The Power Spectral Density of Phase Noise and Jitter: Theory, Data Analysis, and Experimental Results by Gil Engel

Home , Gaussian noise, Jitter, Shot noise, Spectral density

AN-1067

APPLICATION NOTE One Technology Way • P. O. Box 9106 • Norwood, MA 02062-9106, U.S.A. • Tel: 781.329.4700 • Fax: 781.461.3113 • www.analog.com

INTRODUCTION GENERAL DESCRIPTION Jitter on analog-to-digital and digital-to-analog converter sam- There are numerous techniques for generating clocks used in pling clocks presents a limit to the maximum signal-to-noise electronic equipment. Circuits include R-C feedback circuits, ratio that can be achieved (see Integrated Analog-to-Digital and timers, oscillators, and crystals and crystal oscillators. Depend- Digital-to-Analog Converters by van de Plassche in the References ing on circuit requirements, less expensive sources with higher section). In this application note, phase noise and jitter are defined. phase noise (jitter) may be acceptable. However, recent devices The power spectral density of phase noise and jitter is developed, demand better clock performance and, consequently, more time domain and frequency domain measurement techniques costly clock sources. Similar demands are placed on the spectral are described, limitations of laboratory equipment are explained, purity of signals sampled by converters, especially frequency and correction factors to these techniques are provided. The synthesizers used as sources in the testing of current higher theory presented is supported with experimental results applied performance converters. In the following section, definitions to a real world problem. of phase noise and jitter are presented. Then a mathematical derivation is developed relating phase noise and jitter to their frequency representation. The frequency domain representations, or power spectral densities, are shown to directly provide a measure of phase noise/jitter. The theory developed is associated with analog-to-digital and digital-to-analog converters. A spectrum analyzer and an oscilloscope are used to measure a variety of signals. Finally, theory is coupled with experimental results applied to an AD9235 analog-to-digital converter (ADC).

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TABLE OF CONTENTS Introduction ...... 1 Test Equipment ...... 10 General Description ...... 1 Oscilloscopes ...... 10 Revision History ...... 2 Spectrum Analyzers ...... 10 Definitions ...... 3 Proof ...... 11 Phase Noise ...... 3 Lab Results ...... 12 Jitter ...... 3 Signal 1 ...... 12 Power Spectral Density ...... 5 Signal 2 ...... 12 Example 1 ...... 6 Signal 3 ...... 13 Example 2—Phase Noise ...... 6 High Speed Converter ...... 15 Example 3—Jitter ...... 7 Conclusion...... 17 Application to Converters ...... 8 References ...... 17 Example 1 ...... 9

REVISION HISTORY 4/10—Revision 0: Initial Version

Rev. 0 | Page 2 of 20 Application Note AN-1067

DEFINITIONS Phase noise and jitter have various interpretations. In the context The function Φ(t) can be composed of frequency components of this application note, phase noise and jitter are defined as not related to ωt, for example, thermal noise, shot noise, and 1/f follows: noise (flicker noise). However, in most cases, it is modeled as Consider the sinusoidal signal, Gaussian noise (see Frequency Synthesizers Theory and Design Third Edition by Manassewitsch in the References section). sin(ωt + A) (1) Similarly, a sample clock can be considered a periodic square where: wave with rising and falling edges repeating at a fixed time ω = 2πf. interval, τ, such that f is the desired frequency. A is a constant phase offset. τ = 1/f (3) 1.5 1.0

0.8

0.6 1.0

0.4

0.2

t) 0.5 ω 0 sin( –0.2

–0.4 0 –0.6

–0.8 –0.5 –1.0 0 0.5 1.0 1.5 2.0 0 /2 3 /2 2 PERIOD ( ) 0 ≤ ωt ≤ 2 08932-003 08932-001 Figure 3. Sampling Clock Figure 1. Normalized Sinusoidal Signal PHASE NOISE JITTER Phase noise is defined as an arbitrary function Φ(t) such that Jitter can be defined as an additive time variation Δ(t) to the Equation 1 becomes fixed interval τ, giving sin(ωt + A + Φ(t)) (2) τ + Δ(t) = 1/f + Δ(t) (4) 1.5 1.0

0.8

0.6 1.0

0.4

0.2 (t)) 0.5

t + θ 0 ω –0.2 sin(

–0.4 0

–0.6

–0.8 –0.5 –1.0 0 0.5 1.0 1.5 2.0 0 /2 3 /2 2 PERIOD ( ) 0 ≤ ωt ≤ 2 08932-004 08932-002 Figure 4. Sampling Clock with Jitter Figure 2. Sinusoidal Signal with Phase Noise Likewise, Δ(t), is typically characterized as Gaussian noise.

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0 Noise analysis is straightforward above 5 kHz until the active –20 devices are limited at high frequency. Noise below 5 kHz exceeds

–40 the shot noise and thermal noise. This noise varies inversely with frequency and is identified as 1/f noise. Figure 5 shows a –60 typical noise spectrum of an oscillator (see Manassewitsch in –80 the References section).

