NIST Special Publication 250-90 

Calibration Uncertainty for the NIST 

PM/AM Noise Standards 

Archita Hati 

Craig Nelson 

Neil Ashby 

David Howe 

Carrier

USB

LSB

USB

LSB

USB

LSB

USB

LSB

Carrier
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T
NIST Special Publication 250-90

Calibration Uncertainty for the NIST 

PM/AM Noise Standards

Archita Hati

Craig Nelson

Neil Ashby

David Howe

Time and Frequency Division

Physical Measurement Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80305

July 2012

U.S. Department of Commerce

Rebecca Blank, Acting Secretary

National Institute of Standards and Technology

Patrick Gallaher, Director
Certain commercial entities, equipment, or materials may be identified in this 

document in order to describe an experimental procedure or concept adequately. Such 

identification is not intended to imply recommendation or endorsement by the 

National Institute of Standards and Technology, nor is it intended to imply that the 

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Special Publication 250-90

Natl. Inst. Stand. Technol. Spec. Publ. 250-90, 44 pages (July 2012)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2001

_________________________________________

For sale by the Superintendent of Documents, U.S. Government Printing Office

Internet: bookstore.gpo.gov — Phone: (202) 512-1800 — Fax: (202) 512-2250

Mail: Stop SSOP, Washington, DC 20402-0001
iii 

Contents		

1 Introduction ............................................................................................................................. 1

2 AM and PM noise - Basic concepts ........................................................................................ 2

2.1 Characterization of AM and PM noise ............................................................................. 2

2.2 Overview of amplitude, phase and single sideband modulation ...................................... 3

2.2.1 Amplitude modulation .............................................................................................. 3

2.2.2 Phase modulation ...................................................................................................... 3

2.2.3 Single-sideband modulation...................................................................................... 5

3 Description of the noise standard ............................................................................................ 7

4 Measurement setup for calibration of the PM/AM noise standard ......................................... 9

5 Calibration procedure at NIST .............................................................................................. 10

5.1 Measurement of the carrier power via down-conversion from the upper sideband ....... 10

5.2 Measurement of the carrier power via down-conversion from the lower sideband ....... 10

5.3 Measurement of the noise power spectral density down-converted from both the 

upper and the lower sidebands ................................................................................... 11

5.4 Measurement of the noise floor power spectral density ................................................. 12

6 Measurement equation .......................................................................................................... 13

7 Estimation of uncertainty ...................................................................................................... 15

8 Uncertainty budget ................................................................................................................ 18

9 Validation of the measurement ............................................................................................. 20

10 Acknowledgement ................................................................................................................ 20

Appendix A Verification of the baseband measurement equation ..............................................21 

Appendix B Measurement of correction factors, KNL, KRF, KBW and Kβ .....................................25 

Appendix C Error of signal power measurements in the presence of background noise ............28 

REFERENCES .............................................................................................................................31 
iv 

Notations 

Acronym Meaning 

AM Amplitude Modulation 

BPF Bandpass Filter 

BW Bandwidth 

dB Decibel 

dBm Decibel relative to 1 milliwatt 

DDPNMS Direct Digital Phase Noise Measurement System 

FFT Fast Fourier Transform 

IF Intermediate Frequency 

LO Local Oscillator 

LPF Low-Pass Filter 

LSB Lower Sideband 

PDLMS Photonic Delay Line Measurement System 

PM Phase Modulation 

PSD Power Spectral Density 

rf Radio Frequency 

RF Reference Frequency 

SNR Signal-to-Noise Power Ratio 

SSB Single Sideband 

USB Upper Sideband

VSWR Voltage Standing-Wave Ratio 

Symbol Meaning 

α(t) Instantaneous amplitude fluctuations 

β Phase modulation index 

δNL Maximum error of KNL from unity 

δRF Maximum error of KRF from unity

εβ Correction factor due to the small angle modulation approximation 

φ(t) Instantaneous phase fluctuations 

∆φpeak Peak phase deviation 

∆φrms Root-mean-square phase deviation 

γ Noise power in 1 Hz bandwidth 

ν0 Carrier frequency 

νLO Carrier frequency at the LO port of the mixer 

νRF Carrier frequency at the RF port of the mixer 

σβ Fractional uncertainty due to the small angle modulation approximation 

σBeat- Fractional uncertainty in the lower-sideband beat measurement 

σBeat+ Fractional uncertainty in the upper-sideband beat measurement 

σB-FFTAve Uncertainty due to the number of FFT averages for beat measurement 

σBW Fractional uncertainty in the estimated measurement bandwidth for the power 
v 

spectral density function 

σC Combined fractional uncertainty 

σLR Uncertainty due the long-term reproducibility of the PM/AM noise standard 

 between calibration cycles

σN-FFTAve Uncertainty due to the number of FFT averages for noise measurement

σNL Fractional uncertainty due to the FFT analyzer and mixer nonlinearity at baseband

σNoiseOn Fractional uncertainty in the noise measurement with the noise source on 

σRF Fractional uncertainty due to the frequency response of the down-convertor 

 (includes offset signal generator and mixer) at rf 

σSR Uncertainty due to the short-term repeatability between measurements 

ω0 Angular frequency of the carrier signal 

ωm Angular modulation frequency 

BWActual Actual bandwidth 

BWEst Estimated bandwidth 

f Fourier or offset frequency 

fL Lower frequency limit of the integrated phase noise 

fU Upper frequency limit of the integrated phase noise 

gIF Gain of the down-converter at the baseband 

gRF Gain of the down-converter at its two operating frequencies, νRF and νLO

gVSWR Gain of the down-converter due to VSWR mismatch 

gLVL Gain of the down-converter for a specific power level at the RF port 

G Gain of the down-converter

Jn n-th order Bessel function 

k Coverage factor

Kβ Correction factor applied to L(f) due to the small angle modulation approximation 

KBW Correction factor for incorrect estimation of the actual measurement bandwidth

 for the power spectral density function 

KNL Correction factor due to the FFT analyzer and mixer nonlinearity at baseband

KRF Correction factor due to the frequency response of the down-convertor (includes 

 offset signal generator and mixer) at radio frequencies 

L(f) Single sideband phase noise equal to one half of Sφ (f) 

m Amplitude modulation index 

nset Number of repeated measurement sets in a given calibration

NBeat Number of FFT averages for beat measurement

NNoise Number of FFT averages for noise measurement 

O Higher-order terms of Taylor expansion 

Pc Carrier power 

Pcarrier Power level at the RF port of the mixer during carrier power measurement 

PC-Beat Carrier power after down-conversion to baseband 

Pnoise Power level at the RF port of the mixer during the noise measurement

PN-Beat Noise power after down-conversion to baseband 

PSSB-AM Single-sideband AM signal power 

PSSB-PM Single-sideband PM signal power 

Sα (f) One-sided double-sideband power spectral density of amplitude fluctuations 

Sφ (f) One-sided double-sideband power spectral density of random phase fluctuations 
vi 

uC Combined uncorrelated uncertainty 

U Expanded uncertainty 

v(t) Instantaneous voltage fluctuations 

V0 Peak voltage of the carrier signal 

VBaseband Folded double-sideband down-converted voltage of the PM/AM noise standard at 

 baseband 

VBeat- Voltage of the lower-sideband beat frequency 

VBeat+ Voltage of the upper-sideband beat frequency 

VBeatAve Average voltage of the upper- and lower-sideband beats 

VBeat-, M Measured voltage of the lower-sideband beat 

VBeat+, M Measured voltage of the upper-sideband beat

VNoiseOff Voltage of the additive noise at baseband with noise source off 

VNoiseOn Voltage of the additive noise at baseband with noise source on 

VNoiseOff, M Measured voltage of the additive noise at baseband with noise source off 

VNoiseOn , M Measured voltage of the additive noise at baseband with noise source on 

VPNST Signal voltage of the PM/AM noise standard at rf 

VPNST(CarrierOn) Signal voltage of the noise standard when the additive noise is off and the carrier 

 is enabled 

VPNST(NoiseOn) Signal voltage of the noise standard at baseband when the additive noise is 

 enabled and the carrier is off 

VSSB Peak voltage of the single-sideband tone 
 1 

1 Introduction 

In this document, we provide the total (or combined) uncertainty of phase modulation (PM) and 

amplitude modulation (AM) noise measurement of the NIST portable noise standard calculated 

from the individual measurement uncertainties. NIST provides an on-site calibration service 

(77135C) of the accuracy of PM and AM noise and noise floor of the measurement systems at 

the customer's site by use of two separate portable PM/AM secondary noise standards [1]. One 

noise standard is for carrier frequencies of 5 MHz, 10 MHz, and 100 MHz, designated as model 

510100 [2], [3], and the other for carrier frequencies of 10.6 GHz, 21.2 GHz, and 42.4 GHz, 

designated as model 102040 [4]. 

