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The Noise Conversion Method for Oscillatory Systems Yuriy S. Shmaliy, Senior Member, IEEE Abstract—The paper addresses a method for calculating the amplitude and phase power spectral density (PSD) functions of an oscillatory system (resonator, oscillator, bandpass filter, selective circuit, etc.) via the PSDs of its intrinsic noise sources and relevant transformation coefficients. A systematic description of the method is given for the scalar and vector noises. As an illustration, the noise transformation coefficients are derived for a piezoelectric series branch with fluctuating motional inductance, capacity, and losses, in which static capacity is disregarded. We then clarify the rules regarding the shaping of either PSD function. The importance of this method resides in the fact that it enables us to study particular finite ranges of the PSD function without using differential equations.
I. Introduction he noise problem in resonators and oscillators has been under examination for more than 40 years, owing to a number of reasons. In receivers, the phase noise of a local oscillator influences selectivity and limits immunity against a nearby interference signal. In a transmitter, it can affect purity in nearby channels. In digital communications, clock jitter directly affects the bit error rate. The effect of noisy reference signals in radars, GPS-based positioning systems, control systems, etc. also should be mentioned. Extensive fundamental studies of noise in oscillatory systems were carried out by Stratonovich [1], [2] and Malakhov [3]. However, setting aside such rigorous theories, the most influential in the oscillator’s world still has been Leeson’s simple but rather heuristic oscillator phase-noise model presented in 1966 [4]. The latter then was analyzed mathematically in [5]. Nowadays, we use several rigorous mathematical approaches to study oscillatory systems perturbed by noise v(t) depending on its correlation time τv and variance σv2 , by comparison with the system time constant τs and signal power 2Ps , respectively. Regarding τv , τs , and the signal-to-noise ratio (SNR) γ = Ps /σv2 , the most widely used methods are classified as follows. In the below-given brief overview, we observe only those methods that seem to be able to convert the indicated noise to the oscillator performance.
T
•
Johnson’s (thermal) noise (τv τs ). If γ 1, then the statistical linearization and harmonic balance
Manuscript received November 19, 2002; accepted April 26, 2004. The author is with Guanajuato University, Salamanca, Gto 36730, Mexico, and with Kharkiv National University of Radio Electronics, Ukraine (e-mail:
[email protected]).
techniques are efficient. If γ ∼ 1, then the Markov model is the most accurate. −1 • Excess (flicker) noise of slope f (τv ∼ τs ), where f is the Fourier frequency. If γ 1, then the statistical linearization and harmonic balance techniques are efficient. If γ ∼ 1, then the Wiener method (Volterra series method) is valid. • Excess (flicker) noise and random walks with slope f−n , 1 < n, (τv τs ). If γ 1, then the quasistatic method, the statistical linearization, and the harmonic balance technique are accurate. If γ ∼ 1, then the quasistatic method is the best. The Markov model is usually used and uses the FokkerPlank equation when, with low SNRs, the non-Gaussian distributions of the positive valued signal envelope and the phase become crucial for the signal performance. This is the case of a crystal resonator excited with noise drive levels. For such a case, the transformation of the acting noise is inherently nonlinear. Furthermore, the flicker noise of different colors contradicts to the Markov origin, except for that with the slope f−2 of the power spectral density (PSD). Therefore, we avoid discussing the case of γ ∼ 1 in this paper. The statistical linearization is a common technique if the SNR is large, which is the case of precision oscillators operation with normal drives. An enormous number of works exploit this approach. For the oscillatory systems, one of the most comprehensive methods using the system differential equation was proposed and studied by Malakhov in [3]1 . By this method, the system equation reduces to the fluctuation linear differential equations of the first order for the signal amplitude and phase noises. The solutions then are found in the integral forms and converted via the correlation functions to the amplitude and phase PSDs producing the noise transformation coefficients. With the same aim, the operator functions were used by Kuleshov and Janushevsky in 1979–1994 [6] for the crystal resonators2. The sensitivity theory was applied by Tomlin et al. in 2000 [7] for amplifiers. Yet, the algebraic forms were used by Shmaliy in 1999 [8] for crystal resonators and in 2004 [9] for crystal oscillators. In [9], the approach was called the noise conversion method (its essence is discussed in detail below). 1 This fundamental book of 1968 refers to 349 published works on fluctuations in oscillatory systems. 2 The noise transformation coefficients were found for some linear cases within a resonator half bandwidth. The relevant works of these authors in Russian are referenced in [8] and [9].