–100

–120

NORMALIZED POWER(dB) –140 –160 –180 100 105 1010 1015 1020 FREQUENCY (MHz) 08932-005 Figure 5. Single-Sideband Noise Spectrum of an Oscillator

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POWER SPECTRAL DENSITY

Time domain signals have a direct relation to the frequency 1.0 domain through the Fourier transform (see Discrete-Time Signal 0.9

Processing by Oppenheim in the References section). The 0.8

Fourier transform can be viewed as the magnitude and phase 0.7 spectrum of a signal. A signal’s power can also be viewed in the 0.6 frequency domain. The power spectrum or power spectral 0.5 density is given by 0.4

Syy(ω) = Y(ω) × Y’(ω) (5) 0.3 where Y(ω) is the Fourier transform of y(t). POWER NORMALIZED TO 1 0.2 As stated previously in the Definitions section, Φ(t) can be any 0.1 arbitrary undesired signal. To simplify this analysis, Φ(t) is set 0 to a single frequency. Consider the following: 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 (Hz) × 104 08932-006 Φ(t) = θdsinωmt (6) Figure 6. Power Spectral Density, Syy(ω) Such that Equation 2 becomes Figure 6 is a plot of y(t) = sin(ωct + θdsinωmt) (7) Syy(ω) = Y(ω) × Y’(ω) The result is a phase-modulated signal, y(t), with maximum where Y(ω) is the Fourier Transform of y(t). phase deviation in radians, θd, at a frequency, fm, with ωm = 2πfm, and no offset, A = 0. Syy(ω) displays the magnitude of the power at the frequency, f. The power spectral density of the signal, y(t), modulated by a The Jacobi-Anger expansion (see Concise Encyclopedia of single frequency, fm, only has components at fc and fm with Mathematics by Weisstein in the References section) states that Bessel-squared magnitudes.  iz )cos( n in The higher order Bessel coefficients attenuate very quickly. A e   n )( ezJi (8) n  log power scale provides better dynamic range, showing the or higher order components in the same view as the large carrier

 component. The log of Syy(ω) is given by the following equation: e iz )sin(  )( ezJ in (9)  n Lpy(ω) = 10log10(Syy(ω)) (13) n  can be manipulated with the help of Euler’s identity to give and is shown in Figure 7.

 0 0   2n nzJzJz )2cos()(2)()sincos( (10) n1 –20 and –40  –60 z   n12 nzJ  ])12cos[()(2)sinsin( (11) n1 –80 (dB) where the Jn(z) factors are Bessel functions of the first kind. –100

Using trigonometric identities, Equation 7, Equation 10, and –120 Equation 11 can be manipulated to give –140

0 d c y(t) = J (θ )sin(ω t) –160 + J1(θd)[sin(ωc+ ωm)t – sin(ωc − ωm)t] 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 2 d c m c m + J (θ )[sin(ω + 2ω )t – sin(ω − 2ω )t] (Hz) × 104 08932-007 + J3(θd)[sin(ωc+ 3ωm)t – sin(ωc − 3ωm)t] Figure 7. Log of Power Spectral Density, Syy(ω) + … (12) Additional terms are now clearly visible. As the phase deviation From Equation 12, it can be seen that y(t) has a first-order Bessel increases, the magnitude at the carrier frequency decreases and component at the carrier frequency, fc, and Bessel-weighted the magnitude of the modulation terms increases. 500 mrad of signals at multiples of the modulation frequency, fm, offset from phase deviation reduces the carrier power by ~12%. the carrier. For small phase deviations, θd << 1 rad, J0(θd) ≈ 1, J1(θd) ≈ θd/2, The power spectral density, Syy(ω), of the function y(t) for fc = and J2(θd) … Jn(θd) ≈ 0 (see Manassewitsch in the References 32,768 Hz and fm = 1024 Hz with a phase deviation of 500 mrad section). (where mrad means milliradians) is shown in Figure 6. Rev. 0 | Page 5 of 20 AN-1067 Application Note

As the phase deviation approaches zero, the carrier power This is referred to as a single-sideband measurement technique approaches 100%. Furthermore, small phase deviations have a and is usually taken per root Hz (see Manassewitsch in the smaller percentage of the carrier frequency power distributed References section). among the modulation terms. This, in turn, results in a sum of The rms modulation can be expressed in several ways. modulation terms that approximate the power of Φ(t) more accurately. Arms ≈ √Prms in radians (shown in Equation 19) 0 Bessel functions have the following property: A rms ≈ 360 × √Prms/2π (20)

 Equation 20 expresses the phase deviation in degrees. 2 2 0   n βJβJ )(2)(1 (14) n1 To relate phase noise to time jitter, use the following equation: t Taking advantage of the small phase deviation properties, the A rms ≈ τ × √Prms/2π (21) root mean square (rms) power of Φ(t) (for single-tone where τ = 1/fc expresses the phase deviation in time. sinusoidal modulation) is approximately given by EXAMPLE 2—PHASE NOISE  2 Prm ≈   n βJ )(2 (15) Consider a noisy sinusoidal signal sampled with an ideal clock n1 y(t) = sin(ωct + N(t)) or