The PM and AM noise of the noise standard are usually measured twice at NIST, once before 

sending it to the customer (the “out” measurement) and a second time after receiving it back (the 

“in” measurement). The customer uses its PM/AM noise measurement system and measures the 

calibrated noise level of the NIST secondary noise standard. Customer results are compared with 

the average of the “in” and “out” measurements at NIST. If the results are within the 

measurement uncertainty then the calibration is certified and a “PASS” calibration report is 

issued. When there is significant discrepancy in the results, the customer is advised to recheck 

the measurement configuration and re-measure. If the customer results cannot be achieved 

within the measurement uncertainty, a “FAIL” calibration report is then issued. 

The purpose of this document is to describe the calibration uncertainty for the NIST PM/AM 

noise standard. The document is organized as follows. In Section 2, we first introduce the basic 

concepts of AM and PM noise. We briefly discuss the characterization of AM and PM noise in 

Sub-section 2.1, and an overview of different types of signal modulation is presented in Subsection 2.2. Section 3 describes the working principle of the PM/AM noise standard, and in 

Section 4 the experimental set-up for calibrating the standard is presented. In Section 5 a 

detailed step-by-step calibration procedure is provided. The measurement equation, a detailed 

uncertainty calculation and the uncertainty budget for the noise standard are presented in 

Sections 6, 7 and 8, respectively. Finally, in Section 9, a brief discussion for validating the 

calibration results is presented. This document is supported by three appendices. Appendix A 

provides verification that the baseband measurement equation is equal to the original definition 

of single sideband (SSB) PM noise at the radio frequency (rf). Measurement of different 

correction factors involved in the calibration of the noise standard is described in Appendix B. 

Finally, Appendix C presents a calculation for the error of a signal power measurement in the 

presence of background noise 

 
 2 

2 AM and PM noise - Basic concepts 

In this section we briefly describe the basic concept of phase-modulated (PM) noise and 

amplitude-modulated (AM) noise and their characterization in the frequency domain [5-8]. We 

also discuss the basic theory behind the PM/AM noise standard. 

2.1 Characterization of AM and PM noise 

In the real world, an oscillator outputs a signal that fluctuates in both amplitude and phase. This 

signal can be mathematically represented by 

v 1 cos , (t V t t t ) = + 0

    α ω φ ( ) 0 + ( )    

 

(1)

where 02 V V = rms is the peak amplitude, α(t) is the normalized instantaneous amplitude 

fluctuation, ω0 = 2πν0 is the angular frequency of the carrier, φ(t) is the instantaneous phase 

fluctuation, Vrms is the root-mean-square (rms) voltage, and ν0 is the carrier frequency. 

In the frequency domain, the amplitude stability of a signal is characterized by the power spectral 

density (PSD) of instantaneous amplitude fluctuations Sα (f), given by 

( ) ( ) 2

,

f

S f

BW α

α

=

 

(2) 

where ( ) 2

α f is the mean-square normalized amplitude fluctuation at an offset or Fourier 

frequency f (0 < f < ∞) from the carrier and BW is the bandwidth of the measurement system. 

The unit of Sα (f) is 1/Hz. Similarly, in the frequency domain, the phase instability of a signal is 

characterized by the PSD of instantaneous phase fluctuations Sφ (f), given by 

( ) ( ) 2

,

rms

f

S f

BW φ

∆φ

= (3) 

where ( ) 2

rms ∆φ f is the mean-square phase fluctuation at an offset frequency f (0 < f < ∞) from 

the carrier frequency ν0. The unit of Sφ (f) is rad2/Hz. Since f ranges from zero to infinity and 

since both Sα (f) and Sφ (f) include the fluctuations from upper and lower sidebands of the carrier, 

these two quantities are single-sided double-sideband units of measure.

The PM noise unit of measure recommended by the IEEE [5] is L(f), defined as 

( ) ( )

.

2

S f

f

φ

L ≡ (4)
 3 

L(f) includes the fluctuations from only one sideband of the carrier, and hence it is a singlesideband unit of measure. When L(f) is expressed in the form 10log[L(f)], its unit is dBc/ Hz, 

that is dB below the carrier in a 1 Hz bandwidth. The logarithm is computed to the base 10. 

Also, when the integrated PM noise for offset frequencies 0 < f < ∞ is less than 0.01 rad2, L(f) 

can be viewed as the ratio of phase noise power per unit bandwidth in a single sideband to power 

in the carrier [See later Eq.(12)]. 

2.2 Overview of amplitude, phase and single sideband modulation 

Before we discuss the working principle of the NIST PM/AM noise standard, it is worthwhile to 

briefly discuss different types of signal modulation. 

2.2.1 Amplitude modulation 

In amplitude modulation, the carrier amplitude varies with the modulating baseband signal while 

the frequency and phase are not modulated. The simplest AM signal is produced when the 

modulating signal is a sinusoid at frequency ωm [9], [10]. The resultant modulated signal also 

called single-tone AM, is given by 

( ) [ ]

( ) 0

0 0

0

v 1 cos cos

Re 1 ,

2

m m

AM m

i t i t i t

t V m t t

m

V e e e ω ω ω

ω ω

−

= +

  

=  + + 

  

 

(5) 

where m is the amplitude modulation index. The two sidebands are symmetrical about the carrier 

frequency and are equal in magnitude. The ratio of SSB AM power (PSSB-AM) to carrier power 

(Pc) is 

2

.

4

SSB AM

C

P m

P

− =

 

(6) 

For AM, the resultant vector sum of the sidebands is always in phase with the carrier (Figure 1). 

2.2.2 Phase modulation 

In the case of phase modulation, the modulating signal leaves the amplitude of the carrier 

unchanged. Instead, the phase of the carrier varies with the modulating signal [9], [10]. A singletone PM signal can be represented by 

( ) [ ]

{ } 0

0 0

sin

0

v cos sin

Re ,

m

m

i t i t

t V t t

V e e ω β ω

= + ω β ω

=

 

(7) 
 4 

where β is the phase-modulation index (or peak phase deviation, also denoted by ∆φpeak). The 

second exponential term in the curly bracket is a periodic signal with period 2π/ωm and can be 

expanded by the exponential Fourier series 

( ) sin

.

β ω ω

β

∞

=−∞

= ∑

i t m in t m

n

n

e J e (8) 

Jn(β) is the th

n - order Bessel function of the first kind [11]. For small modulation index (β << 

1), 

( )

( )

( )

0

1

1

2

0 1.

β

β β

β

=

=

n = >

J

J

J n

 (9) 

Also, from symmetry, J-n(β) =(-1)nJn(β); therefore, Eq. (7) becomes 

( ) ( ) 0

0

v Re 1 .

2

m m

i t i t i t t V e e e  ω  β ω ω− 

=  + − 

   (10) 

This resembles the AM case in Eq. (5), except that in PM, the phase of the lower sideband is 

reversed, and the resultant vector sum of the sidebands is always in phase quadrature with the 

carrier (Figure 1). 

