diophantine frequency synthesis - applied …seminar/seminar/20060420sotiriadisslides.pdf ·...
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DIOPHANTINEFREQUENCY SYNTHESIS
The Invasion of Number Theory to
Frequency Synthesis Systems
Paul P. [email protected]
Electrical and Computer EngineeringJohns Hopkins University
April 20th 2006
Presented at the APPLIED MATHEMATICS AND STATISTICS department, J.H.U.
1. P. Sotiriadis, “Diophantine Frequency Synthesis”, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (To appear).
2. P. Sotiriadis, “Diophantine Frequency Synthesis; A Number Theory Approach to Fine Frequency Synthesis”, IEEE International Frequency Control Symposium 2006, (June 5th 2006).
3. “Prime-Rational Frequency Synthesis Method and Frequency Synthesizers”,P. Sotiriadis, M.L. Edwards, G. Weaver, S. Cheng, D. Loizos, M. Wesley, C. Haskins, [Patent Pending – with APL].
Refs. [1] and [2] are available upon request: [email protected]
The talk presents part the of material published at:
3W
Who: Paul P. SotiriadisAssistant ProfessorElectrical & Computer EngineeringJHU
What: High Frequency Circuits: Design, Modeling, Optimization- Analog, RF, microwave, interconnects, computational- Integrated, some discrete prototypes
Where: Lab is at Stieff bldg (off campus ~ 1mile, take shuttle)Stieff 150-151Lab’s # 410-516-3801Administrative assistant: Mrs. Catonya Lester : 410-516-4276
Acknowledgements –part 1
• Profs. Daniel Naiman & James Fill
for inviting me to give this talk
• Prof. Daniel Naiman
for the long technical discussions
• Dr. Fred Toscaso
for our ongoing collaboration
Applied Math Involved in my Research
Matrix Theory
CombinatoricsGraph Theory
Nonnegative Matrices
ODEs – PDEsReg. Perturbation
Stochastic ODEsProbability
Statistics
Number Theory
OptimizationVariational Calculus
Frequency Synthesis
Linear Filters
Microwave & RF Circuits
New Analog Circuit Architectures
Nanotechnology – circuit Architec
Weakly Nonlinear Circuits
Specialized Digital circuitscollaboration with Dr. T
orcaso
seminardiscussion with Prof. Naiman
discussion with Prof. Naiman
please EMAIL me…
• If you are interested in using Applied Math to solve some real-world circuits’ problems
• Or, if you find any error(!) in the “Diophantine Frequency Synthesis” paper available to you after the talk.
• * P. Sotiriadis, “Diophantine Frequency Synthesis”, IEEE Transactions on UFFC, (To appear).
• “Prime-Rational Frequency Synthesis Method and Frequency Synthesizers”, P. Sotiriadis, M.L. Edwards, G. Weaver, S. Cheng, D. Loizos, M. Wesley, C. Haskins, patent pending
DIOPHANTINEFREQUENCY SYNTHESIS
Acknowledgements –part 2
• The APL – JHU “Disciplined Ultra Stable Oscillator” TEAM
APL: Dr. Lee EdwardsGreg Weaver
Sheng ChengWes Millard
Chris Haskins
JHU: Dimitri LoizosDr. Paul Sotiriadis
Outline of the Talk
– Frequency Synthesis & Fine Frequency Synthesis?
– Why / where?
– what / why DFS?
– DFS – how things started
– Frequency Synthesis 101
– Diophanthine Frequency Synthesis
– Open questions
What is Frequency Synthesis?
oscillatorFrequencySynthesizer
It generates a periodic signal of FIXED frequency oscf outf
)(tx)(ty
It generates another periodic signal of frequency
)2cos()( tftx oscπ= )2cos()( tfty outπ=E.g.
Automatic or manualdecision on fout and programming
of synthesizer’s parameters
t
t
t
t
E.g. MHzfosc 10=E.g.
E.g. MHzfout 4.25=
What is Fine Frequency Synthesis?
)2cos()( tfty outπ=E.g.
oscillatorFrequencySynthesizer
)(tx)(ty
outfoscf
Fine ≠ good!
)2cos()( tftx oscπ= )2cos()( tfty outπ=E.g.
MHzfosc 10=E.g.
