eecs 270c / winter 2013prof. m. green / u.c. irvine 1 random processes (1) random variable: a...
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EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 1
Random Processes (1)
Random variable: A quantity X whose value is not exactly known.
Probability distribution function PX(x): The probability that a random variable X is less than or equal to a value x.
0.5
1
x
PX(x)
Example 1:
€
X ∈ [−∞,+∞]Random variable
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 2
0.5
1
x
PX(x)
x1 x2
€
P X ∈ [x1,x2 ]( ) =P (x2 )−P (x1)
Probability of X within a range is straightforward:
If we let x2-x1 become very small …
Random Processes (2)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 3
Probability density function pX(x): Probability that random variable X lies within the range of x and x+dx.
€
pX (x) ⋅dx=PX (x+dx)−PX (x)
€
⇒ pX (x) =dPX (x)dx
0.5
1
x
PX(x)
x
pX(x)
dx
€
P X ∈ x1,x2[ ]( ) = pX (x) dxx1
x2∫
Random Processes (3)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 4
Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples.
€
E [X ] ≡X = x⋅pX (x)dx−∞
+∞
∫Mean square value E[X2]: Mean value of the square of a random variable X2 over a large number of samples.
€
E [X 2 ] = x2 ⋅pX (x)dx−∞
+∞
∫
Variance:
€
E (X −X )2[ ] ≡σ2 = x−X( )
2
pX (x)dx−∞
+∞
∫
Standard deviation:
€
σ = E (X −X )2[ ]
Random Processes (4)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 5
Gaussian Function
€
f(x) =1
σ 2πexp
−(x−X )2
2σ 2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
x
€
f(x)
€
X
€
1
σ 2π
€
0.607
σ 2π
€
X −σ
€
X +σ
2σ
€
f(x)dx=1−∞
+∞
∫
1. Provides a good model for the probability density functions of many random phenomena.
2. Can be easily characterized mathematically .
3. Combinations of Gaussian random variables are themselves Gaussian.
€
σ, X( )
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 6
Joint Probability (1)
€
P X ∈ x,x+dx[ ] andY ∈ y,y+dy[ ]( ) =pX (x) ⋅pY (y) ⋅dxdy
€
P (x,y) ≡P X ≤x andY ≤y( )
If X and Y are statistically independent (i.e., uncorrelated):
Consider 2 random variables:
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 7
Consider sum of 2 random variables:
€
Z=X +Y
x
y
€
x+y=z0
€
x+y=z0 +dz
dx
dy = dz
€
P Z∈ z0, z0 +dz[ ]( ) = pX (x)pY (y) dxdystrip
∫∫
€
= pX (x)pY (z0 −x) dx−∞
∞∫ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥dz
€
pZ(z0 )
determined by convolutionof pX and pY.
Joint Probability (2)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 8
*
Example: Consider the sum of 2 non-Gaussian random processes:
Joint Probability (3)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 9
3 sources combined:
*
Joint Probability (4)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 10
4 sources combined:
*
Joint Probability (5)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 11
Central Limit Theorem:Superposition of random variables tends toward normality.
Noise sources
Joint Probability (6)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 12
Fourier transform of Gaussians:
€
pX (x) =1
σ X 2πexp
−(x−X )2
2σ X2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
€
P X (ω) =exp−12σ X
2ω2 ⎛
⎝ ⎜
⎞
⎠ ⎟F
€
P Z∈ z0,z0 +dz[ ]( ) = pX (x)pY (z0 −x) dx−∞
∞∫ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥dz
Recall:
€
pZ(z0 ) = pX (x)pY (z0 −x) dx−∞
∞∫ F
€
P Z(ω) =P X (ω) ⋅PY (ω)
€
=exp−12σ X
2 ω2 ⎛
⎝ ⎜
⎞
⎠ ⎟⋅exp−
12σY
2ω2 ⎛
⎝ ⎜
⎞
⎠ ⎟
€
=exp−12(σX
2 +σy2 )ω 2
⎛
⎝ ⎜
⎞
⎠ ⎟F -1
€
pZ(z) =1
2π σ X2 +σY
2( )
exp−(z−Z)2
2 σ X2 +σY
2( )
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Variances of sum of random normal processes add.
