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5.8: Advanced Discussion of Oscillator Noise

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    This section presents a discussion of oscillator noise, and particularly the rapid increase of phase noise close to the carrier. Noise can be partitioned into amplitude and phase noise components. The nonlinear saturation of an oscillator suppresses amplitude noise so only phase noise is of concern. While usually associated with oscillators, phase noise is also added to a signal by an amplifier.

    There is not a consensus as to the origins of close-to-carrier phase noise. This section begins with observations of oscillator noise in the frequency domain and in the time domain. Then three theories of excess oscillator noise, Leeson’s theory, the linear time-invariant model, and the chaotic map model, are presented.

    Not having a complete model of the physical origin of phase noise means that a simulator cannot reliably predict the phase noise of an oscillator. Also designing an oscillator with good phase noise performance currently relies heavily on experience and projections based on what has been achieved by a designer previously.

    5.8.1 Observations of Oscillator Noise in the Frequency Domain

    The most puzzling noise observed with oscillators is the noise observed at a small frequency offset from the carrier (i.e., the average oscillation signal). To develop an appreciation for the breadth of observations, the signals produced by several different oscillators will be considered. First, Figure \(\PageIndex{2}\)

    clipboard_e966243624a922507cfe10fdf6c310a92.png

    Figure \(\PageIndex{1}\): Common emitter BJT Colpitts oscillators: (a) configuration with a feedback network between the collector and base of the transistor; and (b) alternative configuration.

    clipboard_eea305488ee31ea754e4789044fc3bffe.png

    Figure \(\PageIndex{2}\): Measured phase noise of low-frequency oscillators: (a) instrument noise floor; (b) HP 5087A frequency distribution amplifier at \(5\text{ MHz}\) (used to drive the external reference input of several test instruments using a single high-quality oscillator); (c) TADD-1 frequency distribution amplifier at \(10\text{ MHz}\); (d) TADD-1 frequency distribution amplifier at \(5\text{ MHz}\); (e) Spectracom 8140T frequency distribution amplifier at 10 MHz. Five phase noise regions are identified as \(f^{−5},\: f^{−4},\: f^{−3},\: f^{−1}\), and white noise. The spurious signals are related to injected harmonics of the \(60\text{ Hz}\) power mains. Used with permission of John Ackermann [23].

    is a plot of the phase noise observed at the output of several oscillators and amplifiers operating at \(5\text{ MHz}\) and \(10\text{ MHz}\). Curve (a) is the noise floor of the noise measurement instrument and spurious tones are observed at multiples of \(60\text{ Hz}\), the power mains frequency. Curves (b), (c), (d), and (e) show phase noise varying in straight-line segments. Being a log-log plot, these curves show phase noise varying as \(f^{−5},\: f^{−4},\: f^{−3},\: f^{−1}\), and \(f^{0}\). None of the phase noise plots here show a region with an \(f^{−2}\) dependence, although this is observed with other oscillators.

    Another oscillator to consider is the VCO circuit shown in Figure \(\PageIndex{3}\) [24]. This is a \(50\text{ MHz}\) VCO with a semiconductor varactor being the variable element with a zero-bias capacitance of \(100\text{ pF}\). The capacitance of the varactor is controlled by the voltage, \(V_{b}\). With \(V_{b} = 0\text{ V}\), the phase noise shown in Figure \(\PageIndex{4}\) was observed. The distinct phase noise regions have frequency dependencies of \(f^{0},\: f^{−1},\: f^{−2}\), and \(f^{−3}\). The phase noise of this oscillator is plotted again in Figure \(\PageIndex{5}\) for three different varactor bias voltages. The phase noise characteristics of the oscillator change even though the underlying physical sources of noise do not change (of course). Curve (a), with \(V_{b} = 6\text{ V}\), and Curve (b), with \(V_{b} = 0\text{ V}\), have an \(f^{−1}\) region around \(20\text{ kHz}\) (see Figure \(\PageIndex{4}\) for more details), but the \(f^{−1}\) region is not observed in Curve (c) where \(V_{b} = 18\text{ V}\). One interpretation is that the crossover frequencies have shifted. So what is particularly interesting here is

    clipboard_e48c7da84e065e060f00666e4eb94b088.png

    Figure \(\PageIndex{3}\): Varactor-tuned VCO schematic, from [24].