2 where: Prms ≈ 1 − J0(β) (16) ωc = 2π26,2144. The phase deviation can also be expressed in terms of rms N(t) is Gaussian noise with a standard deviation, σ = 10 mrad. amplitude. The constructed signal is sampled at 4 million samples per Arms ≈ √Prms (17) second for 15 ms, acquiring 65,000 samples. The log of the EXAMPLE 1 power spectral density is normalized to 0 dB and is shown in Figure 8. For a phase deviation, θd, of 100 mrad, 20 2 Prms ≈ 1 − J0(0.1) 0 Prms ≈ 1 – 0.9950094

Prms ≈ 1 – 0.0049906 –20 Arms ≈ 0.0706444 –40 Comparing this result with the power of a sinusoidal signal, –60 e(t) = Asin(ωt)

2 (dB) POWER Pe = A /2 –80

For A = 0.1, the rms power is Pe = 0.005 and Arms = A/√2 = –100 0.0707107, which confirms that, for small phase deviations, –120 the modulating terms sum to provide a good approximation of the rms power. –140 00.20.60.4 0.8 1.0 1.41.2 1.6 1.8 2.0 FREQUENCY (MHz) × 106 This argument can be extended to more complex modulating 08932-008 signals. More complex modulation functions can be treated as a Figure 8. 260 kHz, 10 mrad Phase Noise superposition of many frequency terms, each affecting the The fundamental is at around 260 kHz, and there is noise across spectrum. The power spectral density has additional terms that the spectrum. sum to represent the rms power of the modulating signal. The Using the discrete form of Equation 18, rms power for an arbitrary function, Φ(t), with small amplitude, N 2/ (θd << 1 rad), is given by rms   (),  nnnSyyP c (22) n0 Prms = ∫Syy(ω)dω – Syy(ω = ωc) (18) Equation 18 states that the rms power of a phase modulating Sum the magnitude of the power at all frequencies from 0 to signal is equal to the sum of all the components minus the Nyquist, not including the power at the fundamental. The power at the fundamental (or carrier frequency). resulting noise power is −4 For a sinusoidal signal, y(t), phase modulation produces a symme- Prms = 1.0017 × 10 tric power spectral density, such that the rms power can also be The rms amplitude is Arms = 0.010008 rad. given by Note that the 0.008 mrad discrepancy is several orders of  magnitude smaller than the exact rms noise amplitude of P 2)( dSyy  (19) rms  10 mrad, giving a very good approximation.  c Rev. 0 | Page 6 of 20 Application Note AN-1067

20 The input signal at Time t0 with a phase deviation of Φ(t0) = 0 has Amplitude A0. A noisy input signal with phase deviation 0 Φ(t0) = ΔΦmrad at Time t0 has Amplitude AΦ. By the same –20 token, the input signal sampled at a time deviation, t1 = t0 + Δt, has Amplitude Aτ. –40 1.0 –60 0.8 AΦ

POWER (dB) –80 0.6 ΔΦ A0 0.4 –100

0.2 Δ –120 t) A ω 0 –140 sin( –0.2 t0 00.20.60.4 0.8 1.0 1.41.2 1.6 1.8 2.0 FREQUENCY (MHz) × 106 08932-010 –0.4 Figure 10. 65k FFT of a Phase Noise Modulated 260 kHz Tone –0.6 Sampled at 4 MSPS –0.8 Using Equation 22, sum the magnitude of the power at all –1.0 frequencies from 0 to Nyquist, not including the power at the 0 /2 3 /2 2 fundamental. The resulting noise power is 0 ≤ ωt ≤ 2 08932-009 −4 Figure 9. Effects of Timing and Phase Deviation on Sinusoidal Signal Prms = 1.0031 × 10

Figure 9 shows that there exists a time deviation, Δt, and a phase and the rms amplitude is deviation, ΔΦ, that produce the same amplitude, AΔ. For all intents Arms = 0.010016 rad and purposes, a phase deviation, ΔΦ, with rms amplitude equal Insert the results into Equation 21 to obtain to the jitter, Δt, rms time deviation produces identical results. t −8 t EXAMPLE 3—JITTER A rms = 4.86455 × 10 s or A rms ≈ 49 ns In Example 2, the power spectral density of a signal with phase The results match those obtained in Example 2. noise, N(t), has a Gaussian distribution and standard deviation Broadband noise modulating the clock or input signal results in of σ = 10 mrad. Now consider a signal sampled with a jittery a power spectrum with distributed noise. Furthermore, noise clock having Gaussian noise, η(t). Equation 21 can be used to modulating the input signal or clock produces symmetric noise determine the rms jitter to produce the same effect as 10 mrad about the carrier. The power spectral density can be used to deter- of phase noise. The resulting output is mine the phase noise or jitter associated with specific frequency components or frequency ranges. Large symmetric terms may y(t) = sin(ωc(t + η(t))) highlight specific frequencies that are modulating the signal where the carrier frequency is again 260 kHz and η(t) is and/or clock. The rms power associated with specific frequen- Gaussian noise with a standard deviation of 6.0713 ns. cies can be extracted directly from the power spectral density. The constructed signal is sampled at 4M samples per second for For ranges of frequencies, the following equation can be used:

15 ms, acquiring 65k samples. The log of the power spectral f2 density is normalized to 0 dB. Prms   nSyy )( (23)  fn 1 Or for single sideband,

2 Prms  )(2 dSyy  (24) 1

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APPLICATION TO CONVERTERS Current high speed converters have sampling rates higher than Using Equation 17, this noise power can be related to an rms 100 MSPS at resolutions greater than 12 bits. Signal-to-noise ratios phase deviation. (SNR) better than 70 dBc are routinely achieved with a spurious- Arms ≈ 0.0631 mrad free dynamic range (SFDR) better than 100 dBc. Digital-to-analog converter (DAC) performance is directly impacted by the sampling For a 10 MHz input signal, this corresponds to jitter of t clock jitter. Tones produced by DACs sampled with a noisy A rms ≈ τ × √Prms/2 clock can produce a signal with phase noise. ADCs are affected t −9 −9 A rms ≈ 100 × 10 (√3.98107 × 10 )/2π by noise on both the sampling clock and the input signal. The t −12 results derived in the Example 3—Jitter section can be applied A rms ≈ 1.004 × 10 sec to converters. Table 1 lists converter SNR limits due to quantization noise and Associating the results to an ADC, consider the configuration comparable phase noise rms amplitude. shown in Figure 11. Table 1. Converter SNR Limits L INPUT SIGNAL IDEA CONVERTER Test Setup A sin(ωt) N BITS Bit Theoretical Corresponding 10 dB 6 dB ADC No. SNR Limit (dB) Phase Noise (mrad) (mrad) (mrad) 8 49.96 3.177 1.005 1.592 10 62 0.794 0.251 0.398 12 74.04 0.199 0.063 0.1 SAMPLE CLOCK 14 86.08 0.0497 0.016 0.025 PERIOD 08932-011 Figure 11. ADC Functional Block Diagram 16 98.12 0.0124 0.004 0.006 The ADC samples the input signal, Asin(ωt), at a time instant, t Table 1 also provides the phase noise rms amplitude for test (having Period τ), producing a quantized output of N bits. setups at 10 dB and 6 dB better than the converter. In some cases, a test setup of 6 dB better than the converter is acceptable Assuming the noise on the input signal and the noise on the (especially when 10 dB is difficult to obtain). sampling clock are independent, the total noise is given by the root-sum square (RSS). If the magnitude of the noise is large Equivalent jitter amplitude is easily obtained using Equation 21. enough, the maximum performance of the converter is affected. Converter SNR performance is typically determined using the The quantization noise is directly proportional to the number of power spectral density. The sampling frequency and the num- bits. The maximum error a sample has within the ADC’s range ber of data samples directly determine the frequency resolution. is the least significant bit resolution, QN, divided by 2 (QN/2) (see A 4k FFT for a converter sampling at 32 MHz accumulates enough van de Plassche, Oppenheim, and Delta-Sigma Data Converters data to resolve frequencies down to 8 kHz. Consequently, the Theory Design and Simulation by Norsworthy, Schreier, and Temes power spectral density displays information in 8 kHz intervals. in the References section). The error is defined by the signal Each 8 kHz bin provides the sum of the power of the frequen- being sampled. For randomly changing signals, the quantization cies within that interval and frequencies aliased into that interval. error is uncorrelated and consequently lies anywhere within The magnitude of a component that is 1 kHz from the carrier ±QN/2. If the error is statistically independent of the signal being cannot be determined under these circumstances. The frequency sampled, it can be shown that the maximum SNR that can be resolution is improved by taking larger FFTs. Low frequency achieved is given by phase noise, such as 1/f noise, can be resolved to 32 Hz by taking a 1M FFT for a converter sampling at 32 MHz. SNR = 6.02N + 1.8 (25)

For a 12-bit converter, the theoretical maximum SNR is ~74 dBc. A total quantization noise power of 74 dBc corresponds to Pqn ≈ 10−7.4 Pqn ≈ 39.8107 × 10−9 It is desirable to have a test setup that is 10 dB better than the converter being tested. To test a 12-bit converter, the desired test setup noise power is 84 dBc. Pqn ≈ 10−8.4 Pqn ≈ 3.98107 × 10−9

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EXAMPLE 1 A close-up view (see Figure 13) shows skirting around the For a 12-bit ADC sampling at 32M samples per second with fundamental. The frequency resolution is 8 kHz, and the 2 kHz 20 ps of clock jitter, the input signal is at 4 MHz with a 2 kHz, modulation terms have combined with the fundamental and the 1 mrad phase noise component and 0.5 mrad of Gaussian phase surrounding bins. noise. Acquiring 4k samples using this configuration produces 20 the power spectral density shown in Figure 12. 0 20 –20