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 13
Autocorrelation function RX(t1,t2): Expected value of the product of 2 samples of a random variable at times t1 & t2.
€
RX (t1,t2 ) =E X (t1) ⋅X (t2 )[ ]
For a stationary random process, RX depends only on the time difference
€
RX (τ ) =E X (t) ⋅X (t+τ )[ ] for any t
€
RX (0) =σ2
Note€
τ =t1 − t2
Power spectral density SX(ω):
€
SX (ω) =E X (t) ⋅e−jωtdt−∞
+∞
∫2 ⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
SX(ω) given in units of [dBm/Hz]
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 14
€
RX (τ ) =12π
SX (ω) ⋅ejωτdω
−∞
∞∫
Relationship between spectral density & autocorrelation function:
€
⇒ RX (0) =σ2 =
12π
SX (ω)dω−∞
∞∫
Example 1: white noise
€
SX (ω)
€
RX (τ )
infinite variance(non-physical)
€
SX ω( ) =K
€
RX (τ ) =K2π
⋅δ t( )
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 15
Example 2: band-limited white noise
€
SX (ω)
ω
€
ωp
€
−ωp
€
RX (τ )
€
σ 2 =12Kωp
€
RX (τ ) =σ2e
−ωp τ τ
€
K
€
SX ω( ) =K
1+ω2
ωp2
x
€
pX (x)
€
−σ
€
+σ
For parallel RC circuitcapacitor voltage noise:
€
K =in2
Δf⋅R2 =2kBTR
€
ωp =1RC
€
σVC2 =
kBTC
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 16
Random Jitter (Time Domain)
Experiment:
datasource
CDR(DUT) analyzer
CLK
DATA RCK
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 17
Jitter Accumulation (1)
€
Tosc =1fosc
Free-runningoscillator output
Histogram plots
Experiment:Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence.
NT
τ1 τ2 τ3 τ4
trigger €
σ1
€
σ 2
€
σ 3
€
σ 4
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 18
Observation:As τ increases, rms jitter increases.
€
τ
€
στ2
proportionalto τ2
proportional to τ
€
Jitter Accumulation (2)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 19
Noise Spectral Density (Frequency Domain)
osc osc+
Sv( )
€
Ltotal(Δω) =10 logP1Hz ωosc +Δω( )
Ptotal
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Power spectral densityof oscillation waveform:
€
dBmHz[ ]
Ltotal includes both amplitude and phase noise
Ltotal(Δω) given in units of [dBc/Hz]
€
dBcHz[ ]
Δω (log scale)
€
Ltotal Δω( )
1/ 2 region (-20dBc/Hz/decade)
Single-sideband spectral density:
1/ 3 region (-30dBc/Hz/decade)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 20
Noise Analysis of LC VCO (1)
active circuitry
C L R -R
€
ωr =1
LC
€
Z j ωr +Δω( )[ ] =j ωr +Δω( )L
1−ωr
2 +2ωrΔω + Δω( )2
ωr2
≈ jL⋅ωr
2
2Δω
Consider frequencies near resonance:
€
Q =RωrL
C L
+
_
vcinR
noise from resistor
€
ωrL=RQ
⇒ Z j ωr +Δω( )[ ] ≈ jR2Q
⋅ωr
Δω
€
ωr +Δω
€
ωr
€
Z( jω) = jωL
1−ωωr
⎛
⎝ ⎜
⎞
⎠ ⎟2
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 21
Noise current from resistor:
€
inR2 =
4kTR
⋅ΔfC L
+
_
vcinR
€
vc2 =inR
2 ⋅|Z( jω) |2
=4kTR
Δf ⋅Rωr Δω2Q
⎡
⎣ ⎢
⎤
⎦ ⎥2
=4kTR⋅ωr Δω2Q
⎡
⎣ ⎢
⎤
⎦ ⎥2
⋅Δf
Noise Analysis of LC VCO (2)
€
L Δω{ } =10 ⋅logF ⋅kTPsig
1+ωr
2Q ⋅Δω
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎧
⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪1+
ω1/ f 3
Δω
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Leeson’s formula (taken from measurements):
Where F and ω1/f3 are empirical parameters.