    clipboard_eed030a93fa36d599cf0d59b3086b4594.png

    Figure \(\PageIndex{4}\): Measured phase noise of a \(50\text{ MHz}\) BJT varactor-based VCO with the varactor biased at \(0\text{ V}\) [25, 26]. Three phase noise regions are identified as \(f^{−3}\) (having a slope of \(−9\text{ dB}\)/octave), \(f^{−2}\) (having a slope of \(−6\text{ dB}\)/octave), and \(f^{−1}\) (having a slope of \(−3\text{ dB}\)/octave).

    clipboard_ed42e20d12cc2bfe807e4a2e7a6262503.png

    Figure \(\PageIndex{5}\): Measured phase noise of a \(50\text{ MHz}\) BJT varactor-based VCO at three varactor bias voltages: (a) \(6\text{ V}\); (b) \(0\text{ V}\); and (c) \(18\text{ V}\) [25]. The varactor breakdown voltage is \(30\text{ V}\) Curve (b) was also plotted in Figure \(\PageIndex{4}\).

    that the same physical source of noise can be manifested quite differently at the output of an oscillator when the circuit bias is changed.

    The third phase noise example is for a \(2.4\text{ GHz}\) power oscillator that has the output spectrum shown in Figure \(\PageIndex{6}\) with regions having dependencies of \(f^{−3}\) and \(f^{−0}\), but nothing in between. (The slight increase in noise power spectral density at \(40\text{ kHz}\) offset is due to dynamics of the oscillator’s

    clipboard_ea7c4e13987f9c8be1b2d81f18a7c9686.png

    Figure \(\PageIndex{6}\): Phase noise of a \(2.4\text{ GHz}\) power oscillator with an output power of \(34.5\text{ dBm}\) [27, p. 323]. Two phase noise regions are identified as \(f^{−3}\) (having a slope of \(−9\text{ dB}\)/octave) and white noise (with an \(f^{0}\) dependency).

    clipboard_e5d45eaa90e8038c1a6234660abfa5462.png

    Figure \(\PageIndex{7}\): Long-term stability of a \(10\text{ GHz}\) oscillator measured over a \(24\) hour interval after being on for \(3\) weeks. Used with permission of John Ackermann [23].

    feedback loop.) Finally, the \(5\text{ GHz}\) oscillator considered in Section 5.6 has \(f^{−3}\) and \(f^{−2}\) phase noise regions (see Figure 5.6.15).

    So the whole range of phase noise dependencies on frequency offset are observed, but the universal observation is that the dependency of the noise power spectral density is to a non-positive integer power of frequency (i.e., \(f^{−n},\: n = 0, 1,\ldots \)).

    5.8.2 Observations of Oscillator Noise in the Time Domain

    An important time-domain characterization of noise is referred to as random walk noise. An example of this is the variation of the oscillation frequency over a long period of time. The long-term stability of a \(10\text{ GHz}\) oscillator is shown in Figure \(\PageIndex{7}\). This noise cannot be characterized in the frequency domain and instead is described by its Allan variance, \(\sigma_{y}^{2}(\tau )\), or Allan deviation, \(\sigma_{y}(\tau ) =\sqrt{\sigma_{y}^{2}(\tau)}\), defined as follows.

    If the frequency measured at time \(t\) is \(f(t)\) and the nominal oscillation frequency is \(f_{n}\), then the fractional frequency at time \(t\) is defined as

    \[\label{eq:1}y(t)=\frac{f(t)-f_{n}}{f_{n}} \]

    Then the average fractional frequency over an observation time interval \(\tau\) is defined as

    \[\label{eq:2}\overline{y}(t,\tau )=\frac{1}{\tau}\int_{0}^{\tau}y(t+t_{v})dt_{v} \]

    This leads to the definition of the Allan variance as

    \[\label{eq:3}\sigma_{y}^{2}(\tau)=\frac{1}{2}\left<\overline{y}_{n+1}-\overline{y}_{n}\right> \]

    where \(\tau\) is the observation period and \(\overline{y}_{n}\) is the \(n\)th fractional frequency average over the time interval \(\tau\). Note that there is no dead-time between the \(n\)th and \((n + 1)\)th measurement time intervals.

