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Frequency Reference Stability and Coherence Loss inRadio Astronomy Interferometers Application to the SKADOI:10.1142/S2251171718500010

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Citation for published version (APA):Alachkar, B., Wilkinson, A., & Grainge, K. (2018). Frequency Reference Stability and Coherence Loss in RadioAstronomy Interferometers Application to the SKA. Journal of Astronomical Instrumentation, 7(1), [1850001].https://doi.org/10.1142/S2251171718500010

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Journal of Astronomical Instrumentation

World Scientific Publishing Company

1

Frequency Reference Stability and Coherence Loss in Radio Astronomy Interferometers

Application to the SKA

Bassem Alachkar1,2 , Althea Wilkinson1 and Keith Grainge1 1Jodrell Bank Centre for Astrophysics

School of Physics & Astronomy

University of Manchester

Manchester, UK 2bassem.alachkar@manchester.ac.uk

Received (to be inserted by publisher); Revised (to be inserted by publisher); Accepted (to be inserted by publisher);

The requirements on the stability of the frequency reference in the Square Kilometre Array (SKA), as a radio astronomy

interferometer, are given in terms of maximum accepted degree of coherence loss caused by the instability of the frequency

reference. In this paper we analyse the relationship between the characterisation of the instability of the frequency reference in

the radio astronomy array and the coherence loss. The calculation of the coherence loss from the instability characterisation

given by the Allan deviation is reviewed. A model of a typical frequency distribution system is presented. The verification of the

coherence and frequency reference stability requirements is discussed. Some practical aspects and limitations relevant to the

SKA are analysed.

Keywords: Radio Astronomy- Interferometry- Coherence Loss- Allan Deviation – Frequency Stability.

1. Introduction

The basic principle of radio astronomy interferometry relies on combining the received signals of the array

sensors in a coherent manner. To achieve good coherence of the array, the frequency references at the array

sensors must be synchronised to a common reference. In a connected radio interferometry array, this can be

achieved by a synchronised frequency dissemination system which delivers frequency reference signals to the

receptors, (e.g. Schediwy et al., 2017 and Wang et al., 2015). The local frequency reference signals are used for

sampling and/or for down-converting the frequency of the radio astronomy signals. Any instability of the

frequency references causes loss of coherence in the array. Therefore, the requirements on the stability of the

synchronisation system can be defined in such a way as to limit the coherence loss caused by frequency

instability. The stability of a frequency reference is usually characterised by the Allan deviation, while the

coherence between two signals is calculated by the correlation integral of the signals. This paper analyses the

characterisation and evaluation of frequency distribution system instability and the relationship with the

coherence loss in a radio astronomy array. The application of this analysis is based on the coherence requirements

of the Square Kilometre Array (SKA) (Dewdney et al., 2009).

In the SKA, the Synchronisation and Timing (SAT) system provides frequency and timing signals to the receptors

and other parts of the telescope (Grainge et al., 2017). These signals are required to have certain levels of stability

and accuracy, specified by the SKA requirements. The basic SAT stability and accuracy requirements are derived

from the astronomical science requirements, covering the radio astronomy functions of the telescope for imaging,

pulsar searching and timing, and Very Long Baseline Interferometry (VLBI). The relevant SKA requirements for

frequency distribution are given in terms of coherence loss, so that the maximum coherence loss accepted for

frequencies up to 13.8 GHz and for intervals of 1 s or 60 s is limited to 2% in total. These requirements are driven

by the need for coherence over the correlator integration time (approximately 1 second) and the time for in-beam

2 B. Alachkar

calibration (1 minute). They are expressed in terms of the coherence loss caused by the phase difference between

the two frequency signals delivered at two receptors, or in other words, per baseline.

The focus, in this paper, is on the coherence loss requirements which are related to stability in the time scales in

the range from parts of a second up to a few minutes. There are other requirements of the SKA timing outside this

range. Stability in the orders of few minutes is required for the purpose of calibration (out of beam calibration),

which is done by focusing the receptors on the observed source for a few minutes and then focusing it on a

calibration target for another few minutes. The drift in phase over this calibration period should be limited to a

part of the trigonometric circle (usually less than 1 radian). For larger long-term time scales up to years, stability

is required in the order of nanoseconds for astronomical applications like pulsar timing. The long term stability is

achieved by steering the SKA timescale to the UTC (Coordinated Universal Time) time. For stability on shorter

timescales than parts of a second down to the nanoseconds (the highest frequency of the astronomical radio signal

of the SKA-Mid telescope is 13.8 GHz), there are requirements on the frequency/time stability related to the noise

generated by the jitter on the sampling clock of the analog-to-digital (A/D) convertors. The jitter on the sampling

clock of the A/D convertor introduces noise on the signal and therefore reduces the signal to noise ratio. The jitter

is usually measured by measuring the phase spectrum and integrating it to obtain the RMS value of the phase

fluctuation or equivalently of the time fluctuation. A clean-up oscillator local to the receptor would give very low

jitter down to the tens of femtoseconds.

In section 2, we give the definition of coherence and coherence loss in radio astronomy interferometry, and link it

to the phase noise produced by the frequency reference. In section 3, the model of phase noise is presented and

the instability of a frequency reference is analysed. Allan deviation characterisation is considered as it is the

common tool used for frequency reference characterisation. The main practical cases and aspects of the phase

noise are presented and the relationship between the frequency instability and resulting coherence loss is

analysed. Section 4 introduces the frequency distribution systems, and analyses the model of a typical round-trip

compensation system. The verification of the stability requirements of a frequency distribution system has

operational aspects related to the environmental conditions and the lengths of the fibre links. A basic

understanding of the functional principle that shows the impact of these conditions is necessary in defining the

verification of frequency stability requirements. In section 5, some practical aspects of the coherence and

frequency stability verification are analysed, and examples of simulation results are given. Section 6 gives the

conclusions of this analysis.

2. Coherence Loss in a Radio Astronomy Interferometer

The coherence loss in radio astronomy interferometry caused by frequency reference instability, especially in

VLBI, has been analysed by several authors such as (Rogers & Moran 1981), (Thompson, Moran & Swenson,

2004) and (Kawaguchi, 1983). The basic definitions and equations, and the necessary derivations are given for a

convenient reading of the paper.

The coherence function for an integration time 𝑇 is defined as in (Rogers & Moran 1981) and (Thompson, Moran

& Swenson, 2017):

𝐶(𝑇) = |1

𝑇∫ 𝑒𝑥𝑝[𝑖𝜑(𝑡)]𝑑𝑡

𝑇

0| , (1)

where 𝜑(𝑡) is the phase difference between the two stations of the interferometer. There is no loss of coherence if

the value of 𝐶(𝑇) is 1.The coherence loss 𝐿𝐶 is

𝐿𝐶 = 1 − √⟨𝐶2(𝑇)⟩ (2)

where √⟨𝐶2(𝑇)⟩ is the root mean-square of the coherence 𝐶(𝑇) for an integration time 𝑇.