dBc/Hz
spot noise relative to carrier power
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 22
Oscillator Phase Disturbance
Current impulse Δq/Δt
_+Vosc
t t
ip(t)
Vosc(t) Vosc(t)
Vosc jumps by Δq/C
• Effect of electrical noise on oscillator phase noise is time-variant.• Current impulse results in step phase change (i.e., an integration).
current-to-phase transfer function is proportional to 1/s
ip(t)
€
τ 1
€
τ 2
€
Δφ=0
€
Δφ < 0
ip(t)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 23
Impulse Sensitivity Function (1)The phase response for a particular noise source can be determined at each point τ over the oscillation waveform.
€
Γ(τ ) ≡Δφ(τ )Δq
⋅qmaxImpulse sensitivity function (ISF):
€
=C ⋅Vmax(normalized to signal amplitude)
change in phasecharge in impulse
t
τ
€
Vosc(t)
€
Γ(τ )
€
Vmax
Example 1: sine wave
t
τ
€
Vosc(t)
€
Γ(τ )
Example 2: square wave
Note Γ has same period as Vosc.
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 24
Impulse Sensitivity Function (2)
€
H(s)h(t)
€
iin
€
φout
Recall from network theory:
€
Φout(s)Iin(s)
=H(s)LaPlace transform:
€
φout(t) = h(t,τ ) ⋅iin(τ ) dτ0
t
∫Impulse response:
time-variant impulse response
€
Γ(τ ) ≡Δφ(τ )Δq
⋅qmax ⇒ Δφ(τ ) =Γ(τ )qmax
⋅ΔqRecall:
ISF convolution integral:
€
φ(t) = Γ(τ )qmax0
t
∫ ⋅u(t−τ) ⋅ i(τ ) ⋅dτ[ ] =Γ(τ )qmax0
t
∫ ⋅i(τ ) ⋅dτ
from q
€
=1 forτ ∈ (0,t)
€
Γ(τ ) = ck cos kωoscτ +θk( )k=0
∞
∑
Γ can be expressed in terms of Fourier coefficients:
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 25
Case 1: Disturbance is sinusoidal:
€
i(t) =I0 cos mωosc +Δω( )t[ ] , m = 0, 1, 2, …
€
=I0
2qmax
cksin (k+m)ωosc +Δω[ ]t+θk{ }
(k+m)ωosc +Δω+sin (k−m)ωosc +Δω[ ]t+θk{ }
(k−m)ωosc +Δω
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪k=0
∞
∑
negligible significant only form = k
(Any frequency can be expressed in terms of m and Δω.)
