AM & PM

Posted on Edited on In

Rotaing Phasor

phasor in time domain as time elapses

image-20251105225355996

image-20260619111109416

image-20260619110417243

A. A. Abidi and D. Murphy, "How to Design a Differential CMOS LC Oscillator," in IEEE Open Journal of the Solid-State Circuits Society, vol. 5, pp. 45-59, 2025 [pdf]

—, 30 January, SSCS Technical Webinars, [https://resourcecenter.sscs.ieee.org/education/webinars/sscs_edu_web_abidi_013026]

The oscillation \(\color{red}V_0\cos\omega_0 t\) is represented by the horizontal phasor of magnitude \(\color{red}V_0\)

image-20260620165420109 \[ \boxed{\color{blue}\begin{equation} \left. \begin{aligned} C_{+} &= V_{AM} + V_{PM} \\ C_{-} &= \overline{V}_{AM} - \overline{V}_{PM} \end{aligned} \right\} \quad\Longrightarrow\quad \begin{aligned} V_{AM} &= \tfrac{1}{2}\left(C_{+} + \overline{C}_{-}\right) \\ V_{PM} &= \tfrac{1}{2}\left(C_{+} - \overline{C}_{-}\right) \end{aligned} \end{equation}} \]

AM \[ e^{j\omega_0 t}(V_0 + V_{AM}e^{jy_A}+V_{AM}e^{-jy_A}) = V_0 e^{j\omega_0 t}\cdot \left(1+\frac{2V_{AM}}{V_0}\cos y_A\right) \] modulate amplitude of the oscillation sinusoidally with frequency \(\omega_m\) and depth \(m=\frac{2V_A}{V_0}\)

PM \[ e^{j\omega_0 t}(V_0 + V_{PM}e^{jy_P}-V_{PM}e^{-jy_P})=V_0e^{j\omega_0 t}\left(1 + j\frac{2V_{PM}}{V_0}\sin y_P\right) \approx V_0e^{j\omega_0 t}\cdot e^{j\frac{2V_{PM}}{V_0}\sin y_P} \] modulate the phase of the oscillation with an index of approximately \(\frac{2V_{PM}}{V_0}\) rad

image-20251105222421572

"I" is the in-phase or real axis and "Q" is the quadrature or imaginary axis

  • phasor rotating counter-clockwise represents the upper sideband (USB) \(e^{j(\omega_o+\omega_m )t} = e^{j\omega_o t}\color{red}e^{j\omega_m t}\)

  • phasor rotating clockwise represents the lower sideband (LSB) \(e^{j(\omega_o-\omega_m )t} = e^{j\omega_o t}\color{red}e^{-j\omega_m t}\)

AM modulation

image-20251105224322467 \[\begin{align} x(t)&= (1+2k\cos(\omega_m t)) \cos(\omega_0 t) = \cos(\omega_0 t) + 2k \cos(\omega_m t) \cos(\omega_0 t) \\ &=\mathcal{Re}\{e^{j\omega_0t}+k(e^{j\omega_0t}e^{j\omega_mt}+e^{j\omega_0t}e^{-j\omega_mt})\} =\mathcal{Re}\{e^{j\omega_0t}(\color{red}1+k(e^{j\omega_mt}+e^{-j\omega_mt})\color{black})\} \end{align}\]

image-20251105230141244

PM modulation with incidental AM

image-20260604232437918

image-20260604232710170

Phasors, Modulation & Sidebands [https://www.rfinsights.com/concepts/modulation-and-sidebands/]

image-20260511235100264

Single Sideband Modulation (SSB)

Phase Noise 101: Basics, Applications and Measurements [https://www.qsl.net/ab4oj/test/docs/20180720_KEE7_PhaseNoise.pdf]

