Outline - Enrico Rubiola

In memoriam of Jacques Groslambert
The sampling theorem
in Π and Λ digital dividers
Claudio E. Calosso∇ and Enrico Rubiola
∇ INRIM, Torino, Italy
CNRS FEMTO-ST Institute, Besancon, France
Outline
• Theoretical introduction
• Π and Λ digital dividers
• Experiments
home page http://rubiola.org
Motivations
• Seminal article by W. F. Egan (1990)
• Milestone in the domain, never forget it
• However, TTL and ECL logic families are now obsolete
• Microwave (photonics) –> highest spectral purity
•Transfer the spectral purity to HF/VHF
• Dividers are more comfortable than multipliers
• NIST now uses analog dividers
•Nowadays digital electronics is fantastic
• CPLD & FPGA –> Easy to duplicate
• High number of gates for cheap
• High toggling frequency (1.5 GHz)
W. F. Egan Egan WF, Modeling phase noise in frequency dividers, IEEE T UFFC 37(4), July 1990
E. Rubiola & al, Phase noise in the regenerative frequency dividers, IEEE T IM 41(3), June 1992
A. Hati & al, Ultra-low-noise regenerative frequency divider…, Proc IEEE IFCS, May 2012
2
3
The gearwork model
jitter xo = xi
phase 'i
1
D 'i
⌫o
⌫i
• Keeps the input jitter x(t)
(phase-time fluctuation)
• Scales down
• φ by 1/D
• Sφ by 1/D2
N D teeth
𝞅
N teeth
1
⌫i
D
S (f )
⌫o =
• The noise-free divider
actual (output stage)
1/D2
• Sφ of the output stage adds up
and often dominates
input
1/D2
• In the real divider
[rad]
[rad2/Hz]
gearbox
1/D2
phase 'o =
W. F. Egan Egan WF, Modeling phase noise in frequency dividers, IEEE T UFFC 37(4), July 1990
f
Sampling and aliasing
— Energy conservation applies to the unfiltered signal —
SxII(ƒ)
Input signal
(unfiltered wide-band noise)
ƒ
ƒ
• With white noise, the
PSD increases by B/ƒN
(Bandwidth / Nyquist ƒ)
etc.
s
3fs
ia
ia
2fs
al
as
s
al
alia
ali
fs
s
fs
s
2fs
alia
3fs
main
etc.
• Multiple aliases
overlap to the main
part of the spectrum
ali
as
Reconstructed signal
(aliased)
fN =
1
fs
2
Nyquist frequency
Downsampling increases the (PM) noise spectrum
Low ƒN
N=
2
/fN
ƒ
fN
N=
SxI(ƒ)
SxI(ƒ)
High ƒN
2
/fN
ƒ
fN
4
Aliasing and 1/ƒ noise
SxII(ƒ)
Proportionally lower power
in the higher-ƒ aliases
Input signal
(unfiltered wide-band 1/f noise)
etc.
etc.