    The random walk shown in Figure \(\PageIndex{7}\) is an important clue to unraveling flicker noise. Figure \(\PageIndex{7}\) shows long-term memory and here it is shown that there is memory over several hours. Even on smaller time scales, random walk is apparent and there is a self-similar property—the hallmark of chaotic behavior. Could this random walk effect and \(1/f\) noise arise from the same physical process? Most likely, but there is no accepted theory.

    5.8.3 Excess Oscillator Noise: The Leeson Effect and Flicker Noise

    As seen in Figures \(\PageIndex{2}\) to \(\PageIndex{6}\), oscillators have noise that increases as the offset \(\Delta f\), from the mean oscillation frequency decreases. This noise has separable regions where noise varies as \(\Delta f^{n}\), where \(n\) is an integer ranging from \(0\) to \(−5\). There are transition regions between these discrete states, but there is not a region where \(n\) is a fractional number. Not all of the discrete states are observed because, presumably, either the crossover frequencies have changed order, or the frequency offset, \(\Delta f\), was not low enough.

    In 1966 Leeson [28] examined the effect of feedback on noise in oscillators (see Figure 5.2.1). The phase noise mechanism treated by this analysis is now called the Leeson effect. Leeson showed that white phase noise and white flicker noise (white here meaning independent of frequency) of the amplifier in the feedback loop translate to noise on the oscillation signal with power law dependencies of \(f^{−2}\), called white frequency noise, and \(f^{−3}\), called flicker frequency noise, respectively. These were the dominant “nonwhite” forms of noise observed in his time. However, his analysis did not predict the level of the noise accurately and sometimes was off by an order of magnitude.

    The Leeson effect is briefly summarized here. First, it was observed that nearly every physical system has fluctuations that vary as \(1/f\) at low frequencies. This includes electrical devices such as the amplifier in an oscillator feedback loop. This leads to equal amplitude phase and amplitude noise superimposed on the oscillation. Since noise is small, the amplitude fluctuations are suppressed by the saturation of the active device so that the only noise observed in good designs is phase noise. Leeson determined that the oscillator phase noise has a region with \(\Delta f^{−3}\) dependence that is due to low-frequency \(f^{−1}\) noise (i.e. around DC), a \(\Delta f^{−2}\) region due to white noise in the bandwidth of the oscillator’s tank circuit, and also a white noise region outside the bandwidth of the tank circuit. The basis for the development of Leeson’s oscillator phase noise model is shown in Figure \(\PageIndex{8}\). Mathematically [28],

    clipboard_e293bddedac7c087afeaea2cbbdf24e14.png

    Figure \(\PageIndex{8}\): Derivation of oscillator noise spectra: (a) the noise spectra of an electronic material with noise increasing as the frequency reduces; and (b) the noise spectra close to the oscillation frequency of an oscillator.

    \[\label{eq:4}\mathcal{L}(\Delta f)=\mathcal{L}(\Delta\omega)=\frac{2FkT}{P_{0}}\left[1+\left(\frac{f_{0}}{2Q\Delta f}\right)^{2}\right] \]

    where \(Q\) is the loaded \(Q\) factor of the oscillator’s tank circuit and \(F\) is an empirical factor. \(\mathcal{L}\) has the units of \(\text{radians}^{2}\text{/Hz}\), or in decibels,

    \[\label{eq:5}\mathcal{L}|_{\text{dB}}(\Delta f)=10\log\left\{\frac{2FkT}{P_{0}}\left[1+\left(\frac{f_{0}}{2Q\Delta f}\right)^{2}\right]\right\} \]

    which has the units of \(\text{dB/Hz}\) or more usually expressed as ”decibels below the carrier” of power \(P_{0}\), or \(\text{dBc/Hz}\). This is the power at a specified offset such as the phase noise of a \(5.05\text{ GHz}\) oscillator at an offset of \(1\text{ MHz}\) and with an output power of \(0\text{ dBm}\) being \(−130\text{ dBc/Hz}\) [19].