Further, the ratio of SSB PM power (PSSB-PM ) to carrier power is 

2 2

.

4 4

SSB PM peak

C

P

P

β φ −

∆

= =

 

(11) 

As mentioned in Section 2.1, when the integrated PM noise for offset frequencies 0 < f < ∞ is 

less than 0.01 rad2

, L(f) can be viewed as the ratio of phase-noise power in a single sideband to 

the carrier power. This can also be shown analytically by use of Eqs. (3), (4) and (11), as 

follows: 

( ) ( ) ( ) ( ) ( ) 2 2

1

1

,

2 2 4

rms peak SSB PM

C

BW

f f P f

f S f

BW BW P φ

φ φ −

=

∆ ∆

= = = =

Hz

L

 

(12)

 

when 
 5 

( ) peak

0

2 S d 0.1 rad . f f φ

∞

<< ∫

 

 (13) 

 

However, the practical values for the lower and upper integration limits, fL and fU, are 

respectively the reciprocal of the measurement time and system half-bandwidth. 

The condition of Eq. (13) follows by defining a higher order correction, β

ε , as a normalized 

departure of Eq. (11) from the small angle modulation approximation given by 

( ) ( ) 2

1 0

.

2

J J

β

β β

ε

β

 

=  

 

 (14) 

Then the error ( 1) β

ε − in L(f) for β << 0.1 is 0.3% (0.01 dB) or less. 

2.2.3 Single-sideband modulation 

In this Section, we discuss briefly single-sideband modulation, as this is necessary to explain the 

working principle of the PM/AM noise standard. A single-tone upper SSB signal is represented 

by 

( ) ( )

( ) ( )

0

0

0 0 0

0

0

0

0 0

v cos cos

Re 1

Re 1 ,

2 2

ω ω

ω ω ω ω ω

ω ω ω

− −

= + +

   

= +        

  

    

=   + + + − 

  

  

144424443 144424443

m

m m m m

SSB m

i t SSB i t

i t SSB i t i t SSB i t i t

AM PM

t V t V t

V

V e e

V

V V

V e e e e e

V V

 

(15) 

where VSSB is the peak voltage of the single-sideband tone at (ω0 + ωm). Now comparing Eq. (15) 

with (5) and (10), the second term within the square bracket represents AM, and the last term 

represents PM. Therefore, the SSB signal can be envisioned as a combination of four signals of 

equal amplitude; two of them consist of pure AM, and the other two consist of pure PM. For a 

SSB signal, half of the power is present in AM and other half is present in PM. Eq. (15) can also 

be explained in other words as follows: the two lower sideband terms ( )

i t m e− ω

 are 180°out of 

phase, and thus cancel, and the upper sideband terms ( )

i t m eω

 are in phase, and thus add 

coherently, resulting in a SSB modulation. 

Now if instead of a single tone, there is additive white noise, the theory of single-tone SSB 

modulation can still be applied [12], [13]. Consider an upper noise sideband at “ν0+ f” with 

associated noise power γ in 1 Hz bandwidth. It will produce upper and lower PM and AM noise 
 6 

sidebands, each having average power γ

2

/4. If there is a similar amount of uncorrelated white 

noise power in the lower sideband at “ν0- f”, it also will produce four signals, two AM and two 

PM of average power γ

2

/4. The contribution from the lower and upper sidebands produce on 

average equal amounts of PM and AM noise; however, the phase difference between them varies 

randomly with time. 

SSB

USB

LSB

USB

LSB

SSB

USB

LSB

USB

LSB

Figure 1. Phasor diagrams of the amplitude, phase and SSB modulation. LSB is the lower 

sideband and USB is the upper sideband. The gray vector indicates the resultant of the carrier 

and sidebands. 

 
 7 

3 Description of the noise standard 

The block diagram of the noise standard at 5 MHz, 10 MHz, and 100 MHz is shown in Figure 2. 

The detailed description of this noise standard can be found in [2]. The noise standard has two 

outputs, one clean unmodulated reference and another modulated. The powers of the two output 

signals are approximately +15 dBm for each carrier frequency. 

Figure 2. Block diagram of 510100 PM/AM noise standard, which selects one of three 

frequencies of 5 MHz, 10 MHz, and 100 MHz. BPF is the bandpass filter

A calibrated level of bandpass-limited Gaussian noise is added to the “Modulated Output” of the 

standard to create equal levels of PM and AM noise. The level of added noise is selected such 

that it satisfies Eq. (13), and is at least 40 dB above the noise floor of most measurement 

systems. This calibrated noise level is typically constant in magnitude to better than ± 0.5 dB for 

Fourier frequencies from dc to 10 % of the carrier frequency. For the 5 MHz standard, the 

calibrated noise level is approximately -110 dBc/Hz with a 3.5 MHz bandwidth. The residual 

PM and AM noise between these two outputs is exceptionally low [2]. Typically, the differential 

PM noise between the two output signals is less than -190 dBc/Hz for Fourier offsets of 10 kHz 

and above. 
 8 

The modulated output has precisely equal PM and AM noise, since there is no phase coherence 

between the signal and the noise, as described in 2.2.3 and [12]. The power spectral density 

(PSD) of the SSB PM noise, [L(f)], and AM noise, [1/2 Sα(f)], generated by the noise standard is 

given by 

( ) ( ) 1 [Noise ( - ) + Noise ( + ) 0 ] [ 0 ]

2 4 Carrier Power

PSD f PSD f

f S f α

ν ν

L = = , (16) 

when Eq. (13) is satisfied [2]. One factor of two in the denominator arises because half of the 

additive noise power appears as PM and half as AM. The other factor of 2 is due to the equal 

distribution of noise power between the upper and lower sidebands. 

L(f ) is constant with frequency from dc to approximately the half-bandwidth of the filter, 

assuming that the added noise is constant in amplitude from ν0 - f to ν0 + f. 
 9 

4 Measurement setup for calibration of the PM/AM noise standard 

To measure L( f ) at multiple offset frequencies, f, a digital fast Fourier transform (FFT) 

spectrum analyzer is used to take advantage of its speed, high dynamic range and flexibility in 

measurement bandwidth. To measure beyond the upper frequency capability of the FFT, a 

heterodyne down-converter is used to convert the output of the PM/AM noise standard to an 

intermediate frequency (IF) that can be conveniently measured by the FFT analyzer. 

Figure 3. Block diagram of a heterodyne measurement system used to calibrate the 510100 

PM/AM noise standard

The block diagram of the heterodyne down-converter used to calibrate a PM/AM noise standard 

at NIST is shown in Figure 3. It is composed of a double-balanced mixer (LO level +7 dBm), an 

offset signal generator, and a low-pass filter (LPF). The modulated port of the noise standard is 

connected to the reference frequency (RF) mixer port via a large attenuator. The large attenuator 

at the RF port ensures that the signal from the PM/AM noise standard is far below the saturation 

point of the mixer and that it is operating as a linear frequency down-converter. The small 

attenuators on the local oscillator (LO) and IF ports are used to stabilize the impedance of the 

mixer. A LPF is used after the mixer to reject the sum of the two mixed frequencies and pass the 

difference signal. 
 10 

5 Calibration procedure at NIST 

The offset generator and the PM/AM noise standard are turned on and allowed to stabilize for at 

least 24 hours. The offset signal generator is connected to the LO port of the down-converter 

and adjusted to +10 dBm. If possible, the offset generator and the PM/AM standard are also 

locked to a common frequency reference. The reference port of the PM/AM noise standard is 

terminated with 50 Ω, and the modulated output port is connected to the highly attenuated RF 

port of the down-converter. The noise and the carrier of the PM/AM noise standard can be 

independently enabled. This allows the carrier and the noise to be independently characterized, 

utilizing the full dynamic range of the FFT device. 

