Systems, Modulation and Noise

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AM & PM Noise

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

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

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

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Emad Hegazi , Jacob Rael , Asad Abidi, 2005. The Designer's Guide to High-Purity Oscillators [https://picture.iczhiku.com/resource/eetop/whkgGLPAHoORYxbC.pdf]

Equipartition theorem

[https://www.ieeetoronto.ca/wp-content/uploads/2020/06/DL-VCO-short.pdf]

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AM & PM in Phasor

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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, doi: 10.1109/OJSSCS.2024 [pdf]

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AM-PM conversion in asymmetrical linear system

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

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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]

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

  • 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

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Figure 8 thumb_rev

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]

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

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

phase noise analyzer vs spectrum analyzer

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

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

Phasor representation

img

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?]

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[https://en.lntwww.de/Modulation_Methods/Single-Sideband_Modulation]

Narrowband FM Approximation

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

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 \[ S_x(f) = \mathcal{F}\{R_x(\tau)\} = \frac{P_c}{2}\left[\delta(f+f_0)+\delta(f-f_0)\right]+\frac{P_c}{2}\left[S_\phi(f+f_0)+S_\phi(f-f_0)\right] \] where \(P_c = A^2/2\) is the carrier power of the signal

Modulation of WSS process

Balu Santhanam, Probability Theory & Stochastic Process 2020: Modulation of Random Processes

modulated with a random cosine

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modulated with a deterministic cosine

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Hayder Radha, ECE 458 Communications Systems Laboratory Spring 2008: Lecture 7 - EE 179: Introduction to Communications - Winter 2006–2007 Energy and Power Spectral Density and Autocorrelation

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Sampling of WSS process

Balu Santhanam, Probability Theory & Stochastic Process 2020: Impulse sampling of Random Processes

DT sequence \(x[n]\)

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Owing to \(\phi[0] = \phi_c(0)\), the average power of the sampled version \(x[n]\) is the same as its input \(x_c(t)\)

impulse train \(x_s(t)\)

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That is \[ P_{x_s x_s} (f)= \frac{1}{T_s^2}P_{xx}(f) \] where \(x[n]\) is sampled discrete-time sequence, \(x_s(t)\) is sampled impulse train

Noise Aliasing

apply foregoing observation

Rectangular Pulse Sampling

Balu Santhanam. ece439 Introduction to Digital Signal Processing. Example: Rectangular Pulse Sampling [http://ece-research.unm.edu/bsanthan/ece439/recsamp.pdf]

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reference

Alan V Oppenheim, Ronald W. Schafer. Discrete-Time Signal Processing, 3rd edition [pdf]

R. E. Ziemer and W. H. Tranter, Principles of Communications, 7th ed., Wiley, 2013 [pdf]

John G. Proakis and Masoud Salehi, Fundamentals of communication systems 2nd ed [pdf]

Rhee, W. and Yu, Z., 2024. Phase-Locked Loops: System Perspectives and Circuit Design Aspects. John Wiley & Sons

Lacaita, Andrea Leonardo, Salvatore Levantino, and Carlo Samori. Integrated frequency synthesizers for wireless systems. Cambridge University Press, 2007

Phillips, Joel R. and Kenneth S. Kundert. "Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise." Proceedings of the IEEE 2000 Custom Integrated Circuits Conference. [pdf, slides]

Antoni, J., "Cyclostationarity by examples", Mechanical Systems and Signal Processing, vol. 23, no. 4, pp. 987–1036, 2009 [https://docente.unife.it/docenti/dleglc/a-a-2010-2011-dmsm/ciclostazionarieta.pdf]

Kundert, Ken. (2006). Simulating Switched-Capacitor Filters with SpectreRF. URL:https://designers-guide.org/analysis/sc-filters.pdf

STEADY-STATE AND CYCLO-STATIONARY RTS NOISE IN MOSFETS [https://ris.utwente.nl/ws/portalfiles/portal/6038220/thesis-Kolhatkar.pdf]

Christian-Charles Enz. "High precision CMOS micropower amplifiers" [pdf]