    The derivation of the oscillator noise characteristics from first principles, which led to Equations \(\eqref{eq:4}\) and \(\eqref{eq:5}\), predicts noise levels that are much lower than those observed in practice [29, 30]. As well, the prediction inherent in Equation \(\eqref{eq:4}\) is that by increasing the \(Q\) of the tank circuit the noise level will be reduced. However, this is not always obtained in practice. A further complication is that Equation \(\eqref{eq:4}\) provides no mechanism for the generation of \(1/(\Delta f)\) and \(1/(\Delta f)^{3}\) noise in the oscillator phase noise spectrum. An adhoc modification of Equation \(\eqref{eq:4}\) accounts for this:

    \[\label{eq:6}\mathcal{L}(\Delta f)=\frac{2FkT}{P_{0}}\left[1+\left(\frac{f_{0}}{2Q\Delta f}\right)^{2}\right]\left(1+\frac{f_{c-3}}{|\Delta f|}\right) \]

    Given the inadequacy of this model it is just as well to use

    \[\label{eq:7}\mathcal{L}(\Delta f)=\sum_{i=0}^{-5}b_{i}f_{i}^{n} \]

    where the \(b_{i}\) coefficients are extracted from measurements.

    The Leeson effect can be stated as oscillator phase noise being upconverted white noise around DC.

    5.8.4 Excess Oscillator Noise: Linear Time-Variant Model

    The Leeson effect model described in the previous subsection uses a linear time-invariant model of the oscillator and does not consider down-conversion of noise from frequencies near harmonics. The linear time-variant model, also called the Hajimiri and Lee model, incorporates these higher-order conversion mechanisms [31].

    Noise injected into an oscillator has a different impact depending on whether it is injected at the peak of the oscillating signal or at the zero-crossings. The noise injected at the peaks of the oscillating signal are quenched by the saturating effect of the active device in the oscillator. However, noise at or near the zero-crossings of the waveform introduces jitter and phase noise. This effect on phase noise can be described by an impulse sensitivity function [31]. Consider an impulse injected at phase \(x =\omega_{0}t\), then the time-domain impulse response is

    \[\label{eq:8}h_{\phi}(t,\tau )=\frac{\Gamma(\omega_{0}t)}{q_{\text{max}}}u(t-\tau ) \]

    where \(\Gamma (\: )\) is the impulse sensitivity function, \(q_{\text{max}}\) is the maximum charge displacement on the capacitor forming the tank circuit, \(t\) is the observation time, and \(\tau\) is the time of the excitation. The excess phase of the oscillator (the additional phase induced onto the phase of the carrier) is

    \[\label{eq:9}\phi (t)=\frac{1}{q_{\text{max}}}\int_{-∞}^{t}\Gamma(\omega_{0}t)i(\tau )d\tau \]

    where \(i(\tau )\) is the noise current injected in the oscillator.

    \(\Gamma(\: )\) can be derived approximately for some oscillators such as the CMOS LC oscillator in [32], where it was shown that the impulse sensitivity function can be expressed as a Fourier series with a fundamental component corresponding to the frequency of oscillation. The excess phase at the zero-crossings of the oscillator is

    \[\label{eq:10}\phi (t)=\frac{1}{q_{\text{max}}}\left[c_{0}\int_{-∞}^{t}i(\tau )d\tau +\sum_{m=1}^{∞}c_{m}\int_{-∞}^{t}i(\tau)\cos(n\omega_{0}\tau )d\tau\right] \]

    where \(c_{m}\) are coefficients of the Fourier series. The first term with the \(c_{0}\) coefficient indicates noise that is up-converted from baseband, while the term in the summation is the contribution to the oscillator phase noise due to down-conversion of noise near the harmonic frequencies. With the assumption that the noise at the harmonics is white noise with mean-square current \(\overline{i_{n}^{2}}\), then the noise spectral density is [32, 33]

    \[\label{eq:11}\mathcal{L}(\Delta\omega)=10\log\left(\frac{\overline{i}_{n}^{2}}{\Delta f}\frac{\sum_{m=0}^{∞}c_{m}^{2}}{4q_{\text{max}}^{2}\Delta\omega^{2}}\right) \]

    Here \(\Delta f = 1\text{ Hz}\) for noise in a \(1\text{ Hz}\) bandwidth and \(\Delta\omega\) is the radian offset frequency. Thus white noise at the harmonics is down-converted to \(f^{−2}\) noise at the oscillation frequency, and \(f^{−1}\) noise at baseband is up-converted to \(f^{−3}\) noise at the oscillation frequency.