A typical calibration consists of a set of individual calibration points at different offset 

frequencies from the carrier. This set of offset frequencies is typically logarithmically spaced, 3 

to 5 points per decade, spanning from 1 Hz to 10 MHz or a maximum of 10 % of the carrier 

frequency. For each offset frequency, f, in the calibration set, four measurements are made and 

used to calculate the phase noise generated at that offset frequency. These four measurement 

steps are graphically shown in Figure 4 and described in the following four sub-sections. 

5.1 Measurement of the carrier power via down-conversion from the upper 

sideband 

The carrier power of the PM/AM standard is turned ON and the noise is turned OFF. The 

frequency of the offset generator is set to ν0+ f. The carrier at ν0 from the PM/AM standard and 

the signal from the offset generator at ν0+ f are mixed in the down-converter creating a baseband 

beat at frequency f. The power in the baseband signal at f will be proportional to the carrier 

power of the PM/AM noise standard at ν0. The FFT analyzer range is set to auto to select the 

maximum dynamic range, and a measurement of the signal power at f is made and recorded in 

units of V

2

rms, utilizing a Flattop window. Also a measurement of the signal-to-noise power ratio 

(SNR) between the beat signal and the background noise is recorded. This measurement is 

graphically shown in Figure 4(a). 

5.2 Measurement of the carrier power via down-conversion from the lower 

sideband 

Similar to the previous step, the carrier power of the PM/AM standard stays ON and the noise is 

OFF. The frequency of the offset generator is set to ν0 - f. The carrier at ν0 from the PM/AM 

standard and the signal from the offset generator at ν0 - f are mixed in the down-converter, 

creating a baseband beat at frequency f. The power in the baseband signal at f will be 

proportional to the carrier power of the PM/AM noise standard at ν0. The FFT analyzer is autoranged for maximum dynamic range, and a measurement of the signal power at f is made and 

recorded in units of V

2

rms by use of a Flattop window. Also, a measurement of the signal-tonoise ratio (SNR) between the beat signal and the background noise is recorded. This 

measurement is graphically shown in Figure 4(b). 
 11 

5.3 Measurement of the noise power spectral density down-converted from 

both the upper and the lower sidebands 

The carrier power of the PM/AM standard is then turned OFF and the noise is turned ON. The 

frequency of the offset generator is set to ν0. The double-sideband noise spectrum about ν0 is 

down-converted to baseband by mixing with the offset generator at ν0. The PSD of the baseband 

noise at f is proportional to the PSD of the double-sideband noise of the PM/AM noise standard 

about ν0. The FFT analyzer is auto-ranged for maximum dynamic range, and a measurement of 

the down-converted PSD at f is made and recorded in units of V

2

rms/Hz, by use of a low-leakage 

Hanning window [14]. This measurement is graphically shown in Figure 4(c). 

0 υ0 υ − f υ0 f

0 υ0 υ0 + f f υ − f

υ0 f 0 υ + f

0 υ0 υ0 + f f υ − f

2

( ) V f NoiseOn

2

( ) V f NoiseOff

2

( ) V f Beat+

2

( ) V f Beat−

Figure 4: This diagram graphically represents the four measurement steps involved in 

calibrating the PM/AM noise standard. In (a) and (b), the down-converted signal power at ‘f ’ is 

proportional to the noise standard carrier power at ν0. 
 12 

5.4 Measurement of the noise floor power spectral density 

The carrier power of the PM/AM noise standard is turned OFF and the noise is turned OFF. The 

frequency of the offset generator kept at ν0. Next, a noise floor measurement of the downconversion process is made. The FFT analyzer must NOT be auto-ranged, and the range setting 

should be identical to that used in Section 5.3. Keeping the auto-range at the same level ensures 

that the noise floor of the FFT analyzer is the same for measurements 5.3 and 5.4. The PSD at 

offset frequency f is measured in units of V

2

rms/Hz, utilizing a Hanning window. The PSD of the 

noise floor at f will contain noise contributions from the down-converter components, such as the 

mixer, IF amplifier, the noise of the offset generator, and the input noise of the FFT spectrum 

analyzer. This measurement is graphically shown in Figure 4(d). 

The measurement procedure described above is repeated at least three times for each carrier 

frequency. All measurements are made with a single one-time connection to the measurement 

system, and the same analyzer is used for all measurements. The absolute gain or frequency 

response of the down-converter and the spectrum analyzer are unimportant, because they are 

canceled due to the ratiometric relationship between the measurement of the carrier power and 

additive noise. This same characteristic also removes any inconsistency resulting from voltage 

standing wave ratio (VSWR) mismatch or connector interface non-repeatability. 

The noise-voltage data are combined with the beat-power measurements described above 

according to the baseband measurement Eq. (18) to yield the relative power in either AM, 1/2 

Sa(f), or PM noise. 
 13 

6 Measurement equation 

L(f) in Eq. (16) represents a ratio of power measurements performed at radio frequency (rf). A 

higher order correction, Kβ, due to the small angle modulation approximation is introduced and 

Eq. (16) is rewritten as 

( ) ( ) 1 1 [Noise ( - ) + Noise ( + ) 0 ] [ 0 ]

.

2 4 Carrier Power

PSD f PSD f

f S f

K

α

β

ν ν

L = = (17) 

The correction factor, Kβ, is introduced only to propagate the error due to the approximation in 

the measurement equation. A value of unity will be used. 

The actual measurements described in the above section are made at baseband by use of a downconverter. The following equation is used to calculate the AM or PM noise generated from 

these baseband measurements: 

( ) ( ) ( )

( ) ( )

2

2 2

1 1 1

.

2 4

2

NoiseOn

a

NL RF BW Beat Beat

PSD V f

f S f

K K K K V f V f β − +

 

       

= =  

+  

 

L

 

 (18) 

( )

2 V f NoiseOn is the additive noise down-converted from both the upper and lower ( 0 ν ± f ) 

sidebands to baseband. ( )

2 V f Beat±

 is the carrier power down-converted via either the upper or 

lower sideband to a baseband beat of f. KNL, KRF, and KBW are correction factors needed to 

account for nonlinearity, frequency response, and bandwidth errors in the baseband 

measurement. Appendix A shows the equivalence of Eqs. (17) and (18), and Appendix B 

describes the techniques to measure these correction factors. 

When the bias due to background measurement noise is removed, Eq. (18) can be written as 

follows: 

( )

( ) ( )

( )

( ( ) ( ))

2

2 ,

, 2

,

2 2

, ,

1

1

.

2 1

1

NoiseOff M

NoiseOn M

NoiseOn M

NL RF BW

Beat M Beat M

PSD V f

PSD V f

PSD V f

f

K K K K V f V f

SNR

β

− +

   

 

  − 

    

   

=

 

+   −

 

L

 

(19)

( )

2 V f NoiseOff M,

 and SNR are respectively the noise floor of the measurement system for the 

additive noise measurement and the signal-to-noise power ratio of the carrier measurements. 
 14 

The same SNR is assumed for both VBeat- and VBeat+ measurements. The logarithmic form of Eq. 

(19) is given by: 

 

 

( ) ( ) ( )

( )

( ( ) ( ))

2

2 ,

, 2

,

2 2

, ,

10log 10log 10log 1

1

10log 10log 1 10log(2 ).

NoiseOff M

NoiseOn M

NoiseOn M

Beat M Beat M NL RF BW

PSD V f

f PSD V f

PSD V f

V f V f K K K K

SNR − + β

   

 

=   + −  

 

   

   

 

− + − − −    

L

dBc/Hz (20) 

The first term is the measured noise level with noise source ON, the second term is the bias on 

the measured noise due to the noise floor (measured with the noise source OFF), the third term is 

the average of the measured carrier power down-converted via both the upper and lower 

sidebands, the fourth term is the bias on the carrier power measurements due to the background 

noise, and the last term contains the correction factors. 