0 –40

–20 –60 (dB) –40 –80

–100 –60 (dB) –120 –80 –140 –100 –160 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 –120 (Hz) × 106 08932-014 –140 Figure 14. Higher Resolution FFT Showing Modulation Terms 0 2 4 6 8 10 12 14 16 18 (Hz) × 106 08932-012 Using a 65k FFT, the frequency can be resolved to 500 Hz. Figure 12. 4k FFT of a Modulated 4 MHz Tone Sampled at 32 MSPS Two new symmetric terms are discovered, implying a phase Using Equation 22, calculate a noise power of 6.628−7 W. H o w e v e r, modulation. Once these terms are added into the integrated −7 the theoretical value should be 5.677−7 W, which is the RSS of noise power, calculate a noise power of 5.696 W. the jitter and phase noise and quantization noise.

–20

–40

–60 (dB)

–80

–100

–120

3.8 4.0 4.2 4.4 4.6 4.8 (Hz) × 106 08932-013 Figure 13. Close-Up View of 4 MHz Fundamental Showing Skirting Due to Low Bin Resolution

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TEST EQUIPMENT Phase noise and jitter can be viewed as a time deviation using 4 4 an oscilloscope or as a frequency spectrum using a spectrum 3 3 analyzer. 2 2 OSCILLOSCOPES 1 1 Oscilloscopes fall into two categories: real-time and sampling (see XYZs of Oscilloscopes by Tektronix in the References 0 0 section). –1 –1 QUADRATURE (Q) QUADRATURE (Q)

Real-time oscilloscopes capture a stream of samples on a single –2 –2 trigger event. The cycle-to-cycle deviation is extracted from the data at a fixed threshold. This method is limited by the time inter- –3 –3 val measurement accuracy of the oscilloscope and its internal –4 –4 0–3–2–1012340.2 0.4 –4 jitter. The Tektronix TDS7404 specifies an accuracy of ±8.5 ps IN-PHASE (I) and a typical jitter noise floor of 1.5 ps rms. The Tektronix 0.4

TDS694C has an accuracy of ±15 ps. Increased accuracy can 0.3 be achieved with statistical methods by including the vertical 0.2 resolution of the oscilloscope and large record lengths in the 0.1 processing. Tektronix claims a 1.5 ps jitter measurement accu- 0 racy using the latter technique (see Analyzing Clock Jitter Using –4 –3 –2 –1 0 1 2 3 4

IN-PHASE (I) 08932-015 Excel and Understanding and Performing Precise Jitter Analysis Figure 15. Spectrum Analyzer Envelope Detector Input Distribution by Tektronix in the References section). The magnitude is obtained with an envelope detector and is Sampling oscilloscopes accumulate input signal data with each given by trigger. To obtain a time deviation, the input signal is repeti- 22 tively sampled, acquiring a distribution of points at a horizontal I  vvv Q (26) cross-section. The horizontal and vertical scales are adjusted depending on the magnitude of the time deviation being meas- The noise magnitudes form concentric rings about the center of ured. This method of time deviation measurement is mainly Figure 15. The count within each ring provides a distribution of limited by the trigger jitter. Sampling oscilloscopes have a much the noise magnitude. The distribution function for the noise better time interval accuracy and, more importantly, a sampling envelope is actually a Rayleigh distribution (see Probability, interval as low as 10 fs. The time interval accuracy of a Tektronix Random Variables, and Stochastic Processes by Papoulis in the 11801C is 1 ps + 0.0004% × (position) and the trigger jitter is References section). v2 typically 1.1 ps rms. The TDS8000B Tektronix sampling oscillos-  v 2 cope specifies a trigger jitter of 800 fs (see Automatic Measurement vD )(  e 2 (27) env 2 Algorithms and Methods for the 8000 Series Sampling Oscilloscopes by Tektronix in the References section). Knowing the probability density function, the average of the voltage envelope can be determined using SPECTRUM ANALYZERS   Spectrum analyzers display a signal in terms of its frequency   env )( dvvvDv  (28) content. The spectrum displays a series of measurements within 0 2 the resolution bandwidth (RBW) settings. Spectrum analyzers The average power is given by display the voltage and/or power of a signal in a linear or log  2 2 display. Viewing the power of a signal is analogous to the power v 2 p   env )( dvvD  (29) spectral density plots obtained through Fourier analysis. 0 R R Random noise in electronics has a Gaussian distribution. There- Calculating the power by squaring the average envelope voltage fore, samples within the RBW of the spectrum analyzer have a then dividing by R does not provide the same results as probability distribution; however, the samples are displayed as Equation 29. The result is 1.05 dB smaller. simple magnitudes. The spectrum analyzer actually measures 2 Rv 2  R with the in-phase (I) and quadrature (Q) components (see  log(10)log(10 ) p 42 R Spectrum Analyzer Measurements and Noise by Hewlett Packard in the References section). The I/Q components provide the   log(10 ) magnitude and phase of the signal. Band-passed noise has a 4 Gaussian distribution independently in both the I and Q = −1.05 dB components. Rev. 0 | Page 10 of 20 Application Note AN-1067

