eecs 270c / winter 2013prof. m. green / u.c. irvine 1 random processes (1) random variable: a...

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