Recurrence and Time Synthesis A Tutorial Victor S. Reinhardt June 6, 2000 .

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Recurrence and Time Synthesis Tutorial Organization. Essential ConceptsWhat is a Synthesizer?Basic Concepts of Frequency and Time SynthesisDirect Analog SynthesisAnalog Building Blocks(Digital Building Blocks used to Generate Frequencies)No VCO\'sIndirect SynthesisUses Phase or Frequency Locked VCOsDirect Digital SynthesisUses Digital Processing Techniques to Generate OutputDigital Circuits used to Pro
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Recurrence and Time Synthesis A Tutorial Victor S. Reinhardt June 6, 2000

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Frequency and Time Synthesis Tutorial Organization Basic Concepts What is a Synthesizer? Fundamental Concepts of Frequency and Time Synthesis Direct Analog Synthesis Analog Building Blocks (Digital Building Blocks used to Generate Frequencies) No VCO\'s Indirect Synthesis Uses Phase or Frequency Locked VCOs Direct Digital Synthesis Uses Digital Processing Techniques to Generate Output Digital Circuits used to Process Numbers No VCO\'s

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Basic Concepts What is a Synthesizer? One or More Reference Sources f r1 One or More Input Reference Sources f r1 … f rn Translation to New Frequency f o Phase or Frequency Coherent With References Basic Properties Frequency Range Frequency Resolution Switching Rate/Settling Time DC Power, Weight, Cost, and so on. Synthesizer . . . Yield f o f rN Phase/Frequency Stability (Time Domain, Environmental Effects) Spectral Purity (Frequency Domain, Spurs, Noise)

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Positive Zero Crossings at t n =nT o  n =2n V Amplitude A t  Sine Wave Period T o =2 V Amplitude A t  Pulse Ideal Periodic Waveform Periodic Function F( F) V = A F( F ) F = Phase of Function F( F + 2) = F( F ) In Time Domain F = w o t w o = Angular Frequency w o = 2 p f o f o = 1/T o = Frequency F not a True Observable Measurement Depends on Inverting F( F ) Must Keep Track of Number of Cycles for Multiples of 2 p Best Determined at Zero Crossings where Slope Large

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Non-Ideal Waveform Amplitude and Frequency Now Function of Time Angular Frequency Error  w  w = d f/dt Frequency Error  f  f =  w/2p Fractional Frequency Error y y =  w/w o = d f/f o y = (d f/dt)/w o V Peak Variation Amplitude Error t Zero Crossing Variation Time or Phase Error Force Nearly-Periodic Waveform into Periodic Form V = ( A + a(t) )·F[ w o t + f (t) ] a(t) Amplitude Error f (t) Phase Error

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For Sine Wave Near Zero V = A ( w o t+ f (t) ) V F , t d V f, d t Additive Noise and Phase & Time Error Additive Noise d V Generates Phase Error f( t ) = d V(t)/A f in Radians Equivalent to Noise/Signal Ratio dB( f ) Equivalent to dBc Time Error in Positive Zero Crossing d t = - f/w o = - d V/(A w o ) Note Minus Sign Positive d V Negative d t Positive f For Non-Sine Wave: Effective An is Determined by Slope Near Zero Complex Representation d V Q f A d V I d V I = d V Q = d V

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Clock Reading versus Time Error Basic Clock A Basic Clock Contains a Frequency Reference and a Cycle Counter Zero Crossing Time Error d t = - f/w o Compares Equivalent Zero Crossings at Different Times Clock Reading Error x = f/w o Compares Cycle Counts or Normalized Phases at Same Time Note That x =  y dt But d t = -  y dt f o Frequency Reference Cycle Counter x d t Ideal Source

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dw o K dw r dw r y o = y r w o K w r w r f o K f r f r x o = x r w o K w r w r Ideal Coherent Synthesizer Coherent Frequency Translation by Factor K Multiplies the Input Frequency f r by a Factor K Ideal: Doesn\'t Add Noise Input Phase Error f r Also Multiplied by K The Phase Error Integral of the Angular Frequency Error The y and x of a Reference Oscillator are Independent of the Final Output Frequency f r f o = Kf r Ideal Coherent Synthesizer f r f o = K f r Frequency Reference