c 2004 IEEE 0885–3010/$20.00
shmaliy: method for calculating amplitude and psd functions
The harmonic balance technique is used widely for the oscillatory system operating in the sustaining stage. Its classical use is given in Appendix I of [10]. Especially for the transformation coefficients, the approach was first applied by Sokolinsky in [11] and thereafter developed by Shmaliy in [12]. In such a modulation characteristics method, the double harmonic balance is exploited for the oscillator and modulation frequencies in the system nonlinear differential equation yielding the complex transformation coefficients with the error of the second approximation of the asymptotical methods. The quasistatic method produces consistent results, avoiding the system dynamic equation when the noise acts only within the system bandwidth [1], [2]; therefore, the system is considered to be inertialess. The method was applied by Smythe in [13] to the vibration-induced problems in crystal devices. The Wiener method (Volterra series method) [14] still seems unique in the less understood nonlinear case of τv ∼ τs and γ ∼ 1. For linear systems, this method degenerates to the above listed methods. An analysis of several recently published results [15]– [19] has shown that, with some modifications, the authors mostly follow either the above listed rigorous theories or Leeson’s approximation. It is to be remarked that the noise conversion method still remains a relatively young tool for oscillator problems. Notwithstanding this fact, of all the techniques listed above, this is the one that allows the straightforward engineering calculation of the system amplitude and phase PSD functions via the PSDs of the system intrinsic noises. Furthermore, it clarifies the rules in the PSD shaping, avoiding redundancy of other methods and allowing the derivation of the coefficients of the Leeson approximating polynomial [4]. Even though the method design was motivated by the problems in crystal devices with high quality factor Q, it seems to be promising for other narrowband linear and nonlinear systems. In this paper we give a systematic description of the noise conversion method, concentrating on its use in noisy oscillatory systems with high Q and large SNR, such as oscillators, resonators, and filters. We assume each system to be linearized and described explicitly (or with some reasonable accuracy) by the complex impedance. In Section II, we derive the generic relations to calculate the system amplitude and phase PSD functions for scalar and vector noise with known PSD. To illustrate the method, in Section III we obtain the noise transformation coefficients for the piezoelectric series tuned circuit (resonator, ignoring the static capacity) with fluctuating motional inductance, capacity, and losses. Section IV contains some numerical analysis (consistent with Curtis’s measurements) that clarifies the rules regarding the shaping of either PSD function. Concluding remarks are given in Section V. II. The Noise Conversion Method The essence of the noise conversion method may easily be explained by supposing, for the sake of simplicity, that
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Fig. 1. Noisy oscillatory system: (a) noisy system, (b) noiseless system with input noise, (c) real and imaginary parts of the system noisy complex impedance, (d) system noisy complex impedance with noiseless current.
the oscillatory system [Fig. 1(a)] is the only noise source α that is, for example, in the capacity. We extract α to be the system input [Fig. 1(b)] (the system is then noiseless) and consider, for example, an imaginary component X [Fig. 1(c)] of the system impedance Z [Fig. 1(d)] to be the output. We suppose that the input has an increment δα (f) that is the root mean square deviation (RMSD) of fractional fluctuations estimated in [20] at an arbitrary Fourier frequency f. This increment in turn induces the output increment δA (f) that basically is positive or negative. In the oscillatory system with high Q, such increments are small, δα2 1 and |δA | 1, that allows us to linearize the system and express the output via the input as a linear time invariant operator such as3 δA (f) = HαA (f)δα (f). We then square the left-hand and right-hand sides of this expression, divide them with the bandwidth (BW) of the narrowband fluctuations acting at f (practically [20], the BW is determined by the measurement system), and go 2 (f)Sα (f), where SA (f) and Sα (f) are the to SA (f) = HαA double-sided real PSD functions of the output and input, 2 respectively. The function HαA (f) then appears to be the squared real transfer function or the transformation coeffi2 cient KαA (f) of the input-to-output converter of the PSDs, 2 so that, SA (f) = KαA (f)Sα (f). For the known (measured [21]) Sα (f), one now needs to determine somehow the co2 efficient KαA (f). That may be done in different ways, for example, using the system differential equation [3], the dynamic modulation characteristics [12], or the noise conversion method [8] and [9]. Analogously, we determine contributions of all noise sources to real and imaginary components of the system impedance and then to the system total impedance (amplitude) and phase. The approach is readily extended to the common vector case. Thus, we determine linkage between the system amplitude and phase PSDs and those of the intrinsic noises. Aiming to derive generic relations for the system amplitude and phase PSD functions, we replace the system [Fig. 1(a)] by its complex impedance [Fig. 1(d)]: Z = R + jX = |Z|ejϕ , |Z| = R2 + X 2 ,
(1) (2)
3 Such a form appears if the system is to be reduced to complex impedance ([8] and [9]). Compared to the Wiener method, this oper∞ ator in the time domain is a convolution δ˜A (t) = h (θ)δ˜α (t − −∞ αA θ)dθ, where hαA (t) is the Volterra kernel (impulse response) of the linearized system, and the fluctuations δ˜A (t) and δ˜α (t) are stationary.