    The time-variant model provides a richer description of phase noise on an oscillating signal than does the Leeson model, but it does not describe \(f^{−1}\), or \(f^{−n},\: n> 3\), noise that is observed with oscillators.

    Thus the Hajimiri and Lee model relates to down-conversion of white noise at harmonics of the oscillation frequency to the near carrier noise which is in addition to Leeson’s model of up-converted near-DC white noise. The Hajimiri and Lee, and Leeson models of phase noise led to designers developing microwave oscillators with significantly lower phase design yet there remains appreciable near-carrier phase noise.

    5.8.5 Excess Oscillator Noise: Chaotic Maps and Flicker Noise

    While not firmly established, it is possible that flicker noise originates from nonlinear dynamics and chaos [25, 26, 34, 35, 36, 37, 38, 39, 40, 41]. In this model flicker noise derives from a nonlinear process with delayed feedback. The mathematical foundation is well established [42], describing a phenomenon called intermittency [43] that occurs when a physical process transitions between stable periodic states and chaotic states. Inherent to some forms of intermittency is long-term memory with a \(1/f\) spectrum [44]. This has been established for many physical and biological systems.

    Logistics Map

    A classic example of intermittency, and the first widely accepted, is the following model of population dynamics. If \(t_{n}\) denotes discrete time and (the real number) \(x_{n}\) denotes the ratio of the existing population to the maximum possible population at \(t_{n}\) (so \(x_{n}\) is between \(0\) and \(1\)), then what is called the logistics map provides the population ratio, \(x_{(n+1)}\), at time \(t_{n+1}\). The logistics map is [45]

    \[\label{eq:12}F_{\lambda}(x)=\lambda x_{n}(1-x_{n}) \]

    and so

    \[\label{eq:13}x_{n+1}=F_{\lambda}(x) \]

    Here \(\lambda\) is a positive number representing the combined rate of reproduction and starvation. So environmental conditions determine \(\lambda\), which is constrained so that \(0 <\lambda\leq 4\). Depending on \(\lambda\), the logistics map (i.e., Equation \(\eqref{eq:12}\)) will produce a stable population or a random population depending on the value of \(\lambda\), with \(\lambda = 4\) producing white noise.

    Thermal noise produces random fluctuations in the amplitude and phase of a sinusoidal signal that is being processed in a nonlinear electronic system such as an amplifier or an oscillator. Denoting the thermal amplitude fluctuations by \(a_{t,I} (t)\) and the thermal phase fluctuations by \(\phi_{t,I} (t)\), a sinusoidal signal with mean amplitude \(A\) and an initial phase of zero is

    \[\label{eq:14}x(t)=A[1+a_{t,I}(t)]\cos[\omega t+\phi_{t,I}(t)] \]

    Using the logistics map with \(\lambda = 4\) (which produces white noise) to determine \(a_{t,I} (t)\) and \(\phi_{t,I} (t)\), a sinusoidal signal with thermal (white) noise is as shown in Figure \(\PageIndex{9}\). Using \(a_{t,I} (t)\) and \(\phi_{t,I} (t)\) determined from a Gaussian distribution would yield the same qualitative result. Of course most of this noise would be easy to remove by bandpass filtering but thermal noise will still appear within the finite bandwidth of the signal. It is just easier to visualize the effect of noise by plotting it on this scale.

    Equation \(\eqref{eq:12}\) is a simple nonlinear equation with delayed feedback that mixes \(x\) over time. What is called the rate of mixing describes the extent of correlation to past events and can be thought of as an exponential decay rate. However, with the logistics map, \(F_{\lambda} (x)\) in Equation \(\eqref{eq:12}\), the rate of this mixing is not controllable.

