Space Systems
Statistical Processes for Time and
Frequency
A Tutorial
Victor S. Reinhardt
10/17/01
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 1
Statistical Processes for Time and
Frequency--Agenda
Space Systems
• Review of random variables
• Random processes
• Linear systems
• Random walk and flicker noise
• Oscillator noise
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Space Systems
Review of Random Variables
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Continuous Random Variable
Space Systems
Ensemble of N
Identical Experiments
Unpredictable
Result
xN
Number of
Occurrences
Nx
Nx-dx
Nx+dx
x
x+dx
x
• Random Variable x
x3
– Repeat N identical experiments =
Ensemble of experiments
– Unpredictable (Variable) Result xn
x2
• Nx = Number of of times value
xn between x and x+dx
x1
• Probability density function
(PDF) or distribution p(x)
p( x )dx  lim ( N x / N)
N 
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 4
PDF and Expectation Values
Space Systems
The expectation value of f(x) is the
average of f(x) over the ensemble
defined by p(x)
b
E[f (x)]   f (x)p(x)dx
• Range of random variable
x from a to b

b
a
a
p( x)dx  E[1]  1
• Mean value = [x]
b
[x]  E[x]   xp (x)dx
a
• Standard variance = d2[x]

 d2 [ x ]  E[( x  ) 2 ]
x
a

b
 d2 [ x ]  E[ x 2 ]   2
• Standard deviation = d[x]
d [x]  d2 [x]
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Probability Distributions
Space Systems
• Gaussian (Normal) PDF
Pgauss(x)
p gauss ( x ) 
x
-4 -3 -2 -1 0 1 2 3 4
2   d
– Range = (-, +)
– Mean = 
– Standard deviation = d
• Uniform
Puniform(x)
1
p uniform ( x ) 
D
x
-D/2
e
 ( x  ) 2 /( 2  d2 )
0
D/2
– Range = (-D/2, +D/2) – Mean = 0
– Standard deviation = D/120.5
– Examples: Quantization error,
totally random phase error
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 6
Statistics
Space Systems
• A statistic is an estimate of a parameter like  or 
• Repeat experiment N times to get x1, x2, …… xN
• Statistic for mean [x] is arithmetic mean
m( x )  N
1
N
x
n 1
n
• Statistics for standard variance d[x]
N
– Standard Variance
( known a priori)
s 2p ( x )  N 1  ( x n  ) 2
– Standard Variance
(with estimate of )
s ( x )  ( N  1)
n 1
2
1
• Good Statistics
N
2
(
x

m
)
 n
n 1
– Converge to the parameter as N   with zero error
– Expectation value = parameter value for any N (Unbaised)
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Multiple Random Variables
Space Systems
• x1 and x2 two random variables (1 and 2 not ensemble
indices but indicate different random variables)
– Joint PDF = p(2)(x1,x2)
– Expectation value
(2)
means 2-variable probability
E[f ( x1 , x 2 )]   f ( x1 , x 2 )p ( 2) ( x1 , x 2 )dx1dx 2
– Single Variable PDF
p (1) ( x1 )   p ( 2) ( x1 , x 2 )dx 2
– Conditional PDF = p(x1|x2) is PDF of x2 occuring given that x1 occurred
• Mean & Covariance matrix
R k ,k '  E[( x k  M k )( x k '  M k ' )]
Mk  E[x k ]
• Statistical Independence
(k & k’ = 1,2)
– p(2)(x1,x2) = p(1)(x1)p’(1)(x2)
– Then
R k ,k '  R k ,k  k ,k '
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 8
Ensembles Revisited
Space Systems
Ensemble of N
Identical Experiments
Each with same
PDF p(x)
xN
Each
statistically
independent
x3
• The ensemble for x is a set of
statistically independent
random variables x1, x2, ….. xN
with all PDFs the same =
p(1)(x)
•EThus
[m( x )]  N
1
N
 E[x
n 1
N
n
]  [ x ]
E[s 2p ( x )]  N 1  E[( x n  ) 2 ]   d2 [ x ]
n 1
x2
d [m(x)]  d [x] 1 / N
d [s 2p ( x )]  d2 [ x ] F / N
x1
F  E[( x  ) 4 ] /  d4 [ x ]  1
(F=2 for normal distribution)
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Space Systems
Random Processes
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Random Processes
Space Systems
• A random function in time u(t)
– Is a random ensemble of functions
– That is defined by a hierarchy
probability density functions (PDF)
– p(1)(u,t) = 1st order PDF
– p(2)(u1,t1; u2,t2) = 2nd order joint PDF
– etc
Ensemble Average
E[...]
uN(t)
u2(t)
• One can ensemble average at
fixed times
E[f (u1 , t1 ; u 2 , t 2 )] 
u1(t)
 f (u , t ; u
1
t
Time Average
<…>
1
2
, t 2 )p ( 2) (u1 , t1 ; u 2 , t 2 )du1du 2
• Or time average nth member
 f (u n ( t1 ), u n ( t 2 )) 
lim T
T 
2
T/2
 