The mean-square value of 𝐶(𝑇) is

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 3

⟨𝐶2(𝑇)⟩ = ⟨1

𝑇2 ∫ ∫ 𝑒𝑥𝑝{𝑖[𝜑(𝑡) − 𝜑(𝑡′)]}𝑑𝑡𝑇

0𝑑𝑡′

𝑇

0⟩ (3)

If 𝜑 is a Gaussian random variable, we find:

⟨𝐶2(𝑇)⟩ =1

𝑇2 ∫ ∫ 𝑒𝑥𝑝 [−𝜎2(𝑡,𝑡′)

2] 𝑑𝑡

𝑇

0𝑑𝑡′

𝑇

0 (4)

where 𝜎2(𝑡, 𝑡′) is the variance of the phase difference between the two instants 𝑡 and 𝑡′. If we assume that σ2 is stationary and depends only on 𝜏 = 𝑡′ − 𝑡, then the mean squared coherence can then be

written as:

⟨𝐶2(𝑇)⟩ =2

𝑇∫ (1 −

𝜏

𝑇) 𝑒𝑥𝑝 [

−𝜎2(𝜏)

2] 𝑑𝜏

𝑇

0 (5)

3. Phase Noise in the Frequency Reference and Coherence Loss

The instability of a frequency reference is usually characterized by Allan variance (or Allan deviation). There is a

need to link this characterization to the coherence loss in a radio interferometer resulting from this instability.

This section presents the relationship between the characterization of instability in term of Allan deviation and the

related coherence loss.

3.1. The Frequency Signal Model

The phase 𝜙(𝑡) of an oscillator or frequency reference is defined with reference to another frequency reference

which is usually considered to be of accuracy and stability acceptable for the application. The instantaneous

frequency is the derivative of the phase 𝜙(𝑡) = 2𝜋 𝜐0𝑡 + 𝜑(𝑡) :

𝜐(𝑡) = 𝜐0 + 𝛿𝜐(𝑡) (6)

𝜐0 is the nominal frequency of the frequency reference, and the frequency deviation is:

𝛿𝜐(𝑡) =1

2𝜋

𝑑𝜑(𝑡)

𝑑𝑡 (7)

The instantaneous fractional frequency deviation is:

𝑦(𝑡) =𝛿𝜐(𝑡)

𝜐0 (8)

The deviations of the frequency are described as deterministic systematic deviations or random deviations. In

general, the model of these deviations is given by the following:

𝑦(𝑡) = 𝑦0 + 𝐷𝑡 + 𝜖(𝑡) (9)

where the systematic deviations are covered by the frequency offset y0 and the linear frequency drift 𝐷𝑡, while

the random deviations are covered by ϵ(t). The systematic deviations are usually small and have small effects on

the phase deviation in short periods of time. In the model given by Eq. (9), systematic deviations of higher order

are neglected.

3.2. Allan Variance and Coherence Loss

Allan variance is the most common statistical function used to characterise and classify frequency fluctuations of

a frequency reference (Allan, 1966) (Rutman, 1978). It overcomes convergence and non-stationarity difficulties

that might appear for some phase noise cases (Lindsey & Chie, 1976).

The fractional frequency fluctuation averaged over the time interval 𝜏 is:

�̅�𝑘 =1

𝜏∫ 𝑦(𝑡)𝑑𝑡

𝑡𝑘+𝜏

𝑡𝑘=

𝜑(𝑡𝑘+𝜏)−𝜑(𝑡𝑘)

2𝜋𝜐0𝜏 (10)

Using the sample variance of y̅k, as a measure of the instability of oscillators or frequency references is not

always convenient because it does not converge in many cases of phase noise. This is due to the low frequency

4 B. Alachkar

behaviour. The two-sample or Allan variance σy2(τ) is generally accepted as a measure of instability of the

oscillators or frequency references as it overcomes the convergence problem for most practical cases. The Allan

variance is defined as follows:

𝜎𝑦2(𝜏) =

⟨(�̅�𝑘+1−�̅�𝑘)2⟩

2 (11)

The Allan deviation is 𝜎𝑦(𝜏) .This is usually used to characterise the random deviations ϵ(t) which are related to

the noise in the frequency. The Allan deviation is not affected by the frequency offset y0 because of the

differential operation in its definition (Eq. (11)). The drift 𝐷𝑡 results in an additive contribution of 𝜎𝑦2(𝜏) =

1

2𝐷2𝜏2 to the Allan variance.

The variance of the phase difference in Eq. (5) can be calculated from a series of values of the Allan variance. Eq.

(5) can then be written as (See (Rogers & Moran 1981), (Thompson, Moran & Swenson, 20017)):

⟨𝐶2(𝑇)⟩ =2

𝑇∫ (1 −

𝜏

𝑇) 𝑒𝑥𝑝{−(𝜋𝜐0𝜏)2[𝜎𝑦

2(𝜏) + 𝜎𝑦2(2𝜏) + 𝜎𝑦

2(4𝜏) + ⋯ ]}𝑑𝜏𝑇

0 (12)

The last equation can be simplified in the cases of white phase noise and white frequency noise.

3.3. Phase noise types

In the frequency domain the phase and frequency fluctuations are characterised by their power spectral densities.

If 𝑆𝜑(𝑓) and 𝑆𝛿𝜐(𝑓) are the one-sided (the Fourier frequency goes from 0 to ∞) spectral densities of the phase

and frequency fluctuations respectively, they are related by the following relationship:

𝑆𝛿𝜐(𝑓) = 𝑓2𝑆𝜑(𝑓) (13)

The power spectral density Sy(f) of the fractional frequency deviation is related to Sδυ(f) and Sφ(f) by:

𝑆𝑦(𝑓) =1

𝜐02 𝑆𝛿𝜐(𝑓) =

𝑓2

𝜐02 𝑆𝜑(𝑓) (14)

For most practical applications the spectral densities of the random frequency fluctuations are considered to

follow a power law model (Barnes et al., 1971) and (Rutman, 1978):

𝑆𝑦(𝑓) = ∑ ℎ𝛼𝑓𝛼+2𝛼=−2 (15)

for 0 ≤ 𝑓 ≤ 𝑓ℎ where 𝑓ℎ is an upper cut-off frequency. A cut-off frequency is related to a limitation in the

bandwidth in the reference frequency generator itself, as in the case of the phase-locked loops (PLLs), or in the

measurement systems.

For 𝛼 = −2 to +2, the types of noise are random-walk frequency (𝛼 = −2), flicker frequency (𝛼 = −1), white

frequency (𝛼 = 0), flicker phase (𝛼 = +1), and white phase noise (𝛼 = +2).

Two simple cases which are of practical importance are the cases of white phase noise and white-frequency noise.

White-phase noise

The power spectrum density of the fractional frequency deviation is given by the form 𝑆𝑦(𝑓) = ℎ2𝑓2.