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X , (3) R in which R and X are real and imaginary components of the system impedance, respectively. For the following clarity, it is now important to emphasize that the total impedance |Z| (2) and phase ϕ (3) are associated, respectively, with the system amplitude and phase noises4 . It is a typical situation in oscillators and resonators when the acting noise, by nature, is assumed to be Gaussian white and flicker [3]. Respectively, we assume the noise in (1) to be zero mean stationary Gaussian, additive or colored, low intensity, and with a known PSD Sα (f), or at least with a known part of PSD. We then perform the noisy component α as:
A. Amplitude Power Spectral Density
ϕ = arctan
α = α(1 + δα ),
(4) 2 where α(f) is the noiseless value, δα (f) = ∆α2 (f)/α (f) ≥ 0 is the noise-induced fractional increment, and ∆α2 (f) is the RMSD of fluctuations acting in α at f [20]. We now suppose that δα (f) creates small increments, δr (f) and δx (f), in the real and imaginary components, respectively, of the system impedance. To convert δα to δr and δx , we write R and X as in [8], respectively: R0 R=R 1+ δr , (5) R R0 X =X 1+ δx , (6) X where R0 is the system resistance at the resonance frequency (the piezoelectric resonator motional losses, for example), R and X are the relevant noiseless values, and: R[δα (f)] − R , R0 X[δα (f)] − X , δx (f) = R0 δr (f) =
|Z[δr (f), δx (f)]| − |Z(f)| (10a) R0 ⎡ ⎤ 2 2 R0 R0 1 ⎣ 2 2 2 2 = R 1+ δr +X 1+ δx − R +X ⎦ R0 R X δZ (f) =
(10b) Rδr + Xδx ∼ = KαZ (f)δα (f), = 2 2 R +X
(10c)
where, by (8a) and (8b):
(7b)
(8a) (8b)
where Kαr and Kαx are the transformation coefficients of δα -to-δr and of δα -to-δx , respectively. We notice that the linear transformations (8a) and (8b) make δr (f) and δx (f) to be Gaussian. The next step is to substitute (5) and (6) to (2) and (3) and express either increment in |Z| and ϕ via the increments of R and X. To be able to do so, we recall that all increments are small, that is |δα,r,x,Z | 1 and |∆ϕ| π/2, and thus the system may be linearized. Below we derive the relevant relationships in forms of (8a) and (8b). the external current i(t) = Iej(ωr0 t+θI ) [Fig. 1(d)] to be noiseless and inducing the voltage v(t) = I|Z|ej(ωr0 t+θI +ϕ) . It then follows that the amplitude and phase noises of the system voltage are exhaustedly determined by the fluctuations of the total impedance and the phase of its complex impedance, respectively. 4 Suppose
where |Z| is the relevant noiseless value and |δZ | 1 is the noise-induced fractional increment. Using (2), (5), and (6), and neglecting products of small values, we express δZ in the following forms:
(7a)
are fractional increments in (5) and (6), respectively. Alternative forms of (7a) and (7b) are as follows: δr (f) = Kαr (f)δα (f), δx (f) = Kαx (f)δα (f).
The nonlinear conversion of (7a) and (7b) to the oscillator amplitude must be done with some care. We formally multiply the total impedance (2) with the constant amplitude of an electric current I ≥ 0 and go to the envelope V = |Z|I of the oscillator signal that is positive valued, V ≥ 0 and thus non-Gaussian. We then refer to the known fact [1], [2] that with large SNR, that is δα2 1, the envelope normalizes and its increments become Gaussian without great loss in accuracy. Using this and acting similarly to (5) and (6), we specify the total impedance of the system by: R0 (9) |Z| = |Z| 1 + δZ , |Z|
Kαr R + Kαx X KαZ ∼ , = 2 2 R +X
(11)
is the transformation coefficient of δα -to-δZ . Again, like the case of (8a) and (8b), the linear transformation (10c) makes δZ to be Gaussian. We may now square the lefthand and the right-hand sides of (10c), divide them with the system BW [20], and go to the amplitude double-sided PSD function: 2 SZ (f) = KαZ (f)Sα (f).
(12)
2 where KαZ (f) is determined by the system structure via (11), and Sα (f) is assumed to be the known (or partly known) PSD of the origin noise.