€
φ(t) = I0qmax
ck coskωoscτ +θk( ) ⋅cos mωosc +Δω( )t[ ]{ } dτ0
t∫k=0
∞
∑
€
Γ(τ )
€
≈I0
2qmax
⋅cm⋅sin Δω t+θm( )
Δω
Impulse Sensitivity Function (3)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 26
Impulse Sensitivity Function (4)
€
φ(t) ≈I0
2qmax
⋅cm⋅sin Δω t+θm( )
Δω⇒ φ2 =
I02
8qmax2 ⋅
cm2
Δω( )2
For
€
i(t) =I0 cos mωosc +Δω( )t[ ]
ω
I
2 osc
€
×I0
2qmax
c0ω1
€
×I0
2qmax
c1ω1
€
×I0
2qmax
c2ω1
Current-to-phase frequency response:
oscωoscω1
ω1
ω1 ωosc ω1 2ωoscω1 2ωosc ω1
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 27
ω
Case 2: Disturbance is stochastic:
Impulse Sensitivity Function (5)
MOSFET current noise:
thermalnoise
1/fnoise
€
in2 (f)Δf
=4kTγgm +gm2 Kf
CgfA2/Hz
€
in
€
φ2 Δf ≈in2 Δf8qmax
2⋅cm2
Δω( )2
€
Sφ Δω( )
Δω
€
×c0
€
×c1
€
in2
Δf
ωosc 2ωosc
€
×c2
thermal noise
€
4kTγgm
€
in2
Δf
ωωosc 2ωosc
€
×c0
€
gm2 2π ⋅Kf
Cgω
1/f noise
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 28
Impulse Sensitivity Function (6)
€
Sφ Δω( )
Δω
€
×c0
€
×c1
€
in2
Δf
ωωosc 2ωosc
€
×c2
€
Sφ (Δω) =1
8qmax2
4kTγgm ⋅
ck2
0
∞
∑Δω( )
2+2π gm
2 Kf
Cg
⋅c02
Δω( )3
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
due to 1/f noise
due to thermal noise
Total phase noise:
€
c02 = Γ ( )
2
ck2
k=0
∞
∑ = Γrms( )2
ωn
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 29
Impulse Sensitivity Function (7)
€
Sφ (Δω) =1
8qmax2
4kTγgm ⋅Γrms( )
2
Δω( )2+2π gm
2 Kf
Cg
⋅Γ ( )
2
Δω( )3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
4kTγgm ⋅Γrms( )
2
Δω( )2=2π gm
2 Kf
Cg
⋅Γ ( )
2
Δω( )3
€
⇒
€
Δωn,phase=π
2kT⋅gmγCg
⋅Γ
Γrms
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
noise corner frequency ωn
Δω (log scale)
€
Sφ Δω( ) (dBc/Hz)
€
Δωn,phase
1/(Δω3 region: −30 dBc/Hz/decade
1/(Δω2 region: −20 dBc/Hz/decade
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 30
t
τ
€
Vosc(t)
€
Γ(τ )
t
τ
€
Vosc(t)
€
Γ(τ )
Example 1: sine wave Example 2: square wave
Impulse Sensitivity Function (8)
Example 3: asymmetric square wave
t
τ
€
Vosc(t)
€
Γ(τ )
€
Γ > 0 will generate more 1/(Δω3 phase noise
€
Γrms is higher will generate more 1/(Δω2 phase noise
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 31
Impulse Sensitivity Function (9)
Effect of current source in LC VCO:
Vosc+ _
Due to symmetry, ISF of this noise source contains only even-order coefficients − c0 and c2 are dominant.
Noise from current source will contribute to phase noise of differential waveform.
