Systems, Modulation and Noise

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Modulation of WSS process

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

Chembian Thambidurai, "Power Spectral Density of Pulsed Noise Signals" [link]

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Case Signal Carrier phase Ensemble ACF Stationarity PSD / averaged PSD
Oscillator with small random phase noise \(v(t)=A\sin(\omega_0t+\varphi(t))\) fixed carrier \(\omega_0t\), small random \(\varphi(t)\) \(\displaystyle R_v(t,\tau)\) contains terms like \(\cos(2\omega_0t+\omega_0\tau)\) Cyclostationary, not WSS Time-averaged: \(\displaystyle S_v(f)=\frac{A^2}{4}\left[\delta(f-f_0)+\delta(f+f_0)+S_\varphi(f-f_0)+S_\varphi(f+f_0)\right]\)
Random process × random-phase cosine \(Y(t)=X(t)\cos(\Omega_c t+\Theta)\) \(\Theta\sim U[0,2\pi)\) \(\displaystyle R_{YY}(\tau)=\frac{R_{XX}(\tau)}{2}\cos(\Omega_c\tau)\) WSS, if \(X(t)\) is WSS \(\displaystyle P_{YY}(\Omega)=\frac{P_{XX}(\Omega+\Omega_c)+P_{XX}(\Omega-\Omega_c)}{4}\)
Random process × deterministic cosine \(S(t)=m(t)\cos(\Omega_c t+\phi_0)\) fixed \(\phi_0\) \(\displaystyle R_{SS}(t,\tau)=\frac{R_{mm}(\tau)}{2}\left[\cos(\Omega_c\tau)+\cos(2\Omega_c t+\Omega_c\tau+2\phi_0)\right]\) Cyclostationary, not WSS Time-averaged: \(\displaystyle \tilde P_{SS}(\Omega)=\frac{P_{mm}(\Omega+\Omega_c)+P_{mm}(\Omega-\Omega_c)}{4}\)

modulated with small perturbation

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

Given \(\color{blue}\phi(t)\ll 1\), the autocorrelation still depends on absolute time \(t\). Therefore \(v(t)\) is cyclostationary, and they must take a time average over one carrier period.

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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

modulated with full-cycle random

Given \(\color{blue}\Theta\sim U[0,2\pi]\), after ensemble averaging, the autocorrelation becomes WSS

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Haykin, Simon S., and Michael Moher. Communication Systems. 5th ed. John Wiley & Sons, 2009. - Mixing of a Random Process with a Sinusoidal Process

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sinewaves discrete approximate for continuous white noise

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 [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=10818782]

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suppose \(\color{blue}n(t) = \sum_i N[i]e^{ji\omega_0 t}\) where \(\color{blue}N[i] = \sigma\, e^{j\theta_i} \quad \theta_i \ \text{i.i.d. } \mathcal{U}(-\pi, +\pi]\)

the three related quantities are: \[ |N[i]| = \sigma \quad(\text{deterministic}), \qquad \langle N[i]\rangle = \sigma\,\langle e^{j\theta_i}\rangle = 0, \qquad N[i]\,\overline{N[i]} = \sigma e^{j\theta_i}\cdot \sigma e^{-j\theta_i} = \sigma^2. \] with the definition of the (ensemble) autocorrelation of a complex process \[ R_n(\tau) = \big\langle\, n(t+\tau)\,\overline{n(t)}\,\big\rangle = \sum_i \sum_k \big\langle N[i]\,\overline{N[k]}\big\rangle\, e^{j(i-k)\omega_0 t}\, e^{ji\omega_0 \tau} = \sum_i \sigma^2\, e^{ji\omega_0 \tau} \] The delta-function limit is then a Riemann sum: with \(\color{blue}\sigma^2 = N^2\Delta f\) and \(\omega_0 = 2\pi\Delta f\), \[ R_n(\tau) = \sum_i N^2\,e^{j2\pi (i\Delta f) \tau}\,\Delta f \;\xrightarrow{\;\Delta f \to 0\;}\; N^2\!\int_{-\infty}^{\infty} e^{j2\pi f\tau}\,df = N^2\delta(\tau), \] recovering the ideal white-noise autocorrelation stated in the paper

\[ \color{blue}\boxed{ \text{fixed magnitude + uniform phase} \Rightarrow \text{approximately Gaussian for many tones} } \]

whereas \[ \boxed{ \text{complex Gaussian coefficients} \Rightarrow \text{exactly Gaussian band-limited noise} }. \] If every sinusoid has a fixed amplitude and only its phase is random, \[ N_k = \sqrt{P_k}e^{j\Theta_k}, \] then a finite sum is not exactly Gaussian. However, when many independent components are added, the time-domain distribution becomes approximately Gaussian by the central limit effect.

