APPLICATION NOTE #1. Phase NoiseTheory and Measurement 1 INTRODUCTION

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1 Tommorrow s Phase Noise Testing Today 35 South Service Road Plainview, NY 803 TEL: FAX: APPLICATION NOTE # Phase NoiseTheory and Measurement INTRODUCTION Today, noise measurements have become an essential actor in the characterization o signals. This is easy to understand when one realizes that it is phase noise that: - limits the operating range o radar - degrades the quality o television pictures - limits the precision o satellite positioning - spoils the quality o data transmission The primary characteristic o noise is its randomness, and this is due to the physical mechanisms which generate it. Three leading types o noise are to be ound in all electronic systems: - Thermal Noise: random motion o the carriers in a conductor. - Shot Noise: random low o the carriers through a potential barrier. - Flicker Noise: its origin is not well known. It seems to come rom the macroscopic deects o the materials. To characterize these noise sources, one must reer to the Theory o Random Processes. The mathematical tools will be simpliied by making three assumptions according to the statistical process, i.e. on whether it is: - Stationary (zero mean value) - Ergodic (its statistical mean values are the same as its time mean values) - Gaussian (its amplitude has a Gaussian distribution) NOTE: Other random disturbances, such as random jumps - which do not correspond to the above assumptions, can add themselves to the above type o noise and alter noise measurement results. Application Note # Revised 3/4/95

2 . TIME DOMAIN The study o noise in relation to time gives a random unction. Noise can disturb any physical parameter - this unction can thereore apply to voltage (Volts), current (Amps), phase (radians), requency (Hertz), time (seconds), etc. This unction x(t) can be characterized by its distribution p(x), where each x value represents the probability that x(t)x. It is only possible to describe this time unction statistically. According to the previous assumptions, p(x) is a Gaussian unction (see Figure ) px ( ) e x 2 2σ 2 σ 2π where σ is the standard deviation o p(x). x (t) x max x o + σ - σ p (x) +3 σ -3 σ x min Figure - The Gaussian unction p(x) can be completely characterized by one parameter σ. - It can be calculated by means o a statistical average or a time average rms value o xt () x 2 () t, assuming that the noise is ergodic σ x2() t - p(x) is symmetrical x max x min Application Note # 2 Revised 3/4/95

3 - 99.7% o the unction x(t) is located in the interval ± 3 σ. The value ± 3 σ can represent the peak-to-peak value o the noise. There is no absolute deinition o peak-to-peak noise: ± 4 σ 99.98% o x(t). - This statistical-time representation is useul to calculate bit-error rate. The error unction er (x), which is obtained in bit-error calculations, comes rom the integration o p(x). For some point x o, the probability that x(t) > x o is x o px ( )dx x o - 2 px ( )dx x o 2 - er x o ---- σ.2 FREQUENCY DOMAIN The basic parameter is the distribution o the noise power as a unction o requency. S x () is called the spectral density o phase luctuations, requency luctuations, amplitude luctuations, etc. When one looks into the disturbances that can occur on a signal like phase noise, requency noise, or amplitude noise, S x () is a low requency, continuous spectrum (no discontinuity according to the requency), deined or the positive requencies (single-sideband spectrum). See Figure 2 below. S x () p b σ 2 Hz 99 MHz Figure 2 Units: Phase Noise x radian S x () (radian) 2 /Hz Frequency Noise x Hertz S x () (Hertz) 2 /Hz Amplitude Noise x Volt S x () (Volt) 2 /Hz Application Note # 3 Revised 3/4/95

4 .3 RELATIONSHIP BETWEEN THE TIME AND FREQUENCY DOMAINS Noise power is the main common parameter o the two domains: Time Domain (Figure ) x(t) is deined by its rms value x 2 () t other notations ( x 2 ), x rms p(x) is deined by σ (standard deviation) Frequency domain (Figure 2) S x () enables computation o total noise power P b 0 S x ()d The relationship between the two domains is P b x 2 () t σ 2.4 CHARACTERIZATION OF A SIGNAL Noise, which has just been described, can aect a carrier signal (i.e. a signal which has a signiicantly higher level than the noise aecting it). There are our dierent ways o characterizing this signal: SPECTRAL PURITY The spectrum o an ideal sinusoidal signal v(t)a sin[2π o t] corresponds to a DIRAC unction at requency o S v () A Watt/Hz 2σ( o ) The disturbances due to the requency and amplitude luctuations give rise to a spectral bandwidth and noise sidebands, Figure 3. Application Note # 4 Revised 3/4/95