 
 15 

7 Estimation of uncertainty 

Normalizing the PSD to a 1 Hz bandwidth and rewriting the measurement Eq. (19) gives 

( ) ( ) ( )

( ( ) ( ))

2 2

, ,

2 2

, ,

1

.

2 1

1

NoiseOn M NoiseOff M

NL RF BW

Beat M Beat M

V f V f

f

K K K K V f V f

SNR

β

− +

−

=

 

+   −

 

L

 

(21)

From the Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement 

Results [15][16], the equation for the combined uncorrelated uncertainties (uC) for a 

measurement result, y, is written as 

{

1 2

2

2 2

1

( , ,..., )

( ) ( ).

N

N

C i

i i

Sensitivity

Coefficents

y f x x x

f

u y u x

= x

=

  ∂

=   ∂  ∑ (22) 

 In terms of fractional uncertainties Eq. (22) is 

}

}

Total ith Component

Fractional Fractional Multiplier

Uncertanty Uncertanty

2

2 2 2

2 2 2

1

Multiplier ith Component

Fractional

2 Uncertanty

2

2 2

2

1

( ) ( )

.

N

C i i

i i i

N

i

C

i i

u y f x u x

y x y x

f x

x y

σ σ

=

=

  ∂

=   ∂ 

  ∂

=   ∂ 

∑

∑

64748 678

64748

i

 (23) 

2 σ is used here to represent the fractional variance. The combined uncertainty for L(f) is 

therefore 

( )

( ) ( )

( ) ( ) ( ) ( )

( )

2 2

2 2

, ,

.

2 2 2 2

2 2 2 2 2

2 2

2

,

, ,

NoiseOn M NoiseOff M

NoiseOn M NoiseOff

C NL RF BW

NL RF BW

Beat M

Beat M Beat M

u V u V

V V

u u K u K u K u K

K K K K

u V

V V

β

β

+

+ −

  ∂   ∂

  +   ∂ ∂    

  ∂   ∂   ∂   ∂

= +  +   +   +     ∂ ∂ ∂ ∂        

  ∂   ∂

+  +     ∂ ∂    

L

L L

L L L L

L L

( ) ( )

2

2 2

,

.

u VBeat M u SNR

SNR −

 

 

 

 

 

 

 

 

 ∂     +   

  ∂

 

 

L

 

(24)
 16 

The individual multipliers of (23) are 

( )

2

2 4

, ,

2 2

2 2

,

, ,

4 NoiseOn M NoiseOn M

NoiseOn M NoiseOn M NoiseOff M

V V

V V V

  ∂

  =

∂

  −

L

L

 

(25) 

( )

2

2 4

, ,

2 2

2 2

,

, ,

4 NoiseOff M NoiseOff M

NoiseOff M NoiseOn M NoiseOff M

V V

V V V

  ∂

  =

∂

  −

L

L

 

(26)

2 2 2 2

2 2 2 2

2 2 2 2

1

β

β

    ∂   ∂ ∂   ∂

    =   = =   =

  ∂ ∂ ∂ ∂        

NL RF BW

NL RF BW

K K K K

K K K K

L L L L

L L L L

 

(27) 

( )

2

2 4

, ,

2 2

2 2

,

, ,

4 Beat M Beat M

Beat M Beat M Beat M

V V

V V V

− −

−

− +

  ∂

  =

∂

  +

L

L

 

(28) 

( )

2

2 4

, ,

2 2

2 2

,

, ,

4 Beat M Beat M

Beat M Beat M Beat M

V V

V V V

+ +

+

− +

  ∂

  =

∂

  +

L

L

 

(29) 

( )

2 2

2 2

1

.

1

  ∂

=     ∂ −

SNR

SNR SNR

L

L

 

(30)

The multipliers are simplified by making the following assumptions: 

, ,

, ,

1.

NoiseOn M NoiseOff M

Beat M Beat M

V V

V V

SNR

+ −

>>

≅

>>

 (31) 

The combined fractional uncertainty, σC, obtained by combining the individual fractional 

uncertainties, whether arising from Type A evaluation or Type B evaluation [15], is 

2 2 2

2 4 2 2 2 2 2

,

NoiseOn Beat Beat

C NL RF BW LR

set n

β

σ σ σ

σ σ σ σ σ σ

+ +− + = + + + + +

 

(32) 

where nset is the number of repeated measurement sets in a given calibration. σLR is the 

uncertainty characterizing the long-term reproducibility of the PM/AM noise standard between 

calibration cycles. This includes the combined effect of temperature variations, connector 

mismatch, long-term variations of noise or carrier power, effects due to shock during shipping, 

and other unknown effects. σNoiseOn and σBeat include uncertainties due to the average of a 

random process given by 
 17 

2

2 2 2 1 2 σ σ σ σ ,

−

 

= + =   +

 

 

NoiseOn N FFTAve SR SR NNoise 

(33)

2

2 2 2 2 1 2

. Beat B FFTAve SR SR N SNR Beatσ σ σ − σ

 

= + =   +

 

  

(34)

NNoise and NBeat are the number of FFT averages used for the noise and beat measurements 

respectively. σSR is the short-term repeatability of the PM/AM noise standard during a single 

calibration. This includes the variation due to temperature fluctuations and variability in the 

insertion loss of the internal relays used to enable and disable the carrier and noise source for the 

calibration. Appendix C presents a derivation of (32) describing the error of signal power 

measurements in the presence of background noise.

The expanded uncertainty (U) is equal to kσc, where k is the coverage factor and is chosen to be 2 

for a 95.45 % level of confidence [15]. 

 
 18 

8 Uncertainty budget 

As stated in the Introduction, the AM and PM noise of the noise standard are usually measured 

two times at NIST, once before sending it (“out” measurement) to the customer and a second 

time after receiving it back (“in” measurement). The combined uncertainty includes the 

contributions of different estimated errors associated with the noise-standard calibration. In 

practice, all estimated errors are expressed in the same domain [linear (%) or log (dB)] for the 

calculation of combined uncertainty. Typically, the final measurement results for PM and AM 

noise are reported in the log domain (dBc/Hz). However, in this calibration, not all errors are 

available in the same domain. For example, the uncertainty due to a FFT average is statistically 

calculated in the linear domain, whereas the uncertainty in the linearity of the FFT analyzer is 

provided by the manufacturer in dB. Other errors associated with the noise standard and/or 

calibration that are experimentally measured can be calculated either in dB or %. We chose to 

write the measurement and uncertainty equations in the linear domain and convert any errors 

expressed in dB to the linear domain. When converting dB uncertainties to linear, the upper 

and lower confidence range can be unequal for larger errors. In this document, we will convert 

all the dB errors to their linear values and consider the largest value if they are unequal for the 

uncertainty calculation. Certain assumptions concerning distributions in dB cannot always be 

properly converted to linear, and vice-versa. 

The combined uncertainty is calculated entirely in terms of linear fractional uncertainty and the 

final expanded uncertainty (U) result is converted to dB. The uncertainty budget for the 

calibration of the PM/AM noise standard is given in Table 1. 

Table 1- Uncertainty budget: PM/AM noise measurement of the noise standard 

Sources of Error Estimated 

Fractional 

Error, % 

Effect Type Distribution Divisor Standard 

Fractional 

Uncertainty 

(σ), % 

Symbol 

Number of FFT averages 

for noise measurement 

NNoise = 10,000 

1.0 Random A Normal 1 1.0 σN-FFTAve

Number of FFT averages 

for beat measurement 

NBeat = 100, 

SNR = 1000 

0.0 Random A Normal 1 0.0 σB-FFTAve

Uncertainty in the 

estimated measurement 

bandwidth for the PSD 

function. (0.05 dB) 

1.2 Systematic B Rectangular √3 0.7 σBW

Frequency response of the 

down convertor (includes 

offset signal generator 

and mixer) at rf 

2.3 Systematic B Rectangular √3 1.3 σRF

FFT and mixer linearity at 

baseband 

4.7 Systematic B Rectangular √3 2.7 σNL

Small angle modulation 

approximation. β = 0.02

0.0 Systematic B Fixed value 1 0.0 σβ

Short term noise standard 

repeatability 

2.3 Random A Normal 1 2.3 σSR

Long term noise standard 

reproducibility 

7.2 Systematic B Rectangular √3 4.1 σLR
 19 

By use of Table 1 and Eq. (32) the expanded fractional uncertainty for values shown in this 

example is ± 11.5 % or ± 0.5 dB for nset equal to 6. 