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 32
Impulse Sensitivity Function (10)
ID varies over oscillation waveform
€
in2
Δf=4kTγgm(t)
=(4kTγ) ⋅μCoxWL
⋅VGS (t)−Vt( ) ⎡
⎣ ⎢
⎤
⎦ ⎥
Same period as oscillation
€
in02
Δf=(4kTγ) ⋅μCox
WL
⋅VGS(DC ) −Vt( ) ⎡
⎣ ⎢
⎤
⎦ ⎥Let
Then
€
in2
Δf=in02
Δf⋅α(t)
€
α(t) =VGS (t)−VtVGS(DC ) −Vt
where
€
Γeff(τ ) =Γ(τ ) ⋅α(τ )We can use
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 33
ISF Example: 3-Stage Ring Oscillator
M1A M1B M2A M2B M3A M3B
MS1 MS2 MS3
R1A R1B R2A R2B R3A R3B+
Vout
−
fosc = 1.08 GHzPD = 11 mW
Red curve:Unperturbed oscillation waveform
Blue curve:Oscillation waveform perturbed by impulse
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 34
ISF of Diff. Pairs
ISF by tx6 for differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx6
ISF by tx5 for differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx5
ISF by tx4 for differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx4
ISF by tx3 for differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx3
ISF by tx2 for differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx2
ISF by tx1 for 3stage differential ring osc
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6 7
Radian
ISF by tx1
€
ΓM1A
€
ΓM1B
€
ΓM2A
€
ΓM2B
€
ΓM3A
€
ΓM3B
€
Γrms =1.86Γ =−0.26
for each diff. pair transistor
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 35
ISF of Resistors
€
ΓR1A
€
ΓR2A
€
ΓR3A
€
Γrms =1.72Γ =−0.16
for each resistor
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 36
ISF of Current Sources
ISF by tail tx3 for differential ring osc
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7
Radian
ISF by tail tx3
ISF by tail tx2 for differential ring osc
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7
Radian
ISF by tail tx2
ISF by tail tx1 for differential ring osc
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7
Radian
ISF by tail tx1
€
ΓMS1
€
ΓMS2
€
ΓMS3
ISF shows double frequency due to source-coupled node connection.
€
Γrms =1.00Γ =−0.12
for each current source transistor
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 37
Phase Noise Calculation (Thermal noise)
Using: Cout = 1.13 pF
Vout = 601 mV p-p
qmax = 679 fC
€
L Δf{ } =6⋅Γrms(dp)
2
8π 2Δf 2⋅4kTγgm(dp)
qmax2
+ 6⋅Γrms(res)
2
8π 2Δf 2⋅4kT Rqmax2
+ 3⋅Γrms(cs)
2
8π 2Δf 2⋅4kTγgm(cs)
qmax2
€
322Δf 2
€
122Δf 2
€
70Δf 2
€
⇒ L Δf{ } =514Δf 2
= −112 dBc/Hz @ Δf = 10 MHz
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 38
Phase Noise vs. Amplitude Noise (1)
€
Vosc(t) =Vc +v(t)[ ] ⋅exp j ωosct+φ(t)( )[ ]
ωosct
v
v Spectrum of Vosc would include effects of both amplitude noise v(t) and phase noise (t).
How are the single-sideband noise spectrum Ltotal(Δω) and phase spectral density S(ω) related?
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 39
Phase Noise vs. Amplitude Noise (2)
t t
i(t) i(t)
Vc(t) Vc(t)
0=Δtosc
qt
ωΔ
=Δ
Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude:
Amplitude impulse response exhibits an exponential decay due to the natural amplitude limiting of an oscillator ...
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 40
Δω
€
Lamp Δω( )
€
ωcQ
Δω
+
Δω
€
Ltotal Δω( )
Phase noise dominates at low offset frequencies.
Phase Noise vs. Amplitude Noise (3)
€
Lφ Δω( )
Δω
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 41
€
Vosc(t) = Vc +v(t)( ) ⋅cosωosct+φ(t)( )
≈ Vc +v(t)( ) ⋅ (cosωosct)−φ(t) ⋅ (sinωosct)[ ]
=Vc (cos ωosct)−φ(t) ⋅Vc (sinωosct) +v(t) ⋅ (cosωosct)
Phase & amplitude noise can’t be distinguished in a signal.
Phase Noise vs. Amplitude Noise (4)
noiseless oscillation waveform
phase noise
component
amplitude noise
component
Amplitude limiting will decrease amplitude noisebut will not affect phase noise.
