Phase Noise and Frequency Noise

Foundations of VHF, UHF and Microwave NEMS Resonators

2.1 Theoretical Foundation

2.1.2 Noise Processes

2.1.2.2 Phase Noise and Frequency Noise

Besides the displacement and force noise in resonators, frequency and phase noise are of the most importance, and are particularly crucial for resonant sensors and resonance-based signal generation and processing. A comprehensive introduction to the key concepts and methods of frequency and phase noise in signal sources is given by

Robins in [6]. Here our treatment continues to aim at the intrinsic fundamental limits of frequency and phase noise in NEMS resonators, still following the SHO model context.

(i) Phase Noise

Fig. 2.3 Definition of phase noise per unit bandwidth.

In the frequency domain spectrum signal from a resonator as illustrated in Fig. 2.3, phase noise is a measure of spectral purity and it is defined as the power of noise sideband per unit bandwidth in the units of decibels below the carrier (dBc/Hz),

( ) ( )

⎥⎦

⎢ ⎤

⎣

⎡ +Δ

= Δ

carrier c sideband

P

Hz

S P , 1

log

10 ω ω

φ ω in [dBc/Hz], (2-24)

in which Psideband(ωc+Δω, 1Hz) is the single-side noise power at an offset frequency Δω from the carrier, in a measurement bandwidth of 1Hz, and Pcarrier is the total power under the power spectrum. For a NEMS beam resonator operating at fundamental mode, carrier frequency is simply the resonance frequency, ωc=ω0. The phase noise induced by the thermomechanical fluctuation is

( ) ( )

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +Δ

=

Δ 10log ⁰ ₂

C x

x

S_φ ω S ω ω in [dBc/Hz], (2-25)

where Sx(ω0+Δω) [m²/Hz] is the mean-square displacement per Hertz at Δω offset frequency and 〈xC2〉 is the mean-square displacement at the maximum allowable drive level. For harmonic vibration, 〈xC2〉=aC2/2 with aC being the maximum possible amplitude in linear regime. Let EC=Meffω02

aC2/2; this defines the maximum energy level of the resonator, and hence applying eq. (2-22) we have

( ) ( )

_⎥^⎥_⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛

= Δ

⎥⎦

⎢ ⎤

⎣

⎡

≈ Δ Δ

2 0 2 2

0 10log

log

10 ω

ω ω

ω ω

φ P Q

T k Q

E T S k

C B C

B in [dBc/Hz], (2-26)

where the approximation holds for ω0/Q<<Δω<<ω0, and PC=ω0EC/Q is the maximum drive power level. This clearly shows that thermomechanical-fluctuation-caused phase noise has 1/f ² dependency in the phase noise spectrum. Here we use f as the offset frequency for convenience in the power-law description of the noise spectrum, or in fact we have (ω0/Δω)²=(f0/f)² in eq. (2-26).

(ii) Frequency Noise

The phase noise can also be viewed and measured as frequency noise, because phase is the time-integral of frequency (a.k.a., the famous ‘φ=ωt’). Besides the carrier phase term φc(t)=ωct, let the instantaneous excess phase term be φ(t)=φ0sin(Δωt) assuming a sinusoidal modulation at offset frequency Δω; thus instantaneous frequency varies as δω(t)=dφ/dt=Δωφ0cos(Δωt). Hence if we define the fractional frequency variation y=δω(t)/ω0 [7,8], we have

( ) ( )

0 0

0

1 1

ω ω φ

δω ω φ δω ω φ

= Δ

∂

∂

∂

= ∂

∂

= ∂

∂

≡ ∂

ℜ t

t y

y . (2-27)

Therefore the noise spectrum of the fractional frequency variation is

( ) ( )

²

( )

₂

0 2

Q P

T S k

S S

C B y

y ⎟⎟ Δ =

⎠

⎜⎜ ⎞

⎝

=⎛ Δ Δ ℜ

=

Δ ω

ω ω ω

ω _{φ φ} , in [1/Hz]. (2-28)

Note here that in calculating the Sy(Δω) we use the absolute unit for Sφ(Δω) [1/Hz]

instead of the decibel unit [dBc/Hz]. This result indicates that the thermomechanical- fluctuation-induced fractional frequency variation noise Sy(Δω) is white. The spectrum of the absolute frequency noise is also flat and we have

( )

₀²

( )