It should be noted that this calibration standard creates equal amount of AM and PM noise. 

When this type of standard is used to evaluate AM/PM measurement systems, the AM–to–PM or 

PM-to-AM conversions in the detector cannot be ignored [17]. In a typical PM noise 

measurement, AM–to–PM conversion can typically be 15 dB or worse leading to an error of 0.3 

% (0.1 dB). 
 20 

9 Validation of the measurement 

To further validate the calibration method, we use two different approaches. In the first 

approach, the noise of the standard is measured with different PM noise measurement systems 

and with different measurement techniques, for example, a direct-digital phase noise 

measurement system (DDPNMS) [18], or a photonic delay-line measurement system (PDLMS) 

[19]. The agreement between the results of the DDPNMS and/or PDLMS and the calibrated 

result validate the calibration method. The second approach is to calibrate a phase noise 

measurement system by use of a calibrated noise standard and compare the results with that 

obtained by other calibration methods such as a single-sideband modulation, PM/AM 

modulation, beat frequency, or static phase shift [7], [20]. All of these methods should give 

identical results within the measurement uncertainty. 

10 Acknowledgement 

The authors thank Fred Walls for useful discussions and thoughtful comments on this 

manuscript, and Tom Parker, Mike Lombardi, David Smith, and Danielle Lirette for their helpful 

suggestions in the preparation of this manuscript. 
 21 

Appendix A. Verification of the baseband measurement equation 

The phase noise, L(f) and amplitude noise, Sα(f) generated by the PM/AM noise standard is 

given by 

 

( ) ( ) 1 1 [Noise ( - ) + Noise ( + ) 0 ] [ 0 ]

. [dBc/Hz or 1/Hz]

2 4 Carrier Power a

PSD f PSD f

f S f

Kβ

ν ν

L = =

 

(A1) 

The noise represented by Eq. (A1) is not measured directly at rf, but rather indirectly via downconversion to baseband. The following equation, called the “measurement equation,” describes 

the measurement at baseband: 

( )

( )

( )

2

2

1 1

.

4

NoiseOn

Est

NL RF BW BeatAve

V f

BW

f

K K K K V f β

 

    L =

 

(A2) 

A proof follows that shows that the baseband measurement Eq. (A2) is equivalent to the 

definition (A1) when appropriate approximations and correction terms are defined. ( )

2 V f NoiseOn is 

the measured additive noise down-converted from both the upper and lower ( 0 ν ± f ) sidebands 

to baseband. ( )

2 V f BeatAve is the average of the measured carrier power down-converted via both 

the upper and lower sideband to a baseband frequency of f. BWEst is an estimate of the 

measurement bandwidth used to calculate the PSD function. KNL, KRF, KBW and Kβ, to be defined 

later, are correction factors to account for nonlinearity, frequency response, bandwidth and small 

angle approximation errors in the baseband measurement. 

The simplified function of the heterodyne down-converter is described as follows. An offset 

signal generator is connected to the LO port of a double-balanced mixer, and the PM/AM noise 

standard is connected to the RF port of the mixer via a large attenuator, ensuring linear 

conversion from the RF port to baseband. The gain of the down-converter for a given LO 

frequency (νLO) and RF frequency (νRF) to a baseband IF frequency (f = |νLO - νRF|) is 

( , , ) ( ) ( , ) ( ), G level g g LO RF VSWR IF LO RF RF LO RF LVL ν ν = ν ν ν ν − g g P

 

(A3) 

where, g’s are different gain terms. VSWR g represents any mismatch error due to VSWR [21], 

( ) IF g f represents the baseband frequency response at IF, ( , ) RF LO RF g ν ν represents the rf 

frequency response of the down-converter at its two operating frequencies, and the ( ) LVL g P

represents the response of the down-converter at a specific power, P, at the RF port. 
 22 

When the signals from the offset generator and the noise standard are mixed, two sidebands at

0 ν ± f are translated to a baseband frequency at f. The folded double-sideband downconversion is represented by Eq. (A4) and is shown graphically in Figure 5 [11]. 

2 2 2 2 2

0 0 0 0 0 0

( ) ( , , ) ( ) ( , , ) ( ), V f G f P V f G f P V f Baseband = − ν ν PNST ν ν ν − + + PNST ν +

 

 (A4) 

where VPNST is the signal voltage of the PM/AM noise standard at rf. 

0 υ0 υ − f υ0

f

0 υ0 υ0 + f f υ − f

υ0 f 0 υ + f

Power Power Power

PSD Power Power

0 0 ( , ) RF g f υ υ +

0 0 ( , ) RF g f υ υ −

0 0 ( , ) RF g f υ υ −

0 0 ( , ) RF g f υ υ +

2

( ) V f NoiseOn

2

( ) V f NoiseOff

2

( ) V f Beat+

2

( ) V f Beat−

Figure 5. Graphical representation of the baseband calibration measurements. The red and blue traces 

represent the offset generator and the PM/AM noise standard, respectively. The green spectrum is the 

down-converted baseband signal. The dark gray trace is the noise floor of the measurement system. The 

dotted lines indicate the down-conversion gains, where the light gray dotted line indicates the undesired, 

negligible and hence ignored, down-converted image.

By use of Eq. (A4) the numerator of Eq. (A2) can be written as 

( ) ( )

2 2 2 2 2

0 0 0 0 0 0

( ) ( , , ) ( ) ( , , ) ( ). NoiseOn noise PNST NoiseOn noise PNST NoiseOn V f G f P V = − ν ν ν ν ν − + + f G f P V ν + f

 

 (A5) 

VPNST(NoiseOn) represents the signal voltage of the standard when the additive noise is enabled and 

the carrier is off. Pnoise is the power level at the RF port of the mixer during the noise 
 23 

measurement. The upper and lower sideband gains are factored out into a single term and 

simplified by a first order Taylor expansion when both sidebands are approximately equal as 

follows: 

( )

+ O when ,

2

+ = +

+

=

+

+

= ≅

c x y ax by

ax by

c

x y

a b

c x y

 

(A6)

where O contains the higher-order terms of the Taylor expansion. Therefore, 

( ) ( )

2 2

2 0 0 0 0 2 2

0 0

( , , ) ( , , ) ( ) ( ) ( ) .

2

ν ν ν ν

ν ν

 − + + 

= + − + +  

  

 

noise noise

NoiseOn PNST NoiseOn PNST NoiseOn

G f P G f P V f O V f V f

 

(A7) 

Similarly, 2

( ) V f BeatAve in the denominator of (A2) can be rewritten, when the contribution from 

the unwanted and negligible down-converted image is ignored, as 

 

( )

2 2

2 0 0 0 0 2

0

( , , ) ( , , ) ( ) ( ),

2

carrier carrier

BeatAve PNST CarrierOn

G f P G f P V f Vν ν ν ν

ν

− + +

=

 

(A8)

where VPNST(CarrierOn) represents the signal voltage of the standard when the additive noise is off 

and the carrier is enabled, and Pcarrier represents the power level at the RF port of the mixer 

during measurement of carrier power. Substituting the numerator and denominator in Eq. (A2) 

with Eq. (A7) and Eq. (A8) gives 

 

( )

( ) ( )

( )

2 2

0 0

2

0

( ) ( )

1

.

