However,
such an ideal integrator degrades the stability of the loop. In
practical realizations the stability is recovered by using a serial
combination of a capacitor and a resistor as shown in Fig. 3.
Assuming 2πfcp = 1/RC = ωcp as the 3dB corner frequency
R
ucp (t) = γ(t)
−kpd
R β(t)
= i(t)
Fig. 3.
Charge Pump PLL phase detector
C
(6)
Solving (12) for β̇(t) and plugging it to (4) yields the discrete
time model of the VCO input noise:
1
angle[rad]
of the CP phase detector, the continious time domain DE is
³
´
given as:
(12)
γ̇(t) = −kpd β̇(t) + ωcp β(t) .
γ[n] = (1 − ωGpll ∆t)γ[n − 1] − kpd ωcp ∆tβ[n − 1]
¡√
¢
√
− kpd cvco · Wvco (∆t) + cxtl · Wxtl (∆t) . (13)
IV. C ONCLUSIONS
In [4], [5] a noise analysis for free running VCOs and PLLs
was presented. In these works the PN problem was stated in
terms of stochastic differential equations, where it was shown
that the output phase of the oscillator can be expressed as
a combination of a Wiener process and one component of a
multi-dimensional Ornstein-Uhlenbeck process. The obtained
PSD matched closely the measured PSD. In this contribution a
discrete time solution for the stochastic differential equations
of the phase noise process was presented. The numerically
computed PSD for the discrete time model closely tracks
the analytically obtained PSD and consequently the measured
PSD. Finally, generation of phase noise samples for different
types of oscillators and PLLs is now possible using the
solutions provided in this work.
0
−1
−2
0
200
400
600
800
1000
Sample
Fig. 4.
Free running VCO and Charge Pump PLL PN samples
Single Side PSD (dBc/Hz)
−30
−40
−50
−60
−70
−80
−90
−100 1
10
Fig. 5.
Simulation free running VCO
Analytic free running VCO
Simulation 1st order PLL
Analytic 1st order PLL
2
10
3
Crystal
4
10
10
Offset Frequency
5
10
6
10
Free running VCO and 1st order PLL single side PSD
−50
Single Side PSD (dBc/Hz)
III. N UMERICAL E VALUATION
For numerical evaluation of the given equations the following settings are assumed. The PLL frequency is fo = 200MHz
and a sampling frequency of fs = 10MHz is used. The
oscillator constants are chosen to be cvco = 10−14 s and
cxtl = 10−18 s. In all cases the phase detector constant is set
to 1. The 3dB corner frequency of the different filter types is
set to fGpll = 50kHz for the PLL itself, flp = 16kHz for the
low pass filter and fcp = 16kHz for the phase detector of the
Charge Pump PLL.
Fig. 4 shows PN trajectories for a free running VCO and
a Charge Pump PLL, respectively. As expected, PN samples
coming from a PLL are relatively stable around the mean. In
contrast to this, the variance of the VCO PN samples increases
linearly in time.
The single side power spectrum density (PSD) around the
first harmonic was chosen to compare the given discrete time
equations with the known analytic expression (see (13) in [5]).
The PSD (in dBc/Hz) is defined as:
PSS (fo + f )
L(f ) = 10 log10
(14)
Ptot
with PSS (fo + f ) as the single side power spectrum and Ptot
as the total output signal power. Here f represents the offset
frequency with respect to fo . Given a sinusoidal oscillator
output the PSS (·) can be easily computed by performing a
discrete fourier transform on sin(2πfo · α[n]), where α[n] are
the discrete time shifts due to PN at the PLL output.
Fig. 5 shows the PSDs of a free running VCO and a 1st
order PLL. It can be seen that the simulated PSDs match very
well with the analytic PSDs. The PN spectrum of the Charge
Pump PLL and a second order PLL are shown in Fig. 6.
Charge Pump PLL
Free Running VCO
−60
−70
−80
−90
−100 1
10
Fig. 6.
Simulation 2nd order PLL
Analytic 2nd order PLL
Simulation Charge Pump PLL
Analytic Charge Pump PLL
2
10
3
Crystal
4
10
10
Offset Frequency
5
10
6
10
2nd order PLL and Charge Pump PLL single side PSD
R EFERENCES
[1] S. Wu, P Liu, and Y. Bar-Ness, “Phase Noise Estimation and Mitigation
for OFDM Systems,” IEEE Transactions on Wireless Communications,
vol. 5, no. 12, pp. 3616–3625, 2006.
[2] T. C. W. Schenk, X.-J. Tao, P. F. M. Smulders, and E. R. Fledderus, “On
the Influence of Phase Noise Induced ICI in MIMO OFDM Systems,”
IEEE Communications Letters, vol. 9, no. 8, pp. 682–684, 2005.
[3] D. Petrovic and G. Fettweis W. Rave, “Effects of Phase Noise on OFDM
Systems With and Without PLL: Characterization and Compensation,”
IEEE Transactions on Communications, vol. 55, no. 8, pp. 1607–1616,
2007.
[4] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterisation,”
IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory
and Applications, vol. 47, no. 5, pp. 655–674, 2000.
[5] A. Mehrotra, “Noise Analysis of Phase-Locked Loops,” IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and
Applications, vol. 49, no. 9, pp. 1309–1316, 2002.