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Important Property of S(f) Filter H(f) V(f) = H(f)U(f) U(f) S v (f) = |H(f)| 2 S u (f) dx y(t) = dt d f y(t) = w o - 1 dt w 2 S y (f) = S (f) w o 2 Spectral Density Review A Random Variable u(t) is Wide Sense Stationary if the Autocorrelation Function R is just a Function of t R u ( t ) = T - 1  T u(t+ t )u(t) dt The Spectral Density is the Fourier Transform of R u ( t ) S u (f) =  e j2 p ft R u ( t ) d t For Frequency Translation K S f - yield (f) = K 2 S f - input (f) S y-yield (f) = S y-input (f) S x-yield (f) = S x-input (f) S y (f) = w 2 S x (f)

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Spur Spurs in Time Domain Phasor Diagram Spurious Signal Rotates around Main Phasor at 2 pD f Time Domain Measurements are Sampled at Multiples of t n = nT o Generates Regular Pattern at Aliases of 1/D f 2pD f V o (t) Spur at f o + D f w o f x Discrete Samples When Phasor Crosses Real Axis Phase Error Plot s(t) Allan Variance Counter Histogram t Noise

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Mixers f a f a  f b f b f Switches f 1 f in f 2 f o x f out x . . . . f n Direct Analog Synthesis Multipliers Directly Generates f o Frequency without VCO Multiplicative Devices Multipliers Dividers x Conserved Additive Devices Mixers Others Filters Switches Amps Also Add Their Own Noise x N f Nf f N f Dividers f/M ÷ M f/M x Filters f b f a +f b +f c f Amplifiers x

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... f 2 f 3 f 1 Switch Matrix ... 9f r 2f r f r Reference Generator f r Typical Direct Analog Synthesizer: Divide & Mix f o = f 1 + f 2/10+f 3/100 Two Parts of Synthesizer Switched Reference Section Generates References 0, f r ,… 9f r Switch Refs to LO\'s f 1 , f 2 , f 3 … Divide and Mix Section (3 Stages Shown) Divide f 3 =N 3 f r by 10 Mix with f 2 =N 2 f r and Filter to Produce f 2 +f 3/10 (Bypass Mixer if N 2 =0) Repeat Divide, Mix, and Filter with f 3 =N 3 f r End Result f o = [N 1 + N 2/10+N 3/100 + … ]f r Each N Selects Digit of Output f k =N k f r (N k = 0 to 9) f 1 =N 1 f r + 10 f 2 +f 3/10 f 2 =N 2 f r f 3/10 + 10 f 3 =N 3 f r

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S(f) 1/f Noise White Noise Floor 1/f Knee f Si 1-10 KHz GaAs, InP 0.1-1 MHz 1/f Knees Component Design Parameters Phase Noise Characterization of Devices General Parameters Frequency Response Speed (Switches) DC Power Cost, Weight, & Size Phase Noise (See Left) Phase Stability (Time, Environment) Filters: Phase Shift over Temperature Critical Issue Spurs Mixing IM\'s Switches: On/Of Loss Ratio Determines Spurs Unwanted Multiplier Orders Cascaded Multipliers & Dividers xN 1 xN 2 xN 3 ÷N 3 ÷N 2 ÷N 1 These Most Critical for S (f) Make Lowest Noise and Highest N All x Contributions the Same

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Asynchronous Counter FF ... FF f in f out Clean-up Circuit f in f\' out f out One Shot Synchronous Counter ... f out f in FF Regenerative Delay-t Divider Set R-S FF Q f in f out Delay t Reset Frequency Dividers (Counters) Asynchronous (Ripple) Lowest Power Most Phase Variation (Cascading Delays) Can Use Clean-up Circuit Synchronous High Power Lowest Phase Variation Dual-Modulus Almost Lowest Power Low Phase Variation Limit on Divide Number Regenerative & Analog Dividers Can be Very Simple & Low Noise Limited Frequency Range Susceptible to Cycle Slips