B. Phase Power Spectral Density In line with the total impedance, the transformation of (7a) and (7b) to the phase (3) is also nonlinear. The random phase (3) has a non-Gaussian distribution. However, if SNR becomes large, the phase normalizes [1], [2] and its increments also become Gaussian without great loss
shmaliy: method for calculating amplitude and psd functions
in accuracy. Exploiting this fact, we specify the system phase as: ϕ = ϕ + ∆ϕ,
(13)
where ϕ(f) is its noiseless value at f and the increment |∆ϕ| π/2 is induced by δα (f). To convert δr and δx to ∆ϕ at an arbitrary f, we first introduce two auxiliary variables x and y, then calculate ∆ϕ by: ∆ϕ = arctan x − arctan y,
(14)
X(1 + R0 δx /X) X = , R R(1 + R0 δr /R)
(15)
X . R
(16)
x=
y=
It may now be shown that, for small noise, the following relation holds true, namely xy > −1. This brings (14) to the forms [22]: ∆ϕ(f) = arctan
x−y 1 + xy
Rδx − Xδr ∼ = R0 2 2 = Kαϕ (f)δα (f), R +X
Sϕ (f) =
(22a)
is the system noise PSD matrix of dimensions N × N and Sij (f) is the matrix generic element. It has been an ordinary situation when the cross PSD function Sij (f), i = j of the intrinsic system noises is unknown. Thus, one may use the Cauchy-Schwarz inequality Sij (f) ≤ Si (f)Sj (f) and go to the approximation, knowing that, if noises are uncorrelated, then (22a) takes the diagonal form: ⎤ ⎡ S11 (f) 0 . . . 0 ⎢ 0 S22 (f) . . . 0 ⎥ ⎥. Sα (f) = ⎢ ⎣ ... ... ... ... ⎦ (22b) 0 0 . . . SN N (f)
Sϕ (f) = KTαϕ Sα (f)Kαϕ ,
(18)
(19)
C. Vector Noise The general rule of thumb claims that the oscillatory system is not the only noise source. In view of that, we assume the noise-induced fractional increment to be a vector δ α = [δα1 δα2 . . . δαN ]T of dimensions N × 1, in which δαi , i ∈ [1, N ], is a generic fractional increment and N is the number of noise sources. We assume that at the early stage the transformation coefficients for the amplitude K1Z , K2Z , . . . , KN Z and phase K1ϕ , K2ϕ , . . . , KN ϕ are determined for each of the noise-induced increments separately. We now define the amplitude fluctuations in the linearized system as the sum of all the effects: (20)
T
where KαZ = [K1Z K2Z . . . KN Z ] is the amplitude noise transformation coefficient matrix of dimensions N × 1. Relation (20), as in the scalar case (12), allows transferring to the amplitude PSD function: SZ (f) = KTαZ Sα (f)KαZ ,
⎤ S11 (f) S12 (f) . . . S1N (f) ⎢ S21 (f) S22 (f) . . . S2N (f) ⎥ ⎥, Sα (f) = ⎢ ⎣ ... ... ... ... ⎦ SN 1 (f) SN 2 (f) . . . SN N (f)
(17b)
2 where the transformation coefficient Kαϕ (f) (18) is determined by the system structure through (17b).
δZ = KTαZ δ α ,
⎡
Reasoning along similar lines, we define the phase PSD function by:
is the transformation coefficient of δα -to-∆ϕ. Accordingly, like the case (12), the double-sided phase PSD function becomes: 2 Kαϕ (f)Sα (f),
where:
(17a)
where, by (8a) and (8b): RKαx − XKαr , Kαϕ ∼ = R0 2 2 R +X
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(21)
(23)
where Kαϕ = [K1ϕ K2ϕ . . . KN ϕ ]T is the phase-noise transformation coefficients matrix of dimensions N × 1, and the system noise PSD matrix is given by (22a) or (22b). It should be obvious now that, in the one-dimensional case, (23) degenerates to (19). D. Complex Systems Frequently, the system consists of several noisy subsystems, each of which may separately be modeled as a noisy complex impedance. An example is the oscillator, in which the resonator and amplifier are ordinarily subject to different physical processes. If one has earlier studied each of the noisy subsystems and substituted them by the relevant noisy impedances, then performing the system in terms of its noisy impedance leads to the system amplitude and phase PSDs in a straightforward way. Such an analysis was recently given for the crystal oscillator [9]. Let us add that, though having no peculiarities in the derivation routine, an analysis burden for the complex system may be large. Furthermore, special knowledge about the physical processes may be required, which certainly is out the scope of this paper.