5 P s POWER σ(- o ) dbc Figure 3 The relative level o the noise sidebands deines the spectral purity, usually P b ( Hz) expressed in dbc (db rom the carrier) d 0log P s NOTE: The spectral purity takes into account all the disturbances: phase noise and amplitude noise. The usual measurement instrument is the spectrum Analyzer. AMPLITUDE NOISE AND PHASE NOISE A real signal perturbed by the above two types o noise can be represented by the expression where α(t) and φ(t) represent two random unctions to which correspond spectral density S a () (units: Volt 2 /Hz) and S φ () (units: radians 2 /Hz), respectively. 0 Noise The unction φ(t) represents the phase luctuations around the theoretical phase o the signal φ o 2π o t as illustrated in Figure 4. o P b Hz vt () A[ + at ()] sin( 2π o t + φ() t ) φ(t) φ max φ + φ rms S φ () p ( φ ) φ o P φ φ 2 rms φ min -φ rms Hz 99 MHz Figure 4 φ max and φ min represent the maximum deviation rom the theoretical phase and enables the deinition o a bit-error rate in digital phase modulation. For example, i the modulation has our phase states (+π/4, +3π/4, -π/4, -3π/4), the decision limit will be o ±π/4 around each state. I the φ max and φ min values Application Note # 5 Revised 3/4/95

6 were lower than ±π/4, then the phase noise would not induce transmission errors. φ rms represents the rms value o phase noise. It can also be called mean phase luctuation". The requency luctuations o the carrier signal can be calculated as dφ() t φ() t o () t o πdt o π o + t () Hz where (t) represents the instantaneous luctuations o the carrier requency around its center value o. See Figure 5. (t) max p ( ) S () 0 P 2 rms min rms 99 MHz Figure 5 rms represents the rms value o the requency noise (also reerred to as mean requency luctuation). The relationship between the spectral densities o phase and requency luctuations is S () ' 2 Sφ() Fractional requency luctuations can also be represented yt () () t o With this representation, one is no longer dependent on the center requency, o, The spectral density is S y () 2 2 o Sφ() Application Note # 6 Revised 3/4/95

7 The spectrums S φ (), S (), S y () represent the characterization o the requency instability in the requency domain. In industrial applications, the most commonly used spectrums are S a () and S φ () because they provide an immediate estimation o spectral purity S v (). RELATIONSHIP BETWEEN SPECTRAL PURITY AND SPECTRAL DENSITY: Assume the real signal vt () A[ + at ()] sin[ 2π o t + φ() t ] I the amplitude a(t) and phase noise φ(t) are demodulated, the spectral analysis o these two random unctions gives the ollowing spectral densities S a () and S φ (). See Figure 6. Power δ(- o ) S φ S b φ S a S b a 0 99 MHz Figure 6 A useul approximation (phase noise power << radian 2 ) is to assume that the sidebands that surround the carrier requency o correspond to the double- b band translation o the spectral densities S a () and S φ () --> S a() and b b S φ () Sφ. I the approximation is not valid, the ollowing exact relationship is obtained where convolution and φ 2 total phase noise power Application Note # 7 Revised 3/4/95

8 Sv() A σ( o ) + S b a ( o ) + S b φ ( o ) + 2e φ2 n b ----S n! φ () b S φ () In many cases, particularly with microwave signals, one should take into account the irst phase noise convolution S b φ () S b φ () and the parameter e φ2 S a ().. The amplitude noise is usually low enough to only take into account TIME DOMAIN CHARACTERIZATION OF PHASE NOISE AND FREQUENCY NOISE The requency o the signal can be measured with a requency counter. Frequency instability results in a luctuation o the measurement results o + t () where t is the measuring time o the requency counter (e.g. t ms, 0ms, 00ms, etc.) I the same measurement is made several times, the result will be a series o random data n (T). The longer the measuring time t, the lower the n (T) values. I one does not take into account the drit o the carrier requency (because it results rom a dierent noise process) the average value o the n (T) is zero. The statistical analysis o these n (T) deviations represent the characterization o requency instability in the time domain, the undamental parameter o which is the standard deviation or eective variance σ [ ( T) ] N N N - ( T ) - i N j ( T) i j 2 Hz Note that N must be > 00 Hz The results are oten normalized to the nominal requency o Application Note # 8 Revised 3/4/95