    Logarithmic Map

    There are many maps that will lead to \(1/(\Delta f)\) effects and one of the most convenient to use in modeling flicker noise in electronics is called the loga-

    clipboard_ed09b10ff17e6d1614a97713d23e5de85.png

    Figure \(\PageIndex{9}\): Sinusoidal signal with superimposed white noise calculated using the logistics map rather than calculating noise as a Gaussian process.

    rithmic map [46, 47]:

    \[\label{eq:15}F_{\beta}(x)=\left\{\begin{array}{lll}{x(1+Y(\beta )x|\log(x)|^{1+\beta})}&{\text{if}}&{0\leq x\leq 1/2} \\ {2x-1}&{\text{if}}&{1/2<x\leq 1}\end{array}\right. \]

    and so

    \[\label{eq:16}x_{n+1}=F_{\beta}(x) \]

    \(F_{\beta}(x)\) is defined on the interval \(0 < x\leq 1\) and \(Y (\beta ) = 2(\log 2)^{−(1+\beta )}\) is chosen to ensure that \(\lim_{x→1/2}− f_{\beta}(x)=1\). (Note that the map is discontinuous at \(x = 1/2\).) If \(\Delta t\) is a fixed time interval, then \(x(t + \Delta t) = F_{\beta}(x(t))\).

    The logarithmic map uses only a single parameter, \(\beta\), that controls the rate of mixing and thus the long-term memory property. The solution of the logarithmic map, Equation \(\eqref{eq:15}\), with \(\beta = 0.000005\) is shown in Figure \(\PageIndex{10}\).

    The Fourier transform of the sequence plotted in Figure \(\PageIndex{10}\) is shown in Figure \(\PageIndex{11}\) and its autocorrelation is shown in Figure \(\PageIndex{12}\). The spectrum in Figure \(\PageIndex{11}\) (above \(1\text{ Hz}\)) has an \(f^{−0.5}\) dependence, and since the sequence corresponds to voltage, squaring this yields a \(1/f\) power characteristic. The autocorrelation plot in Figure \(\PageIndex{12}\) shows the slow long-term decay in the correlation with respect to the discrete interval, \(i\), between the sequence points. That is, the logarithmic map describes a process with slowly decaying correlations. The extended correlation is a measure of the rate of mixing. This interpretation corresponds very well to the understanding of physical systems, and in particular to electronic systems. It can be shown [47] that the rate of decay of correlation of this map is bounded as

    \[\label{eq:17} R(n)\leq B(\log n)^{-\beta} \]

    where \(n\) is the \(n\)th time interval. Thus the map is said to describe a logarithmic mixing rate that can be made as slow as desired by varying the value of \(\beta\). It is this long-range dependence that produces \(f^{−1}\) (and \(f^{−2}\), \(f^{−3}\), etc.) noise. In an oscillator these result in phase noise with \(1/(\Delta f)\), \(1/(\Delta f)^{2}\), \(1/(\Delta f)^{3}\), etc. characteristics. In semiconductors, for example, the

    clipboard_e200462908ab28379b3643e628f74e5cc.png

    Figure \(\PageIndex{10}\): Solution of the logarithmic map of Equation \(\eqref{eq:15}\) with (a random initial seed) \(x_{0} = 0.477347,\:\beta = 0.000005\), and \(t_{n+1} − t_{n} = 1\text{ ps}\).

    clipboard_ed483302d5221b62b9685a44371f69f64.png

    Figure \(\PageIndex{11}\): Spectrum of the logarithmic map with \(\beta = 0.000005\).

    clipboard_efdf3cd296ad27dcd2735b722b813fc48.png

    Figure \(\PageIndex{12}\): Correlation plot of the logarithmic map with \(\beta = 0.000005\) for a million-point sequence.

    rate of mixing is influenced by the density of traps [48] and lattice scattering [49] with the lower the density of traps (i.e. better quality semiconductor material) and the lower the amount of scattering, the lower the rate of mixing and hence the lower the level of flicker noise. The delayed feedback in the chaotic-map model is consistent with trapping and the observation that reducing traps improves phase noise performance.

    The long-range mixing indicated by the slow decay of the correlation function (see Figure \(\PageIndex{12}\)) is key to the \(f^{−1}\) response. Another function that yields a long-term correlation is the Ornstein–Uhlenbeck process [50, 51, 52] developed to describe Brownian motion. The autocorrelation function of this process decays exponentially and predicts \(f^{−2}\) noise but not \(f^{−1}\) noise [52]. The Ornstein–Uhlenbeck process decays too rapidly to predict the \(f^{−1}\) response. This is discussed further in [52].

