freq reqs for comm.ppt, V. S. Reinhardt. Page 1 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. The Calculation of Frequency Source Requirements for Digital Communications Systems Victor S. Reinhardt 08/25/04 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control 50th Anniversary Joint Conference, Montreal, August 24-28, 2004

freq reqs for comm.ppt, V. S. Reinhardt. Page 2 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. The Calculation of Frequency Source Requirements for Digital Comm Systems Introduction Frequency sources (oscillators, synthesizers, etc.) are an important part of digital communications systems Paper will discuss the derivation of frequency source requirements from over-all digital comm system parameters Will be tutorial treatment for those not familiar with digital comm theory but familiar with time & frequency theory Frequency source properties directly impact the performance of digital comm systems –Impact link acquisition & loss of acquisition—T&F community familiar with synchronization issues—Will not be covered here –Impact bit error rate (BER) performance--Paper will address this Will utilize quadrature phase shift keyed (QPSK) systems for concrete examples –But theory applicable to other systems

freq reqs for comm.ppt, V. S. Reinhardt. Page 3 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Basic Digital Comm Concepts Signals Carrying Digital Information At the transmitter a carrier is modulated in a regular time sequence of symbols to produce a digital communications signal or waveform A symbol is a temporal waveform in some modulation space representing a single digital word of information At the receiver the signal is sampled at discrete decision epochs to determine a modulation value of the carrier The modulation value is converted into a digital or data word by comparing it to decision thresholds The symbols occur at a symbol rate R s =1/T c (T c = clock period) The bit or data rate R = WR s (W = bits per symbol or word) Example: Unshaped (Rectangular) Symbols in PAM Decision Epochs Time Value Decision Thresholds Symbol 3 Symbol 2 Symbol 1 (1,0) (1,1) (0,1) (0,0) (2-Bit) Digital Words TcTc t3t3 t2t2 t1t1 Carrie r Axis Signal

freq reqs for comm.ppt, V. S. Reinhardt. Page 4 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Shaped Symbols Unshaped (rectangular) symbols are not bandwidth efficient –Sinc functions in freq domain Shaped symbols are sinc-like functions in time domain –Produce more bandwidth efficient trapeziodal functions in freq domain –Do not interfere with each other at decision epochs The price one pays for shaping is more stringent timing — Un- shaped — Shaped t n /T c Symbols in Time Domain Shaped Transmission 1 0 Composite Signal t n /T c — Un- shaped — Shaped f/R s Symbols in Freq Domain

freq reqs for comm.ppt, V. S. Reinhardt. Page 5 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Inter-Symbol Interference (ISI) & Eye Patterns Eye Pattern = Graph of the modulation value vs time at the receiver plotted modulo 1-symbol period (as in a scope trace) Eye opening = region with no value trajectories in it Inter-symbol Interference (ISI) = Contamination at decision epoch of modulation value by adjacent symbols –Ideal Decision epoch—no ISI –Clock errors cause the decision epoch to wander off the best decision epoch increasing the ISI –Sensitivity of ISI to clock timing = Slope of eye opening at decision epoch Even unshaped (square) symbols generate such eye patterns because of receiver and channel filtering necessary to limit signal BW & noise Shaped symbols have narrower eye widths than unshaped ones From: Telecom Glossary 2000, American National Standard for Telecommunications, T , Modulo Symbol Time 0 -+ Eye Pattern Inter-Symbol Interference Ideal Decision Epoch Eye Opening (No Trajectories) Shaping Narrows Eye Width

freq reqs for comm.ppt, V. S. Reinhardt. Page 6 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Types of Digital Modulation Type of carrier: RF carrier or subcarrier, baseband voltage, etc. Parameter modulated: amplitude, phase, frequency, etc. Modulation Order (or number of digital states 2 W ): binary, quadrature, M-ary Shaped or unshaped Coherent, incoherent, differential phase Synchronous & asynchronous data clock timing (used in hardline systems) Binary, M-ary FSK Freq (0)(1) Frequency Shift Keyed Time Pulse Position or Width Modulation PWM Time Pulse Amplitude Shift Keyed or Modulation Amplitude PAM Hybrid Modulation M-ary Quadrature Amplitude Shift Keyed or Modulation Coherent Phase-Frequency Shift Keyed Minimum Shift Keyed (Binary CPFSK) 16-QAM or 16-QASK (4-Bit word) I Q Phase Shift Keyed BPSK, QPSK, 8PSK,.., DPSK (0,0) (0,1) (1,0) (1,1) Complex RF Envelope I Q