9 ms σ [ ( )] 0 x o I ms σ [ ( )] 0 7 o, and o 00 MHz, then σ( ms) 0 Hz, based on the peak value criterion max ± 3 x 0 Hz ± 30 Hz. RELATIONSHIP WITH THE INSTANTANEOUS FREQUENCY FLUCTUATIONS At irst glance, the two parameters seem similar. (t) represents the instantaneous requency luctuations, while (T) represents the requency deviations over a time T, measured with a requency counter. The two parameters are related by the requency noise power σ [ ( T) ] --- P o where P noise power in the bandwidth sin ( πt) ( πt) low pass ilter with a cuto requency o 0.25/T. This is equivalent to a P 0 S () sin ( πt) d ( πt) true variance H() 2 P F Allan --- 2T Figure 7 The requency counter acts like a requency noise integrator. - T 2 - T Application Note # 9 Revised 3/4/95

10 NOTE: A requency spur at /T is not taken into consideration with this measurement, since it is suppressed by the zero value o the transer unction. For example, T ms, F khz, T 00 ms, F 00 Hz. Other variances have been introduced to simpliy the statistical calculations. The Allan variance, or example, is based on only two successive measurements. The time variance, however, requires storage o all N measurements beore calculation. σ Allan [ T ( )] Its transer unction sin 2 (π T )/(π T ) produces a greater selectivity ( max 0.37/ T) and suppresses the contribution o the VLF (very low requency) disturbances, and leads to results which will be dierent than those obtained with time variance calculations. NOTE: The measurement results depend on the type o variance and the measurement time o the requency counter NOTE: These measurement processes do not characterize amplitude noise..5 TIME NOISE OR JITTER N - N - 2 [ i ( T ) i + ( T) ] 2 i For digital transmission systems, it is productive to characterize the jitter o a signal. The phase luctuations give rise to luctuations o the zero crossing o the carrier signal t + φ() t vt () A sin[ 2π o t + φ() t ] A sin 2π o π o vt () A sin[ 2π o ( t + T() t )] υ(t) T(t) T P ( T) S T () t T rms Figure 8 These time luctuations are related to the phase luctuations as ollows P T T(t) is a random unction, where the ollowing parameters are deined: - Distribution Application Note # 0 Revised 3/4/95

11 T() t φ() t π o p ( T) - Mean Fluctuation T rms (standard deviation) - Peak-to-peak Fluctuation ± 3 T rms - Spectral Density S T () Sφ () 2π o seconds 2 /Hz and where jitter power is P seconds 2 T S T () d and the Mean Fluctuation is T rms P T seconds rms Jitter can be normalized to the period o the signal. The unit interval (UI) is then deined One oten considers that the peak-to-peak value o this parameter is the most interesting, since with this value a threshold o disturbance UI pp can be deined. 2 MEASURING METHOD The parameters to be measured are the ollowing: - Spectral Purity - Amplitude Noise - Phase Noise - Frequency Noise - Variances - Jitter UI() t T() t φ() t UI(t) is a unitless quantity T o 2π Application Note # Revised 3/4/95

12 2. SPECTRUM ANALYZER The parameter measured by a spectrum analyzer is spectral purity. See Figure 9. INPUT BPF BPF IF FILTER IF GAIN OSCILLATOR YIG LOG DISPLAY Figure 9 The principle is based on shiting the signal to be measured to an IF (intermediate requency). The critical points are: - Local oscillators - IF iltering - Dynamic range o the logarithmic detection Advantages: - Direct measurement at the signal requency - All the unctions that are necessary or measurements are incorporated in the instrument (demodulation, reerences, etc.) - Easy to use. Disadvantages: SWEEP CONTROL In spite o some recent improvements, such as synthesized oscillators and an increase o the log-ampliier dynamic range (80 --> 00 db), when used speciically or noise measurement, this instrument has many disadvantages: - The phase noise o the local oscillators is too high or most o the sources to be tested (oten, the local oscillators are synchronized oscillators, not synthesized oscillators, so the mean value o the center requency is stable but the noise remains very high). - The instability o the center requency makes it diicult to make measurements close to the carrier. Application Note # 2 Revised 3/4/95