T/2
T / 2 T / 2
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f (u n ( t '1 ), u n ( t '2 ))dt '1 dt '2
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 11
Time Averages and Stationarity
Space Systems
A Stationary Non-ergodic Process
Op Amp Offset Voltage
Ensemble Average
uN(t)
• Time mean
(= 0 for random
M u (t )  u n (t )  Const processes we
will consider)
• Autocorrelation function
R u ( t1 , t 2 )  u n ( t1 ), u n ( t 2 ) 
• Wide sense stationarity
u2(t)
R u (t1  t 2 )  u n ( t1 ), u n (t 2 ) 
u1(t)
• Strict Stationarity
t
Time Average
Ergodic_Theorem:
Stationary
processes are ergodic only if
there are no stationary subsets
of the ensemble with nonzero
probability
– All PDFs invariant under tn  tn - t’
• Ergodic process
– Time and ensemble averages
equivalent
 f (u n (t1 ),...u n (t n ))  E[f (u(t1 ),...u(t n ))]
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Types of Random Processes
Space Systems
• Strict Stationarity: All PDFs invariant under time
translation (no absolute time reference)
– Invariant under tn  tn - t’ (all n and any t’)
– Implies
p(1)(x,t) = p (1)(x) = independent of time
p(2)(x1,t1; x2,t2) = p(2)(x1,0; x2,t2- t1) = function of t2- t1
• Purely random process: Statistical independence
– p(n)(x1,t1; x2,t2; ….xn,tn) = p(1)(x1,t1) p (1)(x2,t2) ….. p (1)(xn,tn)
• Markoff Process: Highest structure is 2nd order PDF
– p(x1,t1;...xn-1,tn-1 | xn,tn) = p(xn-1,tn-1 | xn,tn)
– p(x1,t1 ;...xn-1,tn-1 | xn,tn) is conditional PDF for xn(tn) given that
x1(t1) ;...xn-1,tn-1 have occurred
n
– p(n)(x1,t1; x2,t2; ….xn,tn) = p(1)(x1,t1)
 p(x
k 2
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k-1,tk-1 |
xk,tk)
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 13
Space Systems
Linear Systems
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Linear Systems
Space Systems
u(t)
v(t)
Linear
system
h(t), H(f)
U(f)
V(f)
Time Domain
• In time domain given by
convolution with response
function h(t)

v(t )   h(t  t ' )u(t ' )dt '

h(t-t')
• Fourier transform to
frequency domain (  2f )

t'
t
Frequency Domain
U(f )  [u(t )]   e jt u(t )dt


u(t )  1[U(f )]   e jt U(f )df

dB(|H(f)|)
V(f )  [ v( t )]
log(f)
H(f )  [h ( t )]
• The fourier transform of the
output is
V ( f )  H (f ) U (f )
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Single-Pole Low Pass Filter
Space Systems
t1 = R1C
t2 = R2C
h(t-t')
C
U(f)
R1
t
dB(|H(f)|)
t'
+
R2
G=-1
V(f)
V (f )
t 2 / t1
H (f ) 