2fs
fs
fs
main
3fs
3fs
ƒ
alias
al
alias
alia
s
alias
2fs
ali
s
as
ia
Reconstructed signal
(little aliasing effect)
Small effect on the
overall noise spectrum
ƒ
fN
1
= fs
2
Nyquist frequency
5
6
PM-noise aliasing in the input stage
saturated
in
out
t
analog gain
Convert the input sinusoid into
a square wave, as appropriate
t
equivalent sampling function
• Edge-sampling at 2νi inherent in the sin-to-square conversion
• Full-bandwidth (B) noise is taken in
• The phase-noise Nyquist frequency is νi
• The sampling process increases the noise by B/νi
Eventually, clipping removes the AM noise [Pfaff 1974]
7
Aliasing in Π divider
Regular synchronous divider
The Greek letter Π recalls the square wave
ΠΠΠΠ
input sampling frequency 2⌫i
t
input
jitter is discarded
÷ 10
jitter is
transmitted
jitter is discarded
jitter is
transmitted
jitter is discarded
jitter is
transmitted
jitter is
transmitted
out
t
• Raw decimation without low-pass filter
• Aliasing increases Sφ by D
• Overall, Sφ scales down by 1/D
input
aliasing
Π divider
1/D 2
• The divider takes 1 edge out of D
White S𝞅(f ) = b0
• The gearbox scales Sφ down by
squarewave
sin clock
1/D2
1/D
1
output sampling frequency ⌫o = 2 D
⌫i
Λ divider
8
The Λ divider – Little/no aliasing
New divider architecture
Series of Greek letters ΛΛΛΛΛ recalls the triangular wave
÷ 10
D
• Gearbox and aliasing –> 1/D law
shift register
in
out
• Add D independent realizations
shifted by 1/2 input clock,
• reduce the phase noise by 1/D,
• … and get back the 1/D2 law
squarewave
sin clock
output
White S𝞅(f ) = b0
input
input
aliasing
Π divider
1/D
shift register
1/D
D
2
÷ 10
Λ divider
The names Π and Λ derive from the shape of the weight functions in our article on frequency counters
E. Rubiola, On the measurement of frequency … with high-resolution counters, RSI 76 054703, 2005
Experimental method
9
Large input PM noise is used to emphasize the effect of aliasing
• Intentionally high PM noise
at the input
• The scaled-down input
noise is higher than the
output-stage noise
• Large attenuation/ampli –> noise
input
𝞅
S (f )
!
1/D2
• Digital instruments for phase-noise
measurement can handle
1/D2
ƒinput ≠ ƒreference
gearbox
output stage
f
• Correlation reduces the background
10
Dividers under test
EPM3064A CPLD (Altera MAX 3000 Series, 64 macro-cells, speed grade 7 ns)
in
÷ 10
out
Π divider
– the one everybody knows –
Multi-buffer Π divider
Λ divider
÷ 10
÷ 10
in
D
shift register
in
D-type register
out
out
÷ 10
The outputs are arguably independent
Try to reduce the output-stage noise
D
shift register
White noise:
The clock edges are independent
Correct for aliasing
Results – Test on aliasing
Λ and dividers
(multi-buf Π config)
0.5
–20 dB
dB
dis
cre
pa
nc
y
internal clock
sin input +4 dB
Π divider
–9.3 dB
output
(theory –10)
Λ divider
output
• Flicker region
• Negligible aliasing
• 1/D2 law (–20 dB)
–18.7 dB
(theory –20)
• White region
• Aliasing in the front-end –> +4 dB
• 1/D law and 1/D2 law
11
Phase noise of real dividers
Multibuffer Π divider
÷ 10
in
b
–1
D-type register
=–
out
b
–1
12
0d
B
=–
13
–1
16
–1
0.5
ar
28
.5
.5
dB
dB
dB
tif
ac
ts?
nt
cie ?
uffi ing
ins erag
av
multiple outputs are
expected to reduce the
output-stage noise
– not happened, why? –
Λ and Π dividers
(multibuffer Π config)
b–1 ≈ –156 dB
b0 ≈ –165 dB
experimental problems still present
• Flicker region –> Negligible aliasing
• The multibuffer Π divider is still not well explained
• The Λ divider exhibits low 1/ƒ and low white noise
12
Allan deviation of real dividers
• Slope 1/τ, typical of white and flicker PM noise
• The Λ divider performs 2×10–14 at τ = 1 s, 10 MHz output
13
The bottom line
14
• Aliasing in traditional dividers
• Increases white noise
• Has little effect on flicker
• Flicker in multi-buffer Π divider not understood yet
• The new Λ divider
• Is little/no affected by aliasing
• Exhibits the lowest PM noise
flicker: b–1 ≈ –130 dB
white: b0 ≈ –165 dB
• Features 2×10–14 at τ = 1 s, 10 MHz output
home page http://rubiola.org
Thanks – J. Groslambert, V. Giordano, M. Siccardi, J.-M. Friedt
Grants from ANR (Oscillator IMP and First-TF network), and Region Franche Comte