In this case and for 𝜏 ≫1

2𝜋𝑓ℎ, the Allan variance is given by, see (Rutman, 1978):

𝜎𝑦2(𝜏) =

3ℎ2𝑓ℎ

(2𝜋)2𝜏2 (16)

And the mean squared coherence can be evaluated from (21):

⟨𝐶2(𝑇)⟩ = exp (-ℎ2𝑓ℎ𝜐02) (17)

Hence the coherence loss in this case is:

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 5

𝐿𝐶 = 1 − √exp (−ℎ2𝑓ℎ𝜐02) (18)

The factor ℎ2𝑓ℎ in Eq. (18) can be extracted from the Allan variance function given by Eq. (16).

White-frequency noise

The power spectrum density of the fractional frequency deviation is given by the form 𝑆𝑦(𝑓) = ℎ0.

In this case and for 𝜏 ≫1

2𝜋𝑓ℎ, the Allan variance is given by, see (Rutman, 1978):

𝜎𝑦2(𝜏) =

ℎ0

2𝜏 (19)

In this case the mean squared coherence is given by the following, (Thompson, Moran & Swenson, 2004):

⟨𝐶2(𝑇)⟩ =2(𝑒−𝑎𝑇+𝑎𝑇−1)

𝑎2𝑇2 (20)

where

𝑎 = 𝜋2𝜐02ℎ0 (21)

Hence the coherence loss in this case is:

𝐿𝐶(𝑇) = 1 − √2(𝑒−𝑎𝑇+𝑎𝑇−1)

𝑎2𝑇2 (22)

The factor ℎ0 and therefore 𝑎 in Eq. (21) can be extracted from the Allan variance function given in Eq. (19). The

coherence loss in this case depends on the integration time 𝑇.

4. Analysis of Optical Fibre Frequency Distribution Systems

The aim of the analysis and results developed in this section is to get a basic understanding of the compensated

frequency distribution systems. This understanding is necessary for the verification of the stability requirements

which should take into consideration the variations in the environmental conditions and the fibre lengths.

The frequency distribution system of the SKA1-Mid telescope, which will be built in South Africa, will distribute

the reference frequencies over distances up to around 170 km. The reference frequency from the SKA timescale is

distributed over optical fibres to the receptors by modulating an optical signal on the fibres (Schediwy et al.,

2017). The SKA timescale (SKA clock) frequency signal is generated by a stable active hydrogen maser. The

phase difference instability between the receptors is due to the noise generated on the fibre and optical and RF

components of the link between the SKA timescale and the receptor. Temperature fluctuations and mechanical

stresses and vibration on the fibre modify the optical length of the fibre and affect the travel time of the reference

signal through the fibre, and therefore alter the signal phase. This can be compensated for by measuring the signal

round-trip time (phase change) out to the receptor and back over the same fibre, and applying a phase shift to the

input or the output phase.

The very good stability of the round-trip frequency distribution systems makes them the compensation systems of

choice in applications requiring frequency distribution over long distances with high stability. In this section, we

analyse the model of a typical round-trip frequency distribution system. The focus is on the effect of the phase

noise generated on the fibre on the stability of the frequency signal provided at the remote end. The analysis gives

the basic principle and shows the degree of stability provided, and the compensation limitation. As the system

reacts to the round-trip phase fluctuation measured with a delay corresponding to the signal propagation in the

fibre, the stability is limited to times longer than the delay time. For distances in the order of hundreds of

kilometres, the time delays are in the order of milliseconds. This means that the phase correction and tracking of

the remotely delivered frequency reference will not ensure short term stability in the short time scale of less than

milliseconds. This short term stability can be provided by crystal oscillators which have very good short term

stability. The solution is then to have a local oscillator which has very good short term stability and lock it to the

6 B. Alachkar

central reference (the maser) with the loop of fibre round-trip correction. This local oscillator is called a clean-up

oscillator as it removes the short-term noise accumulated on the optical and electronic components of the link.

A central clock based on an active hydrogen maser with steering to UTC is the solution adopted for the SKA

timescale ensemble. This is to comply with the timing requirements on the timescale of the SKA for pulsar timing

and VLBI applications. The active hydrogen maser provides good stability in the timescale between a second and

thousands of seconds. Longer term stability is achieved by steering to UTC.

This solution permits good stability of the local reference at each receptor on the timescales from fractions of a

second to thousands of seconds and this covers the timescales of the coherence requirements. For shorter

timescales (less than fractions of a second), the stability is assured by passing the frequency signal to a clean-up

oscillator, which filters the remaining high frequency noise on the frequency reference signal.

Round-trip frequency distribution model: The idea of a round-trip noise cancellation frequency distribution system is to measure the round-trip phase shift

on the transmitted signal and use this information in a feed-back system phase-locked loop (PLL) to cancel the

noise at the remote end.

Variants of the design of the frequency distribution systems differ in their structure and the technology used.

Examples of the variations in the structure include the location of the inserted phase correction in the loop

whether it is in the local end or the remote end, and in the way the return phase is coded on the transmitted signal.

Different types of optical and RF components can be used in these systems. RF and optical components are used

to transmit the optical modulated signal, to detect the phase of the RF or optical signals, to provide a phase shift.

Examples of these realisations are the systems adopted for the SKA (Schediwy et al., 2017) and (Wang et al.,

2015). Without considering the technological implementations, the principle of the round trip compensation is

modelled below.

A number of effects limit the stability of the distributed frequency signal. These include phase noise on the

source, RF electronics, receiver, transmitter, nonlinear effects, polarization-mode dispersion, phase noise on the

fibre, and the effect of the propagation delay. If the technological solution and the good design allow the

minimisation of most of those factors, there is still a fundamental limitation related to the delay on the fibre

(Williams et al. 2008). The focus of the analysis in this section is on this limitation.

The reference signals are converted into the optical domain and then transmitted over the optical fibre link where

they acquire phase noise due to environmental disturbances on the link. The phase signals in the model presented

here correspond to the phase deviations.

A typical model of the round-trip PLL compensation is presented in figure 1. The input phase (phase deviation)

𝜑𝑖𝑛(𝑡) is the phase of the frequency reference signal. We can consider 𝜑𝑖𝑛(𝑡) = 0 for a stable source. The phase

fluctuation at the remote end of the fibre 𝜑𝑜(𝑡) is the sum of the accumulated (one way) phase noise on the fibre

𝜑𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑡) and the output compensating phase 𝜑𝑐−𝑜(𝑡). The phases 𝜑1(𝑡) and 𝜑2(𝑡) are the phase shifts

produced on the outgoing and returned signals by a phase shifter such as an AOM (Acousto-optic Modulator) to

compensate for the phase fluctuation on the fibre. The phase shifter is commanded by a signal resulting from

filtering with a corrector filter the signal of the phase difference between the input phase signal and the round-trip

returned phase signal. The phase difference is usually a voltage that is applied to a voltage controlled oscillator

(VCO) or an optical actuator such as an Acousto-optic Modulator (AOM) to modify the phase of the transmitted

signal in response to the difference in phase. Sometimes, the actuator is called a servo.