13 - It is not possible to dierentiate spurious and noise since the spectrum shows only dbm values. - The analysis is linear and related to the distance rom the carrier, which implies a small analysis span (max deviation, min deviation / 50). - The spectrum is not normalized as db/carrier. - There are many measurement inaccuracies (attenuator, gain, IF ilter, bandwidth, amplitude-log linearity, etc.). - Detection is linear and not quadratic. - The act that the video smoothing is operated ater the log-ampliication induces a 2.5 db error on a Gaussian noise. - The carrier signal limits the dynamic range. - The input noise actor is considerable (20 db). - The results are not processed. 2.2 DEMODULATION OF AMPLITUDE AND PHASE NOISE Amplitude Noise: This unction is easily carried out by a wide-band detection diode. An adaptation network is necessary to deine the detection impedance and suppress the direct voltage which could overload the low-noise ampliier. Calibration is perormed with a variable amplitude source. Phase Noise: By maintaining a reerence source in quadrature, phase noise can be demodulated using a double balanced mixer. To maintain quadrature, a phase lock loop is usually used, in which a balanced mixer acts as a phase-sensitive detector (other types o detectors are used to perorm phase control instead o quadrature control). See Figure 0. SOURCE TO BE TESTED LOW NOISE AMPLIFIER LPF LPF CAN REFERENCE LOOP CONTROL PC DISPLAY Disadvantages: Figure 0 - The reerence source must have lower noise than the source to be measured. Application Note # 3 Revised 3/4/95

14 - The phase lock loop must be very precisely controlled, since it eects measurement results, its eect must be compensated or. Advantages: - Easy to set up and use, quick and precise calibration. - Suppression o the carrier signal, which enables a wide measurement dynamic range. - Log-log analysis, enabling a wide dynamic range in level (0-70 db) and requency deviation ( Hz - 0 MHz) - Substantial sensitivity (70 db, residual only limited by the reerence source) - Measurements can be perormed very close to the carrier - Dierentiation o spurious and noise. Noise in db/hz and spurious in db - Separate measurements o phase noise and amplitude noise - Low sensitivity o measurements to the drit o the source to be tested (phase lock loop) - Complete data processing (power o the computer associated with the Set) - Automated measurements 2.3 DEMODULATION OF FREQUENCY NOISE The lowest noise requency demodulation can be obtained through a delay line (the other types o demodulation, e.g. diode demodulation, are not sensitive or stable enough or phase noise measurements). Figure shows the measurement method. τ π - 2 k LPF F(t) SOURCE TO BE TESTED φ Adjustment π 2 - Figure The delay line (τ), introduced into a channel o the balanced mixer, acts as a requency discriminator, or requencies << /τ. The output o the mixer is a signal F(t), which is proportional to the requency luctuations: Application Note # 4 Revised 3/4/95

15 sin ( πt) Ft () ( KG2πT) () t ( πt) Ft () ( KG2πT) ()or t < ----T 0 The demodulation actor depends on the value o the delay T. Advantages: - No reerence source. It is diicult, however, to have a wide-band delay line (e.g. GHz - 8 GHz). Thereore, requency conversion is necessary to use the delay line at a ixed requency, and this brings us back to the problem o the reerence source. - No phase control - The data processing capability can be as high as it is or phase demodulation. Disadvantages: - Requires a reerence source or wide-band operation. - Complicated calibration process. The requency has to be shited to allow or calibration o the demodulation mixer. - Diicult choice between a long delay to obtain increased demodulation sensitivity K2πT and a short delay to get the large span spectral analysis (max. deviation << /T). 2.4 FREQUENCY COUNTING The basic instrument is a requency counter, which is used in its reciprocal mode or increased resolution. The parameters measured are the variances (true, Allan...). A reerence source is necessary to obtain a residual noise compatible with the existing sources. Figure 2 shows the measurement method. SOURCE TO BE TESTED LPF FREQUENCY METER COMPUTER PRINTER REFERENCE Figure 2 Application Note # 5 Revised 3/4/95

16 The measuring process is simple. Perorm a statistical computation o the requency measurement. Advantages: - The test set is easy to set up and use - No calibration - No phase lock loop required Disadvantages: - The measurement time is long and measurements are taken serially - The variances represent an integrated noise and the global values that result are thereore diicult to translate in terms o spectral purity - The measurements are sensitive to spurious (because there are many zeros in the transer unction. Although this is not very important with regards to noise which shows a continuous spectrum, it is critical where spurious are concerned, since they can be overlooked depending on their requencies) - Needs a reerence source 2.5 JITTER MEASUREMENT The eye pattern method allows visualization o the phase margin between the modulation states. Figure 3 shows the measurement method. SIGNAL OSCILLOSCOPE LPF Y EXT SYNCH Figure 3 The demodulation method allows measurement o the peak-to-peak amplitude and the jitter spectral density, as shown in Figure 4. Application Note # 6 Revised 3/4/95

