DIOPHANTINEFREQUENCY SYNTHESIS

The Invasion of Number Theory to

Frequency Synthesis Systems

Paul P. Sotiriadispps@jhu.edu

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: pps@jhu.edu

The talk presents part the of material published at:

3W

Who: Paul P. SotiriadisAssistant ProfessorElectrical & Computer EngineeringJHU

pps@jhu.edu

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.

pps@jhu.edu

• * 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