Further considerations must be taken when using a spectrum Therefore, analyzer in its logarithmic display mode. In the logarithmic display  mode, the input signal passes through a log amplifier. This in log   log env )( dvvDpp turn results in a logged probability density function. Addition- 0  ally, the spectrum analyzer displays an average of the log. Log   env )()log(20 dvvDv (38) processing results in a response to noise that is 2.51 dB lower. 0 v2   PROOF v 22   v)log(20 2 dve Proof of 2.51 dB underestimation of noise follows. 0 

2 Plog = 10 log(v ) or 20 log(v) (30) Let Take the derivative of both sides,  2 v22     v log10)log(20  2  20 1  2  dplog  dv (31) (39) 10ln v  v 2  2    log10)2log(10  2  Plog  2  v  10 20 (32) To obtain and 2  v v  2 plog p  2 )2log(10 2 dve 20  10ln  log  2 dv  10  dplog (33) 0   20  2 (40)  2 v   vv  2    2 Use Liebniz’s rule (see Weisstein in the References section)  log10   22 dve 0  2   dx y  x xfyf )()( (34) The first integral goes to 1 because this is simply the integral of dy the Rayleigh probability density. Let or per Papoulis (see the References section) v u  2 (41) x xf )( 1  y yf )(   yTx }{at (35) dy in the second integral, giving dx  To obtain 2 10 u plog )2log(10   ln dueu (42) 10ln 0 vD )( y pD )(  env v  10at 20 log /dvdp The first term is the log of the average power; the second term log is the negative of the Euler-Mascheroni constant (see Weisstein y y (36) y 20 20 in the References section). The Euler-Mascheroni constant has  D )10(10 20  env v  10at been calculated to 7,000,000 digits and is denoted by Υ. Υ is 10ln/20 approximately equal to 0.5772, making the last term equal to The result is the probability distribution of log power of the −2.5067. input envelope. Most modern spectrum analyzers feature noise measurements The average log power is then given by that apply the necessary correction factors.  In addition to known correction factors, which must be applied log   )( dppDpp loglogloglog (37)  as necessary, spectrum analyzers must be configured correctly to provide accurate results. Smaller input signals are measured From Liebniz’s rule (and Papoulis) more accurately by lowering the reference level. However, lowering fy(y)dy = fx(x)dx the reference level increases the gain of the input IF stage. Care must be taken so that the initial IF stage is not overloaded. Overloading the IF input sections may cause distortion products (see Fundamentals of Spectrum Analysis by Rauscher in the References section). Furthermore, finer frequency and amplitude resolution measurements are obtained by enhancing the resolution bandwidth (RBW) and video bandwidth (VBW), respectively. However, enhanced resolution comes at a cost of longer sweep times. Fortunately, software is available that takes the desired measurements and applies the appropriate correction factors. Rev. 0 | Page 11 of 20 AN-1067 Application Note

LAB RESULTS SML-01 SML-01 (MODULATED) +50mV

C1 Tr In CSA8000B

08932-016 Figure 16. Oscilloscope Setup

SML-01 (MODULATED)

–50mV 2ns/DIV

CH1 10.00mV/0.0V MAIN 2.00000ns 210.00n 08932-019 Figure 19. Close-Up View of Rising Edge Taking a horizontal histogram at the rising edge cross-section In CSA8000B displays a standard deviation in a time of 2.218 ns and a peak-

08932-017 Figure 17. Spectrum Analyzer Setup to-peak deviation of 6.6 ns. 10 A modulated signal generated by a Rohde & Schwarz SML-01 0 was measured using a Tektronix CSA8000B and a Rohde & Schwarz FSIQ7. A second nonmodulated SML-01 set to the same –10 frequency as the first SML-01 is used to trigger the CSA8000B. –20

Phase noise due to large single-tone modulation, large Gaussian –30 noise modulation, and small noise modulation was analyzed. In –40 all cases, the source used is a Rohde & Schwarz SML-01. –50 POWER (dB)

An alternative oscilloscope setup uses a broadband resistive –60 splitter to feed both the trigger and the sampling inputs. This –70 method rejects low frequency noise, which may produce artificially low noise measurement results. –80 –90 SIGNAL 1 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 FREQUENCY (MHz) 08932-020 +500mV Figure 20. Single-Frequency Phase-Modulated Frequency The spectrum analyzer clearly shows a spectrum due to single- frequency phase modulation. The first term to the right of the carrier has a Bessel factor of J1(θd). Because J1(θd) ≈ θd/2, the modulation can be approximated at 200 mrad. C1 C1 SIGNAL 2