III. A Noble Example: Piezoelectric Series Tuned Circuit Here the method demonstrates its very strong inherent worth: if one has learned or measured only a part of the origin noise PSD, then the system amplitude and phase PSD functions may be estimated readily for this part. This is obviously the case of the piezoelectric resonator and oscillator, in which even the standard [20] postulates the various PSD regions to be measured and performed. To
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illustrate the application of the method in this section, we consider the piezoelectric resonator, neglecting its static capacity C0 . Such a model (series tuned circuit) simplifies the routine considerably, but with no significant loss of generality. The difference between the rigorous results [8] and those provided below is a matter of special notation wherever necessary. A. The Noise Transformation Coefficients Let us simulate the piezoelectric resonator by the series turned circuit with the noisy inductance L1 = L10 (1 + δ˜L ), capacity C1 = C10 (1 + δ˜C ), and losses R1 = R10 (1 + δ˜R ), where δ˜L (t) = ∆L1 (t)/L10 , δ˜C (t) = ∆C1 (t)/C10 , and δ˜R (t) = ∆R1 (t)/R10 are the relevant zero
mean
stationary ˜ fractional fluctuations, this is δL (t) = 0, δ˜C (t) = 0,
and δ˜R (t) = 0. Yet, ∆L1 (t), ∆C1 (t), and ∆R1 (t) are their random components. We suppose that all fluctuations are measured and their fractional mean square de2 2 2 viations, δL (f) = ∆L21 (f)/L210 , δC (f) = ∆C12 (f)/C10 , and 2 2 2 5 δR (f) = ∆R1 (f)/R10 , are calculated at an arbitrary f as in [21]. Below, it will be convenient to use the following notations. The system angular resonance and current frequencies are, respectively: 1 , L10 C10 f ω = ω0 + ∆ω = Π0 Q + , 2fr ω02 =
5 For example, to determine δ (f), substitute δ ˜C (t) = ∆C(t)/C10 C with the cosine component √ of the narrowband random processes act2∆C(t,f) ing at f, this is δ˜C (t, f) = cos 2πft. Then write: δ˜2 (t, f) =
∆C 2 (t,f)
=
2 C10 2 (f) becomes δC
∆C 2 (t,f) , 2 C10
∆C 2 (f) 2 C10
=
C10
average it, and go to
2 (f). δC
δLr = δCr = 0, δRr = δR , f δL , δLx ∼ = Q+ 2fr f δCx ∼ δC , = Q− 2fr δRx = 0,
C
2 (t, f) δ˜C
=
Divided by the BW, the function
the PSD of the fluctuating capacity C1 (t). In [8] and [9], we came to the same result in an alternative way: considering δ˜C (t, f), then transferring to its correlation function, and applying the Wiener-Khinchin theorem.
(29)
(30) (31) (32) (33) (34)
where the double subscript (in δLr , for example) means that δr is caused by δL . Using (8a), (8b), and (30)–(34), we determine: KLr = KCr = 0, KRr = 1, ∼Q+ f , KLx = 2fr f , KCx ∼ =Q− 2fr KRx = 0.
(35) (36) (37) (38) (39)
Using (11), (18), (28), (29), and (35)–(39), we derive the transformation coefficients: Qf KLZ ∼ , = KCZ ∼ = 2 fr + f2 ∼ fr KRZ = , f2r + f2
(26)
1 X = XL − XC = ωL10 (1 + δL ) − ωC10 (1 + δC ) (27a) f f f ∼ δL + R10 Q − δC . = R10 + R10 Q + fr 2fr 2fr (27b)
(28)
We now convert the increments δR , δL , and δC to the increments in the resonator total impedance (amplitude) and phase. Based upon (7a), (7b), and (26)–(29), we calculate, using (24) and (25), the relevant increments of real and imaginary components of the system impedance to be6 :
(25)
R = R10 (1 + δR ),
cos 4πft +
R = R10 , f 1 ∼ X = ωL10 − = R10 . ωC10 fr
(24)
where ∆ω is the frequency shift associated with the Fourier frequency f, Π0 = R10 /L10 = 1/Q2R10 C10 is the resonator’s 3 dB bandwidth in radians, Q = ω0 /Π0 is the quality factor, and fr = Π0 /4π is the resonator half 3 dB bandwidth in hertz. In accordance with the method, we first write the real and imaginary parts of the resonator impedance, respectively:
∆C 2 (t,f) 2 C10
Using (26) and (27), we then determine their noiseless parts, respectively:
Qf2 KLϕ ∼ = KCϕ ∼ = 2 r 2, fr + f f f r KRϕ ∼ , =− 2 fr + f2
(40) (41) (42) (43)
which are intended to be used calculating the amplitude and phase PSDs, (21) and (23), respectively. Derivation of the transformation coefficients is a key procedure; therefore, it is a proper place now to compare the results easily derived above with those obtained by some regular methods. Before doing so, we notice that (40)–(43) become equal to those given in Table II in [8] if the resonator κ-factor is set equal to zero (that is if C0 = 0). We find the same result (40) derived with larger 6 All
approximate formulas are derived here for f/fr Q.