4 ( )

PNST NoiseOn PNST NoiseOn

Est

NL RF BW PNST NoiseOn

V f V f

K BW

f

K K K Kβ V

ν ν

ν

 − + + 

 

    L =

 

(A9) 

All common and similar terms are collected in the variable Κ, given by 

 

2 2

2 2

( )

( )

VSWR IF

VSWR IF

g g f K

g g f

=

2 2 2

0 0 0 0

2 2 2

0 0 0 0

( ) ( , ) ( , ) 2

.

( ) ( , ) ( , )

RF RF LVL noise

LVL carrier RF RF

g P g f g f O

g P g f g f

ν ν ν ν

ν ν ν ν

 − + + + 

 

 − + + 

 

 

(A10) 

This clearly shows that the baseband gains as well as VSWR mismatches cancel for the 

measurement Eq. (A2). 

Finally, the four correction factors KNL, KRF, KBW and Kβ are defined. KNL defines the change in 

gain that occurs due to nonlinearity in the down-converter and FFT analyzer due to a difference 

in power level between noise and carrier power measurements: 
 24 

2

( )

.

( )

LVL noise

NL

LVL carrier

g P K

g P

 

=  

 

 

(A11)

KRF is the ratio of the effective power gain between noise and carrier power measurements. 

Ideally, this number should be unity; however, it is affected by the non-flat amplitude response 

of the offset generator. It also contains the asymmetrical frequency response of the mixer at its 

LO and RF ports as well as the higher-order terms of the Taylor expansion of example (A6): 

2 2

0 0 0 0

2 2

0 0 0 0

( , ) ( , ) 2

.

( , ) ( , )

RF RF

RF

RF RF

g f g f O K

g f g f

ν ν ν ν

ν ν ν ν

− + + +

=

− + + 

 (A12) 

KBW is a correction for an imperfect power spectral density (PSD) function given by 

,

Actual

BW

Est

BW K

BW

=

 

 

(A13) 

where BWActual is the true measurement bandwidth. 

Using Eqs. (A11), (A12) and (A13) in Eq. (A9), we see that L(f ) at the baseband measurement 

equation is equal to the original definition of L(f ) at rf (A1) as follows: 

( )

( ) ( )

( )

( ) ( )

( )

2 2

0 0

2

0

2 2

0 0

2

0

( ) ( )

1

4 ( )

( ) ( ) 1

.

4 ( )

PNST NoiseOn PNST NoiseOn

Actual

PNST CarrierOn

PNST NoiseOn PNST NoiseOn

PNST CarrierOn

V f V f

BW

f

K V

PSD V f V f

K V

β

β

ν ν

ν

ν ν

ν

 − + + 

 

   

=

 − + + 

 

=

L

 

(A14)
 25 

Appendix B. Measurement of correction factors, KNL, KRF, KBW and Kβ

(i) Set-up for measuring nonlinearity of the down-converter and the FFT analyzer –

KNL

As discussed earlier, KNL is defined as the change in gain that occurs due to nonlinearity in the 

down-converter and FFT analyzer due to a difference in power level between noise and carrier 

power measurements and given by Eq. (A11). The carrier power is measured with additive noise 

off, while the noise-power measurements are made with the carrier off. 

 

Figure 6. Measurement setup for the estimation of KNL andKRF.

The measurement system used to measure uncertainty due to the FFT and the mixer nonlinearity 

at baseband is shown in Figure 6. The frequency of the offset signal generator is set at ν0 + f and 

the noise standard at ν0. A portion of the signal from the noise standard is coupled out and 

connected to a NIST-calibrated power meter with very low noise floor (-60 dBm). First, the 

carrier power (Pcarrier) and noise power (Pnoise) of the standard are measured with the power 

meter. The noise standard is then replaced by a rf synthesizer at frequency ν0 and power equal to 

Pcarrier. The corresponding down-converted power (PC-Beat) is measured on the FFT analyzer by 

use of a flattop window. Next, the synthesizer power is adjusted to equal the integrated noise 

power of the noise standard (Pnoise), and the down-converted power (PN-Beat) is measured on the 

FFT analyzer. From these measurements KNL can be calculated as: 

( )

( )

2

.

LVL noise N Beat C Beat

NL

LVL carrier noise carrier

g P P P K

g P P P

− −

 

=   =

 

  

 (B1) 
 26 

These measurements are repeated for all the specified offset frequencies ( ±f ) at each of the three 

carrier frequencies. Because the mixer is operating in linear mode, KNL should ideally be equal 

to 1. The maximum observed error from unity, δNL, can be used to construct a rectangular 

distribution for KNL. Therefore, 

1 1

3

NL KNL NL

δ

= ± = ±σ

 , (B2)

where σNL is the fractional uncertainty of KNL. 

(ii) Set-up for measuring frequency response of the down convertor at rf – KRF

KRF is the ratio of the effective power gain between noise and carrier power measurements and 

given by Eq. (A12). The experimental set-up for determining KRF is the same as shown in Figure 

6. In this case, the beat power, PC-Beat(νLO, νRF), is measured for four different LO and RF 

frequency combinations. For all four measurements, the power of the synthesizer is adjusted so 

that its power stays constant at Pcarrier. KRF can then be calculated from 

( ) ( )

( ) ( )

0 0 0 0

0 0 0 0

, ,

.

, ,

C Beat C Beat

RF

C Beat C Beat

P f P f

K

P f P f

ν ν ν ν

ν ν ν ν

− −

− −

− + +

=

− + +

 

(B3) 

These measurements are repeated for all the specified offset frequencies (±f ) at each of the three 

carrier frequencies. KRF should ideally be equal to 1. The maximum observed error from unity, 

δRF , can be used to construct a rectangular distribution for KRF. Therefore, 

1 1

3

RF KRF RF

δ

= ± = ±σ

 , (B4)

where σRF is the fractional uncertainty of KRF. 

(iii) Measurement of PSD correction factor – KBW

KBW as defined in Eq. (A13) is a correction for an imperfect power spectral density function.

 

Since the FFT is digitally implemented, the resolution bandwidth of measurement and associated 

windows are well defined mathematically. Exact details of the FFT implementation depend on 

the manufacturer. The FFT manual should be checked for possible bandwidth uncertainty for a 

given instrument. Verification of the PSD function can also be implemented by use of a well 

characterized filter and a calibrated power meter [14]. In this case, we use KBW equal to unity 

with a fractional uncertainty, σBW = 0.7 %. 
 27 

(iv) Calculation of small angle modulation correction factor – Kβ

Kβ is a correction to the small angle modulation approximation used to calculate L(f ) from a 

single sideband power ratio. An effective peak phase modulation β is obtained by integrating the 

phase noise and utilized to determine Kβ as follows 

( )

U

L

2 S d .

f

f

f f β = ∫ φ

 

(B5) 

Typically, fU is the half-bandwidth of the band-pass filter in the noise standard. 

Kβ is chosen to be unity with a fractional uncertainty σβ. 

( ) ( )

2 2

1 0

2

1 ( 1) 1 1 1 .

4

J J

Kβ β β

β β

ε σ

β

 

= ± − = ±  − = ± 

 

 

 (B6) 
 28 

Appendix C. Error of signal power measurements in the presence of 

background noise 

Consider the fast Fourier transform (FFT) of a signal of amplitude A and frequency f , such that 

the signal occurs in a single-frequency FFT bin (no leakage to neighboring bins) with white 

uncorrelated noise in that bin (and in every bin). Measurements are repeated N times and 

averaged. Let the noise in the th i measurement be represented by i

n , a Gaussian sequence of 

values with zero mean and covariance 2 σ . Then if the signal amplitude in the frequency bin of 

interest is ,

i

S the average of the measurements is 

( )

1

1

.

N

i i

i

M S n

N =

= + ∑ (C1) 

We can regard the expectation value of a limited set of measurement as the simple average: 

1

1

.

N

i

i

M S

N =

= ∑ (C2) 

The expected rms error of the measurement due only to the noise is 

( ) ( ) ( )

2

2

1 , 1

1 1

.