For an exactly Gaussian frequency-domain construction, the spectral coefficients should be complex Gaussian: \[ N_k = N_{I,k}+jN_{Q,k}, \] where \[ N_{I,k},N_{Q,k} \sim \mathcal N(0,\sigma_k^2). \] In that case: \[ \Theta_k\sim\mathcal U(-\pi,\pi], \] but the magnitude is also random and has a Rayleigh distribution

Assume \(\Theta\sim \mathcal U[0,2\pi)\) is a single random phase that remains constant for all \(t\): \[ \color{blue}\boxed{X(t)=A\cos(\omega_0t+\Theta).} \]

Mean \[ \begin{aligned} m_X(t) &=\mathbb E[X(t)]\\ &=\frac{A}{2\pi}\int_0^{2\pi} \cos(\omega_0t+\theta)\,d\theta\\ &=0. \end{aligned} \]

Thus, the mean is independent of \(t\).

Autocorrelation

For \(t_1\) and \(t_2\), \[ R_X(t_1,t_2) = \mathbb E[X(t_1)X(t_2)]. \] Using \(\cos a\cos b=\frac{1}{2}\left[\cos(a-b)+\cos(a+b)\right],\)

we obtain \[ \begin{aligned} R_X(t_1,t_2) &= \frac{A^2}{2} \mathbb E\left[ \cos\big(\omega_0(t_1-t_2)\big) + \cos\big(\omega_0(t_1+t_2)+2\Theta\big) \right]. \end{aligned} \] Since \(\Theta\) is uniform, \[ \mathbb E\left[ \cos\big(\omega_0(t_1+t_2)+2\Theta\big) \right]=0. \] Therefore, \[ \boxed{ R_X(t_1,t_2) = \frac{A^2}{2} \cos\big(\omega_0(t_1-t_2)\big) } \] or, defining \(\tau=t_1-t_2\), \[ \boxed{ R_X(\tau)=\frac{A^2}{2}\cos(\omega_0\tau) }. \] The autocorrelation depends only on the time difference \(\tau\).

Is it WSS?

Yes. The two WSS conditions are satisfied: \[ m_X(t)=0, \] which is constant, and \[ R_X(t_1,t_2)=R_X(t_1-t_2). \] Thus, \[ \boxed{X(t)\text{ is wide-sense stationary.}} \] In fact, because a time shift simply changes the uniformly distributed phase, \[ X(t+t_0) = A\cos\left(\omega_0t+\underbrace{\Theta+\omega_0t_0}_{\text{still uniform modulo }2\pi}\right), \] the process is also strict-sense stationary.

Power spectral density

Using the angular-frequency Fourier-transform convention \[ S_X(\omega) = \int_{-\infty}^{\infty} R_X(\tau)e^{-j\omega\tau}\,d\tau, \] and \[ \mathcal F\{\cos(\omega_0\tau)\} = \pi\left[ \delta(\omega-\omega_0)+\delta(\omega+\omega_0) \right], \] we get \[ \boxed{ S_X(\omega) = \frac{\pi A^2}{2} \left[ \delta(\omega-\omega_0) + \delta(\omega+\omega_0) \right] }. \] Thus, the PSD consists of two spectral lines at \(\omega=\pm\omega_0\).

The total average power is \[ R_X(0)=\frac{A^2}{2}, \] and equivalently, \[ \frac{1}{2\pi}\int_{-\infty}^{\infty}S_X(\omega)\,d\omega = \frac{A^2}{2}. \] For frequency \(f\), where \(f_0=\omega_0/(2\pi)\), \[ \color{blue}\boxed{ S_X(f) = \frac{A^2}{4} \left[ \delta(f-f_0)+\delta(f+f_0) \right] }. \] The random phase makes the ensemble stationary; a sinusoid with a fixed deterministic phase is not normally regarded as a stationary random process because it contains no random ensemble.

modulated with deterministic cosine

the carrier phase/time origin is fixed, not randomized, it not WSS but cyclostationary

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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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Quadrature-Modulated Processes

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

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Dr. Vishal Saxena, ECE518 Memory/Clock Synchronization IC Design: Oscillator Phase Noise [https://www.eecis.udel.edu/~vsaxena/courses/ece504/Handouts/Oscillator%20Phase%20Noise.pdf]

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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

Pulsed Noise Signals

Chembian Thambidurai, "Power Spectral Density of Pulsed Noise Signals" [link]

image-20241208075822212

Above, the output of the multiplier be \(y(t)\) is passed through a ideal brick wall low pass filter with a bandwidth of \(f_0/2\)

When a random signal is multiplied by a pulse function, the resulting signal becomes a cyclo-stationary random process.

As rule of thumb, the spectrum of such a pulsed noise signal

  • thermal noise is multiplied by \(\color{red}D\)

  • flicker noise is multiplied by \(\color{red}D^2\),

where \(D\) is the duty cycle of the pulse signal

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banlimited input (no aliasing)

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wideband white noise input

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flicker noise input

with \(S_x(f)=\frac{K_f}{f}\)

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Assuming \(\Delta f \ll f_0\)

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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]

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