Analysis of Reference Signal Transmission System

Yu Lan Ren ChaoxiInstitute of Electronic Engineering

Radar AcademyWuhan 430019, Hubei, China

Abstract—This paper describes the Allan variance of outputreference signal with phase noise of input reference signal,modulation and demodulation noise, phase noise of localoscillators and channel noise for narrowband and FMnarrowband reference signal transmission systems. Theanalytical results show that the total Allan variance of outputreference signal includes three parts: input Allan variance, IFAllan variance and RF Allan variance, which can be regarded as the foundation of design for IF and RF components of referencesignal transmission systems.

Keywords- Allan variance; Input Allan variance; IF Allanvariance; RF Allan variance.

I. INTRODUCTION

The reference signal transmission systems are widely adopted in radar and communication systems for coherentreceiving, in which the Allan variance of system output reference signal is the key factor. Reference [5] ~ [8] illustratethe Allan variance and phase noise. Reference [2] describes therelation between Allan variance of output reference signal andcarrier-to-noise ratio for noiseless input signal and idealreference signal transmission systems. This paper furtheranalyses the practical reference signal transmission systems,deduces the formulae which describe the Allan variance ofoutput reference signal with input phase noise, modulation anddemodulation noise, phase noise of local oscillators andchannel noise for narrowband and FM narrowband referencesignal transmission systems. The total Allan variance of outputreference signal includes three parts: input Allan variance, IFAllan variance and RF Allan variance, which may be regarded as the foundation of design for IF and RF components ofreference signal transmission systems. Section gives thestructure of a narrowband reference signal transmission systemand analyses its transmission performance, section gives thestructure of a FM narrowband reference signal transmissionsystem and analyses its transmission performance, and section

concludes the analytical results and makes suggestions tothe design of reference signal transmission systems.

II. ANALYSIS OF NARROWBAND REFERENCE SIGNALTRANSMISSION SYSTEM

A reference signal can be described by

02S t Acos f t t

In which A is the amplitude, 0f is the frequency of

reference signal, t is the phase noise of reference signal.

The Allan variance of the reference signal S t can be

described by

22 422

0

1 1 162 2

Sin f f dff

4

20 0

4Sin f

f dff

(1)

In which f is the PSD of phase noise t ofreference signal, is the interval of analysis samples.

Figure 1 shows the practical structure of a narrowbandreference signal transmission system. On the basis of analysisfor its structure, we discuss the relation between the Allan

oS t

InputReference

Signal

Receiver

FrequencyDivider

OutputReference

Signal

Fig.1, The composition block diagram of narrowbandreference signal transmission system

NarrowbandFilter

Transmitter

FrequencyMultiplier

tS t

iS t

rS t

978-1-4244-2251-7/08/$25.00 ©2008 IEEE

variance of output reference signal and various noises in thetransmission system such as input phase noise, channel noiseetc.

As shown in Fig.1, the input reference signal can beexpressed by

0cos 2i c iS t A f t t , in which is the

phase noise of input reference signal.i t

The output of frequency multiplier can be described as

0cos 2t c iS t A nf t n t

The received signal can be considered as the sum

of transmitted signal and white Gaussian noise

rS ttSt tN ,

that is

0cos 2r t

c i

S t S t N t

A nf t n t N tThe output of frequency divider can be expressed as

0cos 2o c iS t A f t t n t , in which n tis a narrowband Gaussian noise and can be described by

0 0cos 2 sin 2c sn t n t f t n t f tIf the carrier-to-noise ratio is large enough, that

is , we havetnAc

0

0 0

0

cos 2

cos 2 sin 2

cos 2

o c i

c s

sc i

c

S t A f t t

n t f t n t f t

n tA f t t

AWhere the total phase noise of output reference signal can

be denoted by

si

c

n tt t

A(2)

Therefore the PSD of can described as t

i RFf f f , in which

if denotes

the PSD of input reference signal, andRFf denotes the

PSD of channel noise s

c

n tA

, it can be calculated by

02

2

0

s

RF

c m mn

c

nA f f f

ff

Aothers

(3)

Assume 2 mf denotes the bandwidth of narrowband

filter, according to equation (1), we may calculate the Allanvariance of the output reference signal as follows

42

20 0

4i RFN

Sin ff f df

f (4)

The Allan variances of output reference signal due toinput noise or RF noise can be calculated respectively as follows

Input Allan variance:4

22

0 0

4m

i

f

I

Sin ff df

f(5)

RF Allan variance:4

22

0 0

40

2 20 0

4

4

m

RF

m

f

RF

f

c

Sin ff df

f

Sin f n dfAf

(6)

Thus we may calculate the total Allan variance of outputreference signal as follows

2 2 2N I RF (7)

III. ANALYSIS OF FM NARROWBAND REFERENCE SIGNALTRANSMISSION SYSTEM

tSm

tSt tSortSrtSot

tM

tSd

tV

PA

InputReference

Signal

LNA

FrequencyDemodulator

OutputReference

Signal

Mixer

Fig.2, The composition block diagram of narrowbandFM reference signal transmission system

FrequencyModulator

OSC MixerOSC

NarrowbandFilter

Figure 2 shows the practical structure of a FMnarrowband reference signal transmission system. On the basisof analysis for its structure, we discuss the relation betweenthe Allan variance of output reference signal and variousnoises in the transmission system such as input phase noise,modulation and demodulation noise, phase noise of the localoscillators and channel noise etc.