MHzfout ...572,487,412.25=
And, in most cases…
we want to be adjustable in Small and Uniform Frequency Steps
(Frequency Resolution)
outf
MHz
MHz
MHzf
MHz
out
59,487,412.25
58,487,412.25
57,487,412.25
56,487,412.25
=
M
M
Why do we use Frequency Synthesis?
- Good (stable, low noise, etc.) oscillators provide only onefrequency*
- Finite (and not large) number of frequencies for which one can find a commercially available good oscillator
- Synthesizers: can generate many frequencies - their output signal “inherits” good characteristics from the oscillator’s signal
- In many applications we need to select among many possible frequencies
* “inconvenient” exceptions exist
Where do we use Frequency Synthesis?
- Almost all electronic products have at least one Frequency Synthesizer
very basic - very complex
( Wireless, Digital, Audio-Video, Computers,….)
- Your computer has several !to generate the many clock signals
Where do we use Fine Frequency Synthesis?
- Atomic Clocks & Time Keeping Systems
- Scientific Instruments (frequency, time, distance etc)
- Medical systems (MRI – NMR in general)
…
Diophantine Frequency Synthesis* (DFS)
What is it ? : A number theoretic approach to fine frequency synthesis
potentially resulting to : � superb frequency resolution, � very clean output signal� fast frequency hoping
using : simple and modular hardware implementations.
its foundation : Diophantine** Equations
working model ? : The APL-JHU team built the first one to demonstrate the mathematical principle.
• * P. Sotiriadis, “Diophantine Frequency Synthesis”, IEEE Transactions on UFFC, (To appear).
* From the great Greek mathematician of antiquity Diophantus, 250 A.D.
• “Prime-Rational Frequency Synthesis Method and Frequency Synthesizers”, P. Sotiriadis, M.L. Edwards, G. Weaver, S. Cheng, D. Loizos, M. Wesley, C. Haskins, patent pending
The Prototype: Math do work!
Applied Physics Laboratory (APL), J.H.U. Spring 2005.
How Everything Started: Concept of Disciplined Operation
From Gregory Weaver’s presentation, APL
(clock)
USOFine
Synthesizer
DigitalController
&Kalman
Estimator
AtomicClock
Phase & Freq.Comparator
Highly Stable & very low phase noise
spacecraft
Earth
Hz710
( )Hzk 77 10510 −⋅⋅+
13 to14 orders of magnitude!
The Basics
of
Frequency Synthesis
Operations on Periodic Signals
• Frequency Addition & Subtraction ( one way of doing it… )
21 fffout −=Σ
1f
2f
+
−
)2cos()( 11 tftx π=
)2cos()( 22 tftx π=
phase-90º
phase-90º ×
)2sin( 1tfπ
)2sin( 2tfπ
+
×( )( )tff 212cos −π
� Symbol:
( )( ) ( ) ( ) ( ) ( )tftftftftff 212121 2sin2sin2cos2cos2cos πππππ +=−
21 fffout +=Σ
1f
2f
+
+
Operations on Periodic Signals
• Waveform Shaping
� Maintain Frequency and Phase (phase offset possible)
t
t
…
Operations on Periodic Signals
• Frequency Division ; done by a Divider = Counter
� Symbol: N÷infN
ff in
out = Ν∈N
t
…for every N= 4 periods (pulses) at the input
Divider
…we get 1 period (pulse) at the output
4÷
Operations on Periodic Signals
• Frequency Multiplication ; using Phase-Locked Loop (PLL)
� symbol: m×inf inout fmf = Ν∈m
t
…for every 1 period (pulse) at the input
Multiplier
…we get 3 period (pulse) at the output
3×
Operations on Periodic Signals
• The Phase-Locked Loop (PLL)
VoltageControlledOscillator
PhaseComparator refout ff 3=reff Low Pass
Filter
3÷
t
t
2
1
3
2
3
1
4
4
5
5
( Nonlinear Dynamical System )
Operations on Periodic Signals
• The Phase-Locked Loop (PLL) - complete -
VoltageControlledOscillator
PhaseComparator outfLow Pass
Filter
m÷
N÷inf
“Phase-Comparator Frequency” :N
fin=
inout fN
mf =
Also Desirable: N : FIXED*
m : Variable
Output “Frequency Step” (Resolution) :
⇒+→ 1mmN
ff in
out =∆
Prescalerfeedback
Desirable: : LARGE*N
f in
Desirable: : SMALLN
f in
* For filtering and stability issues
Phase-Locked Loop (PLL) : summary
“Phase-Comparator Frequency” = “Frequency Step” =N
ff in
out =∆
N
m×outfinf
PLL � frequency multiplication by a RATIONAL number
one PLL is NOT enough !