freq reqs for comm.ppt, V. S. Reinhardt. Page 7 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Bit Error Rate (BER) vs E b /N o Key Comm System Parameter –Rx thermal noise must limited by a filter –For an ideal system the Rx filter’s bandwidth is equal to the symbol rate R s = R/W –The ideal SNR = P rx /(N o R s ) = P b /(N o R) = E b /N o N o = Thermal noise density P b = P rx /W = Power per bit E b = P b /R = P rx /R s = Energy per bit BER vs E b /N o the canonical comm link characterization BER degradation is the extra E b /N o over ideal system to achieve same BER as ideal Error correction coding (ECC) allows up to N bit errors to be corrected in a group or block of bits--Improves BER above a certain E b /N o The bit error rate (BER) is the probability that a received bit is incorrect The BER is a function of the SNR at the digital receiver Uncoded BER Ideal - Actual E b /N o - dB Ideal - Actual Error Correction Coded BER BER Degrad- ation E b /N o - dB

freq reqs for comm.ppt, V. S. Reinhardt. Page 8 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. BER Degradation and ISI Causes of ISI –Symbol distortion –RF carrier phase errors & jitter –Data clock errors & jitter Simple BER degradation Models Worst case model: BER deg = -20Log 10 (1-  V/V) Noise Model: Use theoretical curve with E b /N o  P rx /(N o R s +  V 2 ) Decision thresholds Thermal noise in BW R s (  = N o R s ) causes occasional bit errors BER (uncoded) = ½*Erfc( V/  ) = ½*Erfc((E b /N o ) ½ ) ISI generates non-thermal jitter  V n When V +  V n is closer to decision threshold higher BER with thermal noise Net effect to increase BER for given E b /N o Actual QPSK system (no thermal noise) Sampled values V(±1 ±j)/2 0.5 at decision epoch No ISI (jitter) without thermal noise Jitter  V n Ideal QPSK System

freq reqs for comm.ppt, V. S. Reinhardt. Page 9 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. LO Phase Jitter Requirements in RF Carrier Digital Comm Systems At the transmitter (Tx) an LO and a clock are required At the Receiver (Rx) –a clock recovery loop is always required to track the Rx clock to the Tx clock –a carrier rec loop at the Rx LO required for phase coherent symbols Recovery loops track out relative Rx-Tx LO and clock jitter for fourier frequencies < recovery loop bandwidths This is very important in defining the appropriate jitter statistics in terms of power spectral densities (PSD) Symbol Modulator Data Encode User Data Error Correction Encryption Framing ~~ Symbol Demod -ulator Data (Sampling) Clock ~~ LO RF Xmission Data Decode Recover Loops User Data Data (Sampling) Clock LO Transmitter (Tx)Receiver (Rx) Rx LO recovery loop only for phase coherent symbols Typical RF Carrier Comm System

freq reqs for comm.ppt, V. S. Reinhardt. Page 10 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Carrier Phase Jitter and ISI Phase jitter produces ISI in quadrature systems through I-Q cross- talk Phase jitter much less of an issue in BPSK because there is no Q channel (Just produces loss of power) The definition of the appropriate of phase variance    is determined by the phase coherence properties of the system Phase Jitter Q-Symbol jitter produces cross-talk in I-Channel, etc. Rx Q-Axis Rx I-Axis RMS ISI  V*Sin(     V

freq reqs for comm.ppt, V. S. Reinhardt. Page 11 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. RF Carrier Phase Jitter and Coherent, Incoherent, and Differential Systems Coherent symbols –Tx symbols decoded relative to phase of Rx LO –Rx-Tx LO phase independent over many symbols (recovery loop time constant T p  1/ B p >> T c ) –Must have T p >> T c so thermal noise does not degrade BER through recovery loop (Phase) Incoherent symbols –Inter-symbol phase unimportant –Ex: Freq or amplitude modulation Differential symbols –Data coded so change in symbol phase carries information –Phase matters only from symbol to symbol –No Rx carrier recovery loop needed –BER vs E b /N o worse than for coherent systems Incoherent (i.e., FSK, ASK) Freq Symbols Coherent (i.e., QPSK) Xmitted Symbols Differential (i.e., DPSK) Phase only matters over one symbol Rx & Tx LO phase difference important over many symbols Phase unimportant Decoded Symbol X

