EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 1 Random Processes (1) Random variable: A quantity X whose value is not exactly known. Probability distribution function P X (x): The probability that a random variable X is less than or equal to a value x x PX(x)PX(x) Example 1: Random variable

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine x PX(x)PX(x) x1x1 x2x2 Probability of X within a range is straightforward: If we let x 2 -x 1 become very small … Random Processes (2)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 3 Probability density function p X (x): Probability that random variable X lies within the range of x and x+dx x PX(x)PX(x) x pX(x)pX(x) dx Random Processes (3)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 4 Expectation value E[X]: Expected (mean) value of random variable X over a large number of samples. Mean square value E[X 2 ]: Mean value of the square of a random variable X 2 over a large number of samples. Variance: Standard deviation: Random Processes (4)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 5 Gaussian Function x 22 1.Provides a good model for the probability density functions of many random phenomena. 2.Can be easily characterized mathematically. 3.Combinations of Gaussian random variables are themselves Gaussian.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 6 Joint Probability (1) If X and Y are statistically independent (i.e., uncorrelated): Consider 2 random variables:

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 7 Consider sum of 2 random variables: x y dx dy = dz determined by convolution of p X and p Y. Joint Probability (2)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 8 * Example: Consider the sum of 2 non-Gaussian random processes: Joint Probability (3)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 9 3 sources combined: * Joint Probability (4)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 10 4 sources combined: * Joint Probability (5)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 11 Central Limit Theorem: Superposition of random variables tends toward normality. Noise sources Joint Probability (6)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 12 Fourier transform of Gaussians: F Recall: F F -1 Variances of sum of random normal processes add.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 13 Autocorrelation function R X (t 1,t 2 ): Expected value of the product of 2 samples of a random variable at times t 1 & t 2. For a stationary random process, R X depends only on the time difference for any t Note Power spectral density S X (  ): S X (  ) given in units of [dBm/Hz]

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 14 Relationship between spectral density & autocorrelation function: Example 1: white noise   infinite variance (non-physical)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 15 Example 2: band-limited white noise  x For parallel RC circuit capacitor voltage noise:

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 16 Random Jitter (Time Domain) Experiment: data source CDR (DUT ) analyzer CLK DATA RCK

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 17 Jitter Accumulation (1) Free-running oscillator output Histogram plots Experiment: Observe N cycles of a free-running VCO on an oscilloscope over a long measurement interval using infinite persistence. NT 11 22 33 44 trigger

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 18 Observation: As  increases, rms jitter increases. proportional to  2 proportional to  Jitter Accumulation (2)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 19 Noise Spectral Density (Frequency Domain)  osc  osc +  Sv()Sv() Power spectral density of oscillation waveform : L total includes both amplitude and phase noise L total (  ) given in units of [dBc/Hz]  (log scale) 1/  2 region (-20dBc/Hz/decade) Single-sideband spectral density : 1/  3 region (-30dBc/Hz/decade)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 20 Noise Analysis of LC VCO (1) active circuitry C L R -R Consider frequencies near resonance: C L + _ vcvc i nR noise from resistor

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 21 Noise current from resistor: C L + _ vcvc i nR Noise Analysis of LC VCO (2) Leeson’s formula (taken from measurements): Where F and  1/f 3 are empirical parameters. dBc/Hz spot noise relative to carrier power

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 22 Oscillator Phase Disturbance Current impulse  q/  t _ + V osc t t ip(t)ip(t) V osc (t) V osc jumps by  q/C Effect of electrical noise on oscillator phase noise is time-variant. Current impulse results in step phase change (i.e., an integration).  current-to-phase transfer function is proportional to 1/s ip(t)ip(t) ip(t)ip(t)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 23 Impulse Sensitivity Function (1) The phase response for a particular noise source can be determined at each point  over the oscillation waveform. Impulse sensitivity function (ISF): (normalized to signal amplitude) change in phase charge in impulse t  Example 1: sine wave t  Example 2: square wave Note  has same period as V osc.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 24 Impulse Sensitivity Function (2) Recall from network theory: LaPlace transform: Impulse response: time-variant impulse response Recall: ISF convolution integral: from  q  can be expressed in terms of Fourier coefficients:

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 25 Case 1: Disturbance is sinusoidal:, m = 0, 1, 2, … negligible significant only for m = k (Any frequency can be expressed in terms of m and .) Impulse Sensitivity Function (3)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 26 Impulse Sensitivity Function (4) For  I 2  osc   Current-to-phase frequency response:  osc  osc      osc   2  osc   2  osc  

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 27  Case 2: Disturbance is stochastic: Impulse Sensitivity Function (5) MOSFET current noise: thermal noise 1/f noise A 2 /Hz   osc 2  osc thermal noise   osc 2  osc 1/f noise

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 28 Impulse Sensitivity Function (6)    osc 2  osc due to 1/f noise due to thermal noise Total phase noise: nn

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 29 Impulse Sensitivity Function (7) noise corner frequency  n  (log scale) (dBc/Hz) 1/(  3 region: −30 dBc/Hz/decade 1/(  2 region: −20 dBc/Hz/decade

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 30 t  t  Example 1: sine waveExample 2: square wave Impulse Sensitivity Function (8) Example 3: asymmetric square wave t   will generate more 1/(  3 phase noise is higher  will generate more 1/(  2 phase noise

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 31 Impulse Sensitivity Function (9) Effect of current source in LC VCO: V osc + _ Due to symmetry, ISF of this noise source contains only even-order coefficients − c 0 and c 2 are dominant.  Noise from current source will contribute to phase noise of differential waveform.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 32 Impulse Sensitivity Function (10) I D varies over oscillation waveform Same period as oscillation Let Thenwhere We can use