Marcelino Lazaro, Communication Theory, Chapter 2 Analog Modulations [https://ocw.uc3m.es/pluginfile.php/7097/mod_page/content/15/OCW-CT-C2.pdf] [video]

image-20251104003558243

image-20260619103248132

image-20260619103308475

Real & Complex Modulation

J OHN M. C IOFFI. Complex AWGN and Other Channels [https://cioffi-group.stanford.edu/ee379a/Lectures/L4.pdf]

image-20251123230309624

image-20251123230332297

image-20251123230424242

signal Spectra
\(x_{bb}(t) = x_I(t)+jx_Q(t)\) \(X_{I}(\omega)+jX_Q(\omega)\)
\(x(t)=\color{red}x_I(t)\cdot\cos(\omega_c t)\color{blue}-x_Q(t)\cdot\sin(\omega_c t)\) \(\color{red}\frac{1}{2}X_{I}(\omega+\omega_c)+\color{blue}\frac{1}{2}jX_{Q}(\omega+\omega_c)+\color{red}\frac{1}{2}X_{I}(\omega-\omega_c)+\color{blue}\frac{1}{2}jX_{Q}(\omega-\omega_c)\)
\(x_A=x_{bb}(t)\cdot e^{j\omega_c t}\) \(X_{I}(\omega-\omega_c)+jX_Q(\omega-\omega_c)\)

image-20251123230522862

Matt Guibord. TIPL 4: Real and Complex Modulation [https://www.ti.com/content/dam/videos/external-videos/en-us/2/3816841626001/5576277660001.mp4/subassets/TIPL4708-Real-and-Complex-Modulation.pdf]

image-20251116225706934 \[\begin{align} X_{bb,AM,USB}(t) &= e^{j\omega_m t} = \cos(\omega_m t) + j\sin(\omega_m t)\\ X_{bb,AM,LSB}(t) &= e^{-j\omega_m t} = \cos(\omega_m t) - j\sin(\omega_m t) \end{align}\]

where \(X_{I,AM,USB}(t)=\cos(\omega_m t)\) and \(X_{Q,AM,USB}(t)=\sin(\omega_m t)\); \(X_{I,AM,USB}(t)=\cos(\omega_m t)\) and \(X_{Q,AM,USB}(t)=-\sin(\omega_m t)\);

\(X_{I,AM,USB} = X_{I,AM,LSB}\) and \(X_{Q,AM,USB} = -X_{Q,AM,LSB}\)

\[\begin{align} X_{bb,PM,USB}(t) &= e^{j\omega_m t} = \cos(\omega_m t) + j\sin(\omega_m t)\\ X_{bb,PM,LSB}(t) &= -e^{-j\omega_m t} = -\cos(\omega_m t) + j\sin(\omega_m t) \end{align}\]

where \(X_{I,PM,USB}(t)=\cos(\omega_m t)\) and \(X_{Q,PM,USB}(t)=\sin(\omega_m t)\); \(X_{I,PM,USB}(t)=-\cos(\omega_m t)\) and \(X_{Q,PM,USB}(t)=\sin(\omega_m t)\);

\(X_{I,PM,USB} = -X_{I,PM,LSB}\) and \(X_{Q,PM,USB} = X_{Q,PM,LSB}\)

image-20251116232032760

Amplitude Noise

Deog-Kyoon Jeong. Topics in IC Design: 1.1 Introduction to Jitter [https://ocw.snu.ac.kr/sites/default/files/NOTE/Lec%201%20-%20Jitter%20and%20Phase%20Noise.pdf]

image-20250821202257578

with \(x(t) = A_0\sin (2\pi f_0 t +\phi _0)\), then \(y(t) = x(t) + n_v(t)\)

\[\begin{align} R_y(\tau) &= \mathrm{E}[y(t)y(t+\tau)] \\ &= \mathrm{E}[x(t)x(t+\tau)] + \mathrm{E}[x(t)]\mathrm{E}[n_v(t+\tau)] + \mathrm{E}[x(t+\tau)]\mathrm{E}[n_v(t)] + \mathrm{E}[n_v(t)n_v(t+\tau)]\\ &= \mathrm{E}[x(t)x(t+\tau)] + \mathrm{E}[n_v(t)n_v(t+\tau)] = R_x(\tau) + R_{n_v}(\tau) \end{align}\]

P.E. Allen - 2003. ECE 6440 - Frequency Synthesizers: Lecture 160 – Phase Noise - II [https://pallen.ece.gatech.edu/Academic/ECE_6440/Summer_2003/L160-PhNoII(2UP).pdf]

image-20250609213352109

image-20250609213403991

PM & FM

Dan Boschen What is the difference between phase noise and frequency noise? [https://dsp.stackexchange.com/a/38230/59253]

Phase Noise and Frequency Noise are not two different noise sources, they are artifacts of the same noise, it is just a matter of what units you want to use

image-20251106205014601

image-20251106001007275

AM & PM Sidebands

image-20241012001704081

The spectrum of the narrowband FM signal is very similar to that of an amplitude modulation (AM) signal but has the phase reversal for the other sideband component

