04/19/23

Frequency Domain Methods

Time domain world Frequency domain worldFourier Transform:

F

Inverse Fourier

Transform: F--1

OscilloscopeSpectrum Analyser

This transformation coud be made in real

time by using hardware or in a post procesing

scheme using software

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Time Domain Frequency Domain)2sin( iiii tfAV

time

Am

plitu

de

frequencyA

mpl

itude

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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

frequency

Sy(f)

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Line width problems

frequency

Sy(f)

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Increase time of measurement

frequency

Sy(f)

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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

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Time Domain => Frequency Domain

;frequencynormalized

2

''2

;frequency"eoustantanins"2

1

domain timein the signal))(2sin()(

00

0

0

0

0

0

ttty

dttt

dt

tdt

tvAtx

t

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Noise in frequency domain

dfffSyy

dardSIEEEperfS

fSfBW

fS

dffS

kky

y

RMS

BW

RMS

4

02

0

21

2

2

02

2

sin2

2

1

domainfrequency and

domain in time noisebetween iprelationsh

1139tan,2

1f

L

Spectral density of the phase fluctuations

Spectral density of the frequency fluctuations

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Noise in frequency domain

• 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

dfffSyy kky

4

02

0

21

2 sin2

2

1

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Common types of noise

4

3

2

1

0

)(

f

f

f

f

f

fS

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Common types of noise

22

42

11

31

00

20

11

11

22

02

20

frequency walk random

frequency flicker

frequency white

phaseflicker

phase white

)( noise of Type

fhfh

fhfh

fhfh

fhfh

fhfh

fSfS y

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Types of phase noise

fL

fS 20)(

3:frequency flicker f

4:frequency walk random f

0: phase white f

1: phaseflicker f

2:frequency white f

frequencyLog scale

Log

scal

e

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Types of frequency noise)( fS y

1:frequency flicker f

2:frequency walk random f

2: phase white f

1: phaseflicker f

0:frequency white f

frequencyLog scale

Log

scal

e

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y()

-1

0

Noise type: Whitephase

Flickerphase

Whitefreq.

Flickerfreq.

Randomwalk freq.

-1/2 1/2

Power Law Dependence of y()

As measured byAllan Deviation

1/f noise

-3/2

real noise

2

2

1tt yy

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Frequency Analysisusing a spectrum analyser

• Advantages of frequency analysis– Good detector of modulation/parasitic signals– Easier to look at high frequency noise– Can discriminate between white and flicker

phase noise!

• Disadvantages– Not very good for noise very close to the carrier

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Some examples stressing the differences between time domain and

frequency domain analysis

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White vs flicker phase noise

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White vs Flicker phase noise

Slope = -1Slope -

1

ADev() ADev()

Not very

different

Time domain

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White versus Flicker phase noise

f 0f -1

S(f) S(f)

f f

Frequency domain

No ambiguity

here!

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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.

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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

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Hydrogen maser example II

f0 or f-1

Which one?

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Hydrogen maser example III

f-3

f-2

f0

No f-4 No flicker phase noise f-1

??

ampl

itude

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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

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Another way of looking at data:the moving FFT

• Easy to implement

• Can reveal intermittent problems

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Moving FFT I

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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.

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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.