freq reqs for comm.ppt, V. S. Reinhardt. Page 12 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Definition of phase jitter variance for coherent systems    = 2  0 Rs/2 L  (f) |1-H p (f)| 2 df     2  Bp Rs/2 L  (f) df –H p (f) = recovery loop response function –Assumes channel bandpass filter width = symbol rate R s –L  (f) = sum of SSB  -PSD’s of all LO’s Because of the high pass cut-off from the carrier recovery loop, this standard variance exists even for flicker of frequency noise Rule of thumb for QPSK phase jitter –   should be < 1-3 ° for < 0.1 dB BER degradation Calculating LO Phase Jitter for Coherent Systems For oscillator x N The phase jitter req must be reduced by N to compensate for x N multiplication L  (f) (single sideband noise) f Sum of all LO’s Carrier Recovery Loop BW B p Phase Jitter Integration Region Filter at Symbol Rate R s /2 Recovery Loop tracks out this region

freq reqs for comm.ppt, V. S. Reinhardt. Page 13 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Typical L  (f) Requirements for QPSK LOs (vs Symbol Rate) - 10 Hz Hz - 1 KHz - 10 KHz KHz - 1 MHz - 10 MHz MHz - 1 GHz Symbol Rate R s Composite Spec R s = 10 Hz - 1 MHz The curves above show typical L  (f) requirements vs symbol rate –0.5 ° phase jitter allocated to particular LO –Oscillator model: flicker frequency + white phase –Flicker freq and white phase each contribute equally to jitter –Carrier recovery loop BW optimized for data rate = 0.01 x Data Rate but  100 KHz (assumed hardware limit for VCO modulation rate) For multi-data-rate units, LO’s must satisfy worst case composite spec for all rates covered by that unit

freq reqs for comm.ppt, V. S. Reinhardt. Page 14 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. LO Vibration Sensitivity and Carrier Phase Jitter Vibration induces phase jitter through Freq source g-sensitivity –H g (f) =(  f/f)/  g =  y/  g The vibration PSD S g (f) generates  f/f- PSD S y (f) directly through H g (f) –S y (f) = |H g (f)| 2 S g (f) –S(f) = double-sided PSD’s This can be converted to a phase PSD by adding a (f o /f) 2 factor –S  (f) = |H g (f)| 2 S g (f)*(f o /f) 2 –f o = carrier frequency As before, S  (f) is integrated from B p to R s to produce a phase variance –    =  0 Rs/2 |H g (f)| 2 S g (f)*(f o /f) 2 |1-H p (f)| 2 df –      Bp Rs/2 |H g (f)| 2 S g (f)*(f o /f) 2 df Because of the (f o /f) 2 dependence of S  (f), there is a strong 1/B p dependence in    S g (f) f Vibration Spectrum |H g (f)| 2 f Oscillator g sensitivity Structural Resonances f Vibration Induced Phase Noise S  (f) (f o /f) 2 factor because vib generates frequency sidebands

freq reqs for comm.ppt, V. S. Reinhardt. Page 15 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Typical Vibration Levels in a Commercial Aircraft From: PHASE NOISE PERFORMANCE OF SAPPHIRE MICROWAVE OSCILLATORS IN AIRBORNE RADAR SYSTEMS, T. Wallin, L. Josefsson, B. Lofter, GigaHertz 2003, Proceedings from the Seventh Symposium, November 4–5, 2003, Linköping, Sweden, Linköping ISSN (www), Issue: No. 8, URL: S g (f) – dBg 2 /Hz Fourier Frequency - dBHz With Vibration Damper Without Vibration Damper S g Level g 2 /Hz Double Sideband Spectrum Damper Response f/f res - dB Response – dB f res = 14.3 Hz Q = 3 Vibration levels at a crystal oscillator with and without a vibration damper