U(f ) jt 2  1
t2 e  t / t2
h ( t )  ( t )
t1 t 2
( t ) 10ifif tt00
(Causal filter)
-2
-1
0
1
log(f/B3)
2
1
B3 
2t2
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(3-dB bandwidth)
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 16
Spectral Density of a Random
Process
Important Property of S(f)
U(f)
System
H(f)
V(f) =
H(f)V(f)
Sv(f) = |H(f)|2Su(f)
v(t) = du
dt
V(f) = j U(f)
Sv(f) = 2Su(f)
1
Su(f) = 2 Sv(f)

Space Systems
• Requires wide sense
stationary process
• The spectral density is the
fourier transform of the
autocorrelation function
Su (f )  [R u ( t )]
• For linearly related
variables given by
V ( f )  H (f ) U (f )
• The spectral densities have
a simple relationship
Sv (f ) | H(f ) |2 Su (f )
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Average Power and Variance
Space Systems
• Autocorrelation Function back from Spectral Density

R v (t )   [Sv (f )]   e jtSv (f )df

• Average power (intensity)

2
R v (0)  v(t )   Sv (f )df
1

• Average power in terms of input

R v (0)   | H(f ) |2 Su (f )df

• For ergodic processes

 [v]  R v (0)   | H(f ) |2 Su (f )df
2
d

(Mean is assumed zero)
• Where d2 the standard variance is
 d2 [ v]  E[ v 2 ]
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 18
White Noise
Space Systems
H(f) for Ideal Bandpass Filter
1
B
B
f
-fo



0
fo
| H(f ) |2 df  2B  2Bn
• For Single-Pole LP Filter
1

Bn 
 B3dB
4t 2 2
• Bn  B3-dB as number of poles
increases
• For Thermal (Nyquist) Noise
No = kT
• Uncorrelated (zero mean)
process
No
R u (t) 
( t )
2
• Generates white spectrum
No
S u (f ) 
2
• At output
 d2 ( v) | H(f o ) |2 N o Bn
• Bn is noise bandwidth of
system
Bn  0.5 | H(f o ) |
2
Bn  0.5 | H(f o ) |
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



2
| H(f ) |2 df


h(t ' ) 2 dt '
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 19
Band-Limited White Noise
& Correlation Time
Filter
No/2
S(f)
White
Noise
Band
Limited
Noise
B3
R(t-t')
log(f)
Space Systems
• White noise filtered by
single pole filter
– t1 = t2 = to
– Called Gauss-Markoff Process
for gaussian noise
• Frequency Domain
N
1
S v (f )  o
2 | jt0  1 |2
• Time Domain
N o e |t  t '|/ t0
R v (t  t' ) 
2 2t o
Dt
0
t-t'
• Correlation Time = to
– Correlation width = Dt = 2to
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 20
Spectrum Analyzers and Spectral
Density
• Model of Spectrum Analyzer
Model of Specrum Analyzer
In
X
e j o t
Res
Filter
Br
Br 
Det
1
2 Dt
Video Out
Filter
Bv P(f o )
1
Bv 
2T
Radiometer Formula (finite Br)
d [P(f o )]

P (f o )
1

Br T
2B v

Br
2Dt
T
T/Dt Independent Samples
Dt
Dt
Dt
Dt
Dt
Dt
Dt
Space Systems
Dt
Averaging Time T
Dt = Correlation Width = 1/(2Br)
2
2
Same as d [s p ( x )]  d [ x ] F / N
– Downconverts signal to baseband
– Resolution Filter: BW = Br
– Detector
– Video Filter averages for T = 1/(2Bv)
• Spectrum Analyzer Measures
Periodogram
(Br0)
_
ST (f o )  T 1 | U T (f o ) |2
– uT(t) = Truncated data from t to t+T
– Fourier Trans UT (f )  [u T (t)]
• Wiener-Khinchine Theorem
– When T  
– Periodogram  Spectral Density
_
lim E[ST (f o )]  S(f o )
T 
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 21
Response Function and Standard
Variance for Time Averaged Signals
y = (f-fo)/fo
v(t) = <y(t)>t,t
Response Function
1/t
h1(t’)
t
t+t
Sy(f)
|H1(f)|2
0
f
Space Systems
• Finite time average over t
v(t )  y  t ,t 
t t
1
t t