+500mV

–500mV 10ns/DIV

CH1 100.0mV/0.0V MAIN 10.0000ns 210.00n 08932-018 Figure 18. SML-01 at 10 MHz, Modulated 200 mrad at 101 kHz C1

–500mV 20ns/DIV

CH1 100.0mV/0.0V MAIN 20.0000ns 210.00n 08932-021 Figure 21. SML-01 at 10 MHz with 67 mrad RMS of Gaussian Noise Modulation Rev. 0 | Page 12 of 20 Application Note AN-1067

+50mV +50mV

C1 C1

–50mV 2ns/DIV –50mV 2ps/DIV

CH1 10.00mV/0.0V MAIN 2.00000ns 210.00n 08932-022 CH1 1.000mV/0.0V MAIN 2.00000ps 20.302n 08932-025 Figure 22. Close-Up View of Rising Edge Figure 25. Close-Up View of Rising Edge A horizontal histogram at the rising edge cross-section displays In this case, the deviation is so small that it is below the trigger a standard deviation in a time of 1.005 ns and a peak-to-peak jitter of the oscilloscope. The result shows a standard deviation deviation of 7.92 ns. The oscilloscope trigger input accuracy of 837.3 fs and a peak-to-peak deviation of 5.56 ps. decreases at lower slew rates. 10

10 0

0 –10

–10 –20

–20 –30

–30 –40

–40 –50 POWER (dB) –50 POWER (dB) –60

–60 –70

–70 –80

–80 –90 999.0 999.4 999.8 1000.2 1000.6 1001.0 –90 FREQUENCY (MHz) 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 08932-026 FREQUENCY (MHz) Figure 26. Full-Scale 1 GHz Signal with 1.6 mrad RMS of Phase Noise 08932-023 Figure 23. 10 MHz Signal with Noise Out to 1 MHz from Carrier The spectrum analyzer is centered at 1 GHz and set to a span of Figure 23 shows a spectrum due to broadband noise out to 2 MHz. The dynamic range can be enhanced by changing the 1 MHz from the carrier. Using a single sideband phase noise frequency range so that the large carrier is outside the span. measurement (see Equation 19), the calculated phase noise has –70 an rms amplitude of 0.0676 radians, corresponding to about 1.075 ns of time jitter. –80 SIGNAL 3

+500mV –90

–100 POWER (dB)

–110 C1

–120 1000.0 1000.4 1000.8 1001.2 1001.6 1002.0 FREQUENCY (MHz) 08932-027 Figure 27. Single Sideband 1 GHz Signal with 1.6 mrad RMS Phase Noise

–500mV 200ps/DIV

CH1 100.0mV/0.0V MAIN 200.000ps 20.5072 08932-024 Figure 24. SML-01 Set to Produce a Clean 1 GHz

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With the range now starting 5 kHz from the carrier, Figure 27 0 shows that there is broadband phase noise to 1 MHz from the –20 carrier. A reference level that is too low causes the distortions appearing in Figure 27. To obtain accurate results, the measure- –40 ments must be made in smaller intervals. Measurements within –60 10 kHz of the carrier must have a reference level that will not –80 overload the input IF stage. (dB) With a single sideband measurement (see Equation 19), the –100 phase noise is measured at 1.6 mrad. This corresponds to 255 fs, –120 far below the trigger jitter resolution of the oscilloscope. –140

–160 10 100 1k 10k 100k 1M 10M (Hz) 08932-028 Figure 28. Single Sideband 1 GHz Signal with 1.6 mrad RMS Phase Noise and Nonoverloaded Input Stage

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HIGH SPEED CONVERTER Phase noise and jitter were introduced to an AD9235 application. Figure 30 shows symmetric peaks about the fundamental, The results obtained were verified using a spectrum analyzer indicating a modulation. The bin width for the FFT is ~1057 Hz. and correlated to the developed theory. The AD9235 is a high Counting bins, the main peaks are found in the ninth and 10th speed analog-to-digital converter that features a 65M sample bins from the fundamental, implying a modulation between clock and SNR around 70 dBc. The experiment was executed on 9.5 kHz and 10.5 kHz. The modulation term is spread out due a CTS5340 tester. Rohde & Schwarz SMGUs generated both the to a sampling rate that is not a direct multiple of 10 kHz and a input tone and clock. The input frequency was set to 2.4 MHz, nonwindowed FFT. For small signal modulation (a safe assump- and the clock input was set to 259.995 MHz divided down to tion because the modulation peaks are more than 50 dBc), the produce a sampling rate, fs ≈ 65 MSPS. The sampling rate was first modulation Bessel term, J1(θd), is approximately θd/2. Sum decimated to produce an effective sampling rate, fes ≈ fs/15. the power for eight highest peaks around the modulation term The application normally generates an SNR around 70 dBc. to obtain −5 A typical power spectral density, using a 4k FFT with the PJ1 ≈ 2.0324 × 10 fundamental aliased into Bin 1827, is shown in Figure 29. θd/2 = √PJ1 ≈ 0.0045 0

θd = 0.009 Even though the modulation terms are spread throughout the spectrum, most of the modulation energy is centered around –40 fc ± fm. Summing eight terms results in an approximation within 10% of the actual value.