Note:Nk
nk
N
n
⋅⋅≠
� Lets try to use 2 PLLs
Desirable: : LARGEN
f in
Desirable: : SMALLN
f in
2 PLLs ????
outf1
1
N
m×
inf Σ
2
2
N
m×
1f
2f
+
+
� Suppose that are sufficiently SMALL so
that the phase-comparator frequencies :
of both PLLs are sufficiently large.
21 , NN
1N
fin
2N
fin,
�
� Suppose that are FIXED (we choose them)21 , NN �
� What is the Frequency step (Resolution) of ?outf
…if we can make this Small,….. we have (almost) all we want!
Putting
Number Theory
to
Work
2 PLLs - The New Idea
in
inout
fNN
NmNm
fN
m
N
mf
21
1221
2
2
1
1
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
� Allow for the moment to take non-positive values too. 21, mm
Theorem : Given an integer a, the Diophantine equation m1 N2+ m2 N1 = a
has a solution (m1, m2) if and only if gcd(N1,N2)|a.
ΖΖΖ ),gcd( 2121 NNNN =+
,...2,1,0,),gcd(
21
21 ±±=⋅= rfNN
NNrf inout
the (output) Frequency Step
*
*
outf1
1
N
m×
inf Σ
2
2
N
m×
1f
2f
+
+
),gcd( 21 NNra ⋅=
2 PLLs - The New Idea
,
),gcd(
21
21
21
NN
fr
fNN
NNrf
in
inout
⋅=
⋅=
,...2,1,0 ±±=r
(output) Frequency step << Phase-comparator frequencies!
21NN
fin
21
,N
f
N
f inin
…if : Pairwise Relatively Prime, i.e.21 , NN 1),gcd( 21 =NN
<<
Summarizing:
1
1
N
m×
inf Σ
2
2
N
m×
1f
2f
+
+
-15 -10 -5 0 5 10 15
-2
0
2
a
n1
-15 -10 -5 0 5 10 15-5
0
5
a
n2
-15 -10 -5 0 5 10 15-1
-0.5
0
0.5
1
a
f out
2 PLLs - Example
,15
15
)5,3gcd(
in
inout
fr
frf
⋅=
⋅=
)1( Hzfin =
outf3
1m×
inf Σ
52m×
1f
2f
+
−
r
r
r
m1
m2
Generalization: k - PLLs
inf2
2
N
m×
k
k
N
m×
2
1
N
m×
MM
outfΣ⋅⋅ ⋅ ink
kk
ink
kout
fNNN
EmEmEm
fN
m
N
m
N
mf
⋅+++=
⎟⎟⎠
⎞⎜⎜⎝
⎛+++=
L
K
K
21
2211
2
2
1
1
∏≠
=ij
ji NEwhere:
Theorem : Given an integer a, the Diophantine equation m1E1+ m2E2 +…+ mkEk= a
has a solution (m1, m2 ,…,mk) if and only if gcd(E1,E2,…,Ek)|a.
*
,...2,1,0,),...,,gcd(
21
21 ±±=⋅= rfNNN
EEErf in
k
kout
L
The (output) Frequency Step!
Generalization: k - PLLs
1,...,,gcd21
=⎟⎟⎠
⎞⎜⎜⎝
⎛ ∏∏∏≠≠≠ ki
ii
ii
i NNN
Proposition : If N1, N2 ,…,Nk are pairwise relatively prime, i.e.
gcd(Ni,Nj)=1 for all i≠j, then:
E1 E2 Ek…
we can solve m1E1+ m2E2 +…+ mkEk= a for every a.⇒...