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routine by the modulation characteristics method [25] and an equal result reported by Malakhov [3] for the parallel turned circuit. The formula that is equal to (41) is yielded by Malakhov’s method [3] and by analyzing the system impulse response [18, (35)]. We were not able to find the result similar to (42) and just mention that the relative formula for the parallel turned circuit was shown by Malakhov [3]. The result of the second order effect (43) does not appear in the observed works because of either limitations of the used methods or large derivation routine. B. Amplitude Double-Sided Power Spectral Density In the following, we consider a typical case when the double-sided PSDs of the intrinsic fluctuations of the piezoelectric resonator are by SL (f) = SC (f) = given αX f−1 and SR (f) = αR f−1 +2kT /P , where αX and αR are the flicker coefficients, k is the Boltzmann constant, T is temperature in Kelvins, and P is the power dissipated in R1 . Based on our early results [8], we suppose the noise sources to be uncorrelated. Using (40)–(43), we go, by (21) to (44) (see next page), It now follows that, within the half bandwidth, with f2 f2r , the amplitude PSD shapes by: SZ (f) ∼ =
2kT 2αX Q2 |f| + αR f−1 + . 2 fr P
(45)
If the first and last terms in (45) are neglected, according to the values of αX and αR given by Janushevsky [23], then the formula becomes as in [8, (29)]. In the second limiting case, beyond the half bandwidth, one may consider f2r f2 , and then (44b) degenerates to: 2kT 2 −2 SZ (f) ∼ f f , = 2αX Q2 f−1 + αR f2r f−3 + P r
(46)
which shows that, depending on the term weights, the function slope may vary from f−1 to f−3 . It should be noted now that the flicker noise of the neglected static capacity C0 may result in a substantial increase in the amplitude noise [8], in which case (46) loses its generality for crystal resonators.
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In (48), two last terms may be neglected [23], then (48) reduces to [8, (30)] if the coefficient 4 is substituted by 2 (uncorrelated noises) and κ = 0, having neglected the static capacity. In (49), beyond the half bandwidth, the PSD’s slope may vary from f−2 (additive thermal noise dominates) through f−3 (loss’s flicker noise dominates) to f−5 (reactance’s flicker noise dominates). However, it is important to remark that the static capacity of a crystal resonator affects the picture so that the thermal-additive noise effect attenuates even more, to the slope f−4 . Here, an increase in the weights of the converted flicker noises forces the slope7 to range beyond the half bandwidth from f−2 to f−3 [8].
IV. Numerical Analysis The aim of this section is to model the amplitude and phase PSD functions of the crystal resonator and compare the result to those measured by Curtis [24] and simulated in [8] at the excitation resonance frequency f0 = ω0 /2π = 125 MHz with αX = 5.1 × 10−22 and αR = 5 × 10−12. The matched load RL = R10 in the measurement set doubles the resonator bandwidth, producing QL = Q/2 = 3.13 × 104 and fL = 2fr = 1997 Hz. The phase noise floor is measured to be −170 dBc/Hz that corresponds to8 P = 820 µW with T = 300 K. To fit the measurement, we operate below with the one-sided PSD and its logarithmic measure [20], respectively: 1 [S(f) + S(−f)], 2 L(f) = 10 log[S(f)].
S(f) =
(50) (51)
A. Amplitude Spectral Density
Similar to (44a) and (44b), the resonator double-sided phase PSD is calculated by (23) to be: 2αX Q2 f4r f−1 αR f2r |f| f2r f2 2kT Sϕ (f) = + + 2 2 2 P . 2 2 2 2 2 2 (fr + f ) (fr + f ) (fr + f ) (47)
Fig. 2 illustrates the shaping of the amplitude PSD. The resonator’s thermal noise (a) inherently exhibits its constant value of −170 dBc/Hz within the loaded half bandwidth, then is attenuated beyond it by the resonator transfer function with slope f−2 . It follows that, in the range f < fL , the main shaper appears to be the loss’s flicker noise9 (c). Beyond the bandwidth, the amplitude PSD owes mostly to the flicker noise of the motional inductance and capacity for the slope f−1 (d). The load thermal noise turns the PSD up to the noise floor of −170 dBc/Hz (e). Fig. 3 shows what happens if the reactance’s flicker noise is assumed to be negligible, αX = 0. With zero flicker coefficient of losses (a), the amplitude PSD is only
Again, we may distinguish two limiting cases. Examining (47) within and beyond the half bandwidth, we go to the approximations: αR 2kT Sϕ (f) ∼ (48) = 2αX Q2 f−1 + 2 |f| + 2 f2 , fr fr P 2kT 2 −2 Sϕ (f) ∼ f f . = 2αX Q2 f4r f−5 + αR f2r f−3 + P r (49)
7 There had been a discussion in the literature about the actual phase PSD’s slope of the crystal resonator beyond the half bandwidth (see a brief review in [8]). Some authors had brought forward evidence for near f−2 , and others had shown the measurement with almost f−3 . 8 The first reviewer gave another evaluation of the resonator noise performance, namely −173 dBc/Hz with P = 1260 µW that seems to be more realistic for Curtis’s measurement. 9 The loss’s flicker coefficient is readily measured via S (f) at some Z frequency f1 within the half bandwidth, in which the amplitude PSD −1 has the slope f , to be αR = f1 SZ (f1 ).