(dB) The power spectral density of a fundamental with added phase

–80 noise is shown in Figure 31. 0

–120 0 0.54 1.08 1.62 2.16 –40 FREQUENCY (MHz) 08932-029 Figure 29. FFT of AD9235 ADC Application, Sample Rate Decimated to

~65 MSPS/15 and Input Frequency Set to 2.4 MHz (dB) The signal Φ(t) = 0.01 sin2π10,000t was modulated onto the –80 fundamental. The resulting power spectral density is shown in Figure 30. 0

–120 0 0.54 1.08 1.62 2.16 FREQUENCY (MHz) 08932-031 Figure 31. FFT of AD9235 ADC Application with –40 Phase Noise Modulating the Fundamental The CTS5340 tester sends a filtered signal to the application (dB) board. The equivalent noise bandwidth for the 2.4 MHz filter is

–80 approximately 300 kHz. The bandlimited noise is clearly seen around the fundamental, indicating phase modulated noise on the 2.4 MHz input tone. The phase noise results in an SNR of 58 dBc.

–120 0 0.54 1.08 1.62 2.16 FREQUENCY (MHz) 08932-030 Figure 30. FFT of AD9235 ADC Application with a 10 kHz Tone Modulating the Fundamental

Rev. 0 | Page 15 of 20 AN-1067 Application Note

0 In this case, the power spectral density noise floor has been

–20 slightly raised, resulting in an SNR of 64 dBc. The clock source is a Rohde & Schwarz SML-01 set to 259.995 MHz. A single –40 sideband measurement of the SML-01 clock source is shown in

–60 Figure 34. 0 –80 (dB) –20 –100 –40 –120 –60 –140 –80 –160 (dB) 10k 100k 1M –100 (Hz) 08932-032 Figure 32. Single-Sideband Noise Spectrum of Fundamental –120 with 1.5 mrad of Phase Noise –140 A single sideband measurement of the input signal shows 1.5 mrad of phase noise. –160 10 100 1k 10k 100k 1M 10M 100M 2 (Hz) Prms = A 08932-034

−6 Figure 34. Phase Noise Prms = 2.25 × 10 = −56.478 dB The clock source has 66.9 mrad of phase noise. This matches the source phase noise with the application test t 6 results. A rms = 0.0669/(2π259.995 × 10 ) ≈ 40.953 ps The results of jitter added to the sampling clock are shown in Figure 33. The phase noise on the sampling clock corresponds to about 0 41 ps of clock time jitter. The jitter can be related to radians of phase noise on the 2.4 MHz input signal.

t −12 6 A rms = (40.953 × 10 ) × (2π2.4 × 10 ) ≈ 0.618 mrad –40 41 ps of clock jitter corresponds to 0.618 mrad of phase noise on the 2.4 MHz fundamental.

(dB) 2 Prms = A rms

–80 −7 Prms = 3.814 × 10 = −64.187 dB The noise on the sampling clock matches the application test results.

–120 0 0.54 1.08 1.62 2.16 FREQUENCY (MHz) 08932-033 Figure 33. FFT of AD9235 ADC Application Sampled with Jitter Clock

Rev. 0 | Page 16 of 20 Application Note AN-1067

CONCLUSION The theory presented in this application note provides a direct Rauscher, Christoph. 2001. Fundamentals of Spectrum Analysis. relationship between phase noise and jitter and their frequency Germany: Rohde & Schwarz. domain representation. Analysis of phase noise and jitter in the Tektronix. 2009. XYZs of Oscilloscopes. Application Note. frequency domain highlights the content of the noise signals. Furthermore, measurements in the frequency domain provide Tektronix. 2001. Analyzing Clock Jitter Using Excel. Application enhanced resolution at higher frequencies. Note. REFERENCES Tektronix. 2004. Understanding and Performing Precise Jitter Analysis. Application Note. Hewlett Packard. 1998. Application Note 1303. Spectrum Analyzer Measurements and Noise. Tektronix. 2000. Automatic Measurement Algorithms and Methods for the 8000 Series Sampling Oscilloscopes. Manassewitsch, Vadim. 1987. Frequency Synthesizers Theory Application Note. and Design Third Edition. Wiley-Interscience. van de Plassche, Rudy. 1994. Integrated Analog-To-Digital and Norsworthy, Steven, Richard Schreier and Gabor Temes. 1997. Digital-To-Analog Converters. The Netherlands: Kluwer Delta-Sigma Data Converters Theory Design and Simulation. Academic Publishers. IEEE Press. Weisstein, Eric. 1999. Concise Encyclopedia of Mathematics. Oppenheim, Alan and Ronald W. Schafer. 1989. Discrete-Time Florida: CRC Press. Signal Processing. Prentice Hall. Papoulis, Athanasios. 1984. Probability, Random Variables, and Stochastic Processes. McGraw-Hill.

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NOTES

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NOTES

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