,...2,1,0,21
21
2211
2
2
1
1
±±=⋅=
⋅+++=⎟⎟⎠
⎞⎜⎜⎝
⎛+++=
rNNN
fr
fNNN
EmEmEmf
N
m
N
m
N
mf
k
in
ink
kkin
k
kout
L
L
K
K
The (output) Frequency Step!
Generalization: k - PLLs
Summarizing: • Frequency synthesis: NEW Idea
Theorem: If N1, N2 ,…, Nk are pairwise relatively prime then fout can take all values:
i.e. the frequency step is: fin / (N1N2 · · · Nk )
K
L
,2,1,0,21
out ±±=⋅= rNNN
frf
k
in
inf2
2
N
m×
k
k
N
m×
1
1
N
m×
MM
outfΣ⋅⋅ ⋅
�
k
in
NNN
f
L21
� N1 , N2 ,… , Nk can be Fixed & SMALL
� The output - Frequency Step
can be EXTREMELY SMALL �
�� Phase-Comparator frequencies : LARGEi
in
N
f
Example: 4 PLLs
2511m×
inf Σ253
2m×
2553m×
2564m×
99 104,...,104 ⋅⋅−=r
9out 104 ⋅⋅≅ inf
rf
E.g. if
- Range of :
- Frequency Step :
MHzf in 1=
outf MHzMHz 1,...,1 +−
Hzf 6out 10250 −⋅≅∆
N1N2N3N4 = 4,145,475,8409104 ⋅≅
251, 253, 255, 256 : Pairwise Relatively Prime
-100 -80 -60 -40 -20 0 20 40 60 80 100
-200
0
200
n1
-100 -80 -60 -40 -20 0 20 40 60 80 100
-200
0
200
n2-100 -80 -60 -40 -20 0 20 40 60 80 100
-200
0
200
n3
-100 -80 -60 -40 -20 0 20 40 60 80 100
-200
0
200
a
n4
-100 -80 -60 -40 -20 0 20 40 60 80 100-0.02
0
0.02
a
f out
Example: 4 PLLs
2553m×
2511m×
MHz1 Σ253
2m×
2564m×
outf
Hz02.0±
r
m1
m2
m3
m4
Finding the “Right” Solution
⎟⎟⎠
⎞⎜⎜⎝
⎛= ∏
≠ ijji NE
Note that (1) has many solutions:
Not all solutions of are “convenient”.
�We want Small for all i’s.
aEmEmEm kk =+++ K2211
|| im
(1)
Proposition: Let N1, N2 ,…, Nk be pairwise relatively primes, so gcd(E1,E2,…,Ek)=1. Then, there exists a (fixed) k×k integermatrix C, such that, det(C)=1 and for every a, the complete set of solutions of (1) is:
( ) ( ){ }Ζ∈⋅∈ − iT
kk aCmmm ρρρρ :,...,,,,...,, 12121
Finding A “Right” and Convenient Solution
Corollary : If N1, N2 ,…, Nk are pairwise relatively prime, then given an integer, a, we can find a solution (m1 , m2 ,…,mk ) of (1)such that –Ni ≤ mi ≤ Ni , for all i’s.
Moreover, we can do so, in a very computationally efficient way.
aEmEmEm kk =+++ K2211 (1)
Finding A Convenient Solution
aEmEmEm kk =+++ K2211 (1)
Proof : 1) Solve
2) ==> (ax1, ax2,…,axk) is a solution of (1)
3) Note: (1) <=>
and set: yi= axi mod Ni, i=1,2,…,k.
Then:
with
4) A desired solution is: y1-N1, y2-N2,…, yq-Nq, yq+1,…, yk.
Note that |yi| ≤ Ni and |yi-Ni| ≤ Ni for all i’s.
12211 =+++ kk ExExEx K
kk
k
NNN
a
N
m
N
m
N
m
L
K
212
2
1
1 =+++
qNNN
a
N
y
N
y
N
y
kk
k +=+++L
K
212
2
1
1
{ }kq ,..,1,0∈
Fixing the Sign and Relative variation
infoutf
1
11
N
mm +×
MM
Σ⋅⋅ ⋅
2
22
N
mm +×
k
kk
N
mm +×
In addition to choosing N1,N2,…,Nk
we also choose fixed
so that:
1) the relative variation of
the feedback dividers:
(pullability of the VCOs)
is sufficiently SMALL.
kmmm ,...,, 21
i
ii
m
Nm ±
ink
kout f
N
m
N
m
N
mf ⎟⎟
⎠
⎞⎜⎜⎝
⎛+++= K
2
2
1
12) The “central” frequency:
is the appropriate one.