Assume the modulation frequency of PM and AM are same, \(\omega_m\)

\[\begin{align} x(t) &= (1+A_m\cos{\omega_m t})\cos(\omega_0 t + P_m \sin\omega_m t) \\ &= \cos(\omega_0 t + P_m \sin\omega_m t) + A_m\cos{\omega_m t}\cos(\omega_0 t + P_m \sin\omega_m t) \\ &= X_{pm}(t) + X_{apm}(t) \end{align}\]

\(X_{pm}(t)\), PM Only \[ X_{pm}(t) = \cos\omega_0 t - \frac{P_m}{2}\cos(\omega_0 - \omega_m)t + \frac{P_m}{2}\cos(\omega_0 + \omega_m)t \] \(X_{apm}(t)\), AM & PM \[\begin{align} X_{apm}(t) &= A_m \cos{\omega_m t} (\cos\omega_0 t-P_m\sin\omega_m t\sin\omega_0 t) \\ &= \frac{A_m}{2}[\cos(\omega_0 + \omega_m)t + \cos(\omega_0 -\omega_m)t] - \frac{A_mP_m}{2}\sin(2\omega_m t)\sin(\omega_0 t) \\ &= \frac{A_m}{2}\cos(\omega_0 + \omega_m)t + \frac{A_m}{2}\cos(\omega_0 -\omega_m)t - \frac{A_mP_m}{4}\cos(\omega_0 - 2\omega_m)t + \frac{A_mP_m}{4}\cos(\omega_0 + 2\omega_m)t \end{align}\]

That is \[\begin{align} x(t) &= \cos\omega_0 t + \frac{A_m-P_m}{2}\cos(\omega_0 - \omega_m)t + \frac{A_m+P_m}{2}\cos(\omega_0 + \omega_m)t \\ &\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space - \frac{A_mP_m}{4}\cos(\omega_0 - 2\omega_m)t + \frac{A_mP_m}{4}\cos(\omega_0 + 2\omega_m)t \end{align}\]

For general case, \(x(t) = (1+A_m\cos{\omega_{am} t})\cos(\omega_0 t + P_m \sin\omega_{pm} t)\), i.e., PM is \(\omega_{pm}\), AM is \(\omega_{am}\)

\[\begin{align} x(t) &= \cos\omega_0 t - \frac{P_m}{2}\cos(\omega_0 - \omega_{pm})t + \frac{P_m}{2}\cos(\omega_0 + \omega_{pm})t \\ &\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space + \frac{A_m}{2}\cos(\omega_0 - \omega_{am})t + \frac{A_m}{2}\cos(\omega_0 + \omega_{am})t \\ &\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space - \frac{A_mP_m}{4}\cos(\omega_0 - \omega_{pm}-\omega_{am})t + \frac{A_mP_m}{4}\cos(\omega_0 + \omega_{pm}+\omega_{am})t \\ &\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space + \frac{A_mP_m}{4}\cos(\omega_0 + \omega_{pm}-\omega_{am})t - \frac{A_mP_m}{4}\cos(\omega_0 - \omega_{pm}+\omega_{am})t \end{align}\]

Therefore, sideband is asymmetric if \(\omega_{pm} = \omega_{am}\) same

Asymmetrical Linear System

Golara, S. (2015). Identifying Mechanisms of AM-PM Distortion in Large Signal Amplifiers. UCLA [https://escholarship.org/uc/item/4jp786z8]

image-20251010234110656

Bob Nelson. Phase Noise 101: Basics, Applications and Measurements [[https://www.qsl.net/ab4oj/test/docs/20180720_KEE7_PhaseNoise.pdf])https://www.qsl.net/ab4oj/test/docs/20180720_KEE7_PhaseNoise.pdf]

image-20251104003200956

Hegazi, Emad, Asad Abidi, and Jacob Rael. The Designer's Guide to High-purity Oscillators. [New York]: Kluwer Academic Publishers, 2005. [pdf]

image-20251105231301581

Equipartition theorem

Carlo Samori, Phase Noise in LC Oscillators: From Basic Concepts to Advanced Topologies [https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-VCO-short.pdf]

Enrico Rubiola. The Measurement of AM-PM Noise, and the Origin of Noise in Oscillators [https://rubiola.org/pdf-slides/2010T-ANL-Noise-and-oscillators.pdf]

image-20251106214759580

image-20251104233701576

Stationary noise can also be decomposed into AM and PM components, but there will always be equal amounts of both.