17 SIGNAL OSCILLOSCOPE BPF LPF NORMALIZED BAND WIDTH SPECTRUM ANALYZER Figure 4 The resolution o this technique is low in comparison to the preceding methods, but suicient to test contemporary digital links. 3 NOISE IN FREQUENCY GENERATORS 3. NOISE IN THE CIRCUITS Noise sources can be separated into two categories, ultimate and excess, according to their origin: Ultimate noise Thermal Noise and Shot Noise are reerred to as ultimate because they derive rom the physics o materials, and do not depend on the quality o the components. They can never be suppressed; one can only optimize their action. These types o noise can also be expressed mathematically. Excess noise Flicker Noise and Popcorn Noise are reerred to as excess because they depend on the quality o the components, oten on the cleanness o their suraces. The same manuacturing process can produce components with very dierent noise levels. There is no mathematical expression that describes these types o noise. THERMAL NOISE Thermal noise is a power source e(t). Its spectral density is white noise, which is constant in relation to the requency where, S b () 4KTR Volts Hz K Boltzmann actor.38 X 0-23 joule/degrees Kelvin T Absolute temperature (in degrees Kelvin) R Resistance value in ohms Application Note # 7 Revised 3/4/95

18 Noise power P b o Noise level per bandwidth B S b () d 4KTRB Volts 2 e 2 Noise level in bandwidth B 4KTR Volts rms Hz e 2 For example, i R KΩ, then 4KTRB Volts rms and in a MHz bandwidth e nV Hz e 2 4µV Noise power transmitted to a load having the same impedance P b P a KTB watts 4R P a expressed in dbm, P a -74 dbm or a bandwidth o Hz. RELATIONSHIP BETWEEN SIGNAL AND NOISE The ollowing is or a Hz BW only: - For a signal o 0 dbm, S/N 74 db - For a signal o +0 dbm, S/N84 db SHOT NOISE Shot Noise is modeled as a current source i(t). Its spectral density is a white noise, S i () 2 q I Amps 2 /Hz q is the charge on the electron.602 x 0-9 Coulombs - I is the direct current DC in Amperes Noise Power P b 2 q I B Application Note # 8 Revised 3/4/95

19 Noise Current per unit bandwidth For example, I ma i 2 2qI Amps rms Hz i Amps rms Hz With an impedance o KΩ v 2 R i V rms Hz compared to thermal noise e V rms kω Hz FLICKER NOISE This noise can be represented as either a voltage or a current source. It is ound in all components, and is characterized by its spectral density variation slope The slope o this spectrum is oten expressed in db per decade: 0 db/decade, or in db per octave: 3 db/octave. With a Log-Log representation, it is easy to recognize this type o noise. There is at present no physical theory to explain the mechanism o licker noise. POPCORN NOISE This type o noise varies in time through random quantiied jumps, which generate a VLF (very low requency) spectral density. When this type o noise is present in a component, it can be concluded that this component has a major deect, and it must be eliminated or reduced by testing and selection. SPECTRAL DENSITY OF THE NOISE OF ELECTRONIC CIRCUITS To estimate the total noise o a circuit, one must add all the noise sources that have been described above (in power or spectral density). For example, 2 2 K K 2 S total () 4KT( R + R ) + 2q ( I R + I2 R ) I the noise level has not been iltered, the noise spectral density will have the general outlook o white noise or licker noise K 2 S () K Application Note # 9 Revised 3/4/95

20 Reer to igure 5: S () k - 4KTR+2qIR2+... EXCESS NOISE ULTIMATE NOISE NOISE FLOOR Log c Figure 5 A cut-o requency c called licker cut-o requency can be deined as, In silicon technology: In GaAs technology: c S () K KHz c 0kHz c 00MHz 3.2 NOISE IN OSCILLATORS An oscillator can be modeled as an ampliier with a band-pass ilter, as shown in Figure 6 NOISE G Signal S o () B o Figure 6 The ilter is deined by its center requency o and its band-pass B, as a unction o its quality coeicient Q Application Note # 20 Revised 3/4/95

21 B o --- Q Three types o noise will disturb the signal:. The ampliier s noise sources generate phase noise φ(t). 2. Since the center requency o is deined by the resonator, the parametric noise (variation o a parameter which deines the o value) modulates the oscillator by generating a requency noise 3. I the requency o the oscillator can be controlled by an external input, a noise signal applied to this input modulates the oscillator in requency, thus generating requency noise. NOISE GENERATED BY THE AMPLIFIER o () t o + t () The study o circuit noise has shown us that the spectral density o noise sources can be represented by c S b () K comprising white noise and licker noise with units o Volt 2 /Hz. This noise modulates the signal s phase going through the ampliier and the phase noise which is generated depends on the level (A) o the signal S φ () K c 2A radian 2 Hz The resonator ilters the noise with the ollowing transer unction [ H ()] 2 resulting in the phase noise spectral density S o () S φ () [ H ()] 2 and thus: + o Q 2 K S o () A 2 c o Q 2 Two major cases are to be ound in most oscillators, directly related to their quality coeicient: Application Note # 2 Revised 3/4/95