Finally
infoutf
1
11
N
mm +×
MM
Σ⋅⋅ ⋅
2
22
N
mm +×
k
kk
N
mm +×
inout ff − inout ff +outf� Ranges from: to
� with Frequency Step (resolution) :k
in
NNN
f
L21
The phase-comparator frequencies
of the PLLs are:
and they can be sufficiently LARGE
while…
k
ininin
N
f
N
f
N
f,,,
21
K
Example 1: Fixed Frequency DFS
Problem: 10 MHz signal is available, a 9.285,739,4 MHz signal is needed.
? 9.285,739,4 MHz10 MHz
512
4221×
Σ495
712×
397
1305×
10 MHz 9.285,739,359 MHz++
−
Hz
MHz
1.0397495512
10
≅⋅⋅
Resolution:
Phase-comparator Frequencies:
20kHz, 20kHz, 25kHz≅
Example 2: Variable Frequency DFS
Problem: 1 MHz available, 2 MHz - 4 MHz with step <1Hz is needed.
Phase-comparator
Frequencies: 10 kHz≅
100
5500 1m+×
Σ10 MHz
−+
+101
4040 2m+×
103
1854 3m+×
outf103103
101101
100100
3
2
1
≤≤−≤≤−≤≤−
m
m
m
Hz1≅Resolution:Range: to MHzMHz 42
Output Frequency outf
Solving the Diophantine Equations
aEmEmEm kk =+++ K2211
Euclid’s algorithmaNmNm =+ 1221
Decompose it intoa sequence of k-1equations on twovariables
• Using MATLAB : Prefer Variable Precision Arithmetic (VPA)
• Binary Tree decomposition of allows for minimizing the size of all integers used in the intermediate calculations
• Fast algorithms allow for searching for “best sets” ofpairwise relatively prime integers
aEmEmEm kk =+++ K2211
�
�
Diophantine Frequency Synthesis
• The essence of the problem:
( ) inkout fpppFf ⋅= ,...,, 21
• By varying the parameters , p1 , p2,…, pk , within acceptable
intervals, we generate a SET, S, of frequencies fout
ℜ
bf
1],[
max +− iiff
ffba
af
S
1],[
maxmax+< −
−≡ii
ff
ab
ff ff
ff
baba
“Quality”
…of F & constraints on p1 , p2,…, pk
Diophantine Frequency Synthesis• Extensions
- Problems to be Solved - 1
Given: x>0 (real) minimize: xN
m
N
m −+2
2
1
1
Maxi NNN ≤<min
Maxi mmm ≤<min
“Extension” of the continuedfraction approximation???
inout fN
m
N
mf ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
2
2
1
1Quality of by varying numerators & denominators ?
Maxi NNN ≤<min
Maxi mmm ≤<min
Diophantine Frequency Synthesis• Extensions
- Problems to be Solved - 2
Extend Problem 1 to k- Fractions
Diophantine Frequency Synthesis• Extensions
- Problems to be Solved - 3
inout fN
m
N
mf ⋅⋅=
2
2
1
1Quality of by varying numerators & denominators ?
Maxi NNN ≤<min
Maxi mmm ≤<min
Given: x>0 (real) minimize: xN
m
N
m −⋅2
2
1
1
Maxi NNN ≤<min
Maxi mmm ≤<min
Something like continuedfraction approximation???
Diophantine Frequency Synthesis• Extensions
- Problems to be Solved - 4
Extend Problem 3 to k- Products
Diophantine Frequency Synthesis• Extensions
- Problems to be Solved - 5
xNNN k
−⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+ 1
11
11
121
L
ink
out fNNN
f ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛±⎟⎟
⎠
⎞⎜⎜⎝
⎛±⎟⎟
⎠
⎞⎜⎜⎝
⎛±= 1
11
11
121
LQuality of ?
Maxi NNN ≤<min
Given: x>0 (real) minimize:
Maxi NNN ≤<min
choose anycombination ofsigns (and keep it fixed)
Thank You