- [[#Introduction: Why Phase Noise Matters]] - [[#Understanding Oscillators]] - [[#Ideal Oscillators]] - [[#Non-Ideal (Real) Oscillators: Adding Noise]] - [[#VCOs in the System]] - [[#Frequency Dividers and Phase Noise Reduction]] - [[#Well Then…]] - [[#Conclusions + Resources]] - [[#Connections | Questions:]] # Introduction: Why Phase Noise Matters Systems require stable frequency sources for ideal operation. The oscillator in the system dictates the definition of a hertz so it is important for the oscillator to be stable. Phase noise is a term used to describe the measurement of the short-term frequency stability. Less phase noise is good, more phase noise is bad. An important metric of frequency stability is **phase noise**. In general: - **Less phase noise = better performance** - **More phase noise = worse performance** It is important to accurately quantify this level of phase instability and improve it when possible. The use of a **frequency divider** improves the phase noise of a signal by 6 dB. More generally, a frequency multiplier or divider of factor $N$ improves the phase noise by $20\log_{10}N$ Lets explore the physics and math behind it. # Understanding Oscillators ## Ideal Oscillators The ideal oscillator would produce a perfect sine wave with no phase noise: $v(t)=A\cos(\omega t+\phi)$ Where: - $A$: Amplitude - $\omega = 2\pi f$: Angular frequency - $\phi$: Phase shift - $t$: Time Viewed in the frequency and time domains: ![[88acd68b-6989-462f-a460-7c03b3600a44.png]] Figure 1: Sine Wave with Frequency 5 Hz ![[74eb7d1c-bc99-4c35-b7c4-c4b014b48063.png]] Figure 2: Fourier Transform of a Sine Wave with a Frequency 2500 Hz ## Non-Ideal (Real) Oscillators: Adding Noise The non-ideal oscillator: $V(t)=A(t)\cos(\omega t+\phi (t))$ Where now both $A(t)$ and $\phi (t)$ are time-dependent due to noise. In practice, **phase fluctuations** are more detrimental than amplitude fluctuations. This is because system timing and frequency behavior are directly tied to phase. Consider a band where the drummer is keeping the beat. So long as the beat is consistent, the band will stay on tempo, even if the beat gets quieter or louder. ![[a83f1240-0302-4daf-be78-a05bf2733b56.png]] Figure 3: Sine wave with frequency 5 Hz and phase noise ![[5235f4d2-f55d-47d7-8f8d-51fe9fd9b0c0.png]] Figure 4: Fourier Transform of a sine wave with frequency 2500 Hz and phase noise ## VCOs in the System In RF and digital communication systems, the **Voltage-Controlled Oscillator (VCO)** is a critical component. A VCO produces a sine wave with the **frequency being a function of the input voltage**: $f(t)=f_0+K_vV_{in}(t)$ Where: - $f_0$: Center frequency of the oscillator - $K_v$: VCO gain - $V_{in}(t)$: Control voltage input The input voltage controls the frequency, and thus the phase. Even **small voltage variations** can lead to significant **phase noise.** This makes phase noise suppression techniques - like frequency division - especially important. In **Phase-Locked Loops (PLLs)**, for example, the VCO is the noise-limiting component and frequency division in the feedback path helps reduce system phase noise. # Frequency Dividers and Phase Noise Reduction Starting with the definition of the non-ideal oscillator; It goes through the frequency multiplier/divider of factor $N$ which modifies both the frequency and the phase: _NB: The amplitude term has been ignored in this context as they do not significantly impact the relative phase noise performance._ $v(t)=\cos(N\omega t+N\phi (t))$ Using trig identities: $v(t)=\cos(N\omega t)\cos(N\phi (t))-\sin(N\omega t)\sin(N\phi(t))$ Since the phase deviation is generally small, we can use the small angle approximation where $\sin(x) \cong x$ $v(t)=\cos(N\omega t)-N\phi (t)\sin(N\omega t)$ We can see that the instantaneous voltage of the sidebands, the second term, is now directly proportional to $N$ $v(t) \propto N$ Recall our good friend Ohm $P \propto v^2 \propto N^2$ Converting power to log units $10 \log_{10}N^2=20\log_{10}N$ This confirms the original claim - dividing frequency by 2 improves the phase noise by **6 dB**. # Well Then… **If dividing improves phase noise, why not always divide as much as possible?** What trade-offs arise in RF circuits that depend too heavily on frequency dividers for phase noise performance? # Conclusions + Resources Stable frequency sources are required for ideal operations of circuits. Less phase noise means better performance, and more phase noise means worse. Ideal oscillators produce a perfect sine wave with no phase noise whilst non-ideal oscillators have a time-dependent, random, phase term. The use of a frequency divider of factor N improves the phase noise of a signal by 6 dB and more generally by $20 \log_{10}N$. A great white paper on an introduction to phase noise: [https://www.allaboutcircuits.com/uploads/articles/RnS-Understanding-phase-noise-fundamentals_wp.pdf](https://www.allaboutcircuits.com/uploads/articles/RnS-Understanding-phase-noise-fundamentals_wp.pdf) ![[Connect#Connect]]