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 33 ISF Example: 3-Stage Ring Oscillator M1AM1A M1BM1B M2AM2A M2BM2B M3AM3A M3BM3B MS1MS1 MS2MS2 MS3MS3 R1AR1A R1BR1B R2AR2A R2BR2B R3AR3A R3BR3B + V out − f osc = 1.08 GHz PD = 11 mW Red curve: Unperturbed oscillation waveform Blue curve: Oscillation waveform perturbed by impulse

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 34 ISF of Diff. Pairs for each diff. pair transistor

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 35 ISF of Resistors for each resistor

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 36 ISF of Current Sources ISF shows double frequency due to source-coupled node connection. for each current source transistor

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 37 Phase Noise Calculation (Thermal noise) Using:C out = 1.13 pF V out = 601 mV p-p q max = 679 fC = −112  f = 10 MHz

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 38 Phase Noise vs. Amplitude Noise (1)  osc t  v vv Spectrum of V osc would include effects of both amplitude noise v(t) and phase noise  (t). How are the single-sideband noise spectrum L total (  ) and phase spectral density S  (  ) related?

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 39 Phase Noise vs. Amplitude Noise (2) t t i(t)i(t) i(t)i(t) Vc(t)Vc(t)Vc(t)Vc(t) Recall that an input current impulse causes an enduring phase perturbation and a momentary change in amplitude: Amplitude impulse response exhibits an exponential decay due to the natural amplitude limiting of an oscillator...

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 40  + Phase noise dominates at low offset frequencies. Phase Noise vs. Amplitude Noise (3) 

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 41 Phase & amplitude noise can’t be distinguished in a signal. Phase Noise vs. Amplitude Noise (4) noiseless oscillation waveform phase noise component amplitude noise component Amplitude limiting will decrease amplitude noise but will not affect phase noise.  osc Sv()Sv() phase noise amplitude noise 

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 42 Sideband Noise/Phase Spectral Density noiseless oscillation waveform phase noise component

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 43 Jitter/Phase Noise Relationship (1) Recall R  and S  (  ) are a Fourier transform pair: NT 11 22 33 44 autocorrelation functions

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 44 Jitter/Phase Noise Relationship (2)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 45 Let Consistent with jitter accumulation measurements! Jitter/Phase Noise Relationship (3) Jitter from 1/(  noise: 2 Let Jitter from 1/(  noise: 3 ^ ^ ^ ^ ^

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 46 Jitter/Phase Noise Relationship (4) ff (dBc/Hz) dBc/Hz per decade Let f osc = 10 GHz Assume phase noise dominated by 1/(   ) 2 Setting  f = 2 X 10 6 and S  = : Let  = 100 ps (cycle-to-cycle jitter):    = 0.02ps rms (0.2 mUI rms) Accumulated jitter: 2 MHz

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 47 More generally: ff (dBc/Hz) fmfm NmNm -20 dBc/Hz per decade Jitter/Phase Noise Relationship (5)  rms jitter increases by a factor of 3.2 Let phase noise increase by 10 dBc/Hz:

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 48 Jitter Accumulation (1) K pd phase detector loop filter K vco VCO  in  out  vco  fb Open-loop characteristic: Closed-loop characteristic:

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 49 Jitter Accumulation (2) Recall from Type-2 PLL:  |G| zp  |1 + G| -40 dB/decade  (dBc/Hz) 1/(  3 region: −30 dBc/Hz/decade 1/(  2 region: −20 dBc/Hz/decade  1 80 dB/decade  As a result, the phase noise at low offset frequencies is determined by input noise...

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 50 f osc = 10 GHz Assume 1-pole closed-loop PLL characteristic Jitter Accumulation (3) ff (dBc/Hz)  f 0 = 2 MHz dBc/Hz per decade

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 51 Jitter Accumulation (4) (log scale)  For large  :   = 0.02 ps rms cycle-to-cycle jitter  f 0 = 2 MHz f osc = 10 GHz For small  :   = 1.4 ps rms Total accumulated jitter

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 52 The primary function of a PLL is to place a bound on cumulative jitter:  (log scale) proportional to  (due to thermal noise) proportional to   (due to 1/f noise)  Jitter Accumulation (5) Free-running VCO PLL

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 53 L(  ) for OC-192 SONET transmitter Closed-Loop PLL Phase Noise Measurement

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 54 Other Sources of Jitter in PLL Clock divider Phase detector Ripple on phase detector output can cause high-frequency jitter. This affects primarily the jitter tolerance of CDR.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 55 Jitter/Bit Error Rate (1) Histogram showing Gaussian distribution near sampling point 1UI Bit error rate (BER) determined by  and UI … LR Eye diagram from sampling oscilloscope

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 56 R 0T Probability of sample at t > t 0 from left- hand transition: Probability of sample at t < t 0 from right- hand transition: Jitter/Bit Error Rate (2)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 57 Total Bit Error Rate (BER) given by: Jitter/Bit Error Rate (3)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 58 t 0 (ps) log BER Example: T = 100ps (64 ps eye opening) (38 ps eye opening) log(0.5) Jitter/Bit Error Rate (4)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 59 Bathtub Curves (1) The bit error-rate vs. sampling time can be measured directly using a bit error-rate tester (BERT) at various sampling points. Note: The inherent jitter of the analyzer trigger should be considered.

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 60 Bathtub Curves (2) Bathtub curve can easily be numerically extrapolated to very low BERs (corresponding to random jitter), allowing much lower measurement times. Example: BER with T = 100ps is equivalent to an average of 1 error per 100s. To verify this over a sample of 100 errors would require almost 3 hours! t 0 (ps)

EECS 270C / Winter 2013Prof. M. Green / U.C. Irvine 61 Equivalent Peak-to-Peak Total Jitter BER , T determine BER BER determines effective Total jitter given by: Areas sum to BER