N N

i i j

i i j

E M M n n n

N = N =

 

= − = =     ∑ ∑ (C3) 

Given the Gaussian uncorrelated property of the noise, we can write 

2

,

i j ij n n =σ δ (C4) 

which is zero unless i j = . There are N such terms in the sum in Eq. (C3), so 

1 2 E N .

N N

σ

= = σ (C5) 

Thus the measurement can be reported with a statistical error given by 

1

1

.

N

i

i

S

N N

σ

=

∑ ± (C6) 

Suppose, however, that the FFT is squared and then averaged. We then have for the 

measurement 

( )

2 2 2 * *

1 1

1 1

( ) .

N N

i i i i i i i i

i i

M S n S S n S n n

N N = =

= + = + + + ∑ ∑ (C7) 

The middle terms drop out, because there is no correlation between the signal and the noise. In 

the last sum, the average of the noise is 2 σ ,

by Eq. (C4). The measurement is then 
 29 

2 2

1

1

.

N

i

i

M S

N

σ

=

= + ∑ (C8) 

If the noise power is known, it can be subtracted off and the measurement of the average signal 

power reported as 

2 2

1

1

.

N

i

i

M S

N

σ

=

− = ∑ (C9) 

Apart from the propagation of error that occurs in this subtraction, there will be some error 

associated with the measurement itself. This can be denoted by E, where 

( )2

2 E M M = − . (C10) 

Writing this out, 

2

2 2 2 2

1 1

1 1

.

N N

i i i

i i

E S n S

N N

σ

= =

  

= + − +   

  

∑ ∑ (C11) 

There is a rather subtle change in the order in which the operations are performed. If we again 

assume no correlation between signal and noise, the terms involving the signal in the above 

equation all cancel out or vanish. Then 

( ) ( )

2

2 2 2 2

2 2 2 4

2

1 , 1 1

1 1 1

σ 2 σ σ .

= = =

   

= − =   − + 

   

∑ ∑ ∑

N N N

i i j i

i i j i

E n n n n

N N N

 (C12) 

Evaluating each of these terms one by one, in the first sum there are N terms with i j = . These 

will contribute the amount 

1 4

.

i

n

N

(C13) 

There are additional contributions in the first sum with i j ≠ , where there are 

2 N N N N − = − ( 1) terms, each contributing ( )

2

2 σ . The first sum therefore contributes a total of 

4

2 2

4

2 2

, 1

1 ( 1)

.

N i

i j

i j

n N N

n n

N N N

σ

=

−

∑ = + (C14) 

In the second term of Eq. (C12) , there are N terms each contributing 4 σ ,

 so the second term 

contributes 

2 4 4 N 2 .

N

− = − σ σ (C15) 

The last term contributes 

4

+σ . (C16) 

Collecting all the terms, the error is given by 

4 4

2 4 4 4 4

2

( 1) 1

2 .

i i

n N N n

E

N N N N

σ σ σ σ

−

= + − + = − (C17) 

Thus, we have to estimate the fourth moment. In a normalized Gaussian distribution that is a 

function of x, the probability of getting a value x within the increment dx is 
 30 

2

2

2

1

( ) .

2

x

P x dx e dx σ

πσ

−

= (C18) 

The fourth moment is 

2

2

4

4 2 4

3 .

2

σ σ

πσ

∞

−

−∞

= = ∫

x

x

x e dx (C19) 

Thus the error is 

4 4 4

2

2

3 2

;

2

σ σ σ

σ

= − =

=

E

N N N

E N

(C20) 

The measurement of signal power can then be reported with an uncertainty as: 

2

2 2

M .

N

σ

− ± σ (C21) 

 

Expressing Eq. (C21) in terms of SNR gives 

2

1 2

1 .

σ

=

 ± 

 − 

 

 

M

SNR

N

M

SNR

 (C22) 
 31 

REFERENCES 

[1] “Calibration Services User Guide,” 

http://ts.nist.gov/MeasurementServices/Calibrations/pm_am_noise.cfm#pm_am. [Online]. 

Available: http://ts.nist.gov/MeasurementServices/Calibrations/pm_am_noise.cfm#pm_am. 

[2] F. L. Walls, “Secondary standard for PM and AM noise at 5, 10, and 100 MHz,” IEEE 

Transactions on Instrumentation and Measurement, vol. 42, no. 2, pp. 136-143, Apr. 1993. 

[3] F. L. Walls, “Calibration system for determining the accuracy of phase modulation and amplitude 

modulation noise measurement apparatus,” U.S. Patent 5,172,06415-Dec-1992. 

[4] F. G. Ascarrunz and F. L. Walls, “A standard for PM and AM noise at 10.6, 21.2 and 42.4 GHz,” in 

Frequency Control Symposium, 1996. 50th., Proceedings of the 1996 IEEE International., 1996, pp. 

852-853. 

[5] IEEE, “IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time 

Metrology — Random Instabilities,” IEEE Std 1139-1999, 1999. [Online]. Available: ISBN 0-7381-

1754-4. [Accessed: 06-Aug-2010]. 

[6] D. B. Sullivan, D. W. Allan, D. A. Howe, and F. L. Walls, “Characterization of Clocks and 

Oscillators,” in NIST Technical Note 1337, 1990. 

[7] F. L. Walls and E. Ferre-Pikal, “Measurement of Frequency, Phase noise and Amplitude noise,” in 

Wiley Encyclopedia of Electrical and Electronics Engineering, 1st ed., vol. 12, 24 vols., WileyInterscience, 1999, pp. 459-473. 

[8] E. Rubiola, Phase Noise and Frequency Stability in Oscillators. Cambridge University Press, 2010. 

[9] S. I. Baskakov, Signals and circuits. Mir Publishers, 1990. 

[10] B. P. Lathi, Modern Digital and Analog Communications Systems, International 2 Revised ed. 

Henry Holt & Company, 1983. 

[11] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, 

and Mathematical Tables. Dover Publications, 1965. 

[12] F. L. Walls, “Correlation between upper and lower sidebands,” IEEE Transactions on Ultrasonics, 

Ferroelectrics and Frequency Control, vol. 47, no. 2, pp. 407-410, Mar. 2000. 

[13] R. P. Scott, C. Langrock, and B. H. Kolner, “High-dynamic-range laser amplitude and phase noise 

measurement techniques,” Selected Topics in Quantum Electronics, IEEE Journal of, vol. 7, no. 4, 

pp. 641-655, 2001. 

[14] F. L. Walls, D. B. Percival, and W. R. Irelan, “Biases and variances of several FFT spectral 

estimators as a function of noise type and number of samples,” in Frequency Control, 1989., 

Proceedings of the 43rd Annual Symposium on, 1989, pp. 336-341. 

[15] B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressingthe Uncertainty of NIST 

Measurement Results,” NIST Technical Note 1297, 1994. 

[16] BIPM, “Evaluation of measurement data — Guide to the expression of uncertainty in 

measurement.” 2008. 

[17] E. Rubiola and R. Boudot, “The effect of AM noise on correlation phase-noise measurements,” 

IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol. 54, no. 5, pp. 926-

932, May 2007. 

[18] J. Grove, J. Hein, J. Retta, P. Schweiger, W. Solbrig, and S. R. Stein, “Direct-digital phase-noise 

measurement,” in Frequency Control Symposium and Exposition, 2004. Proceedings of the 2004 

IEEE International, 2004, pp. 287-291. 

[19] E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise 

measurement of microwave oscillators,” Journal of the Optical Society of America B, vol. 22, no. 5, 

pp. 987-997, May 2005. 

[20] F. L. Walls, “Method and apparatus for wide band phase modulation,” U.S. Patent 4,968,90806-

Nov-1990. 

[21] Hewlett Packard, “Fundamentals of RF and Microwave Power Measurements,” Application Note, 

no. 64-1, 1978. 



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