y(t ' )dt '
• Response Fn for average
t for t  t' t  t
h1,t ,t (t ' ) 10/otherwise
sin( t)
H1 (f )   je  jf ( 2 t  t)
ft
• Variance of with H1
2
 sin( ft)
2
2
1  d [ v]  
Sy (f )df
 (ft) 2
• For ft  1 | H1 (f ) |2  1
• So 12 diverges when
Sy (f )   as f  0
( for non-stationary noise)
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 22
Response Function for Zero DeadTime Sample (Allan) Variance
y = (f-fo)/fo
v(t) = <y(t)>t+T,t - <y(t)>t,t
Response Function
1/t
t
h2(t’)
t+t
t+2t
-1/t
Sy(f)1/f2
|H2(f)|2 f2
0
f
Space Systems
• Response for difference of
time averaged signals
h 2, t ,t ( t ' )  12 (h1, t , 2 t ( t ' )  h1, t ,t ( t ' ))
2
sin(

f
t
)
H 2 (f )   je  j2 f ( t  t)
ft
• Variance with H2 (Allan
variance)
4
 sin( ft)
22  d2 [ v]  
Sy (f )df
2
 (ft)
• For
ft  1 | H 2 (f ) |2  (ft) 2
• So 22 doesn’t diverge for
Sy (f )  Kf n (n  2) as f  0
( for noise up to random run)
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 23
Graphing to Understand System
Errors
Satellite Ranging
y(t-T)
X
Dy
Round Trip
Time T
Average
Dy for Time t
y = f/f
Satellite
~
y(t)
Meas
Error 
(t > T)
h(t)
T
t+T+t
1/t
t
t
-1/t
t
t+T
t+t
v(t) = <y(t)>t+T,t - <y(t)>t,t
Space Systems
• Can represent system
error as

 [v]   | H(f) |2 Sy (f )df
2
d

• h(t) includes
– Response for measurement
– Plus rest of system
• Graphing h(t) or H(f) helps
understanding
• Example: Frequency error
for satellite ranging
– Ranging: d2(t,T) = 22(T,t) =
Allan variance with dead time t
and averaging time T reversed
– Radar: d2(t,T) = 22(t,T) = no
resversal of T and t
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Space Systems
Random Walk and Flicker Noise
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Integrated White Noise--Random
Walk (Wiener Process)
Random Walk Increases as t½
18
12
v(t)
6
Space Systems
• Let u(t) be white noise
R u ( t  t ' )  0.5N 0( t  t ' )
• And
t
v(t )   u(t ' )dt '
0
• Then
R v ( t, t ' )  0.5N 0 t 
– where t< = the smaller of t or t’
0
• Note Rv is not stationary
(not function of t-t’)
-6
-12
±1(t)
-18
0
10
t
20
30
– This is a classic random walk
with a start at t=0
– The standard deviation is a
function of t
d [ v( t )]  0.5N 0 t
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 26
Generating Colored Noise from
White Noise
White
Noise
S u (f ) 
No
2
Wiener Colored
Filter
Noise
h(t-t’)
v(t)
|H(f)|2
White
Noise
u(t)
Wiener
Filter
log(f)
Space Systems
• A filter described by h(t-t’)
is called a Wiener filter
– Must know properties of filter for
all past times
• To generate (stationary)
colored noise can Wiener
filter white noise
N
S v (f )  o | H (f ) |2
2
t
No
| H (f ) | 2
2