22 . High Q coeicient oscillators, such as o For 2Q c example, a crystal oscillator: o 0 MHz, Q 0,000, c khz o Hz «2Q c KHz 2. Low quality coeicient or high center requency o, such as o or 2Q > c example, a DRO: o 3 GHz, Q 000, and c 0 khz o MHz» 2Q c 0KHz Figure 7 shows the spectral densities o the phase luctuations obtained in both o the above cases. HIGH Q LOW Q S () φ log log H 2 () log log S () o o c log c o log Figure 7 st case o Q < c The ollowing rules are determined: Application Note # 22 Revised 3/4/95

23 - / 3 - licker requency noise, 30 db / decade or 9 db / Octave - / - licker requency noise, 0 db / decade or 3 db / Octave - - white phase noise, called the noise loor 2nd case: o Q > c The ollowing rules are obtained: - / 3 - licker requency noise, 30 db / decade or 9 db / Octave - / 2 - blank requency noise, 20 db / decade or 6 db / Octave - - white phase noise, called the noise loor A Log-Log representation is absolutely necessary to recognize these dierent slopes. The study o these slopes enables the analysis o the oscillator noise: - loaded quality coeicient - oscillation level - ampliier noise The main diiculty with this analysis lies in the act that this type o noise can be mistaken or the parametric and requency control noise sources. PARAMETRIC NOISE Some o the components o the band-pass ilter such as the varicaps, crystals, trimming capacitors, etc. have luctuations which directly aect the oscillating requency. Assume a luctuation o the capacitor given by - L 2 2πo -- C i C C o + Ct () o () t o + t () where (t) is the requency noise introduced by C(t). This is a parametric eect: the level o the noise is not related to the oscillation level, ilter Q, or the ampliier noise. Generally, this requency noise has a licker type spectral density S () K -- i.e. a phase noise Application Note # 23 Revised 3/4/95

24 Sφ() S () K This noise merges with the licker requency noise given by the ampliier, which has the same slope. This type o noise is also characterized by a substantial variation rom one component to another. For example, a 0 MHz crystal oscillator has a noise level between -5 dbc and -35 dbc at 0 Hz oset rom the carrier, depending on its manuacturing process. However, it can vary by as much as 0 db between the crystals that are produced by the same manuacturing process. OSCILLATOR CONTROL NOISE I noise is added to the control voltage, it will modulate the oscillator, generating a requency noise. I the slope o the oscillator control is K Hz/Volt, b(t) is the voltage noise input with a spectral density S b (). The ollowing is obtained: - the requency luctuations: (t) K b (t) - The spectral density o these luctuations: S () K 2 S b () - The spectral density o phase noise NOTE: Sφ() S b () or K S φ () K S 2 b () For example, an oscillator with a MHz/Volt slope and a control resistance o ΚΩ. Application Note # 24 Revised 3/4/95

25 S φ () 3-28 db -68 db 2 o S b () c 0 khz MHz Figure 8 log The noise level generated by the resistance is b 2 4x0 9 V rms Hz The noise spectral density is S b () 6x Volt2 68dB Hz volt The phase noise spectral density is S φ () K S 2 b () Sφ() ( 0 6 ) Sφ() At 0 KHz, S φ (0 KHz).6 X 0-3, corresponding to: S φ (0 KHz) -28 db Spectral purity S v (0 KHz) -28 db - 3 db -3 db I the control noise is white ( > c, Figure 8), the phase noise obtained is white requency noise with a slope o 20 db per decade. I the control noise is licker ( < c, Figure 8), the phase noise obtained is a licker requency noise with a slope o 30 db per decade. 3.3 NOISE IN THE PHASE LOCK LOOP Application Note # 25 Revised 3/4/95

26 We shall investigate the phase noise spectral density o a Voltage Controlled Oscillator (VCO) locked to a high stability reerence. Figure 9 shows a simpliied model o a phase lock loop. VCO C K o S LOOP FILTER H (p) COUNTER (/N) A B K c PHASE DETECTOR REFERENCE Figure 9 where (p) is the requency domain, (Laplace notation) ( jw) K c is the gain o the phase comparator (Volt/Radian) K o is the slope o the VCO (Hz/Volt) N is the dividing actor φ o (t) is the noise o the ree running VCO φ S (t) is the noise o the controlled VCO φ R (t) is the noise o the reerence source The phase luctuations o the VCO are divided by the counter N φ a ( p) Nφ s ( p) The output level o the phase comparator is proportional to the dierence in phase o the two channels φ s ( p) V b ( p) K c φ R ( p) N The output level is iltered and applied to the varicap o the VCO s requency control: Application Note # 26 Revised 3/4/95