freq reqs for comm.ppt, V. S. Reinhardt. Page 16 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Typical LO H g Required vs Data Rate Using this vib data (scaled by peak Sg without damper), one can generate the above curves of required H g vs symbol rate –Assumes: 0.25° allocated to vibration induced phase jitter, B p = 0.01R s, f o = 10 GHz, and constant H g vs freq Note (because of strong B p dependence in    ) : (1) H g regs more stringent for lower symbol rates, (2) vibration damper helps more at higher symbol rates & can make things worse at lower rates With Vibration DamperNo Vibration Damper S g =0.003S g =0.01S g =0.03S g =0.1

freq reqs for comm.ppt, V. S. Reinhardt. Page 17 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Clock Jitter Requirements for Data and Sampling Clocks Decision epoch jitter from data clocks –Clock jitter requirement value determined by eye pattern behavior Sampling or aperture clock jitter in A/Ds & D/As (in digitally implemented Tx’s and Rx’s) –Jitter in aperture clock causes non-thermal SNR degradation in A/D’s and D/A’s (creates amplitude jitter) –Reduces effective number of bits (ENOB) –Causes BER degradation A/D Analog Input Demod Recovery Loops Typical Digital Implementation Sampling or Aperture Clock SNR of N-Bit word degraded by clock jitter Decision Threshold Data Clock Jitter Modulo time Effective eye Opening reduced Symbol Period Decision Epoch Jitter ISI

freq reqs for comm.ppt, V. S. Reinhardt. Page 18 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Decision Epoch Jitter from Data Clocks Analysis of decision jitter similar to that of phase jitter –  x  = 2  0 Rs/2 L x  (f) |1-H p (f)| 2 df –  x   2  Bp Rs/2 L x (f) df = (  T c ) 2 – x =  /(2  R s ) = clock reading error – L x (f) = sum of SSB x-PSD’s of clocks – Rec loop: H p (f) = response B p = BW – Rule of thumb:  should be < % for < 0.1 dB DER deg Data clock phase jitter   –   = 2  R s  x = 2  (in radians) – L  (f) = sum of SSB  -PSD’s of clocks –    = 2  0 Rs/2 L  (f) |1-H p (f)| 2 df –     2  Bp Rs/2 L  (f) df –Rule of thumb:   should be < 1-3 ° for < 0.1 dB BER degradation –Same curves as LO L  (f) vs R s (for same phase jitter and B p ) Clock Jitter Reqs vs Symbol Rate

freq reqs for comm.ppt, V. S. Reinhardt. Page 19 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Effect of Sampling Clock Jitter in Digital Implementations Digital implementations use A/D and D/A converters to convert between analog and digital domains Jitter t j in aperture clock generates random amplitude noise in digitizing a signal with carrier frequency f Phase noise generated  = 2  f SW  t j =  V/A Limits SNR of digital output to   Can be converted to an effective number of bits (ENOB) of the converter (with assumptions about the size of A) From: Analog Devices, Mixed-Signal and DSP Design Techniques, Section 2, Sampled Data Systems, p35 Modulated Sinewave Input at Frequency f SW Time Jitter  t j Amplitude Jitter  V Phase Jitter  = 2  f SW  t j 2A

freq reqs for comm.ppt, V. S. Reinhardt. Page 20 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. SNR due to Aperture (Sampling) Clock Jitter for Full Scale Sinewave Input From: Analog Devices, Mixed-Signal and DSP Design Techniques, Section 2, Sampled Data Systems, p ENOB Sinewave Frequency - dBHz SNR - dB 1 ns 0.1 ns 10 ps 1 ps 0.1 ps Clock Jitter

freq reqs for comm.ppt, V. S. Reinhardt. Page 21 Copyright 2005 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized. Summary--Conclusions T&F specs for frequency sources in comm systems can be derived by understanding the relationship between BER degradation and frequency source phase and clock jitter Recovery loops act as high pass filters that allow the use of standard variances even in the presence of flicker of frequency noise The critical jitter statistics are derived from PSD’s by integrating from the loop recovery BW to the symbol rate –Spurs must be included in jitter integrations (not covered in talk) Quadrature systems have more stringent phase jitter requirements because of I-Q crosstalk Frequency source vibration requirements are more critical for low data rate systems