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Dual Modulus Counter ÷ P/P+1 f in ÷ Out P/P+1 Control ÷A Reset A Counter Reset M Counter ÷M f out Dual Modulus Counter Dual Modulus Counter High Speed Dual Modulus ( ÷ P/P+1 ) Prescaler 2 Low Speed ( ÷M, ÷A ) Counters f out = f in/(MP+A) M  P, A = 0 to P-1 Minimum Divide Ratio = P(P-1) Operation Prescaler Starts with ÷(P+1) Prescaler Switches to ÷P when A Count Reached An and M Counters Reset when M Count Reached (Thus Must Have M  A ) Prescaler Switches Back to ÷(P+1) For Contiguous Divide Numbers A = 0 to P-1 (so Must Have M  P-1 )

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t Frequency Multipliers f Nf Nonlinear Device Filter 1. Resistive Diode and Mixer Broadband & Loss Low Efficiency for High Harmonics 2. Step Recovery Diode & Varactor Narrowband (to Match 5  Input Z) Higher Efficiency for High Harmonics 3. Transistor Highest Efficiency (Gain) Too High Drive Can Cause Slow Damage from Avalanche Breakdown 2 & 3 Susceptible to Parametric Oscillations Sharpness of Distortion Features ( t ) Determine Amplitude of High Harmonics Good Efficiency Limit Nf  1/t Device Degradation Due to Overdrive

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f R f LO f IF Harmonics of f LO Harmonics of f R IF to Spur Ratios (dB) Mixers Many Types of Mixers Single Device Single, Double, Triple Balanced SubHarmonic (Doubles LO Input) Single Sideband Higher Order Mixers Suppress Spurious Mixing Products f goad = Nf lO - Mf R ( N,M ) = Spur Order Major Issue: Keeping Spurs Away From f IF (WJ-M9E)

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Loop Filter f o = T - 1 (f r ) VCO Freq Control Frequency Translation Error Signal Phase or Frequency Discriminator T(f o ) f r Indirect Synthesis f o = Nxf r Loop Filter VCO ÷N Error Signal f o/N f r Example: Divider Loop Indirect Synthesis Utilizes Phase or Frequency Locked VCO to Act as: Operation Inverter VCO Output f o Goes Through Frequency Translation T(f o ) Phase or Frequency Discriminator Compares f r to T(f o ) and Generates Error Signal Through Loop Filter and VCO Frequency Control, Error Signal Driven to Zero so f r = T(f o ) Thus VCO Output is Inverse of T f o = T - 1 (f r ) Tracking Filter Uses Bandwidth Properties of Loop to Filter Reference Signal

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f = f o - V o/s V i = f r - f VCO f r f V i G(f) V o V o = G(f) V i H(f) 1-H(f) 1 f n f n Basic Phase Locked Loop Definitions Open Loop Gain G(f) Output Phase Error f Reference Phase Error f r VCO (Free Running) Phase Error f o Closed Loop Response H(f) = f/f r = G(f)/(s + G(f)) H(f) has Low Pass Response with Knee at f n 1-H(f) has High Pass Response with Knee at f n Output Phase Error f = H(f) f r + (1-H(f)) f o Reference Characteristic f << f n VCO Characteristic f >> f n Idealized PLL

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S (f) Free Running VCO S (f) Reference S (f) Optimum PLL S (f) f Optimum f n Optimum Loop Bandwidth Free Running VCO: Higher Near In Noise Lower White Noise Floor Reference Lower Near In Noise Higher White Noise Floor Optimum Loop Bandwidth f n for Integrated Noise is Where Curves Cross May Have Other Reasons not to Choose this f n Such as Settling Time Requirement

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Oscillator Noise Characteristics Loss = L Reso

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