The transfer function 𝐹(𝑆) between the input phase difference and the generated phase shift by the actuator

usually includes a multiplicative factor 𝑘/𝑆, which represents the transformation from frequency to phase of the

VCO, as the output of the oscillator is a shift in frequency proportional to the input voltage. The corrector filter is

usually of the Proportional-Integral (PI) type, as the integral action I achieves good precision of the tracking loop

and the proportional action allows determining the stability degree and speed (bandwidth) of the loop.

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 7

The measured round-trip effect of the phases 𝜑1(𝑡) and 𝜑2(𝑡), that are generated by the phase shifter, at the local

side of the fibre is 𝜑shift(𝑡) = 𝜑1(𝑡 − 2𝜏𝑑) + 𝜑2(𝑡) (see figure 1). The time 𝜏𝑑 =𝐿

𝑐/𝑛 is the time delay over the

fibre of length 𝐿, and 𝑛 is the refractive index of the optical fibre medium (𝑐/𝑛 ≈ 2 × 108m/s is the speed of

light in the fibre).

The phase 𝜑shift(𝑡) is the output generated by the PLL. The total round-trip phase is the sum of 𝜑shift(𝑡) and

𝜑𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑡) the accumulated noise phase on the fibre of the round-trip signal. The round-trip phase is applied to

the PLL with a negative sign to counteract the phase fluctuation of the fibre transmitted signal.

Using the Laplace transform, we can write (see figure 1):

𝜙1(𝑆) = 𝜙2(𝑆) = 𝐹(𝑆)𝜙𝑐−𝑖𝑛(𝑆) (23)

Figure 1. A simplified model of a ‘typical’ round trip frequency distribution system

Figure 2. The block diagram of the control system (PLL) of the frequency distribution typical model presented in figure 1

Optical Source

Phase shifter

(AOM)

𝜑𝑐−𝑜(𝑡)

𝜑𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑡)

𝜑𝑜(𝑡)

𝜑𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑡)

𝜑1(𝑡)

+

+

−

Corrector + VCO (𝐹(𝑆))

𝑒−𝑆𝜏𝑑

𝑒−𝑆𝜏𝑑

Fibre (noise + delay)

−

𝜑2(𝑡)

𝜑𝑐−𝑖𝑛(𝑡)

𝜑𝑟(𝑡)

𝜑𝑠ℎ𝑖𝑓𝑡(𝑡)

𝜑𝑖𝑛(𝑡)

𝜑𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑡) 𝜑𝑠ℎ𝑖𝑓𝑡(𝑡)

−

−

𝜑𝑐−𝑖𝑛(𝑡)

𝜑𝑟(𝑡)

𝐻(𝑆)

8 B. Alachkar

𝐹(𝑆) is the transfer function of the control chain including the VCO and the corrector filter. It is of the form:

𝐹(𝑆) = 𝐾 𝐹𝐶(𝑆)

𝑆 (24)

The term 1

𝑆 is related to the VCO as it transforms a voltage signal to a frequency shift which by integrating (

1

𝑆 )

gives the phase shift. 𝐾 indicates the gain (or attenuation) in the control chain which includes the VCO gain

factor. 𝐹𝐶(𝑆) is the transfer function of the corrector filter, which is usually of the form PI.

The shift in phase measured at the local end is:

𝜙𝑠ℎ𝑖𝑓𝑡(S) = 𝜙1(𝑆)𝑒−2𝑆𝜏𝑑 + 𝜙2(𝑆) = 𝐹(𝑆)(1 + 𝑒−2𝑆𝜏𝑑)𝜙𝑐−𝑖𝑛(𝑆) (25)

The transfer function of the open loop of the PLL is (see figures 1 and 2):

𝐻(𝑆) = 𝜙𝑠ℎ𝑖𝑓𝑡(S)

𝜙𝑐−𝑖𝑛(𝑆)= 𝐹(𝑆)(1 + 𝑒−2𝑆𝜏𝑑) (26)

Then the output phase at the remote side of the fibre is (see Annex A for the calculation details):

𝜙𝑜(𝑆) = 𝜙𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑆) −𝐻(𝑆)

1+𝐻(𝑆)

𝜙𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑆)

(𝑒+𝑆𝜏𝑑+𝑒−𝑆𝜏𝑑) (27)

In the Fourier domain (27) can be written as:

�̃�𝑜(𝑓) = �̃�𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑓) −ℎ̃(𝑓)

1+ℎ̃(𝑓)

1

2cos (2𝜋𝑓𝜏𝑑)�̃�𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑓) (28)

ℎ̃(𝑓) = 𝐻(2𝜋𝑖𝑓) = 2𝑒−2𝜋𝑖𝑓𝜏𝑑𝐹(2𝜋𝑖𝑓)cos (2𝜋𝑓𝜏𝑑) is the frequency response (complex gain) of the open loop of

the PLL.

In Eq. (28), we see that for the frequencies where the gain |ℎ̃(𝑓)| ≈ 0, the phase noise at the end of the fibre is the

noise accumulated on the fibre on the outgoing signal, and this is not compensated. When the gain |ℎ̃(𝑓)| is high,

the phase noise is more compensated. The limitation on the open loop gain |ℎ̃(𝑓)|is constrained by the stability

considerations. For the system to be stable and to avoid oscillations, the open loop gain ℎ̃(𝑓) should be less than 1

when its phase approaches −𝜋. This limits the band of compensation to the low frequencies to less than 𝑓 =1

4𝜏𝑑

because of the delay term 𝑒−2𝜋𝑖𝑓𝜏𝑑 in ℎ̃(𝑓) and the integral term 1

𝑆 in 𝐹(𝑆) which adds a phase of −𝜋/2. A low

degree of stability may result in a bump in the spectrum around the frequency 𝑓 =1

4𝜏𝑑 .

For high gain |ℎ̃(𝑓)| ≫ 1 , low frequencies where 𝑓 ≪1

2𝜋𝜏𝑑 , and if the phase noise on the fibre is considered

spatially-uncorrelated, the power spectral density (PSD) of the remote side phase 𝑆𝑜(𝑓) can be written as (see

Appendix A):

𝑆𝑜(𝑓) =(2𝜋𝑓)2

(𝑐/𝑛)2 ∫ ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ 𝑥2 𝑑𝑥𝐿

0 (29)

𝛿�̃�(𝑓, 𝑥) is the elementary spectral component of the phase on the fibre at the position 𝑥 from the local side.