C. Phase Double-Sided Spectral Density
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⎤T Qf ⎢ f2r + f2 ⎥ ⎥ ⎢ ⎢ Qf ⎥ ∼ ⎥ ⎢ SZ (f) = ⎢ 2 2⎥ ⎢ fr + f ⎥ ⎦ ⎣ fr 2 2 fr + f ⎡
⎤ Qf ⎡ ⎤ ⎢ f2 + f2 ⎥ αX f−1 0 ⎥ ⎢ r 0 −1 Qf ⎥ ⎢ ⎥⎢ 0 αX f 0 ⎥ ⎢ ⎣ 2 + f2 ⎥ −1 2kT ⎦ ⎢ f r ⎥ ⎢ 0 0 αR f + ⎦ ⎣ fr P 2 2 fr + f 2 −1 2 2 2αX Q |f| αR fr f f 2kT = 2 + 2 + 2 r 2 . fr + f2 fr + f2 fr + f P
Fig. 2. Shaping of the amplitude PSD function: (a) effect of a resonator thermal additive noise, (b) effect of a motional inductance and capacity flicker noise, (c) effect of a losses flicker noise, (d) total PSD with a noiseless load, (e) total PSD.
⎡
(44a)
(44b)
Fig. 4. Amplitude PSD function for the estimated value of the resistance’s flicker coefficient αR = 5 · 10−12 [8] and different values of the reactance’s flicker coefficient: (a) αX = 0, (b) αX = 1.5 · 10−22 , (c) αX = 5.1 · 10−22 [8], (d) αX = 2 · 10−21 , (e) αX = 1 · 10−20 , (f) αX = 5 · 10−20 .
due to the resonator and load thermal noise. It is almost a white noise, having, however, a nonuniformity at f = fL . Increasing αL gradually leads to the slope f−1 (b)– (f) within a half bandwidth, and to the slope f−3 beyond it. Once again, in the far carrier range, the noise floor is −170 dBc/Hz, being due to the load thermal noise. Fig. 4 illustrates what one should expect if αR is set equal to the value estimated in [8] based upon Curtis’s measurements [24], αR = 5 × 10−12 [this is curve (c) in Fig. 4], then αX is gradually increased. The amplitude PSD starts with two slopes f−1 and f−3 (a). It then tends to the actual shape (c) obtained for the parameters estimated in [8]. The case (d) proves that the slope f−3 cannot occur at all. Beyond the half bandwidth the slope f−3 is substituted by f−1 (d)–(f). Conversely, within the half bandwidth the PSD passes through two extremes (f). B. Phase Spectral Density Fig. 3. Amplitude PSD functions for reactance’s zero flicker noise, αX = 0, and different values of the resistance’s flicker noise coefficient: (a) αR = 0, (b) αR = 1 · 10−12 , (c) αR = 5 · 10−12 [8], (d) αR = 2 · 10−11 , (e) αR = 9 · 10−11 , (f) αR = 5 · 10−10 .
Fig. 5 exhibits the shaping of the phase PSD for the same particular case of the crystal resonator. Again, the losses thermal noise makes only an insignificant contribution (a) and may be neglected. In contrast to the amplitude PSD, here the main shaper within the half bandwidth is
shmaliy: method for calculating amplitude and psd functions
Fig. 5. Shaping of the phase PSD function: (a) effect of a resonator additive thermal noise, (b) effect of a motional inductance and capacity flicker noise, (c) effect of a losses flicker noise, (d) total PSD with a noiseless load, (e) total PSD.
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Fig. 7. Phase PSD function for estimated value of the reactance’s flicker coefficient αX = 5.1 · 10−22 [8] and different values of the resistance’s flicker coefficient: (a) αR = 0, (b) αR = 1 · 10−12 , (c) αR = 5 · 10−12 [8], (d) αR = 2 · 10−11 , (e) αR = 9 · 10−11 , (f) αR = 5 · 10−10 .
αX shapes the phase PSD with slope f−1 within the half bandwidth, and with f−5 beyond it. Fig. 7 illustrates the influence of the losses flicker noise if αX is set to be equal to the value estimated in [8], αX = 5.1 × 10−22 . Assuming that αR is initially zero (a), we look for the slope f−1 within the half bandwidth, and for f−3 beyond it. Increasing αR leads to the shape (c), which was measured by Curtis in Fig. 12 in [24]. Then, once αR keeps rising, the PSD reaches the flat part (d) and a brightly pronounced nonuniformity with two extremes appears near the bound frequency fL , (e) and (f).
V. Conclusions
Fig. 6. Phase PSD function for resistance’s zero flicker noise coefficient, αR = 0, and different values of the reactance’s flicker coefficient: (a) αX = 0, (b) αX = 5 · 10−25 , (c) αX = 5 · 10−24 , (d) αX = 5 · 10−23 , (e) αX = 5.1 · 10−22 [8], (f) αX = 5 · 10−21 .
the flicker noise of the reactances10 (b). The losses flicker noise produces the same effect as that of the reactance’s flicker noise in the amplitude case (Fig. 4). One substantial difference between the amplitude and phase PSDs may now be observed: in the phase case (Fig. 5), the PSD’s slope beyond the half bandwidth may reach even f−5 if the reactance’s flicker noise dominates all the others. It seems, however, that this is impractical because the proper measurement is not known. Fig. 6 shows what one should expect if the reactance’s flicker noise dominates. The phase noise induced by the losses and load is almost negligible here (a). Increasing 10 The reactance’s flicker coefficient is readily measured via S (f) ϕ at some frequency f2 within the half bandwidth, in which the phase −1 PSD has the slope f , to be αX = f2 Sϕ (f2 ).