As shown in Fig.2, the input reference signal can beexpressed as

cos 2m m iM t A f t t , where denotes

the phase noise of input reference signal.i t

The output of frequency modulator can be described as

cos 2m c c mS t A f t t t , where m tdenotes the phase noise of frequency modulator, and

t 2 cos

cos

t t

F m

t

m i

iK M d f d

d

The output of local oscillator in transmitter can be

expressed as , wherecos 2ot o otS t f t t

tot is the phase noise of the oscillator.The output of mixer in transmitter can be written as

0cos 2t c c m otS t A f f t t t tThe received signal can be considered as the sum of

transmitted signal and white Gaussian noisetSt tN , thatis

tNtStS tr

0cos 2r c c m otS t A f f t t t t

N t

If the signal-to-noise ratio is large enough, that

is , the

output signal with noise from the narrowband filter can bedescribed by

The output of local oscillator in receiver can be expressed

as , where is the

phase noise of the oscillator.

cos 2or o orS t f t t or t

The output of mixer in receiver can be described as

cos 2d c c m ot orS t A f t t t t t

n tWhere is a narrowband Gaussian noise and can be

described bytn

cos 2 sin 2c c sn t n t f t n t f tcIf the carrier-to-noise ratio is large enough, that

is , we havetnAc

cos 2

cos2 sin2d c c m ot or

c c s c

S t A f t t t t t

n t f t n t f t

cos 2 sc c m ot or

c

n tA f t t t t t

ASince the output voltage of frequency demodulator is

directly proportional to the instantaneous frequency deviationof its input signal, that is

12

cos2 2 2 2

sm ot or d

c

m ot or sm i d

c

n tdV t t t t t N tdt A

t t t n tf t t N

At

Where dN t denotes demodulation noise.

It can be proved that is also a narrowband

Gaussian noise and

tns'

2 2 2m ot or

d

t t tN t can

be also considered to be approximately narrowband Gaussiannoise.

Thus, the sum of them can be described by

2 2 2 2

cos2 2 2 2

sin2 2 2 2

m ot or sd

c

mc otc orc scdc m

c

ms ots ors ssds m

c

t t t n tN t

A

t t t n tN t t

A

t t t n tN t t

A

2 2 2 2m ot or s

dc

t t t n tf N t

A

cos

cos2 2 2 2

sin2 2 2 2

o m i

mc otc orc scdc m

c

ms ots ors ssds m

c

V t f t t

t t t n tN t

A

t t t n tN t t

A

t

It can be simplified as

cos2

2 2 2

ms dso m i

ots ors ss

c

t N tV t f t t

f f

t t n tf f fA

The total phase noise can be described as

2 2 2ms ds ots ors ss

ic

t N t t t n tt t

2f f f f fA(8)

Assume that if

msf

dsf

otsf

orsf

ssf expresses the power

spectral density of i t ms t dsN t ots t

respectively, we haveors t ssn t

2 2

2 2 2

2

2 2 2

ms ds

i

ots ors ss

i IF RF

c

f ff f

f f

2

f f f

f f f

f f f

A (9)

Whereif denotes the PSD of phase noise from

input reference signal,IFf denotes the PSD of phase

noise from frequency modulation and demodulation given by

22ms ds

IF 2

f ff

f f(10)

AndRFf denotes the PSD of phase noise from RF local

oscillators and channel noise given by

2 2 22 2 2ots ors ss

RF

c2

f f ff

f f f A(11)

The Allan variance of the output signal can becalculated by

tV0

42

20

4i IF RFFM

m

Sin ff f f

fdf (12)

The Allan variances resulted from Input noise, IF noise

and RF noise can be calculated respectively as followsInput Allan variance:

42

20

4iI

m

Sin ff df

f(13)

IF Allan variance:4

22

0

4IFIF

m

Sin ff df

f (14)

RF Allan variance:4

22

0

4RFRF

m

Sin ff df

f(15)

Thus we may calculate the total Allan variance of outputreference signal as follows

2 2 2 2FM I IF RF (16)

IV. CONCLUSIONS

Equation (7) and (16) show that whether the referencesignal transmission system adopts narrowband or FMnarrowband mode, the Allan variance of output signal includesthree parts: the Allan variance due to input noise; the Allanvariance due to IF modulation and demodulation noise; andthe Allan variance duo to RF phase noise of local oscillatorand channel noise. We may design and control the systemparameters by above equations to meet the practicalrequirements; the design foundation of the reference signal RFtransmission system may be founded in reference [2].

REFERENCES

[1] Yu Lan, “A Comparison of pulse Transmission Systems”, Proceedings ofICCT 2006.

[2] Yu Lan, “A Comparison of Reference Source Signal TransmissionSystems”, Proceedings of ICCT 2006.

[3] Fan C X. “Theory of Communications”. China: National Defense and Industry Publishing House.

[4] Lu D J. “Stochastic Processes and Its Applications”. China: TsinghuaUniversity Publishing House.

[5] D.W. Allan “Statistics of atomic frequency standards” Proc. IEEE, vol.54,pp.221-230, Feb 1966.

[6] L.S. Cutler and C.L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards” Proc.IEEE, vol.54, pp.136-154, Feb 1966.

[7] J.A. Barnes et al., “Characterization of frequency stability” IEEE Trans.Instrum. Meas., vol. IM-20, pp.105-120, May 1971.

[8] Paul Lesage and Theophane AYI, “Characterization of frequencystability: Analysis of the Modified Allan Variance Properties of ItsEstimate” IEEE Trans. Instrum. Meas., vol. IM-33, pp.332-336,December 1984.