Narrowband FM Signal PSD

Chembiyan T. Jitter and Phase Noise in Phase Locked Loops [link]

\[ y(t) = A\cos(2\pi f_0t+\phi_n(t)) \approx A \cos(2\pi f_0 t) - A \phi_n (t)\sin(2\pi f_0 t) \]

image-20241228020953646 \[ R_x(\tau) = \frac{A^2}{2}\cos(2\pi f_0\tau) + \frac{A^2}{2}R_\phi(\tau)\cos(2\pi f_0\tau) \] The PSD of the signal \(x(t)\) is given by \[ S_x(f) = \mathcal{F}\{R_x(\tau)\} = \frac{P_c}{2}\left[\delta(f+f_0)+\delta(f-f_0)+S_\phi(f+f_0)+S_\phi(f-f_0)\right] \] where \(P_c = A^2/2\) is the carrier power of the signal

Nicola Da Dalt , Understanding Jitter and Phase Noise: 3.1.3 Voltage to Excess Phase Transformations: Random Noise

image-20260621100923112

Narrowband Noise

image-20251116151034742

in-phase & quadrature component

image-20251116151212165

image-20251116150946685

\[\begin{align} n(t)\cdot 2\cos(\omega_c t) &= n_I(t) + n_I(t)\cos(2\omega_c t) - n_Q(t)\sin(2\omega_c t) \\ n(t)\cdot -2\sin(\omega_c t) &= n_Q(t) - n_I(t)\sin(2\omega_c t) - n_Q(t)\cos(2\omega_c t) \end{align}\]

image-20251116152134926

nI_nQ.drawio

[https://arnabiitk.wordpress.com/wp-content/uploads/2013/02/solution-manual-for-communication-systems-haykin.pdf]

image-20251116154806955

image-20251116161009110

If the narrowband noise \(n(t)\) is Gaussian and its power spectral density \(S_N (t )\) is symmetric about the mid-band frequency \(f_c\), then the in-phase component \(n_I(t)\) and quadrature component \(n_Q(t)\) are statistically independent

image-20251116170549277

image-20251116171255430

envelope and phase components

image-20251116174252304

AN-PN Conversion

G. Giust, Influence of Noise Processes on Jitter and Phase Noise Measurements [https://www.signalintegrityjournal.com/articles/800-influence-of-noise-processes-on-jitter-and-phase-noise-measurements]

—. "Methodologies for PCIe5 Refclk Jitter Analysis,", PCI-SIG Electrical Workgroup Meeting (Jan. 19, 2018)

—. How to Identify the Source of Phase Jitter through Phase Noise Plots [https://www.sitime.com/company/newsroom/blog/how-identify-source-phase-jitter-through-phase-noise-plots]

AN10072 Determine the Dominant Source of Phase Noise, by Inspection [https://www.sitime.com/support/resource-library/application-notes/an10072-determine-dominant-source-phase-noise-inspection]

  • AM alone doesn't introduce jitter (e.g., doesn't change zero-crossings) nor impact phase

  • AM changes slew rate, and so influences the conversion of amplitude noise to jitter when amplitude noise (BB) is present

image-20250719122931298

Figure 8 thumb_rev

additive & parametric noise

Enrico Rubiola. The Measurement of AM-PM Noise, and the Origin of Noise in Oscillators [https://rubiola.org/pdf-slides/2010T-ANL-Noise-and-oscillators.pdf]

—, February 7, 2025. Phase Noise - Art, Science and Experimental Methods [https://rubiola.org/pdf-lectures/Scient-Instrum-Files/!-Phase-noise.pdf]

image-20251116190237879

image-20251110234817907

image-20251110233928508

image-20251104010027135

image-20251105002402811

image-20251104010925436

image-20251105231715117

AM & PM Noise Separation

Ken Kundert. Introduction to RF Simulation and its Application [https://designers-guide.org/analysis/rf-sim.pdf]

[https://designers-guide.org/analysis/am-pm-conv.pdf]

image-20251117235457764

Separation.drawio

For AM & pacphase=90deg, complex exponentials with frequency \(\omega_m\) \[ \alpha_+(t) = je^{j\omega_m t} = e^{j(\omega_mt+\pi/2)} \] where \(A_+=j\)