If we, also, assume that the elementary PSD ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ is uniform along the fibre:

⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ = 𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓)

𝐿= 𝐷(𝑓) (30)

𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓) is the PSD of the phase noise accumulated on one-way transmission of the fibre. 𝐷(𝑓) is the density

of the PSD 𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓) per unit of length of the fibre. Then 𝑆𝑜(𝑓) can be written as:

𝑆𝑜(𝑓) =(2𝜋𝑓)2𝐷(𝑓)𝐿3

3(𝑐/𝑛)2 =1

3(2𝜋𝑓𝜏𝑑)2 𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓) (31)

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 9

Eq. (31) is valid in the case of uniform spatially-uncorrelated phase noise on the fibre, and low frequencies

(𝑓 ≪1

2𝜋𝜏𝑑) and high gain |ℎ̃(𝑓)| ≫ 1. In this case 𝑆𝑜(𝑓) is proportional to 𝐿3, and therefore the Allan deviation

𝜎𝑦(𝜏), corresponding to this spectrum in low frequencies and 𝜏 ≫ 𝜏𝑑, is proportional to 𝐿3/2. Eq. (31) shows that

the round-trip compensation changes the spectrum in the low frequencies at the remote end and removes 1/𝑓 or

1/𝑓2 behaviours that have bad effect on the long term stability. If, for example, the phase noise on the fibre is of

white frequency noise type (1/𝑓2 type), the output phase noise becomes white phase noise in the low frequencies.

The level and type of phase noise accumulated on the fibre, depends on the environmental conditions of

temperature and mechanical stresses and vibrations. The optical, RF, and electronic components of the system

have their additional sources of phase noise which add to the noise accumulated on the fibre. The systems are

designed to minimise these sources of noise.

5. Verification of the Frequency Distribution Stability and Coherence Loss Requirements

In this section we describe some practical aspects of the verification of the stability of the frequency distribution

systems and the corresponding coherence loss.

5.1. Allan deviation for different phase noise types

Given the Allan deviation of the frequency reference, we can calculate the coherence loss between two receptors

(i.e. for a baseline) of an interferometer caused by the instability of the frequency reference, by using Eq. (12). Eq.

(12) can be applied in the general case. If the Allan deviation characterisation shows white phase noise or white

frequency noise behaviour, the coherence loss can be calculated directly using Eq. (18) and (22). In the general

case, when the Allan deviation curve suggests that the noise falls in neither specific phase noise regime, least

squares methods can be applied to the data to calculate a ‘best’ fit of the function 𝜎𝑦2(𝜏), which can then be used

in Eq. (12).

5.2. The phase difference between two receptors

The loss of coherence due to the instability of the frequency references between two stations or receptors of the

interferometer array depends on the fluctuations of the phase difference 𝜑(𝑡) of the two frequency references. The

stability characterisation of one frequency reference is usually given with reference to another much more stable

frequency reference. In the case of the frequency distribution systems, the phase difference considered in the

stability characterisation (e. g. Allan deviation) is between the frequency reference delivered at the receptor and a

common central frequency reference. Different link lengths cause different levels of instability. The Allan

variance or the variance in phase difference between the frequency references in the two receptors is the sum of

the corresponding variances between each of these points and the central reference, assuming that the variations

on the two links causing the instability are independent.

5.3. Frequency conversion of local oscillator or sampling clock reference

The frequency reference at the astronomy receptor might be used as a local oscillator to be mixed with the

astronomical signals for converting the frequency before sampling, or used directly as a reference for the

sampling clock of the analog-to-digital convertor.

In the first case, the phase deviation of the local oscillator is added directly to the phase of the astronomical

signal. In Eq. (12), the frequency υ0 is the local oscillator frequency, which is the frequency reference and the

Allan deviation is the Allan deviation of this frequency reference.

In the second case, when the frequency reference is used to provide the sampling clock of sampling frequency 𝐹𝑆,

we will see that in Eq. (12) the frequency υ0 is the astronomical observation frequency and the Allan deviation is

again the Allan deviation of the frequency reference.

10 B. Alachkar

Consider a frequency component 𝑥(𝑡) = 𝐴 𝑒𝑥𝑝[𝑖(2𝜋𝑓0𝑡)] of the astronomical observation signal. The sampled

signal 𝑥𝑆(𝑛), resulting from sampling 𝑥(𝑡) at the sampling instants tn =n

FS generated by an ideal frequency

reference, is: 𝑥𝑆(𝑛) = 𝐴 𝑒𝑥𝑝 [𝑖(2𝜋𝑓0

𝐹𝑆𝑛)]. The phase deviation ∆φ in the real frequency reference will cause a

shift in the sampling instant so that it becomes: 𝑡𝑛′ =

𝑛

𝐹𝑆+ ∆𝑇, where ∆𝑇 =

∆𝜑

2𝜋𝐹𝑆. The sampled signal becomes:

𝑥𝑆′ (𝑛) = 𝐴 𝑒𝑥𝑝 [𝑖 [(2𝜋

𝑓0

𝐹𝑆𝑛) +

𝑓0

𝐹𝑆∆𝜑 ]].

This is equivalent to introducing a phase of 𝑓0

𝐹𝑆∆𝜑 to the signal 𝑥(𝑡). If σy(τ) is the Allan deviation

corresponding to the phase deviation ∆φ of a frequency reference with a nominal frequency FS, then the same

value 𝜎𝑦(𝜏) will be the Allan deviation corresponding to the phase deviation 𝑓0

𝐹𝑆∆𝜑 of a frequency reference with

a nominal frequency 𝑓0.

We conclude that the phase deviation introduced by sampling a signal of frequency 𝑓0with a sampling frequency

reference with Allan deviation 𝜎𝑦(𝜏) will result in a loss of coherence equivalent to the coherence loss caused by

an oscillator of frequency 𝑓0 and Allan deviation 𝜎𝑦(𝜏), and therefore Eq. (12) is applicable with 𝜐0 = 𝑓0, the

observation frequency, and 𝜎𝑦(𝜏) is the Allan deviation of the frequency reference.

5.4. Coherence loss calculation from phase measurements:

Numerically, the coherence loss is calculated from the discrete phase sequence 𝜑𝑑(𝑛) = 𝜑(𝑛𝑇𝑆), which results

from the sampling of the continuous phase signal 𝜑(𝑡). 𝑇𝑆 is the sampling period. The integral (1) is calculated by

the sum:

�̂�(𝑇) = |1

𝑁∑ 𝑒𝑥𝑝[𝑖𝜑𝑑(𝑛)]𝑁

𝑛=1 | (32)

�̂�(𝑇) is the estimation of the coherence on an interval of length 𝑇 , where 𝑇 = 𝑁𝑇𝑆.

