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Introduction: This application note Spectral Density of Phase Fluctuations
discusses various techniques for the S(f)
measurement of phase noise using
frequency domain analysis. Typical This is the most basic measure of phase
applications areas are characterization of noise. It is the square of the Fourier
phase noise in crystal oscillators, phase transform of the time varying phase (t)
noise of synthesized sources and phase normalized to a 1 Hz bandwidth. The
noise in PLL or other measurement spectrum is normalized because when
systems. These methods should give measuring a broadband signal such as
good estimates of phase noise for noise, the amplitude of the spectrum
frequencies upto 2 GHz. Frequency changes with the bandwidth over which
stability in the microwave region is not the measurement is made. Normalizing
covered. allows measurements with different
bandwidths to be compared. If the noise
Concepts of frequency stability: is Gaussian in nature, the amount of
noise in other bandwidths may be
Instantaneous frequency can be defined approximated by scaling the spectral
as the time rate of change of phase, i.e. density by the bandwidth. Using the
d(t ) above example of phase modulation,
2 (t ) ( t )
dt S(f) can be written as
where (t) is the instantaneous phase of 2 rms ( f )
S ( f ) =
the oscillator, and the angular frequency B
is where rms = / 2 and B is the
2f .
bandwidth (in Hz) used to measure
The instantaneous output voltage of a rms. S(f) describes the phase noise at
DUT may be written as an offset f on both sides of the carrier.
V ( t ) = [Vo + (t )] sin[2v o t + (t )]
where Vo and vo are the nominal Single Sideband Phase Noise L(f)
amplitude and frequency of the output,
and it is assumed that This is the most common way to express
(t ) a phase noise spectrum. L(f) is defined
<< 1 and as the ratio of power in a single 1 Hz
Vo
sideband at an offset frequency f to the
.
(t ) total signal power. Typically, when
<< 1 measuring phase noise, almost all of a
2v o t signal's power is in the carrier, so carrier
for substantially all time t. Then, power can be substituted for total power,
(t ) = 2v o t + (t ) and giving
Power in a single 1Hz sideband
1 . L( f ) =
v( t ) = v o + (t ) Carrier power
2
The units of L(f) are dBc/Hz ( decibels
relative to carrier per Hz bandwidth).
One can see that this quantity can be
easily measured using an RF spectrum



1
analyzer(provided it has enough
dynamic range). Residual FM is the rms of the frequency
integrated over a certain band
In the case of small total phase
modulation(total rms<<1 radian), 1
b

S v ( f )df
b-a
which is the norm when making phase Residual FM
noise measurements, one can relate L(f) a

to S(f) using phase modulation theory.
S ( f ) 2 rms where a is the lower frequency and b is
L( f ) = 10 log = 10 log the upper frequency of the band of
2 2B interest. Residual FM is commonly used
L(f) describes the phase noise at an to specify frequency stability in
offset f on one side of the carrier. communication systems. Standard bands
used are 50Hz - 3kHz, 300Hz - 3kHz
Spectral Density of Frequency and 20Hz - 15kHz. Due to averaging
Fluctuations Sv(f) over frequency, residual FM does not
convey information on the frequency
Phase and frequency fluctuations are dependence of the noise in the band.
directly related. Sv(f) is the square of the
Fourier transform of the frequency
fluctuations v(t) normalized to a 1 Hz
measurement bandwidth.

2
Sv ( f ) rms
[Hz2/Hz]
BW

This measure is useful for characterizing
noise on FM signals.

