# Frequency Domain Methods - PowerPoint

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```					            Frequency Domain Methods

8/30/2010
Time Domain  Frequency Domain
Vi  Ai sin( 2  f i  t  i )
Amplitude

Amplitude
time                     frequency

8/30/2010
Why go to the Frequency Domain
• Frequency analysis can show characteristics
of oscillator:
– Noise processes
– Side bands (modulation, parasitic)
• Spectrum analyzer easy to use to show
noise far from the nominal frequency.
– Limited by the bandwidth of the measuring
system.

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Detection of parasitic signals
• Parasitic signals simply adds to the signal
• Parasitic signal modulates the signal

• In either case, if the signal is far enough
from the carrier (greater than the resolution
of the spectrum analyser available) it can be
resolved.

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Sideband detection

Sy(f)

frequency

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Line width problems

Sy(f)

frequency

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Increase time of measurement

Sy(f)

frequency

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Solution to limited spectrum
analyzers
• Record the data for a very long time using a
time measurement system
• Feed your data to a proper analyzer
software.
• Convert the time data into frequency data.
• Interpret the results

8/30/2010
Time Domain => Frequency Domain
x(t )  A sin( 2  v(t ))  signal in the time domain
1 d t 
 t    0              " instantaneous" frequency;
2 dt
t
 t   0   2  t '  0 dt '
0

 t   0  t 
y t                      normalized frequency;
0       2 0

8/30/2010
Noise in frequency domain
RMS      S  f  df
2                                Spectral density of the phase fluctuations

BW
2
   2
 0                                Spectral
S  f         Sy  f 
RMS

BW  f 
density of the
                                   frequency
fluctuations

Lf   S  f , per IEEE S tandard 1139
1
2
relationsh ip between noise in time domain
and frequency domain

 y      yk 1  yk                S  f  sin 4 f  df
1                        2
 0 2 
2                        2

2                               0

8/30/2010
Noise in frequency domain

      yk 1  yk                S  f  sin 4 f  df
1                        2
2 
2                     2
y
2                      0  0
• Not very useful to calculate the Allan
variance from the spectral density of the
noise
• Very useful to detect anomalies in the noise
pattern of a device

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Common types of noise S ( f )
0
f
1
f
2
f
3
f
4
f
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Common types of noise

Type of noise          S ( f )       2
0   Sy  f     
0               2
white phase               h2 f             h2 f
1               1
flicker phase             h1 f             h1 f
2                  0
white frequency           h0 f             h0 f
3                 1
flicker frequency         h1 f            h1 f
4                     2
random walk frequency     h 2 f            h 2 f

8/30/2010
Types of phase noise
S ( f )  0
2

L f    Log scale
random walk frequency : f 4

flicker frequency : f 3

white frequency : f 2

flicker phase : f 1
white phase : f 0

Log scale     frequency

8/30/2010
Types of frequency noise
Sy ( f )
random walk frequency : f 2
Log scale

white phase : f 2

flicker frequency : f 1
flicker phase : f 1

white frequency : f 0
Log scale   frequency

8/30/2010
Power Law Dependence of
y()
real noise
As measured by
y()             Allan Deviation
1/f noise
1
 yt   yt 2                 -1
2                   -3/2                                 1/2
-1/2
0

Noise type: White     Flicker     White    Flicker Random
phase     phase       freq.     freq.  walk freq.

8/30/2010
Frequency Analysis
using a spectrum analyser
– Good detector of modulation/parasitic signals
– Easier to look at high frequency noise
– Can discriminate between white and flicker
phase noise!
– Not very good for noise very close to the carrier

8/30/2010
Some examples stressing the
differences between time domain and
frequency domain analysis

8/30/2010
White vs flicker phase noise

8/30/2010
Time domain
White vs Flicker phase noise

Slope @ -1
Slope = -1

Not
very
different

                             

8/30/2010
Frequency domain
White versus Flicker phase noise
No
S(f)             S(f)     ambiguity
here!

f -1
f0

f                  f

8/30/2010
White and flicker phase noise
• They are often present over the same time
scale and are difficult to separate.
• ADev is unable to do it.
• FFT will tell quickly if white phase noise is
present, which is very likely for most
oscillators on short time interval.
• This is true generally at high frequency
offset from the nominal frequency.

8/30/2010
Hydrogen maser example I
• Case of a “sick” hydrogen maser
• It has excess white or flicker phase noise.
• ADev method of evaluation reveals higher
than normal noise at short term. Unable to
sort out white from flicker noise.
• FFT of phase signal sorts out the type of
noise

8/30/2010
Hydrogen maser example II

f0 or f-1

Which
one?

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Hydrogen maser example III

f-3

No f-4                    No flicker phase
amplitude

noise f-1

??

f-2
f0

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Hydrogen maser example IV
• Frequency analysis has resolved the type of
noise affecting the performance of the
maser.
• Frequency analysis has also revealed the
presence of parasitic signals.
– Some of it is due to some 4 seconds cycle
operation within the phase comparator itself

8/30/2010
Another way of looking at data:
the moving FFT
• Easy to implement
• Can reveal intermittent problems

8/30/2010
Moving FFT I

8/30/2010
Moving FFT II
Moving FFT over sixty days of phase residuals of two hydrogen
masers reveals strange parasitic signal.

Modulation
period =
one week

Parasitic
frequency
not stable

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Parasitic signal

It turns out that this signal
is generated in the path
between one maser and
the phase comparator.
There are three buffer
amplifiers and distribution
boxes along the path.
The one week amplitude
modulation tends to point
out to interference with
normal activities in the
building.

8/30/2010
Conclusion
• Frequency domain methods should be used
as well as time domain methods
• Both methods are complement of each other
• Never miss the opportunity to look at your
data from all angles possible.

8/30/2010

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