The sequence of the measured phase signal is divided into 𝐾 segments of length 𝑇, the coherence �̂�𝑘(𝑇) is

calculated on each segment 𝑘, and then the estimation of the coherence loss �̂�𝐶 is calculated as following:

�̂�𝐶(𝑇) = 1 − √1

𝐾∑ �̂�𝑘(𝑇)2𝐾

𝑘=1 (33)

5.5. Coherence Loss calculation from Allan deviation characterisation:

From a set of values of σy2 given for a number of averaging times τm, the Allan deviation function σy

2(τ) is

estimated by a least squares fitting. σy2̂(τ) is the least squared fitting function, which is defined by a limited

number of parameters. Then the coherence loss is estimated numerically by the equation:

⟨𝐶2(𝑇)⟩̂ =2

𝑁∑ (1 −

𝑛

𝑁) 𝑒𝑥𝑝 {−(𝜋𝜐0𝑛∆𝜏)2[𝜎𝑦

2̂(𝑛∆𝜏) + 𝜎𝑦2̂(2𝑛∆𝜏) + 𝜎𝑦

2̂(4𝑛∆𝜏) + ⋯ ]}𝑁−1𝑛=0 (34)

where 𝑇 = 𝑁 ∆𝜏.

5.6. Cases of white frequency noise and white phase noise

The coherence requirements for the SKA limit the coherence loss to 2% at an integration time of 60 seconds. If

the Allan deviation characterisation of the frequency distribution system shows that the system has either white

frequency noise or white phase noise, then with the integration time, the value of the Allan deviation at 𝜏 =1 second is sufficient to determine the coherence loss.

In figures 3 and 4, the coherence loss at integration time of 1 s and 60 s is given in the cases of white frequency

noise and white phase noise as a function of the Allan deviation value at 𝜏 = 1 second between two receptors, for

an operating frequency 13.8 GHz. The coherence loss is independent of the integration time in case of the white

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 11

phase noise and it increases with the integration time in the case of white frequency noise. Figure 5 gives the

coherence loss as a function of the integration time for different values of Allan deviation at 𝜏 = 1 second in the

two cases of white phase noise and white frequency noise, for an operating frequency 13.8 GHz.

5.7. Simulations

We show here one example of white phase noise and another example of white frequency noise, which cause a

coherence loss of 2% for an integration time 𝑇 = 60 𝑠 and at the astronomical frequency 13.8 GHz, the highest

frequency of the SKA Mid telescope. The frequency reference which has to be delivered by the frequency

dissemination system is taken as 4 GHz. It is then used to sample signals in different bands. In the band of the

highest frequency 13.8 GHz, the frequency reference 4 GHz will be multiplied by 8 before being applied to the

A/D converter for this signal chain.

White phase noise:

We show in figure 6 an example of white phase noise that causes a phase deviation of RMS= 0.2 radians at 13.8

GHz and about 2% coherence loss (𝐿𝐶 = 0.0198). Figure 6 gives the phase and frequency deviation signals.

Figure 7 gives the Allan deviation corresponding to this case. Figure 8 gives the coherence loss calculated directly

from the phase and using a least squares fit of the Allan deviation.

White frequency noise:

We show in figure 9 an example of white frequency noise for a frequency reference at 4 GHz, which gives 2%

coherence loss for an astronomical frequency 13.8 GHz and an integration time of T=60 seconds. Figure 9 gives

the phase and frequency deviation signals. Figure 10 gives the Allan deviation corresponding to this case. Figure

11 gives the coherence loss calculated directly from the phase and using a least squares fit of the Allan deviation.

5.8. Limitations of the calculation of the coherence loss from Allan deviation:

We consider here two types of limitations of the calculation of the coherence loss from Allan deviation. The first

limitation is due to the fact that a linear drift in the phase deviation is not reflected in the Allan deviation, while it

has an effect on the coherence loss. The other difficulty is encountered when the phase deviation has a complex

structure that makes a simple least squares fitting with a simple model inaccurate. This can happen, for example,

in the case of a sinusoidal component in the phase deviation. A sinusoidal component in the phase deviation could

be caused by vibrations or interference from other frequencies occurring within the frequency reference generator

(Filler, 1988).

Linear phase drift

One of the limitations of using the Allan deviation in estimating the coherence loss caused by the phase noise is

that the Allan deviation is insensitive to a drift in phase which, obviously, causes coherence loss. A linear phase

drift is caused by an offset (DC component 𝑦0) in the frequency reference, see Eq. (9). If the phase drift is of the

form:

𝜑(𝑡) = 𝐴𝑡 (35)

where 𝐴 = 2𝜋𝑦0.

Then the coherence and coherence loss are given by:

𝐶(𝑇) = |𝑠𝑖𝑛𝑐(𝐴𝑇

2)|, 𝐿𝐶 = 1 − |𝑠𝑖𝑛𝑐(

𝐴𝑇

2)| (36)

where 𝑠𝑖𝑛𝑐(𝑥) =sin (𝑥)

𝑥.

12 B. Alachkar

Fig. 3. The coherence loss for integration time T=1 second as function of the Allan deviation value at 𝜏 = 1 second,

in the two cases of white frequency noise and white phase noise, for the operating frequency 13.8 GHz.

Fig. 4. The coherence loss for the integration time T=60 seconds as function of the Allan deviation value at 𝜏 = 1 second,

in the two cases of white frequency noise and white phase noise, for the operating frequency 13.8 GHz.

Fig. 5. The coherence loss as a function of the integration time for different values of Allan deviation at 𝜏 = 1 𝑠,

in the two cases of white frequency noise and white phase noise, for the operating frequency 13.8 GHz.

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 13

Fig. 6. Frequency deviation and phase of a white phase noise for a frequency reference at 4 GHz, which gives 2%

coherence loss for an astronomical frequency 13.8 GHz.

Fig. 7. Allan deviation of a white phase noise signal (the signal of figure 6) calculated from the simulation data and from a

least squares fitting.

Fig. 8. Coherence Loss of the white phase noise signal of figure 6, calculated directly from the phase signal and from a

least squares fitting of Allan deviation.

14 B. Alachkar

Fig. 9. Frequency deviation and phase of a white frequency noise for a frequency reference at 4 GHz, which gives 2%

coherence loss for an astronomical frequency 13.8 GHz and an integration time of T=60 seconds.

Fig. 10. Allan deviation of a white frequency noise signal (the signal of figure 9) calculated from the simulation data and

from a least squares fitting.

Fig. 11. Coherence Loss of the white phase noise signal of figure 9, calculated directly from the phase signal and from a

least squares fitting of Allan deviation.

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 15

Fig. 12. Coherence loss caused by a phase drift of 0.1 rad/min.

A drift 𝐴 of 0.1 rad/min causes a coherence loss of about 0.1 % for an integration time of about 100 s. This effect

does not appear in the Allan deviation characterisation alone. This value of linear phase drift is the maximum

limit acceptable for the SKA. The requirement is to limit the phase drift to less than 1 radian over 10 minutes.

This requirement is related to the out-of-beam calibration.

Sinusoidal component in the phase deviation:

If the phase is modulated by a sinusoidal component of frequency 𝑓𝑚 then that the phase deviation includes a

component of the form:

𝜑(𝑡) = 𝐵𝑐𝑜𝑠(2𝜋𝑓𝑚𝑡) (37)

its corresponding Allan deviation is given by:

𝜎𝑦(𝜏) = 𝐵𝑠𝑖𝑛2(𝜋𝑓𝑚𝜏)

𝜋𝜐0𝜏 (38)

𝜐0 is the reference frequency.