Sv(f)
Colored
Noise
S v (f ) 
vcolored (t )   h(t  t ' )u white(t ' )dt '
log(f)
– Can turn convolution into
differential (difference) equation
(Kalman filter) for simulations
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 27
Wiener Filter for Random Walk
Space Systems
C
  
 = R1/R2
U(f)
h(t-t')
t1 = R1C
R2
+
R1
V(f)
G=-1
h
t'
t
1
H  (f ) 
jt1  
1
 lim H  (f )
jt1 0
e  t / t1
h  ( t )  ( t )
t1
h ( t )  lim h  ( t )
dB(|H(f)|)
  
H
-2
-1
0
1
log(f)
2
v( t )  lim

t
 0  
Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source
reference is listed on each page, section, or graphic utilized.
 0
h  ( t  t ' )u ( t ' )dt '
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 28
Wiener Filter for Flicker Noise
Space Systems
Heavyside Model of
Diffusive Line
R
R
Z
R  r
C
C
C  
Impedance Analysis
R
Z
Z
C
White
Current
Noise
Flicker
Voltage
Noise
v(t)
i(t)
R
• Impedance of diffusive
line
R
r
Z

j C
j
• White current noise
generates flicker voltage
noise
Sv (f ) | Z(f ) |2 Si (f )
S v (f ) 
R
C
C
Nir 1
4 f
– Ni = Current noise density
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 29
Multiplicative Flicker of Phase
Noise
Sv
0
• Nonlinearities in RF
amplifier produce AM/PM
f
AM/PM converts low frequency
amplitude fluctuations into
phase fluctuations about carrier
• Low frequency amplitude
flicker processes
modulates phase around
carrier through AM/PM
• Modulation noise or
multiplicative noise is
what appears around
every carrier
Sf
fo
Space Systems
f
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 30
An Alternative Wiener Filter for
Flicker Noise
N Independent White
Current Noise Sources
SI ( f )  I
Filter
t10
Filter
t2
2
Filter
tN
Sum (Integrate) Over Outputs
Flicker Noise
I2R 2
Sflic ker (f ) 
4f
Space Systems
• Single-Pole Filters T. C. = t
C
White Current
Source I(f)
t = RC = -1
R = Constant
+
SI(f)=I2
R
V(f)
G=-1
R
R
H t (f ) 

jt  1 j  
• Independent current sources
SI ( f )  I 2
 2I2R 2
St ( f )  2
  2
• Integrate outputs over t
Sflic ker (f )  

0
2 2
 I R d
 2 I 2 R 2dt
I2R 2
 2 2 
2
2
0  
 
4f
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 31
A Practical Wiener Filter for Flicker
Noise
• Single-pole every decade
1
H m (f ) 
f m  10 m
(1  jf / f m )
• With independent white noise
inputs
1.471
Sin  m (f ) 
fm
• Spectrum
0
Results for m = 0 to 8
10
Sf(f)
20
30
40
S v (f)
m
50
60
1
0
0.4
5
4
3
2
1
1.471
Sf ( f )  
2
2
1

f
/
f
fm
m
m
6
Error in dB from 1/f
dBerr 0.2
n
Mn
0
Max
Min
0.2
0.4
0
2
4
Logfreq
n
Space Systems
6
• For time domain simulation
turn convolutions into
difference equations for each
filter and sum
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 32
Space Systems
Oscillator Noise
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reference is listed on each page, section, or graphic utilized.
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 33
Properties of a Resonator
Space Systems
• High frequency
approximation (single pole)
|YR|2
Df
YR (f ) 
-5
0
2Q y
5
1
2( f  f 0 )
1 j
Df
fo
Df 
Q
Df = 3-dB full width
Phase (radians)
• Phase shift near fo
df/dy = 2Q
-5
0
2Q y
5
2(f  f 0 )
2Q(f  f 0 )
R  