Spectral Density of Fractional
Frequency Fluctuations Sy(f)

To compare the frequency noise of
sources with different carrier
frequencies, one measures fractional
frequency fluctuations y(t) = v(t)/vo.
Sy(f) is defined as the square of the
Fourier transform of the fractional
frequency fluctuations y(t) referenced to
a 1 Hz measurement bandwidth.

y rms ( f )
2
Sv ( f ) 1
S ( f ) =
B v 2 Hz
o

Residual FM


2
Measurement Techniques for Frequency Stability

Heterodyning Techniques




'
fh




Phase noise measurements are done by Assume that cos[(t)] has essentially no
mixing the device under test with a very power in Fourier frequencies f in the
low phase noise reference oscillator region f f'h. The effect of the low-pass
which is "loosely" phase locked at 90 to filter is to remove the second term on the
the device under test. Let the "ideal" right, i.e.
reference be (V V )
v ' (t ) = Or O 1 + cos ( t )
2` VO
Vr ( t ) = VOr sin 2v o t
This separation of terms by the filter is
.
Let the DUT's output signal be (t )
correct only if << 1 for all t.
2v o
V ( t ) = [VO + ( t )] sin[2v ot + (t )]
The relative phase of the oscillators is
The output of the mixer is then adjusted to be in approximate
quadrature, i.e.
V (t ).Vr (t ) = VOr (VO + )
.
(t ) = (t ) +
2
[sin 2vo t ][sin(2vo t + )] where (t ) << 1. Now, we can use the
'


following approximation
= v( t )
(V V )
= Or O 1 +

cos ( t ) = sin ' (t ) ' (t ) and

2` VO

v ' (t ) = VOrVO ' ( t ) + VOr ' (t ) ( t )
[cos - cos(4v t + )]o 2 2



3
(t ) The output is then fed through an SR560
If << 1 for all t, then Low Noise Voltage Amplifier, with gain
VO
set to 100, 300kHz bandwidth and a 6

v ' (t ) = VOrVO ' (t ) i.e. dB/octave slope for the built in low pass
2 filter. An SR760 FFT Analyzer is used
v ( t ) is proportional to the phase
'
to characterize the spectral density of the
fluctuations. For different phase values, output signal. A feedback is provided by
mixtures of amplitude and phase noise another SR560 with the gain set to 10,
are observed. and the low pass filter set to a 1 Hz
bandwidth at 6 dB/octave. Two analog
A VCO as a reference and a low noise meters are used to tune the VCO, one for
amplifier as the feedback element serve course tuning and one for fine tuning.
as the phase locked loop to maintain From the measurement of power spectral
quadrature. The output is analyzed by an density (Vrms/Hz), the single sideband
FFT spectrum analyzer, like the SR760 phase noise is given by
or SR770, which can display the power dBc Vn(rms / Hz
spectral density(PSD) of this noise = 20 log -3
Hz K
signal, normalized to a 1 Hz bandwidth.
The 3 dB correction accounts for the fact
that when mixed to zero beat note, we
A practical system for setting up
measure the combined power of the
such a measurement is shown below. It
upper and the lower sidebands. Noise of
consists of the DUT which is a10MHz
the reference should also be corrected
oscillator, a reference oscillator REF,
for. If the reference is much quieter than
and a mixer from Mini Circuits, model
the DUT, no correction is required. The
ZAD-3SH. The mixer output is low pass
only other item which must be measured
filtered by a 1MHz filter which
essentially filters out the high frequency is K, the phase detector sensitivity.
component of 2w.



SR560 LNA SR760 FFT
DUT




X100 Meter
RF


LO X10 Meter
1000pf 5000pf 5000pf 499
SR560 LNA
1 MHz LPF

REF
VCO




4
This must be measured for each device Cross Correlation Phase Noise
tested, as it will depend on the matching Method
of the DUT into the mixer, the output
level, the output impedance of the DUT, The noise floor of a typical phase noise
the LO drive level etc. measurement system is determined by
the phase detector and the amplifiers that
To measure K(V/radian), disconnect the follow. If we measure phase noise using
feedback, replace the SR760 with an two such test systems, then the output
SR620 Time Interval Counter, and noise is uncorrelated except for the
characterize the beat note between DUT component due to the phase noise
and the LO. Measure the period (T) and between the oscillators. This technique
the rise time (tr) to rise between improves the noise floor of
thresholds separated by V. Then measurement, but takes longer since
V