ωosc
Sv(ω)
phase noise
amplitude noise
ω
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 42
Sideband Noise/Phase Spectral Density
€
Vosc(t) =Vc ⋅cosωosct+φ(t)( )
≈Vc ⋅ (cos ωosct)−φ(t) ⋅ (sinωosct)[ ]
€
Vc ⋅ (cosωosct)−Vc ⋅φ(t) ⋅ (sinωosct)
€
PphasenoisePsignal
=
12Vc
2 ⋅φ2
12Vc
2
=φ2
€
Lphase Δω( ) =12⋅Sφ Δω( )
noiseless oscillation waveform
phase noise
component
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 43
Jitter/Phase Noise Relationship (1)
€
στ2 ≡
1ωosc
2⋅E φ(t+τ )−φ(t)[ ]
2 ⎧ ⎨ ⎩
⎫ ⎬ ⎭
€
=1
ωosc2
⋅ E φ2 (t+τ )[ ] +E φ2 (t)[ ] −2E φ(t) ⋅φ(t+τ )[ ]{ }
€
Rφ (τ ) =12π
S (Δω) ⋅ej(Δω )τd(Δω)
−∞
∞∫Recall R and S(Δω) are a Fourier transform pair:
€
⇒ στ2 =
2ωosc
2⋅Rφ (0)−Rφ (τ )[ ]
€
Tosc =1fosc
NT
τ1 τ2 τ3 τ4
€
σ 1
€
Rφ (0)
€
Rφ (0)
€
2Rφ (τ )autocorrelation functions
€
σ 2
€
σ 3
€
σ 4
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 44
€
Rφ (0) =12π
Sφ (Δω)d(Δω)−∞
∞
∫
Rφ (τ ) =12π
Sφ (Δω) ⋅ej (Δω )τd(Δω)
−∞
∞
∫
Jitter/Phase Noise Relationship (2)
€
στ2 =
1πωosc
2⋅ Sφ (Δω) 1−e j(Δω )τ
( )−∞
∞
∫ d(Δω)
=1
πωosc2
⋅ Sφ (Δω) 1− (cos Δωτ )−j (sin Δωτ )[ ]−∞
∞
∫ d(Δω)
=4
πωosc2
⋅ Sφ (Δω) ⋅sin2 Δωτ
2
⎛
⎝ ⎜
⎞
⎠ ⎟
0
∞
∫ d(Δω)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 45
Let
€
Sφ (Δω) =a
(Δω)2
€
στ2 =
4πωosc
2⋅
a(Δω)2
⋅sin2(Δω)τ
2
⎛
⎝ ⎜
⎞
⎠ ⎟
0
∞
∫ d(Δω)
€
=4
πωosc2
⋅aπτ4
€
=a
ωosc2
⋅τ
Consistent with jitter accumulation measurements!
Jitter/Phase Noise Relationship (3)
Jitter from 1/(Δω noise:2
Let
€
Sφ (Δω) =b
(Δω)3
€
στ2 =
4πωosc
2⋅
b(Δω)3
⋅sin2(Δω)τ
2
⎛
⎝ ⎜
⎞
⎠ ⎟
ε
∞
∫ d(Δω)
€
=ζ ⋅τ 2
Jitter from 1/(Δω noise:3
^
^
^
^
€
=afosc2
⋅τ wherea≡(2π )2 ⋅a^
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 46
Jitter/Phase Noise Relationship (4)
Δf
€
Sφ Δf( ) (dBc/Hz)
-100
-20dBc/Hzper decade
• Let fosc = 10 GHz• Assume phase noise dominated by 1/(Δω)2
€
Sφ Δf( ) =a
(Δf)2
€
Sφ 2 ⋅10 6( ) =
a
2 ⋅10 6( )
2=10−10 ⇒ a=400
Setting Δf = 2 X 106 and S =10-10:
Let τ = 100 ps (cycle-to-cycle jitter):
σ = 0.02ps rms (0.2 mUI rms)
€
στ2 =
afc2⋅τ =
400
10 ⋅10 9( )
2⋅τ = 4 ⋅10−18
[ ] ⋅τ
Accumulated jitter:
€
στ = 2 ⋅10−9[ ] ⋅ τ
2 MHz
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 47
More generally:
f
€
Sφ Δf( ) (dBc/Hz)
Δfm
Nm
-20 dBc/Hzper decade
€
στ =Δfmfosc
⎛
⎝ ⎜
⎞
⎠ ⎟⋅10Nm 20⋅ τ ps[ ]
€
στ
Tosc=Δfm⋅10
Nm 20⋅ τ UI[ ]
€
στ2 =
afosc2 ⋅τ =
Δfmfosc
⎛
⎝ ⎜
⎞
⎠ ⎟2
⋅10Nm 10⋅τ
€
Sφ Δf( ) =a
(Δf)2=(Δfm )
2 ⋅10Nm 10
(Δf)2
Jitter/Phase Noise Relationship (5)