27 φ s ( p) V c ( p) Hp ( )K c φ R ( p) N Hp φ vco ( p) K o V c ( p) K o K ( ) φ c s ( p) φr ( p) p N The VCO adds its internal noise: Hp φ vco ( p) K o K ( ) φ c s ( p) φr ( p) p φ N o ( p) Because the system is locked, the luctuations o the controlled VCO (φ S (p)) are also present Hp ( ) φ φ s ( p) K o K c s ( p) φ p R ( p) φ N o ( p) Assume that K K o K c N φ s ( p) p φ ( p + KH( p) ) o ( p) + NKH( p) φ ( p + KH( p) ) R ( p) φ s ( p) H o ( p)φ o ( p) + NH R ( p)φ R ( p) Or, expressed in terms o spectral density S φs () H o () 2 S φo () + N 2 H R () 2 S φr () To obtain a second-order loop (proportional and integral gain), the standard transer unctions are Application Note # 27 Revised 3/4/95

28 H o () 2 ( 4 nξ + n ) ( 4 nξ 2n ) + n ) H R () ( 4 nξ 2n ) 2 4 ( + + n ) where : n is the natural cut-o requency o the loop G τ RC G is the DC loop gain K c K o HO ( ) N τ RC is the integrator time constant value ξ x 2πRC Where: ξ is the loop damping actor - 2 b RC b is the cut-o requency o the continuous gain: G RC is the cut-o requency o the integral loop: τ RC n is the geometric average o the two cut-o requencies b and RC 2 G n G τ RC b RC RC The transer unction H R () corresponds to a low-pass iltering o the reerence noise multiplied by N. See Figure 20. The transer unction H o () corresponds to a high-pass iltering o the VCO noise. See Figure 20. Application Note # 28 Revised 3/4/95

29 H R () H O () 2 RC η b log Figure 20 The spectral density o the output signal, Figure 2, copies the noise o the reerence multiplied by N (or N 2 in terms o spectral density) close to the carrier, and copies the noise o the ree-running VCO ar rom the carrier. Between these two inluences a noise platorm is oten observed, which arises either rom the noise o the VCO iltered by the loop (shown in Figure 2) or rom the noise o the phase comparator, or the divider. S φs () 20 log n CONTROLLED VCO 3 2 FREE-RUNNING VCO 3 0 NOISE PLATFORM 2 NOISE FLOOR REFERENCE b log Figure 2 Application Note # 29 Revised 3/4/95

30 4 CHARACTERIZATION OF FREQUENCY GENERATORS The noise o a requency generator is characterized mainly by the spectral density o its phase noise S φ () and amplitude noise S a (). This representation is useul because it enables calculation, on the basis o these spectrums, o all the other parameters which characterize the noise o a requency generator. Take the spectrums, shown in Figure 22, or a carrier requency o 0 MHz. The amplitude noise and phase noise have the same noise loor: -40 db. SPECTRAL DENSITY -50 dbc 3 S φ () -40 dbc S a () o Hz khz 00 khz log 4. SPECTRAL PURITY Figure 22 Spectral purity is expressed in dbc. It is the relation between the power o the signal P s and the total noise power P b, per unit bandwidth ( Hz), and at a requency m rom the carrier P b Hz dbc 0log P s The Phase Noise contribution is called L(). A irst approximation gives Application Note # 30 Revised 3/4/95

31 L () example: Noise loor o L() -40 db -3 db -43 dbc The Amplitude Noise contribution is called M() example: Noise loor o M() -40 db -3 db -43 dbc The spectrum at the carrier requency o is S v () L() + M() example: Noise loor o S v () -43 dbc -43 dbc -40 dbc This spectrum is symmetrical around the carrier requency (two-sided) *Spectral Densities immediately become spectral purity, i one considers that: phase noise << radian 2. (The complete expression is given in Section.) 4.2 PHASE NOISE POWER -S 2 φ ()whichresultsinl ()dbc M () S φ ()db -S 2 a ()which resultsinm()dbc 3dB S a ()db 3dB By integrating the phase noise spectral density noise power obtained can be represented as Sφ() into a bandpass B, the For the example presented in Figure 22 Hz-00 khz: Pφ 5.54 µ rad 2 Hz- khz: Pφ 5.54 µ rad 2 khz-00 khz: Pφ 0.00 µ rad 2 B Pφ Sφ()d radians 2 o The square root o this power gives the rms power o phase noise, which corresponds to the mean luctuation o the signal s phase due to the noise: φ rms Pφ radian rms Hz-00 khz: φ rms 2.35 m rad rms Hz- khz: φ rms 2.35 m rad rms khz-00 khz: φ rms m rad rms Application Note # 3 Revised 3/4/95