    So the solution of Equation \(\eqref{eq:16}\) (the logarithmic map), shown in Figure \(\PageIndex{10}\), has complicated dynamics with long periods of stability with rapid transitions between stable and rapidly varying levels. The sequence of \(x_{n}\)s depends on the starting condition (i.e., \(x_{0}\)), but no matter how it starts, the power spectrum of the solution has an inverse frequency dependence (i.e., it is exactly \(f^{−1}\)). The logarithmic map, as with all chaotic maps, describes a nonlinear process with delayed feedback. This matches the situation in physical, biological, chemical, and financial systems. Since nearly every physical process can be described as a (perhaps weak) nonlinear process with delayed feedback, the widespread observation of \(1/f\) fluctuations is not surprising. So the basis of \(1/f\) noise is the most basic of physical processes.

    Intermittency (described by chaotic maps) results in random fluctuations in the amplitude and phase of a sinusoidal signal processed by a nonlinear electronic system such as an amplifier or oscillator. Denoting the amplitude intermittency by \(a_{I}(t)\) and the phase intermittency by \(\phi_{I}(t)\), a sinusoidal signal with mean amplitude \(A\) and an initial phase of zero is

    \[\label{eq:18}x(t)=A[1+a_{I}(t)]\cos[\omega t+\phi_{I}(t)] \]

    This signal is shown in Figure \(\PageIndex{13}\) with the logarithmic intermittency fluctuations \(a_{I}(t)\) and \(\phi_{I}(t)\) as shown in Figure \(\PageIndex{10}\), calculated using different seeds. The effect of intermittency fluctuations is greatly exaggerated here for visualization purposes. In practice, the fluctuations at the scale shown in Figure \(\PageIndex{13}\) could be eliminated using a bandpass filter. However, the fluctuations are self-similar (another property of chaotic processes) and is repeated at all scales. In a bandpass electronic system the in-band amplitude fluctuations are suppressed by device nonlinearity, but the phase fluctuations appear as phase noise on an oscillator.

    5.8.6 Summary

    Three oscillator phase noise models were presented. The Leeson model is based on up-conversion of white noise from baseband producing noise around the oscillator carrier with an \(f^{−2}\) dependency. The Hajimiri and Lee model is based on a linear time-varying model of the oscillator with \(f^{-1}\) noise at baseband resulting in oscillator phase noise with a \(1/(\Delta f)^{3}\) dependency, and white noise at the baseband and harmonics resulting in noise around the oscillating frequency with an \(1/(\Delta f)^{2}\) dependency. Upconversion of noise has been demonstrated as a mechanism that describes

    clipboard_e5f2849fdd64ea3d566cdf3af306c12c4.png

    Figure \(\PageIndex{13}\): Sinusoidal signal with superimposed intermittency noise. Using Equation \(\eqref{eq:18}\) with \(A = 0.9, a_{I} (t)\) scaled to the interval \([0, 0.09]\), \(a_{I} (0) = 0.477347\) (before scaling), \(\phi_{I} (t)\) scaled to the interval \([0, 5]\) radians, and \(\phi_{I} (0) = 0.00915926\) (before scaling).

    some of the observed oscillator noise. Neither the Leeson nor the linear timevarying models describe the full set of observations of phase noise with \(1/(\Delta f)^{5},\ldots 1/(\Delta f)\) dependencies.

    The chaotic map model is physically appealing and describes the origin of flicker noise as the time-delayed feedback of the output of a nonlinear process. This can produce a chaotic response called intermittency that embodies long-term memory. It has been shown through simulation that this model predicts the \(1/(\Delta f)^{3},\: 1/(\Delta f)^{2},\: 1/(\Delta f)^{1},\) and \(1/(\Delta f)^{0}\) dependencies of phase noise. It is also consistent with the random walk seen in time-domain observations of oscillator noise. However, the chaotic map-based model has not yet led to a compact formula for phase noise similar to Leeson’s formula. The development of a compact phase noise model (e.g., like Leeson’s model) will not be simple as integer calculus and transfer function-based analyses cannot be directly used with a chaotic map. However, it is clear that the description of phase noise is getting close to a satisfying physical explanation.

    Phase noise can also be induced by vibrations [53, 54], and spurious signals from the environment (such as those coupled from the power mains) can also appear as phase noise.


    5.8: Advanced Discussion of Oscillator Noise is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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