⁰² ₂

Q P

T S k

S

C B y

ω ω ω

ω Δω = Δ =

Δ , in [Hz²/Hz]. (2-29)

Beyond the consideration of thermomechanical limited phase noise, some of the above analyses hold for more general cases. At least we can arrive at the following relations:

( ) ( ) ( ) ( ) (

ω ω ω

ω

ω ω _ω

φ Δ

= Δ Δ Δ

=

Δ S S_Δ

S _{2 y 2}

2

0 1

)

. (2-30)

Here to avoid confusion we note: (i) ω=ω(t) is the time-dependent instantaneous frequency of the resonator; (ii) ω0 is the resonance frequency and also the carrier frequency (ωc=ω0); (iii) Δω is the offset frequency from the carrier, i.e., it is the time-independent Fourier frequency that appears in any spectral density (sometimes for convenience in speaking of the power laws of the spectral density we use f as Δω).

The relations in eq. (2-30) clarify that the power law of the phase noise spectrum is 2 orders lower than that of the frequency noise. We illustrate this in Fig. 2.4. For instance, flat (white) and 1/f frequency noise spectra translate into phase noise spectra with 1/f ² and 1/f ³ power laws, respectively.

1m 10m 100m 1 10 100 1k 10^-9

10^-8 10^-7 10^-6

1m 10m 100m 1 10 100 1k

B-120 B-100 B-80 B-60 B-40 B-20 A+0 A+20 A+40 A+60

τ ^1/2 τ ^-1/2

σ A(τ)

Averaging Time, τ [s]

(c) (b)

offset frequency, f [Hz]

1/f ² 1/f ³

S φ(f)[dBc/Hz]

1/f ⁴

1/f ² 1/f

S y(f)or S f(f)[dB/Hz]

white

(a)

Fig. 2.4 Illustration of power-laws of the spectra of (a) frequency noise and (b) phase noise, and (c) dependency of Allan deviation on the averaging time. (Here A and B are offset constants; and for the axes labels, only the decades and differences are meaningful.)

(iii) Time-Domain Characterization of Phase Noise: Frequency Stability

In the time domain, phase noise is more suitably described as frequency stability (or equivalently, frequency instability). There is a very important quantity for this measure—the Allan deviation [7-9]—which is widely used in the resonator, frequency control, communication and timekeeping instruments, and other communities.

Sometimes the square of Allan deviation—Allan variance, is also used. Allan deviation is the deviation of variations between every two adjacent measured average fractional frequencies, as a function of the averaging time interval. This definition can be readily used to characterize the frequency stability of the NEMS beam resonator with nominal resonance frequency ω0=2πf0:

( ) ( ) ∑ ( ) ∑

= +

=

+ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

= − ^N

i

i i N

i

i i

A f

f f N

N ₁

2

0 1 1

2

0 1

1 2

1 1

2 1

ω ω τ ω

σ . (2-31)

Here f_i is the measured average frequency in the ith time interval. By examining the Allan deviation with various averaging time intervals, it is possible to attain comprehensive understanding of the frequency stability performance within the ranges of interest.

The conversion relation between the phase noise spectral density and Allan deviation is

( ) ( )

^τ _ωτ

( )

^{ω ωτ}

( )

^ω

σ _φ ⎟ Δ

⎠

⎜ ⎞

⎝ Δ ⎛ Δ

=

∫

^{∞S d}

A 0

4

02 sin 2

2 . (2-32)

Now consider the frequency stability limit set by thermomechanical fluctuations, with eqs. (2-26) and (2-32) we determine the Allan deviation to have a σA(τ)~τ^-1/2 dependency:

( )

^{τ π} _τ ^π_{ω τ}

σ E Q

T k P

T k

Q _C

B C

B A

0

1 =

= . (2-33)

Likewise, we obtain σ_A(τ)~τ^1/2 for 1/f ² drifting frequency noise spectrum (which has phase noise following 1/f ⁴), and σ_A(τ) independent of τ (~τ⁰) for 1/f frequency noise (with its phase noise spectrum following 1/f³). We have illustrated this in Fig. 2.4 together with the noise spectral density power laws.

Often both the frequency-domain and time-domain measures of phase noise (or frequency noise) are used. For practical reasons, phase noise spectra are usually used to characterize short-term (very small τ and very large Δω) frequency instability, and Allan deviation is more often employed when it becomes more and more difficult to directly measure the very-close-to-carrier (very smaller Δω) phase noise.