Df
fo
R  2Qy
f  f0
y
fo
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 34
Simple Model of an Oscillator
Space Systems
Resonator
Loss = GR = YR
Loaded Q = QL
Near
Resonance
fR = -2QLy
• Amplifier and resonator in
positive feedback loop
• Amplifier
Thermal
– Amp phase noise
Flicker of
Phase
Sfamp (f) = FkT/Pin (1+ ff/f)
Oscillation Conditions
|GaGR| = Loop Gain  1
S f Around Loop = 0
– Thermal noise + flicker noise
• Resonator (Near Resonance)
fR = -2QLy [ y = (f - fo)/fo ]
Pin
Ga, fa
Noise
White Noise
Density =
FkT
Amp
Gain = Ga
Phase Shift = famp
Noise Figure = F
Flicker Knee = ff
• Oscillation Conditions
– Loop Gain = |GaGL|  1
– Phase shift around loop = 0
fR + famp = 0
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Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 35
Leeson’s Equation
Space Systems
Resonator Phase vs f/f
Response
• Phase Shift Around Loop = 0
famp = 2QLy = - fR
Phase -->
– Thus the oscillator fractional frequency
y must change in response to amplifier
phase disturbances famp
• Amp Phase Noise is Converted
to Oscillator Frequency Noise
fR = -fa
Sy-osc(f) = 1/(2QL)2 Sf-amp(f)
y
• But y = o-1df/t so
Sf-osc(f) = (fo2/f2) Sy-osc(f)
y = f/f -->
The Oscillator f/f must shift
to compensate for
the amp phase disturbances
• And thus we obtain Leeson’s
Equation
Sf-osc(f) = ((fo/(2QLf))2+1)(FkT/Pin)(1+
ff/f)
Converted Noise
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reference is listed on each page, section, or graphic utilized.
Original Amp Noise
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 36
Oscillator Noise Spectrum
Space Systems
• Oscillator noise
Spectrum
Oscillator Noise Spectrum
Sf(f)
Sf(f) = K3/f3 + K2/f2 + K1/f + K0
– Some components may
mask others
K3/f3
QL
• Converted noise
Converted
Noise
K2/f2
Amp Noise
K1/f
K0
– K2 = FkT/Pin (fo/(2QLf))2
– K3 = FkT/Pin(ff/f) (fo/(2QLf))2
– Varies with (fo/(2QL)2 and
FkT/Pin
f
Leeson’s Equation
Sf-osc(f) = (fo/(2QLf))2+1)(FkT/Pin)(1+ ff/f)
• Original amp noise
– Ko= FkT/Pin
– K1= FkT/Pin(ff/f)
– Only function of FkT/Pin
– and flicker knee
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reference is listed on each page, section, or graphic utilized.
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 37
References
Space Systems
• R. G. Brown, Introduction to
Random Signal Analysis and
Kalman Filtering, Wiley, 1983.
• D. Middleton, An Introduction to
Statistical Communication
Theory, McGraw-Hill, 1960.
• W. B, Davenport, Jr. and W. L.
Root, An Introduction to the
Theory of Random Signals and
Noise, Mc-Graw-Hill, 1958.
• A. Van der Ziel, Noise Sources,
Characterization, Measurement,
Prentice-Hall, 1970.
• D. B. Sullivan, D. W. Allan, D. A.
Howe, F. L. Walls, Eds,
Characterization of Clocks and
Oscillators, NIST Technical Note
1337, U. S. Govt. Printing office,
1990 (CODEN:NTNOEF).
• B. E. Blair, Ed, Time and
Frequency Fundamentals, NBS
Monograph 140, U. S. Govt.
Printing office, 1974
(CODEN:NBSMA6).
• D. B. Leeson, “A Simple Model
of Feedback Oscillator Noise
Spectrum,” Proc, IEEE, v54,
Feb., 1966, p329-335.
Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source
reference is listed on each page, section, or graphic utilized.
Statisitcs Tutorial V. S. Reinhardt 10/17/01 Page 38