Then, the complex exponentials with frequency \(-\omega_m\) shall be \[ \alpha_-(t) = [\alpha_+(t)]^* = -je^{-j\omega_m t} = e^{-j(\omega_mt+\pi/2)} \] where \(A_-=A_+^* =-j\)

image-20251125214622501

image-20251118000021345

image-20251117233740882

image-20251118213331072

image-20251118213700865

image-20251118223254553

\(\phi_c=0\) in above simulation

Ken Kundert. Re: Question about phase noise simulation result [https://designers-guide.org/forum/YaBB.pl?num=1309258199/15#15]

Spectre Circuit Simulator RF Analysis Theory — Measuring AM, PM and FM Conversion

image-20251104004721935

noise profile sidebands contribution
stationary uncorrelated \(S_{AM} = S_{PM}\)
cyclostationary correlated \(S_{AM} \gt S_{PM}\) or \(S_{AM} \lt S_{PM}\)

sine wave + white noise

image-20251111231609124

equal amounts of AM and PM noise in both USB and LSB

sine wave + AM

image-20251112002405912

肥肥牛是只虎. PSS+Pnoise仿真：基本设置 [https://mp.weixin.qq.com/s/etyQ2UkfisPkvbc44XFw4w]

image-20251104004017859

image-20251104010500397

Phase Noise Measurement

Phase Noise Measurement Solutions [https://www.keysight.com/vn/en/assets/7018-02528/technical-overviews/5990-5729.pdf]

Greg Bonaguide. Advances in Phase Noise Measurement Techniques [https://ieee.li/pdf/viewgraphs/advances_in_phase_noise_measurement_techniques.pdf]

U. L. Rohde, A. K. Poddar and A. M. Apte, "Getting Its Measure: Oscillator Phase Noise Measurement Techniques and Limitations," in IEEE Microwave Magazine, vol. 14, no. 6, pp. 73-86, Sept.-Oct. 2013 [https://sci-hub.jp/10.1109/MMM.2013.2269860]

The three most widely adopted techniques are direct spectrum, phase detector, and two-channel cross-correlation.

While the direct spectrum technique measures phase noise with the existence of the carrier signal, the other two remove the carrier (demodulation) before phase noise is measured.

Though direct spectrum technique method may not be useful for measuring very close-in phase noise to a drifting carrier, it is convenient for qualitative quick evaluation on sources with relatively high noise

image-20251105003812138

1
2
3
4
5
6
7
8
9
import numpy as np

# at 5KHz
n_pm = -140     # dBc
n_am = -142     # dBc

# calculate the total noise by PM + AM
n_tot = 10*np.log10(10**(n_pm / 10) + 10**(n_am / 10))
print(n_tot)    # -137.8755739720566

image-20251105233926270

reference

Ken Kundert, Measuring AM, PM & FM Conversion with SpectreRF [https://designers-guide.org/analysis/am-pm-conv.pdf]

Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise [https://designers-guide.org/theory/cyclo-preso.pdf], [https://designers-guide.org/theory/cyclo-paper.pdf]

Dan Boschen. Creating uneven sidebands with AM + PM modulation? [https://dsp.stackexchange.com/a/61670/59253]

—. Creating uneven sidebands with AM + PM modulation? [https://dsp.stackexchange.com/a/61670/59253]

—. Qualitative Explanation of Fourier Transform [https://dsp.stackexchange.com/a/78911/59253]

Timing 201 #1: The Case of the Phase Noise That Wasn't - Part 1 [https://community.silabs.com/s/share/a5U1M000000knpiUAA/timing-201-1-the-case-of-the-phase-noise-that-wasnt-part-1?]

Application Notes: Spectrum Analysis Amplitude and Frequency Modulation [https://www.keysight.com/us/en/assets/7018-06742/application-notes/5954-9130.pdf]

Haykin, Simon S., and Michael Moher. Communication Systems. 5th ed. John Wiley & Sons, 2009.

Marcelino Lazaro, Communication Theory: Bachelor in Telecommunication Technologies Engineering [https://ocw.uc3m.es/file.php/326/Bibliography/OCW-CT-Book.pdf]

L.W. Couch, Digital and Analog Communication Systems, 8th Edition, Pearson, 2013. [pdf]

Sklar, Bernard, and Fredric J. Harris. 2021. Digital Communications: Fundamentals and Applications. 3rd ed. New York: Pearson