€
στ
Tosc→ Δfm⋅10
Nm+10( ) 20⋅ τ = Δfm⋅10Nm 20⋅ τ( )⋅10
0.5
rms jitter increases by a factor of 3.2
Let phase noise increase by 10 dBc/Hz:
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 48
Jitter Accumulation (1)
Kpd
phasedetector
loopfilter
€
F (s) Kvco
VCO
€
÷N€
+in out
vco
fb
€
φoutφε
=G(s) =Kpd ⋅F (s) ⋅Kvco
2πs⋅1N
Open-loop characteristic:
€
φout =NG(s)1+G(s)
⋅φin +1
1+G(s)⋅φvcoClosed-loop characteristic:
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 49
Jitter Accumulation (2)
€
G(s) =IchKvco
N⋅
1s2 (C +Cp)
⋅1+sCR1+sCeqR
Recall from Type-2 PLL:
Δω
|G|
z pω
|1 + G|
-40 dB/decade
Δω
€
Sφ Δω( ) (dBc/Hz)
€
Δωn,phase
1/(Δω3 region: −30 dBc/Hz/decade
1/(Δω2 region: −20 dBc/Hz/decade
Δω
€
φoutφvco
jΔω( )
2
1
80 dB/decade
ω
As a result, the phase noise at low offset frequencies is determined by input noise...
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 50
• fosc = 10 GHz• Assume 1-pole closed-loop PLL characteristic
Jitter Accumulation (3)
f
€
Sφ Δf( ) (dBc/Hz)
Δf0 = 2 MHz
-100-20dBc/Hzper decade
€
Sφ Δf( ) =
a
Δf0( )2
1+ΔfΔf0
⎛
⎝ ⎜
⎞
⎠ ⎟2≈
a
Δf0( )2, Δf <<Δf0
a
Δf( )2, Δf >> Δf0
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
€
⇒ στ2 =
22π ⋅fosc
2⋅Rφ (0)−Rφ (τ )[ ]
=afosc2
⋅1−e−2π⋅f0 τ
2π ⋅Δf0
€
Rφ (τ ) = Sφ (Δf) ⋅ej(2 πΔf )τ ⋅d(Δf) =
a2π ⋅Δf0( )
⋅e−2π⋅f0τ
−∞
∞
∫
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 51
Jitter Accumulation (4)
€
στ2 =
afosc2
⋅1−e−2π⋅f0 τ
2π ⋅Δf0≈
afosc2
⋅τ
afosc2
⋅1
2π(Δf0 )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
, τ <<1
2π(Δf0 )
, τ >>1
2π(Δf0 )
€
στ2
(log scale)
τ
€
1(2π) ⋅(2 )MHz
€
slope=afosc2
€
a=4×10 2
For large τ:
στ = 0.02 ps rms cycle-to-cycle jitterΔf0 = 2 MHz
fosc = 10 GHz
For small τ:
στ = 1.4 ps rms Total accumulated jitter
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 52
The primary function of a PLL is to place a bound on cumulative jitter:
τ
€
στ2
(log scale)
€
στ2
(log scale)
proportional to (due to thermal noise)
proportional to 2(due to 1/f noise)
τ
Jitter Accumulation (5)
Free-running VCO
PLL
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 53
L( ) for OC-192 SONET transmitter
Closed-Loop PLL Phase Noise Measurement
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 54
Other Sources of Jitter in PLL
• Clock divider
• Phase detectorRipple on phase detector output can cause high-frequency jitter. This affects primarily the jitter tolerance of CDR.