The formula (38) can be proved using Eq. (10) and (11). 𝜎𝑦2(𝜏) can be written as:

𝜎𝑦2(𝜏) =

1

2

𝐵2

(2𝜋𝜐0𝜏)2⟨[𝑐𝑜𝑠(2𝜋𝑓𝑚(𝑡 + 2𝜏)) − 2𝑐𝑜𝑠(2𝜋𝑓𝑚(𝑡 + 𝜏)) + 𝑐𝑜𝑠(2𝜋𝑓𝑚(𝑡))]2⟩ (39)

then

𝜎𝑦2(𝜏) =

2𝐵2

(2𝜋𝜐0𝜏)2⟨[𝑐𝑜𝑠(2𝜋𝑓𝑚(𝑡 + 𝜏))𝑐𝑜𝑠(2𝜋𝑓𝑚𝜏) − 𝑐𝑜𝑠(2𝜋𝑓𝑚(𝑡 + 𝜏))]2⟩ (40)

and

𝜎𝑦2(𝜏) =

2𝐵2

(2𝜋𝜐0𝜏)2⟨

(1+𝑐𝑜𝑠(4𝜋𝑓𝑚(𝑡+𝜏))

24𝑠𝑖𝑛4(𝜋𝑓𝑚𝜏)⟩ (41)

then we find Eq. (38) as ⟨𝑐𝑜𝑠(4𝜋𝑓𝑚(𝑡 + 𝜏))⟩ = 0. The last term might not vanish if the instants 𝑡 are distributed

in a way so that this term does not average to 0. The instants 𝑡 determine the beginning of the averaging segments

in the Allan deviation calculation.

Figure 13 shows the Allan deviation curves of two phase noise signals, the first corresponding to the phase

deviation of the sum of a white phase noise of 0.2 radian at 13.8 GHz and a sinusoidal component of a frequency

10 Hz and an RMS value of 0.2 radians (amplitude of 0.2√2 radians) at 13.8 GH, and the second to the white

noise component alone. A sinusoidal component in the phase of RMS value σ would cause the same value of

coherence loss that is caused by a white phase noise of variance 𝜎2 (standard deviation σ). The coherence loss

caused by a combination of phase noise and a sinusoidal component of 𝜎 = 0.2 radians each causes the same

16 B. Alachkar

coherence loss as is seen with a white phase signal of σ = 0.2√2 ≅ 0.2828 radians, which is about 0.039 as is

shown in figure 14.

A sparse sampling of the same Allan deviation function would obscure the effect of the sinusoidal component. A

calculation of the coherence loss based on a least squares fit of the Allan deviation over a sampling of a few

points of the curve would result in significant error in the estimated values.

Fig. 13. Allan deviation corresponding to phase deviation of the sum of a white phase noise of 0.2 radian at 13.8 GHz and a

sinusoidal component of 10 Hz and amplitude of 0.2√2 radians at 13.8 GHz.

Fig. 14. The coherence loss caused by the sum of a white phase noise of 0.2 radian at 13.8 GHz and a sinusoidal

component of 10 Hz and amplitude of 0.2√2 radians at 13.8 GHz.

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 17

6. Conclusions

In this paper, we have analysed the verification of the requirements, in a radio astronomy interferometer, of the

coherence loss and the stability of the frequency distribution systems. We have analysed the relationship between

the instability of a frequency reference and the coherence loss caused by it. A particular case of the coherence

requirements of the SKA was considered as an example. Theoretical and practical aspects, in addition to

simulation results have been presented. A model of a typical frequency distribution system is given. We

highlighted the importance of a careful analysis of the types of phase noise that could be masked in Allan

deviation and will have an effect on the coherence loss.

Appendix A

Calculation of the phase output compensated by the round-trip PLL:

The generated phases by the phase shifter are (see figure 1):

𝜙1(𝑆) = 𝜙2(𝑆) = 𝐹(𝑆)𝜙𝑐−𝑖𝑛(𝑆) (A1)

The input control signal of the corrector filter is (See figure 1):

𝜙𝑐−𝑖𝑛(𝑆) = −𝜙𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑆) − 𝜙𝑠ℎ𝑖𝑓𝑡(𝑆) (A2)

When the loop is closed, we can write:

𝜙𝑐−𝑖𝑛(𝑆) =1

1+𝐻(𝑆)(−𝜙𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑆)) (A3)

The output compensation phase can be written as (see figure 1):

𝜙𝑐−𝑜(𝑆) = 𝑒−𝑆𝜏𝑑𝜙1(𝑆) = 𝑒−𝑆𝜏𝑑𝐹(𝑆)𝜙𝑐−𝑖𝑛(𝑆) (A4)

From Eq. (26):

𝐻(𝑆) = 𝐹(𝑆)(1 + 𝑒−2𝑆𝜏𝑑) (A5)

Then, using (A3) and (A5), (A4) can be written as:

𝜙𝑐−𝑜(𝑆) = −𝐻(𝑆)

1+𝐻(𝑆)

𝜙𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑆)

(𝑒+𝑆𝜏𝑑+𝑒−𝑆𝜏𝑑) (A6)

Then the output phase at the remote side of the fibre is:

𝜙𝑜(𝑆) = 𝜙𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑆) −𝐻(𝑆)

1+𝐻(𝑆)

𝜙𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑆)

(𝑒+𝑆𝜏𝑑+𝑒−𝑆𝜏𝑑) (A7)

And in the frequency domain:

�̃�𝑜(𝑓) = �̃�𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑓) −ℎ̃(𝑓)

1+ℎ̃(𝑓)

1

2𝑐𝑜𝑠 (2𝜋𝑓𝜏𝑑)�̃�𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑓) (A8)

Calculation of the PSD of the noise accumulated on the fibre:

If 𝛿𝜑(𝑡, 𝑥) is the element phase noise generated at the time 𝑡 and the location 𝑥 of the fibre (0 ≤ 𝑥 ≤ 𝐿 ), then we

can write the noise generated on one way on the outgoing transmitted signal:

𝜑𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑡) = ∫ 𝛿𝜑 (𝑡 − (𝜏𝑑 −𝑥

𝑐/𝑛) , 𝑥) 𝑑𝑥

𝐿

0 (A9)

18 B. Alachkar

And the noise accumulated on the round trip (outgoing and return) of the transmitted signal:

𝜑𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑡) = ∫ [𝛿𝜑 (𝑡 − (𝜏𝑑 −𝑥

𝑐/𝑛) , 𝑥) + 𝛿𝜑 (𝑡 − (2𝜏𝑑 −

𝑥

𝑐/𝑛) , 𝑥)] 𝑑𝑥

𝐿

0 (A10)

In frequency domain these equations can be written as follows:

�̃�𝑓𝑖𝑏𝑟𝑒_1𝑤(𝑓) = ∫ 𝑒+2𝜋𝑖𝑓

𝑥

𝑐/𝑛 𝑒−2𝜋𝑖𝑓𝜏𝑑 𝛿�̃�(𝑓, 𝑥) 𝑑𝑥𝐿

0 (A11)

�̃�𝑓𝑖𝑏𝑟𝑒_𝑅𝑇(𝑓) = 2 ∫ 𝑐𝑜𝑠 (2𝜋𝑓(𝜏𝑑 −𝑥

𝑐/𝑛)) 𝑒−2𝜋𝑖𝑓𝜏𝑑 𝛿�̃�(𝑓, 𝑥) 𝑑𝑥

𝐿

0 (A12)

If we assume that the elementary phase noise 𝛿𝜑(𝑡, 𝑥) is uncorrelated along the fibre, then the PSD of the phase

noise for one-way and round-trip are respectively:

𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓) = ⟨|�̃�𝑓𝑖𝑏𝑟𝑒_1𝑤(𝑥, 𝑓)|2

⟩ = ∫ ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ 𝑑𝑥𝐿

0 (A13)

𝑆𝜑,𝑓𝑖𝑏𝑟𝑒_𝑅𝑇(𝑓) = 4 ∫ (cos (2𝜋𝑥

𝑐/𝑛𝑓) )

2⟨|𝛿�̃�(𝑓, 𝑥)|

2⟩ 𝑑𝑥

𝐿

0 (A14)

Using (A8), we find in the case when the gain |ℎ̃(𝑓)| is very high (|ℎ̃(𝑓)| ≫ 1):

�̃�𝑜(𝑓) = �̃�𝑓𝑖𝑏𝑟𝑒−1𝑊(𝑓) −1

2cos (2𝜋𝑓𝜏𝑑)�̃�𝑓𝑖𝑏𝑟𝑒−𝑅𝑇(𝑓) (A15)

and using (A11) and (A12):

�̃�𝑜(𝑓) = ∫ 𝑒+2𝜋𝑖𝑓

𝑥

𝑐/𝑛 𝑒−2𝜋𝑖𝑓𝜏𝑑 𝛿�̃�(𝑓, 𝑥) 𝑑𝑥𝐿

0−

1

cos (2𝜋𝑓𝜏𝑑)∫ cos (2𝜋𝑓(𝜏𝑑 −

𝑥

𝑐/𝑛)) 𝑒−2𝜋𝑖𝑓𝜏𝑑 𝛿�̃�(𝑓, 𝑥) 𝑑𝑥

𝐿

0 (A16)

then:

�̃�𝑜(𝑓) = 𝑒−2𝜋𝑖𝑓𝜏𝑑 ∫ 𝛿�̃�(𝑓, 𝑥)[𝑒+2𝜋𝑖𝑓

𝑥

𝑐/𝑛 −1

cos (2𝜋𝑓𝜏𝑑)cos (2𝜋𝑓(𝜏𝑑 −

𝑥

𝑐/𝑛)) ] 𝑑𝑥

𝐿

0 (A17)

and then:

�̃�𝑜(𝑓) = 𝑒−2𝜋𝑖𝑓𝜏𝑑 ∫ 𝛿�̃�(𝑓, 𝑥)[cos (2𝜋𝑓𝑥

𝑐/𝑛) + 𝑖 sin (2𝜋𝑓

𝑥

𝑐/𝑛) −

1

cos (2𝜋𝑓𝜏𝑑)(cos(2𝜋𝑓𝜏𝑑) cos (2𝜋𝑓

𝑥

𝑐/𝑛) +

𝐿

0

sin(2𝜋𝑓𝜏𝑑) sin (2𝜋𝑓𝑥

𝑐/𝑛))] 𝑑𝑥 (A18)

Then:

�̃�𝑜(𝑓) = 𝑒−2𝜋𝑖𝑓𝜏𝑑 ∫ 𝛿�̃�(𝑓, 𝑥)[𝑖 − tan(2𝜋𝑓𝜏𝑑)] sin (2𝜋𝑓𝑥

𝑐/𝑛) 𝑑𝑥

𝐿

0 (A19)

Then, the PSD of the phase 𝜑𝑜(𝑡) at the remote end of the fibre (at the receptor) can be written:

𝑆𝑜(𝑓) = ⟨|�̃�𝑜(𝑓)|2⟩ = ⟨|∫ 𝛿�̃�(𝑓, 𝑥)[𝑖 − tan(2𝜋𝑓𝜏𝑑)] sin (2𝜋𝑓𝑥

𝑐/𝑛) 𝑑𝑥

𝐿

0|

2⟩ (A20)

If we assume that the elementary phase noise 𝛿𝜑(𝑡, 𝑥) is uncorrelated along the fibre, 𝑆𝑜(𝑓) can be written as:

𝑆𝑜(𝑓) = [1 + tan2(2𝜋𝑓𝜏𝑑)] ∫ ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ sin2 (2𝜋𝑓𝑥

𝑐/𝑛) 𝑑𝑥

𝐿

0 (A21)

then

𝑆𝑜(𝑓) =1

cos2(2𝜋𝑓𝜏𝑑)∫ ⟨|𝛿�̃�(𝑓, 𝑥)|

2⟩ sin2 (2𝜋𝑓

𝑥

𝑐/𝑛) 𝑑𝑥

𝐿

0 (A22)

For low frequencies where 𝑓 ≪1

2𝜋𝜏𝑑, and so cos(2𝜋𝑓𝜏𝑑) ≈ 1, and sin (2𝜋𝑓

𝑥

𝑐/𝑛) ≈ 2𝜋𝑓

𝑥

𝑐/𝑛 , 𝑆𝑜(𝑓) can be

written as:

Frequency Stability and Coherence Loss in Radio Astronomy Interferometers 19

𝑆𝑜(𝑓) =(2𝜋𝑓)2

(𝑐/𝑛)2 ∫ ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ 𝑥2 𝑑𝑥𝐿

0 (A24)

If we, also, assume that the elementary PSD ⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ is uniform along the fibre:

⟨|𝛿�̃�(𝑓, 𝑥)|2

⟩ = 𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓)

𝐿= 𝐷(𝑓) (A25)

𝐷(𝑓) is the density of the PSD per unit of length of the fibre. Then 𝑆𝑜(𝑓) can be written as:

𝑆𝑜(𝑓) =(2𝜋𝑓)2𝐷(𝑓) 𝐿3

3(𝑐/𝑛)2 =1

3(2𝜋𝑓𝜏𝑑)2 𝑆𝜑,𝑓𝑖𝑏𝑟𝑒(𝑓) (A26)

Acknowledgements

This work is being carried out for the SKA Signal and Data Transport (SaDT) consortium as part of the Square

Kilometre Array (SKA) project. Fourteen institutions from eight countries are involved in the SaDT consortium,

led by the University of Manchester. The SKA project is an international effort to build the world’s largest radio

telescope, led by SKA Organisation with the support of 10 member countries.

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