In this paper we presented an efficient tool, which we called in [9] the noise conversion method, for calculating the oscillatory system amplitude and phase PSD functions via the intrinsic system noises with known PSDs. The first significant merit of the approach is a straightforward PSD calculation through the transformation coefficients derived from the system structure. After the latter are obtained in a way similar to that used for the crystal resonator (40)– (43) (or [8]) and oscillator [9], one may merely substitute the system noise PSD into (21) and (23), and calculate the output straightforward. The second advantage is that this technique allows the calculation of particular finite ranges of the amplitude and phase PSDs such as those predicted by Leeson [4] and postulated by the IEEE Standard [20]. This engineering tool seems to be simplest among all now known (see concluding remarks for Section III-A). Summarizing, we would like to notice again that the method has an important constraint that is large values of SNR 1 necessary for statistical linearization. Thus, both the signal envelope and phase are assumed to be Gaussian. With this constraint, the method is invariant to the noise correlation time allowing for analyzing the acting
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noises with arbitrary shapes of PSD functions. However, even though the technique has desirable features for engineering needs, a mathematical analysis of errors associated with the introduced constraint must be provided; this is currently under investigation.
Acknowledgment The author would like to thank Dr. Raymond Filler of U.S. Army Communications-Electronics Command and the anonymous reviewers for valuable comments. The author is also grateful to Prof. Christian Constanda of the University of Tulsa for assistance in reading this paper.
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[16] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE J. Solid-State Circuits, vol. 31, pp. 331–343, Mar. 1996. [17] A. Dec, L. Toth, and K. Suyama, “Noise analysis of a class of oscillators,” IEEE Trans. Circuits Syst. II: Analog and Digital Sign. Proc., vol. 45, no. 6, pp. 757–760, 1998. [18] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, 1998. [19] R. Brendel, N. Ratier, L. Couteleau, G. Marianneau, and P. Guillemot, “Analysis of noise in quartz crystal oscillators by using slowly varying functions method,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 46, no. 2, pp. 356–365, 1999. [20] Standard definitions of physical quantities for fundamental frequency and phase time metrology–random instabilities, IEEE SRD 1139-1999. [21] H. D. Ascarrunz, A. Zhang, E. S. Ferre-Pikal, and F. L. Walls, “PM noise generated by noisy components,” in Proc. IEEE Int. Freq. Contr. Symp., 1998, pp. 210–217. [22] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 1980. [23] G. D. Janushevsky, “On fluctuations conversion in passive fourport circuit with crystal resonator,” in Proc. Freq. Stability Conf., vol. VIMI, 1986, pp. 246–249. [24] G. S. Curtis, “The relationship between resonator and oscillator noise, and resonator noise measurement techniques,” in Proc. IEEE Annu. Symp. Freq. Contr., 1987, pp. 420–428. [25] V. Y. Barzhin, F. F. Kolpakov, and Y. S. Shmaliy, “Dynamic characteristics of controlled crystal oscillators,” Radioelectron. Commun. Syst., vol. 24, no. 12, pp. 67–72, 1981.
Yuriy S. Shmaliy (M’96–SM’00) was born January 2, 1953. He received the B.S., M.S., and Ph.D. degrees in 1974, 1976, and 1982, respectively, from Aviation Institute of Kharkiv, Ukraine, all in electrical engineering. In 1992 he received the Doctor of Technical Sc. degree from the State Railroad Academy of Kharkiv. From 1978 to 1981, he was involved in postgraduate studies in the Aviation Institute of Kharkiv. In March 1985, he joined the Kharkiv Military University. He served as professor beginning in 1986 and had certificate of Professor awarded in January 1993. In June 1992, he established the “Sichron” Center, working in a field of time and frequency, and joined as director-collaborator. In 1999, he joined Kharkiv National University of Radioelectronics, and since November 1999 he has been with Guanajuato University of Mexico, Salamanca, Mexico, as a professor. Dr. Shmaliy has written 161 papers and holds 80 patents. From 1985 to 1991, he headed the interbranch USSR’s Annual Seminar of “Quartz Frequency Stabilization.” He was awarded a title, Honorary Radio Engineer of the USSR, in 1991; he was listed in Marquis Who’s Who in the World in 1998; and he was listed in Outstanding People of the 20th Century, Cambridge, England, in 1999. He is a member of several professional societies and technical program committees of international symposia. His current interests include the statistical theory of precision resonators and oscillators, optimal estimation, and stochastic signal processing for time and frequency.