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 55
Jitter/Bit Error Rate (1)
Histogram showing Gaussian distribution
near sampling point
1UI
Bit error rate (BER) determined by and UI …
L R€
2σ R
€
2σ L
Eye diagram fromsampling oscilloscope
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 56
R
€
2σ
0 T
€
T2
€
t0
€
T −t0
€
PL =1
σ 2π⋅ exp−
x2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
t0
∞∫ dx
€
PR =1
σ 2π⋅ exp−
T −x( )2
2σ 2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
t0
∞∫ dx
€
pL (t) =1
σ 2π⋅exp−
t2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
pR (t) =1
σ 2π⋅exp−
T −t( )2
2σ 2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Probability of sample at t > t0 from left-hand transition:
Probability of sample at t < t0 from right-hand transition:
€
2σ
Jitter/Bit Error Rate (2)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 57
Total Bit Error Rate (BER) given by:
€
BER =PL +PU =1
σ 2π⋅ exp−
x2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
t0
∞∫ dx+1
σ 2π⋅ exp−
x2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
T−t0
∞∫ dx
€
=12erfc
t02σ
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟+erfc
T −t02σ
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
€
where (erfct) ≡2
π⋅ exp
t
∞∫ −x2( )dx
€
PL =1
σ 2π⋅ exp−
x2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
t0
∞∫ dx
€
PR =1
σ 2π⋅ exp−
T −x( )2
2σ 2
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
t0
∞∫ dx=1
σ 2π⋅ exp−
x2
2σ 2
⎡
⎣ ⎢
⎤
⎦ ⎥
T−t0
∞∫
Jitter/Bit Error Rate (3)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 58
€
•
€
•
€
•
€
•
t0 (ps)
log BER
€
σ =5 ps
€
σ =2.5 ps
€
σ =2.5 ps:
€
BER ≤10−12 fort0 ∈ 18ps, 82ps[ ]
€
σ =5 ps:
€
BER ≤10−12 fort0 ∈ 36ps, 74ps[ ]
Example: T = 100ps
(64 ps eye opening)
(38 ps eye opening)
log(0.5)
Jitter/Bit Error Rate (4)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 59
Bathtub Curves (1)
The bit error-rate vs. sampling time can be measured directly using a bit error-rate tester (BERT) at various sampling points.
Note: The inherent jitter of the analyzer trigger should be considered.
€
J rmsRJ
( )measured
2= J rms
RJ( )
actual
2+ J rms
RJ( )
trigger
2
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 60
Bathtub Curves (2)
Bathtub curve can easily be numerically extrapolated to very low BERs (corresponding to random jitter), allowing much lower measurement times.
Example: 10-12 BER with T = 100ps is equivalent to an average of 1 error per 100s. To verify this over a sample of 100 errors would require almost 3 hours!
€
•
€
•
€
•
€
•
t0 (ps)
EECS 270C / Winter 2013 Prof. M. Green / U.C. Irvine 61
Equivalent Peak-to-Peak Total Jitter
BER
10-10
10-11
10-12
10-13
10-14
€
J PPRJ
σ, T determine BERBER determines effectiveTotal jitter given by:
€
J PPRJ
€
J TJ = n⋅σ( ) + J PPDJ
€
12nσ
€
p(t)
€
12nσ
Areas sumto BER
€
12.7 ⋅σ
€
13.4 ⋅σ
€
14.1⋅σ
€
14.7 ⋅σ
€
15.3 ⋅σ