32 Multiplying this value by 6 gives us the peak-to-peak (pp) luctuation o this phase: φ pp 6 Pφ radian pp Hz-00 khz: φ pp 4 m rad pp By calculating 0 log (Pφ / radian 2 ) the noise power obtained is in db: Application Note # 32 Revised 3/4/95

33 example Figure 22: Hz - 00 khz: Pφ db Hz - khz: Pφ db khz - 00 khz: Pφ -90 db NOTE: The phase noise power and mean phase luctuation depend (in this example) on the lower part o the spectrum ( < khz), i.e. closer to the carrier. 4.3 FREQUENCY NOISE POWER By multiplying the phase noise spectral density by the requency square, one gets the requency noise spectral density S () 2 Sφ() Hertz Hertz Hertz Thus, the requency noise power can be calculated P S ()d 2 Sφ()d Hertz 2 Hz-00 khz: P 3.35 Hz 2 Hz- khz: P 80 µhz 2 khz-00 khz: P 3.35 Hz 2 The square root o this power gives the mean luctuation o the requency o the signal, around its center value o rms P Hertz rms Hz-00 khz: rms.83 Hz rms Hz-00 khz: rms 8.95 m Hz rms khz-00 khz: rms.83 Hz rms Multiplying this value by 6 gives us the peak-to-peak (pp) requency luctuation pp 6 P Hertz pp Hz-00 khz: pp Hz pp This value represents the maximum luctuation o the center requency, due to the noise. NOTE: The requency noise power and mean requency luctuation depend (in this example) on the higher part o the spectrum ( > khz), i.e. ar rom the carrier. Application Note # 33 Revised 3/4/95

34 4.4 FREQUENCY STABILITY The stability o the center requency o the signal can be characterized by the variance o the measurement results obtained with a requency counter. This measurement method is one o the oldest used to characterize requency stability. The main interest o this method was that it was easy to set up. All the dierent types o variance can be calculated on the basis o the phase noise spectral density True variance Allan variance σ [ () t ] o π o T Sφ ()sin 2 ( πt)d 0 σ [ () t ] o Sφ()sin 4 ( πt)d π o T 0 The above expressions correspond to a noise power calculation in the bandwidths: sin 2 (π T ) and sin 4 (π T ) (T Time o measure) The values are expressed as a unction o time o measurement T. See Figure 23. H L () S φ () sin 2 (πτ) T khz Figure 23 Application Note # 34 Revised 3/4/95

35 Example igure 22: True Variance: (Tms) 0.77 x 0-9 Allan variance: (Tms) 0.95 x 0-9 The true variance corresponds to low-pass iltering o the requency luctuations, with a cut-o requency c 0.25/T. I this noise power is calculated with an ideal low-pass ilter with c 0.25/ ms 250 Hz, the ollowing result is obtained This result is very close to the calculated true variance: 0.77 X JITTER Jitter, or mean luctuation o the edge o the signal due to phase noise, is given by the relation: example Figure 22: rms 7.9mHz rms,i.e. rms X0 9 o T rms π o Sφ()d seconds rms 0 Hz-00 khz: o 0 MHz : T rms 37.5 picoseconds rms Hz- khz: T rms 37.5 picoseconds rms khz-00 khz: T rms 0.5 picoseconds rms Normalized Jitter: UI pp (unit interval peak-to-peak) T rms UI pp T o π φ ()d example Figure 22: Hz - 00 KHz: UI pp The jitter depends (in this example) on the low-requency luctuations ( < khz), o the phase noise, i.e. close to the carrier Application Note # 35 Revised 3/4/95

36 5 CONCLUSION - Today s noise measurement instruments, which measure the noise on a carrier signal, are based on phase and amplitude demodulation. - This method is quick and easy to set up and allows the user to make precise and high resolution measurements. - Processing the signal on phase noise and amplitude gives access to all the parameters which characterize a signal (spectral purity, noise power, requency instability, jitter.) - This method requires a reerence source, as in act, all noise measurement methods do. Application Note # 36 Revised 3/4/95