Systems, Modulation and Noise

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Modulation of WSS process

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

modulated with a random cosine

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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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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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Quadrature-Modulated Processes

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

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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]

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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.